Linking continuous and discrete intervertebral disc models through homogenisation

At present, there are two main numerical approaches that are frequently used to simulate the mechanical behaviour of the human spine. Researchers with a continuum-mechanical background often utilise the finite-element method (FEM), where the involved biological soft and hard tissues are modelled on a macroscopic (continuum) level. In contrast, groups associated with the science of human movement usually apply discrete multi-body systems (MBS). Herein, the bones are modelled as rigid bodies, which are connected by Hill-type muscles and non-linear rheological spring-dashpot models to represent tendons and cartilaginous connective tissue like intervertebral discs (IVD). A possibility to benefit from both numerical methods is to couple them and use each approach, where it is most appropriate. Herein, the basic idea is to utilise MBS in simulations of the overall body and apply the FEM only to selected regions of interest. In turn, the FEM is used as homogenisation tool, which delivers more accurate non-linear relationships describing the behaviour of the IVD in the multi-body dynamics model. The goal of this contribution is to present an approach to couple both numerical methods without the necessity to apply a gluing algorithm in the context of a co-simulation. Instead, several pre-computations of the intervertebral disc are performed offline to generate an approximation of the homogenised finite-element (FE) result. In particular, the discrete degrees of freedom (DOF) of the MBS, that is, three displacements and three rotations, are applied to the FE model of the IVD, and the resulting homogenised forces and moments are recorded. Moreover, a polynomial function is presented with the discrete DOF of the MBS as variables and the discrete forces an moments as function values. For the sake of a simple verification, the coupling method is applied to a simplified motion segment of the spine. Herein, two stiff cylindrical vertebrae with an interjacent homogeneous cylindrical IVD are examined under the restriction of purely elastic deformations in the sagittal plane.     

A combined numerical and experimental technique for estimation of the forces and moments in the lumbar intervertebral disc

Evaluation of the loads on lumbar intervertebral discs (IVD) is critically important since it is closely related to spine biomechanics, pathology and prosthesis design. Non-invasive estimation of the loads in the discs remains a challenge. In this study, we proposed a new technique to estimate in vivo loads in the IVD using a subject-specific finite element (FE) model of the disc and the kinematics of the disc endplates as input boundary conditions. The technique was validated by comparing the forces and moments in the discs calculated from the FE analyses to the in vitro experiment measurements of three corresponding lumbar discs. The results showed that the forces and moments could be estimated within an average error of 20%. Therefore, this technique can be a promising tool for non-invasive estimation of the loads in the discs and may be extended to be used on living subjects.     

Cellular mechanobiology of the intervertebral disc: New directions and approaches

The more we learn about the intervertebral disc (IVD), the more we come to appreciate the intricacies involved in transmission of forces through the ECM to the cell, and in the biological determinants of its response to mechanical stress. This review highlights recent developments in our knowledge of IVD physiology and examines their impact on cellular mechanobiology. Discussion centers around the continually evolving cellular and microstructural anatomy of the nucleus pulposus (NP) and the annulus fibrosus (AF) in response to complex stresses generated in support of axial load and spinal motion. Particular attention has been given to cells from the immature NP and the interlamellar AF, and assessment of their potential mechanobiologic contributions to the health and function of the IVD. In addition, several innovative approaches that have been brought to bear on studying the interplay between disc cells and their micromechanical environment are discussed. Techniques for “engineering” cellular function and technologies for fabricating more structurally defined biomaterial scaffolds have recently been employed in disc research. Such tools can be used to elucidate the biological and physical mechanisms by which different IVD cell populations are regulated by mechanical stress, and contribute to advancement of preventative and therapeutic measures.     

