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.     

Embedding of human vertebral bodies leads to higher ultimate load and altered damage localisation under axial compression

Computer tomography (CT)-based finite element (FE) models of vertebral bodies assess fracture load in vitro better than dual energy X-ray absorptiometry, but boundary conditions affect stress distribution under the endplates that may influence ultimate load and damage localisation under post-yield strains. Therefore, HRpQCT-based homogenised FE models of 12 vertebral bodies were subjected to axial compression with two distinct boundary conditions: embedding in polymethylmethalcrylate (PMMA) and bonding to a healthy intervertebral disc (IVD) with distinct hyperelastic properties for nucleus and annulus. Bone volume fraction and fabric assessed from HRpQCT data were used to determine the elastic, plastic and damage behaviour of bone. Ultimate forces obtained with PMMA were 22% higher than with IVD but correlated highly (R² = 0.99). At ultimate force, distinct fractions of damage were computed in the endplates (PMMA: 6%, IVD: 70%), cortex and trabecular sub-regions, which confirms previous observations that in contrast to PMMA embedding, failure initiated underneath the nuclei in healthy IVDs. In conclusion, axial loading of vertebral bodies via PMMA embedding versus healthy IVD overestimates ultimate load and leads to distinct damage localisation and failure pattern.     

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.     

Quantifying the centre of rotation pattern in a multi-body model of the lumbar spine

Understanding the kinematics of the spine provides paramount knowledge for many aspects of the clinical analysis of back pain. More specifically, visualisation of the instantaneous centre of rotation (ICR) enables clinicians to quantify joint laxity in the segments, avoiding a dependence on more inconclusive measurements based on the range of motion and excessive translations, which vary in every individual. Alternatively, it provides motion preserving designers with an insight into where a physiological ICR of a motion preserving prosthesis can be situated in order to restore proper load distribution across the passive and active elements of the lumbar region. Prior to the use of an unconstrained dynamic musculoskeletal model system, based on multi-body models capable of transient analysis, to estimate segmental loads, the model must be kinematically evaluated for all possible sensitivity due to ligament properties and the initial locus of intervertebral disc (IVD). A previously calibrated osseoligamentous model of lumbar spine was used to evaluate the changes in ICR under variation of the ligament stiffness and initial locus of IVD, when subjected to pure moments from 0 to 15 Nm. The ICR was quantified based on the closed solution of unit quaternion that improves accuracy and prevents coordinate singularities, which is often observed in Euler-based methods and least squares principles. The calculation of the ICR during flexion/extension revealed complexity and intrinsic nonlinearity between flexion and extension. This study revealed that, to accommodate a good agreement between in vitro data and the multi-body model predictions, in flexion more laxity is required than in extension. The results showed that the ICR location is concentrated in the posterior region of the disc, in agreement with previous experimental studies. However, the current multi-body model demonstrates a sensitivity to the initial definition of the ICR, which should be recognised as a limitation of the method. Nevertheless, the current simulations suggest that, due to the constantly evolving path of the ICR across the IVD during flexion–extension, a movable ICR is a necessary condition in multi-body modelling of the spine, in the context of whole body simulation, to accurately capture segmental kinematics and kinetics.     

Intervertebral disc creep behavior assessment through an open source finite element solver

Degenerative Disc Disease (DDD) is one of the largest health problems faced worldwide, based on lost working time and associated costs. By means of this motivation, this work aims to evaluate a biomimetic Finite Element (FE) model of the Intervertebral Disc (IVD). Recent studies have emphasized the importance of an accurate biomechanical modeling of the IVD, as it is a highly complex multiphasic medium. Poroelastic models of the disc are mostly implemented in commercial finite element packages with limited access to the algorithms. Therefore, a novel poroelastic formulation implemented on a home-developed open source FE solver is briefly addressed throughout this paper. The combination of this formulation with biphasic osmotic swelling behavior is also taken into account. Numerical simulations were devoted to the analysis of the non-degenerated human lumbar IVD time-dependent behavior. The results of the tests performed for creep assessment were inside the scope of the experimental data, with a remarkable improvement of the numerical accuracy when compared with previously published results obtained with ABAQUS^®. In brief, this in-development open-source FE solver was validated with literature experimental data and aims to be a valuable tool to study the IVD biomechanics and DDD mechanisms.     