A comparison between porcine,ovine, and bovine intervertebral disc anatomy and single lamella annulus fibrosus tensile properties

This project aimed to compare gross anatomical measures and biomechanical properties of single lamellae from the annulus fibrosus of ovine and porcine lumbar vertebrae, and bovine tail vertebrae. The morphology of the vertebrae of these species differ significantly both from each other and from human, yet how these differences alter biomechanical properties is unknown. Geometric parameters measured in this study included: 1) absolute and relative intervertebral (IVD) and vertebral body height and 2) absolute and relative intervertebral disc (IVD) anterior‐posterior (AP) and medial‐lateral (ML) widths. Single lamella tensile properties included toe‐region stress and stretch ratio, stiffness, and tensile strength. As expected, the bovine tail IVD revealed a more circular shape compared with both the ovine and porcine lumbar IVD. The bovine tail also had the largest IVD to vertebral body height ratio (due to having the highest absolute IVD height). Bovine tail lamellae were also found to be strongest and stiffest (in tension) while ovine lumbar lamellae were weakest and most compliant. Histological analysis revealed the greatest proportion of collagen in the bovine corroborating findings of increased strength and stiffness. The observed differences in anatomical shape, connective tissue composition, and tensile properties need to be considered when choosing an appropriate model for IVD research. J. Morphol. 277:244–251, 2016. © 2015 Wiley Periodicals, Inc.     

Effect of endplate calcification and mechanical deformation on the distribution of glucose in intervertebral disc: a 3D finite element study

The intervertebral disc (IVD) is avascular, receiving nutrition from surrounding vasculature. Theoretical modelling can supplement experimental results to understand nutrition to IVD more clearly. A new, 3D finite element model of the IVD was developed to investigate effects of endplate calcification and mechanical deformation on glucose distributions in IVD. The model included anatomical disc geometry, non-linear coupling of cellular metabolism with pH and oxygen concentration and strain-dependent properties of the extracellular matrix. Calcification was simulated by reducing endplate permeability (～79%). Mechanical loading was applied based on in vivo disc deformation during the transition from supine to standing positions. Three static strain conditions were considered: supine, standing and weight-bearing standing. Minimum glucose concentrations decreased 45% with endplate calcification, whereas disc deformation led to a 4.8-63% decrease, depending on the endplate condition (i.e. normal vs. calcified). Furthermore, calcification more strongly affected glucose concentrations in the nucleus compared to the annulus fibrous region. This study provides important insight into nutrient distributions in IVD under mechanical deformation.     

Impact of material and morphological parameters on the mechanical response of the lumbar spine – A finite element sensitivity study

Finite element models are frequently used to study lumbar spinal biomechanics. Deterministic models are used to reflect a certain configuration, including the means of geometrical and material properties, while probabilistic models account for the inherent variability in the population. Because model parameters are generally uncertain, their predictive power is frequently questioned. In the present study, we determined the sensitivities of spinal forces and motions to material parameters of intervertebral discs, vertebrae, and ligaments and to lumbar morphology. We performed 1200 model simulations using a generic model of the human lumbar spine loaded under pure moments. Coefficients of determination and of variation were determined for all parameter and response combinations. Material properties of the vertebrae displayed the least impact on results, whereas those of the discs and morphology impacted most. The most affected results were the axial compression forces in the vertebral body and in several ligaments during flexion and the facet-joint forces during extension. Intervertebral rotations were considerably affected only when several parameters were varied simultaneously. Results can be used to decide which model parameters require careful consideration in deterministic models and which parameters might be omitted in probabilistic studies. Findings allow quantitative estimation of a model׳s precision.     

Molecular profiling of the developing mouse axial skeleton: a role for Tgfbr2 in the development of the intervertebral disc

Background  

Very little is known about how intervertebral disc (IVD) is formed or maintained. Members of the TGF-β superfamily are secreted signaling proteins that regulate many aspects of development including cellular differentiation. We recently showed that deletion of Tgfbr2 in Col2a expressing mouse tissue results in alterations in development of IVD annulus fibrosus. The results suggested TGF-β has an important role in regulating development of the axial skeleton, however, the mechanistic basis of TGF-β action in these specialized joints is not known. One of the hurdles to understanding development of IVD is a lack of known markers. To identify genes that are enriched in the developing mouse IVD and to begin to understand the mechanism of TGF-β action in IVD development, we undertook a global analysis of gene expression comparing gene expression profiles in developing mouse vertebrae and IVD. We also compared expression profiles in tissues from wild type and Tgfbr2 mutant mice as well as in sclerotome cultures treated with TGF-β or BMP4.     

In vitro axial preload application during spine flexibility testing: towards reduced apparatus-related artefacts

Presently, there is little consensus about how, or even if, axial preload should be incorporated in spine flexibility tests in order to simulate the compressive loads naturally present in vivo. Some preload application methods are suspected of producing unwanted “artefact” forces as the specimen rotates and, in doing so, influencing the resulting kinematics. The objective of this study was to quantitatively compare four distinct types of preload which have roots in contemporary experimental practice. The specific quantities compared were the reaction moments and forces resulting at the intervertebral disc and specimen kinematics. The preload types incorporated increasing amounts of caudal constraint on the preload application vector ranging from an unconstrained dead-load arrangement to an apparatus that allowed the vector to follow rotations of the specimen. Six human cadaveric spine segments were tested (1-L1/L2, 3-L2/L3, 1-L3/L4 and 1-L4/L5). Pure moments were applied to the specimens with each of the four different types of compressive preload. Kinematic response was measured using an opto-electronic motion analysis system. A six-axis load cell was used to measure reaction forces and moments. Artefact reaction moments and shear forces were significantly affected by preload application method and magnitude. Unconstrained preload methods produced high artefact moments and low artefact shear forces while more constrained methods did the opposite. A mechanical trade-off is suggested by our results, whereby unwanted moment can only be prevented at the cost of shear force production. When comparing spine flexibility studies, caution should be exercised to ensure preload was applied in a similar manner for all studies. Unwanted moments or forces induced as a result of preload application method may render the comparison of two seemingly similar studies inappropriate.     

The strain-generated electrical potential in cartilaginous tissues: a role for piezoelectricity

The strain-generated potential (SGP) is a well-established mechanism in cartilaginous tissues whereby mechanical forces generate electrical potentials. In articular cartilage (AC) and the intervertebral disc (IVD), studies on the SGP have focused on fluid- and ionic-driven effects, namely Donnan, diffusion and streaming potentials. However, recent evidence has indicated a direct coupling between strain and electrical potential. Piezoelectricity is one such mechanism whereby deformation of most biological structures, like collagen, can directly generate an electrical potential. In this review, the SGP in AC and the IVD will be revisited in light of piezoelectricity and mechanotransduction. While the evidence base for physiologically significant piezoelectric responses in tissue is lacking, difficulties in quantifying the physiological response and imperfect measurement techniques may have underestimated the property. Hindering our understanding of the SGP further, numerical models to-date have negated ferroelectric effects in the SGP and have utilised classic Donnan theory that, as evidence argues, may be oversimplified. Moreover, changes in the SGP with degeneration due to an altered extracellular matrix (ECM) indicate that the significance of ionic-driven mechanisms may diminish relative to the piezoelectric response. The SGP, and these mechanisms behind it, are finally discussed in relation to the cell response.     

The involvement of interleukin-1 and interleukin-4 in the response of human annulus fibrosus cells to cyclic tensile strain: an altered mechanotransduction pathway with degeneration

Introduction  

Recent evidence suggests that intervertebral disc (IVD) cells derived from degenerative tissue are unable to respond to physiologically relevant mechanical stimuli in the 'normal' anabolic manner, but instead respond by increasing matrix catabolism. Understanding the nature of the biological processes which allow disc cells to sense and respond to mechanical stimuli (a process termed 'mechanotransduction') is important to ascertain whether these signalling pathways differ with disease. The aim here was to investigate the involvement of interleukin (IL)-1 and IL-4 in the response of annulus fibrosus (AF) cells derived from nondegenerative and degenerative tissue to cyclic tensile strain to determine whether cytokine involvement differed with IVD degeneration.     

Hyperosmotically induced volume change and calcium signaling in intervertebral disk cells: the role of the actin cytoskeleton

Loading of the spine alters the osmotic environment in the intervertebral disk (IVD) as interstitial water is expressed from the tissue. Cells from the three zones of the IVD, the anulus fibrosus (AF), transition zone (TZ), and nucleus pulposus (NP), respond to osmotic stress with altered biosynthesis through a pathway that may involve calcium (Ca(2+)) as a second messenger. We examined the hypothesis that IVD cells respond to hyperosmotic stress by increasing the concentration of intracellular calcium ([Ca(2+)](i)) through a mechanism involving F-actin. In response to hyperosmotic stress, control cells from all zones decreased in volume and cells from the AF and TZ exhibited [Ca(2+)](i) transients, while cells from the NP did not. Extracellular Ca(2+) was necessary to initiate [Ca(2+)](i) transients. Stabilization of F-actin with phalloidin prevented the Ca(2+) response in AF and TZ cells and decreased the rate of volume change in cells from all zones, coupled with an increase in the elastic moduli and apparent viscosity. Conversely, actin breakdown with cytochalasin D facilitated Ca(2+) signaling while decreasing the elastic moduli and apparent viscosity for NP cells. These results suggest that hyperosmotic stress induces volume change in IVD cells and may initiate [Ca(2+)](i) transients through an actin-dependent mechanism.     