Modeling the role of IGF-1 on extracellular matrix biosynthesis and cellularity in intervertebral disc

The insulin-like growth factor-1 (IGF-1) is a well-known anabolic agent for intervertebral disc (IVD), promoting both proteoglycan (PG) biosynthesis and cell proliferation. Accordingly, it is believed that IGF-1 may play a central role in IVD homeostasis. Furthermore, the exogenous administration of IGF-1 has been proposed as a possible therapeutic strategy for disc degeneration. The objectives of this study were to develop a new computational framework for describing the mechanisms regulating IGF-mediated homeostasis in IVD, and to apply this numerical tool for investigating the effectiveness of exogenous administration of IGF-1 for curing disc degeneration. A diffusive–reactive model was developed for describing competitive binding of IGF-1 to its binding proteins and cell surface receptors, with the latter reaction initiating the intracellular signaling mechanism leading to PG production and cell proliferation. Because PG production increases cell metabolic rate, and cell proliferation increases nutritional demand, nutrients transport and metabolism were also included into the model, and co-regulated, together with IGF-1, IVD cellularity. The sustainability and the effectiveness of IGF-mediated anabolism were investigated for conditions of pathologically insufficient nutrient supply, and for the case of exogenous administration of IGF-1 to degenerated IVD. Results showed that pathological nutrients deprivation, by decreasing cellularity, caused a reduction of PG biosynthesis. Also, exogenous administration of IGF-1 was only beneficial in well-nourished regions of IVD, and exacerbated cell mortality in malnourished regions. These findings remark the central role of nutrition in IVD health, and suggest that adequate nutritional supply is paramount for achieving a successful IGF-based therapy for disc degeneration.     

Effect of Cartilage Endplate on Cell Based Disc Regeneration: A Finite Element Analysis

This study examines the effects of cartilage endplate (CEP) calcification and the injection of intervertebral disc (IVD) cells on the nutrition distributions inside the human IVD under physiological loading conditions using multiphasic finite element modeling. The human disc was modeled as an inhomogeneous mixture consisting of a charged elastic solid, water, ions (Na⁺ and Cl⁻), and nutrient solute(oxygen,glucose and lactate) phases. The effect of the endplate calcification was simulated by a reduction of the tissue porosity (i.e., water volume faction) from 0.60 to 0.48. The effect of cell injection was simulated by increasing the cell density in the nucleus pulposus (NP) region by 50%, 100%, and 150%. Strain-dependent transport properties(e.g., hydraulic permeability and solute diffusivities) were considered to couple the solute transport and the mechanical loading. The simulation results showed that nutrient solute distribution inside the discis maintained at a stable state during the day and night. The physiological diurnal cyclic loading does not change the nutrient environment in the human IVD. The cartilage endplate plays a significant role in the nutrient supply to human IVD. Calcification of the cartilage endplate significantly reduces the nutrient levels in human IVD. Therefore, in cell based therapy for IVD regeneration, theincreased nutrient demand as a result of cell injection needs to be addressed. Excessive numbers of injected cells may cause further deterioration of the nutrient environment in the degenerated disc. This study is important for understanding the pathology of IVD degeneration and providing new insights into cell based therapies for low back pain.     

Finite element analyses of human vertebral bodies embedded in polymethylmethalcrylate or loaded via the hyperelastic intervertebral disc models provide equivalent predictions of experimental strength

Quantitative computer tomography (QCT)-based finite element (FE) models of vertebral body provide better prediction of vertebral strength than dual energy X-ray absorptiometry. However, most models were validated against compression of vertebral bodies with endplates embedded in polymethylmethalcrylate (PMMA). Yet, loading being as important as bone density, the absence of intervertebral disc (IVD) affects the strength. Accordingly, the aim was to assess the strength predictions of the classic FE models (vertebral body embedded) against the in vitro and in silico strengths of vertebral bodies loaded via IVDs. High resolution peripheral QCT (HR-pQCT) were performed on 13 segments (T11/T12/L1). T11 and L1 were augmented with PMMA and the samples were tested under a 4° wedge compression until failure of T12. Specimen-specific model was generated for each T12 from the HR-pQCT data. Two FE sets were created: FE-PMMA refers to the classical vertebral body embedded model under axial compression; FE-IVD to their loading via hyperelastic IVD model under the wedge compression as conducted experimentally. Results showed that FE-PMMA models overestimated the experimental strength and their strength prediction was satisfactory considering the different experimental set-up. On the other hand, the FE-IVD models did not prove significantly better (Exp/FE-PMMA: R²=0.68; Exp/FE-IVD: R²=0.71, p=0.84). In conclusion, FE-PMMA correlates well with in vitro strength of human vertebral bodies loaded via real IVDs and FE-IVD with hyperelastic IVDs do not significantly improve this correlation. Therefore, it seems not worth adding the IVDs to vertebral body models until fully validated patient-specific IVD models become available.     

The influence of serum,glucose and oxygen on intervertebral disc cell growth <Emphasis Type="Italic">in vitro</Emphasis>: implications for degenerative disc disease

Introduction  

The avascular nature of the human intervertebral disc (IVD) is thought to play a major role in disc pathophysiology by limiting nutrient supply to resident IVD cells. In the human IVD, the central IVD cells at maturity are normally chondrocytic in phenotype. However, abnormal cell phenotypes have been associated with degenerative disc diseases, including cell proliferation and cluster formation, cell death, stellate morphologies, and cell senescence. Therefore, we have examined the relative influence of possible blood-borne factors on the growth characteristics of IVD cells in vitro.     

Finite element simulation of an artificial intervertebral disk using fiber reinforced laminated composite model

Degeneration of intervertebral disk (IVD) has been increased in recent years. The lumbar herniation can be cured using conservative and surgical procedures. Surgery is considered after failure of conservative treatment. Partial discectomy, fusion, and total disk replacement (TDR) are also common surgical treatments for degenerative disk disease. However, due to limitations and disadvantages of the current treatments, many studies have been carried out to approach the best design of mimicking natural disk. Recently, a new method of TDRs has been introduced using nature deformation of IVD by reinforced fibers of annulus fibrosis. Nonetheless, owing to limitations of experimental works on the human body, numerical studies of IVD may help to understand load transfer and biomechanical properties within the disks with reinforced fibers. In this study, a three-dimensional (3D) finite element model of the L2-L3 disk vertebrae unit with 12 vertical fibers embedded into annulus fibrosis was constructed. The IVD was subjected to compressive force, bending moment, and axial torsion. The most important parameters of disk failures were compared to that of experimental data. The results showed that the addition of reinforced fibers into the disk invokes a significant decrease of stress in the nucleus and annulus. The findings of this study may have implications not only for developing IVDs with reinforced fibers but also for the application of fiber reinforced IVD in orthopedics surgeries as a suitable implant.     

Practical Implementation of Accurate Finite-Element Calculations for Electromagnetic Scattering by Nanoparticles

Plasmonics - The finite-element method (FEM) is increasingly used as a numerical tool to support experimental and theoretical studies of the optical properties of nanoparticles, in contexts such as...     

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.     

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.     

Lumbar spine angles and intervertebral disc characteristics with end-range positions in three planes of motion in healthy people using upright MRI

Understanding changes in lumbar spine (LS) angles and intervertebral disc (IVD) behavior in end-range positions in healthy subjects can provide a basis for developing more specific LS models and comparing people with spine pathology. The purposes of this study are to quantify 3D LS angles and changes in IVD characteristics with end-range positions in 3 planes of motion using upright MRI in healthy people, and to determine which intervertebral segments contribute most in each plane of movement. Thirteen people (average age = 24.4 years, range 18–51 years; 9 females; BMI = 22.4 ± 1.8 kg/m²) with no history of low back pain were scanned in an upright MRI in standing, sitting flexion, sitting axial rotation (left, right), prone on elbows, prone extension, and standing lateral bending (left, right). Global and local intervertebral LS angles were measured. Anterior-posterior length of the IVD and location of the nucleus pulposus was measured. For the sagittal plane, lower LS segments contribute most to change in position, and the location of the nucleus pulposus migrated from a more posterior position in sitting flexion to a more anterior position in end-range extension. For lateral bending, the upper LS contributes most to end-range positions. Small degrees of intervertebral rotation (1–2°) across all levels were observed for axial plane positions. There were no systematic changes in IVD characteristics for axial or coronal plane positions.     

Caveolin-1 expression and stress-induced premature senescence in human intervertebral disc degeneration

Introduction  

Chronic and debilitating low back pain is a common condition and a huge economic burden. Many cases are attributed to age-related degeneration of the intervertebral disc (IVD); however, age-related degeneration appears to occur at an accelerated rate in some individuals. We have previously demonstrated biomarkers of cellular senescence within the human IVD and suggested a role for senescence in IVD degeneration. Senescence occurs with ageing but can also occur prematurely in response to stress. We hypothesised that stress-induced premature senescence (SIPS) occurs within the IVD and here we have investigated the expression and production of caveolin-1, a protein that has been shown previously to be upregulated in SIPS.     

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.     