Influence of experimental protocols on the mechanical properties of the intervertebral disc in unconfined compression

The lack of standardization in experimental protocols for unconfined compression tests of intervertebral discs (IVD) tissues is a major issue in the quantification of their mechanical properties. Our hypothesis is that the experimental protocols influence the mechanical properties of both annulus fibrosus and nucleus pulposus. IVD extracted from bovine tails were tested in unconfined compression stress-relaxation experiments according to six different protocols, where for each protocol, the initial swelling of the samples and the applied preload were different. The Young's modulus was calculated from a viscoelastic model, and the permeability from a linear biphasic poroviscoelastic model. Important differences were observed in the prediction of the mechanical properties of the IVD according to the initial experimental conditions, in agreement with our hypothesis. The protocol including an initial swelling, a 5% strain preload, and a 5% strain ramp is the most relevant protocol to test the annulus fibrosus in unconfined compression, and provides a permeability of 5.0 ± 4.2e(-14)m(4)/N[middle dot]s and a Young's modulus of 7.6 ± 4.7 kPa. The protocol with semi confined swelling and a 5% strain ramp is the most relevant protocol for the nucleus pulposus and provides a permeability of 10.7 ± 3.1 e(-14)m(4)/N[middle dot]s and a Young's modulus of 6.0 ± 2.5 kPa.     

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.     

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.     

Prediction of Individual Muscle Forces Using Lagrange Multipliers Method - A Model of the Upper Human Limb in the Sagittal Plane: II. Numerical Experiments and Sensitivity Analysis

Using the method of Lagrange multipliers an analytical solution of the optimization problem formulated for a two-dimensional, 3DOF model of the human upper limb has been described in Part I of this investigation. The objective criterion used is the following: [formula: see text], where F(i) -s are the muscle forces modelled and c(i) -s are unknown weight factors. This study is devoted to the numerical experiments performed in order to investigate which sets of the weight factors may predict physiologically reasonable muscle forces and joint reactions. A sensitivity analysis is also presented. The influence of: the gravity forces, different external loads applied to the hand, changes of the weight factors and of joint angle on the optimal solution is studied. A general conclusion may be drawn: using the above mentioned objective criterion, practically all motor tasks performed by the human upper limb may be described if the c(i) -s are properly chosen. These weight factors generally depend on the joint moments and must be different (their magnitudes as well as their signs) for agonistic muscles and for their antagonists.     

Notochordal conditioned media from tissue increases proteoglycan accumulation and promotes a healthy nucleus pulposus phenotype in human mesenchymal stem cells

Introduction  

Notochordal cells (NCs) are influential in development of the intervertebral disc (IVD) and species that retain NCs do not degenerate. IVD repair using bone marrow derived mesenchymal stem cells (MSCs) is an attractive approach and the harsh microenvironment of the IVD suggests pre-differentiation is a necessary first step. The goal of this study was to use soluble factors from NCs in alginate and NCs in their native tissue to differentiate human MSCs to a young nucleus pulposus (NP) phenotype.     

Structural analysis for the forces in the human spinal column and its musculature

A method of structural analysis is proposed for a model of the thoracic and lumbar spine which involves body weight, external load, muscle forces, intervertebral reactions and disc and ligament elasticity. The procedure consists of three phases: (i) for an initial position the equations of equilibrium are solved using a linear programming technique which minimizes the total force in the system and produces intervertebral reactions and muscle forces, (ii) for an incremental change in configuration a structural analysis is performed to obtain a further set of intervertebral reactions, (iii) with the net values of intervertebral reactions, from combining the results of the previous two phases, a further application of the linear programming technique is made to obtain the muscle forces in the deformed positions. The results are presented for a subject, in a single step of forward flexion.     