Quantitative analysis of exogenous IGF-1 administration of intervertebral disc through intradiscal injection

Exogenous administration of IGF-1 has been proposed as a therapy for disc degeneration. The objectives of this study were to develop a numerical model for quantitatively analysing exogenous administration of IGF-1 into the intervertebral disc (IVD) via intradiscal injection and to investigate the effects of IGF-1 administration on distribution of glucose and oxygen in the IVD. In this study, the reversible binding reaction between IGF-1 and IGF binding proteins was incorporated into the mechano-electrochemical mixture model. The model was used to numerically analyse transport of IGF-1, glucose, oxygen and lactate in the IVD after IGF-1 administration. The enhancement of IGF-1 on lactate production was also taken into account in the theoretical model. The numerical analyses using finite element method demonstrated that the binding reactions significantly affect the time-dependent distribution of IGF-1 in the IVD. It was found that the region affected by IGF-1 was smaller and the duration of the therapeutic IGF-1 level was longer in the degenerated disc with a higher concentration of IGF binding proteins. It was also found that the IGF-1 injection can reduce glucose concentration and increase lactate accumulation (i.e., lower pH) in the IVD and these influences were regulated by the IGF-1 binding reactions. This study indicated the complexity of intradiscal administration of growth factors, which needs to be fully analysed in order to achieve a successful outcome. The new theoretical model developed in this study can serve as a powerful tool in analysing and designing the optimal treatments of growth factors for disc degeneration.     

椎间盘微环境对干细胞疗法修复椎间盘的影响的研究概况

椎间盘(IVD)退变是一种常见的病理状态,保守治疗往往失败,IVD变性的患者最后往往需要手术干预。已经提出的几种治疗方案中,只有椎间盘切除术和关节融合术被证明可以达到预料的效果。生物治疗的目的是预防和控制椎间盘变性,改善椎间盘的功能、髓核和纤维环细胞的合成代谢和修复能力,并抑制基质降解。目前,临床应用仍处于起步阶段。间充质干细胞和基因治疗在预防和治疗IVD变性的作用还需要进一步的研究。最近的研究向我们展示了一种新的保护椎间盘的结构和功能的治疗策略:间充质干细胞(MSCs)移植,尤其是骨髓间充质干细胞(BM-MSCs)。而了解MSCs是否可以以及如何在有排拆性的退化的椎间盘中存活并繁殖是十分重要的。因此,本文着重讨论内源性蛋白酶、细胞因子、低氧、低营养、机械负荷及渗透压的调节对移植的MSCs的影响。     

Nucleus pulposus repair with cultured autologous elastic cartilage derived chondrocytes

Low back pain is one of the most common medical conditions in the Western world. Disc degeneration, an inevitable process of ageing, is one of the major causes of low back pain. Autologous chondrocyte transplantation (ACT) is an increasingly popular method of addressing pathological disorders of cartilage. The purpose of our study was to determine whether autologous chondrocytes from elastic cartilage could survive and synthesise a cartilage specific matrix in the intervertebral disc of rabbits. Sixteen lumbar intervertebral discs (IVD) of New Zealand White rabbits were analysed. In 6 IVD, the nucleus pulposus was evacuated and replaced with tissue engineered autologous chondrocytes from auricular cartilage. In the second group, only the nucleus pulposus was evacuated from 6 IVD, with no chondrocytes implantation. Four non-operated IVD were used as a control. Six months after the operation, the animals were euthanized and the IVD were analysed histologically. Autologous cartilage implants were well tolerated by the host for up to six months in vivo. There was only hyaline-like cartilage in the place of the nucleus pulposus. We could not detect any elastic fibres in the new cartilage matrix. In IVD from which only the nucleus pulposus was evacuated and no chondrocytes were implanted, just fibrous tissue was found instead of nucleus pulposus. The overall histological analysis of new cartilage produced after implantation in our study confirmed the hypothesis that ACT from auricular cartilage can be implanted into the IVD instead of the nucleus pulposus and that a significant percentage of implanted chondrocytes survive and produce hyaline-like cartilage.     

HR-pQCT-based homogenised finite element models provide quantitative predictions of experimental vertebral body stiffness and strength with the same accuracy as μFE models

This study validated two different high-resolution peripheral quantitative computer tomography (HR-pQCT)-based finite element (FE) approaches, enhanced homogenised continuum-level (hFE) and micro-finite element (μFE) models, by comparing them with compression test results of vertebral body sections. Thirty-five vertebral body sections were prepared by removing endplates and posterior elements, scanned with HR-pQCT and tested in compression up to failure. Linear hFE and μFE models were created from segmented and grey-level CT images, and apparent model stiffness values were compared with experimental stiffness as well as strength results. Experimental and numerical apparent elastic properties based on grey-level/segmented CT images (N=35) correlated well for μFE (r2=0.748/0.842) and hFE models (r2=0.741/0.864). Vertebral section stiffness values from the linear μFE/hFE models estimated experimental ultimate apparent strength very well (r2=0.920/0.927). Calibrated hFE models were able to predict quantitatively apparent stiffness with the same accuracy as μFE models. However, hFE models needed no back-calculation of a tissue modulus or any kind of fitting and were computationally much cheaper.