Effects of geometric individualisation of a human spine model on load sharing: neuro-musculoskeletal simulation reveals significant differences in ligament and muscle contribution

In spine research, two possibilities to generate models exist: generic (population-based) models representing the average human and subject-specific representations of individuals. Despite the increasing interest in subject specificity, individualisation of spine models remains challenging. Neuro-musculoskeletal (NMS) models enable the analysis and prediction of dynamic motions by incorporating active muscles attaching to bones that are connected using articulating joints under the assumption of rigid body dynamics. In this study, we used forward-dynamic simulations to compare a generic NMS multibody model of the thoracolumbar spine including fully articulated vertebrae, detailed musculature, passive ligaments and linear intervertebral disc (IVD) models with an individualised model to assess the contribution of individual biological structures. Individualisation was achieved by integrating skeletal geometry from computed tomography and custom-selected muscle and ligament paths. Both models underwent a gravitational settling process and a forward flexion-to-extension movement. The model-specific load distribution in an equilibrated upright position and local stiffness in the L4/5 functional spinal unit (FSU) is compared. Load sharing between occurring internal forces generated by individual biological structures and their contribution to the FSU stiffness was computed. The main finding of our simulations is an apparent shift in load sharing with individualisation from an equally distributed element contribution of IVD, ligaments and muscles in the generic spine model to a predominant muscle contribution in the individualised model depending on the analysed spine level.

     

Trunk muscle activation and associated lumbar spine joint shear forces under different levels of external forward force applied to the trunk.

High anterior intervertebral shear loads could cause low back injuries and therefore the neuromuscular system may actively counteract these forces. This study investigated whether, under constant moment loading relative to L3L4, an increased externally applied forward force on the trunk results in a shift in muscle activation towards the use of muscles with more backward directed lines of action, thereby reducing the increase in total joint shear force. Twelve participants isometrically resisted forward forces, applied at several locations on the trunk, while moments were held constant relative to L3L4. Surface EMG and lumbar curvature were measured, and an EMG-driven muscle model was used to calculate compression and shear forces at all lumbar intervertebral joints. Larger externally applied forward forces resulted in a flattening of the lumbar lordosis and a slightly more backward directed muscle force. Furthermore, the overall muscle activation increased. At the T12L1 to L3L4 joint, resulting joint shear forces remained small (less than 200N) because the average muscle force pulled backward relative to those joints. However, at the L5S1 joint the average muscle force pulled the trunk forward so that the increase in muscle force with increasing externally applied forward force caused a further rise in shear force (by 102.1N, SD=104.0N), resulting in a joint shear force of 1080.1N (SD=150.4N) at 50Nm moment loading. It is concluded that the response of the neuromuscular system to shear force challenges tends to increase rather than reduce the shear loading at the lumbar joint that is subjected to the highest shear forces.     

Effects of mechanical compression on metabolism and distribution of oxygen and lactate in intervertebral disc

The objective of this study was to examine the effects of mechanical compression on metabolism and distributions of oxygen and lactate in the intervertebral disc (IVD) using a new formulation of the triphasic theory. In this study, the cellular metabolic rates of oxygen and lactate were incorporated into the newly developed formulation of the mechano-electrochemical mixture model [Huang, C.-Y., Gu, W.Y., 2007. Effect of tension-compression nonlinearity on solute transport in charged hydrated fibrosus tissues under dynamic unconfined compression. Journal of Biomechanical Engineering 129, 423-429]. The model was used to numerically analyze metabolism and transport of oxygen and lactate in the IVD under static or dynamic compression. The theoretical analyses demonstrated that compressive loading could affect transport and metabolism of nutrients. Dynamic compression increased oxygen concentration, reduced lactate accumulation, and promoted oxygen consumption and lactate production (i.e., energy conversion) within the IVD. Such effects of dynamic loading were dependent on strain level and loading frequency, and more pronounced in the IVD with less permeable endplate. In contrast, static compression exhibited inverse effects on transport and metabolism of oxygen and lactate. The theoretical predictions in this study are in good agreement with those in the literature. This study established a new theoretical model for analyzing cellular metabolism of nutrients in hydrated, fibrous soft tissues under mechanical compression.