A nonlinear homogenized finite element analysis of the primary stability of the bone–implant interface

Stability of an implant is defined by its ability to undergo physiological loading–unloading cycles without showing excessive tissue damage and micromotions at the interface. Distinction is usually made between the immediate primary stability and the long-term, secondary stability resulting from the biological healing process. The aim of this research is to numerically investigate the effect of initial implantation press-fit, bone yielding, densification and friction at the interface on the primary stability of a simple bone–implant system subjected to loading–unloading cycles. In order to achieve this goal, human trabecular bone was modeled as a continuous, elasto-plastic tissue with damage and densification, which material constants depend on bone volume fraction and fabric. Implantation press-fit related damage in the bone was simulated by expanding the drilled hole to the outer contour of the implant. The bone–implant interface was then modeled with unilateral contact with friction. The implant was modeled as a rigid body and was subjected to increasing off-axis loading cycles. This modeling approach is able to capture the experimentally observed primary stability in terms of initial stiffness, ultimate force and progression of damage. In addition, it is able to quantify the micromotions around the implant relevant for bone healing and osseointegration. In conclusion, the computationally efficient modeling approach used in this study provides a realistic structural response of the bone–implant interface and represents a powerful tool to explore implant design, implantation press-fit and the resulting risk of implant failure under physiological loading.     

Probabilistic failure analysis of bone using a finite element model of mineral–collagen composites

Microdamage accumulation is a major pathway for energy dissipation during the post-yield deformation of bone. In this study, a two-dimensional probabilistic finite element model of a mineral–collagen composite was developed to investigate the influence of the tissue and ultrastructural properties of bone on the evolution of microdamage from an initial defect in tension. The probabilistic failure analyses indicated that the microdamage progression would be along the plane of the initial defect when the debonding at mineral–collagen interfaces was either absent or limited in the vicinity of the defect. In this case, the formation of a linear microcrack would be facilitated. However, the microdamage progression would be scattered away from the initial defect plane if interfacial debonding takes place at a large scale. This would suggest the possible formation of diffuse damage. In addition to interfacial debonding, the sensitivity analyses indicated that the microdamage progression was also dependent on the other material and ultrastructural properties of bone. The intensity of stress concentration accompanied with microdamage progression was more sensitive to the elastic modulus of the mineral phase and the nonlinearity of the collagen phase, whereas the scattering of failure location was largely dependent on the mineral to collagen ratio and the nonlinearity of the collagen phase. The findings of this study may help understanding the post-yield behavior of bone at the ultrastructural level and shed light on the underlying mechanism of bone fractures.     

Investigation of impact loading rate effects on the ligamentous cervical spinal load-partitioning using finite element model of functional spinal unit C2–C3

The cervical spine functions as a complex mechanism that responds to sudden loading in a unique manner, due to intricate structural features and kinematics. The spinal load-sharing under pure compression and sagittal flexion/extension at two different impact rates were compared using a bio-fidelic finite element (FE) model of the ligamentous cervical functional spinal unit (FSU) C2–C3. This model was developed using a comprehensive and realistic geometry of spinal components and material laws that include strain rate dependency, bone fracture, and ligament failure. The range of motion, contact pressure in facet joints, failure forces in ligaments were compared to experimental findings. The model demonstrated that resistance of spinal components to impact load is dependent on loading rate and direction. For the loads applied, stress increased with loading rate in all spinal components, and was concentrated in the outer intervertebral disc (IVD), regions of ligaments to bone attachment, and in the cancellous bone of the facet joints. The highest stress in ligaments was found in capsular ligament (CL) in all cases. Intradiscal pressure (IDP) in the nucleus was affected by loading rate change. It increased under compression/flexion but decreased under extension. Contact pressure in the facet joints showed less variation under compression, but increased significantly under flexion/extension particularly under extension. Cancellous bone of the facet joints region was the only component fractured and fracture occurred under extension at both rates. The cervical ligaments were the primary load-bearing component followed by the IVD, endplates and cancellous bone; however, the latter was the most vulnerable to extension as it fractured at low energy impact.     

Is the sheep a suitable model to study the mechanical alterations of disc degeneration in humans? A probabilistic finite element model study

Intervertebral disc degeneration is one major source of low back pain, which because of its complex multifactorial nature renders the treatment challenging and thus necessitates extensive research. Experimental animal models have proven valuable in improving our understanding of degenerative processes and potentially promising therapies. Currently, the sheep is the most frequently used large animal in vivo model in intervertebral disc research. However, despite its undoubted value for investigations of the complex biological and cellular aspects, to date, it is unclear whether the sheep is also suited to study the mechanical aspects of disc degeneration in humans.A parametric finite element (FE) model of the L4–5 spinal motion segment was developed. Using this model, the geometry and the material properties of both the human and the ovine spinal segment as well as different appearances of disc degeneration can be depicted. Under pure and combined loads, it was investigated whether degenerative changes to both the human and the ovine model equivalent caused the same mechanical response.Different patterns of degeneration resulted in large variations in the ranges of motion, intradiscal pressure, ligament and facet loads. In the human, but not in the ovine model, all these results differed significantly between different degrees of degeneration.This FE model study highlighted possible differences in the mechanical response to disc degeneration between human and ovine intervertebral discs and indicates the necessity of further, more detailed, investigations.     

Does screw–bone interface modelling matter in finite element analyses?

The effect of screw-bone interface modelling strategies was evaluated in the setting of a tibial mid-shaft fracture stabilised using locking plates. Three interface models were examined: fully bonded interface; screw with sliding contact with bone; and screw with sliding contact with bone in an undersized pilot hole. For the simulation of the last interface condition we used a novel thermal expansion approach to generate the pre-stress that the bone would be exposed to during screw insertion. The study finds that the global load-deformation response is not influenced by the interface modelling approach employed; the deformation varied by less than 1% between different interaction models. However, interface modelling is found to have a considerable impact on the local stress-strain environment within the bone in the vicinity of the screws. Frictional and tied representations did not have significantly different peak strain values (<5% difference); the frictional interface had higher peak compressive strains while the tied interface had higher tensile strains. The undersized pilot hole simulation produced the largest strains. The peak minimum principal strains for the frictional interface were 26% of those for the undersized pilot hole simulation at a load of 770 N. It is concluded that the commonly used tie constraint can be used effectively when the only interest is the global load-deformation behaviour. Different contact interface models, however, alter the mechanical response around screw holes leading to different predictions for screw loosening, bone damage and stress shielding.     

Dynamic permeability of the lacunar–canalicular system in human cortical bone

A new method for the experimental determination of the permeability of a small sample of a fluid-saturated hierarchically structured porous material is described and applied to the determination of the lacunar–canalicular permeability \((K_\mathrm{LC})\) in bone. The interest in the permeability of the lacunar–canalicular pore system (LCS) is due to the fact that the LCS is considered to be the site of bone mechanotransduction due to the loading-driven fluid flow over cellular structures. The permeability of this space has been estimated to be anywhere from \(10^{-17}\;\) to \(10^{-25}\; \hbox {m}^{2}\) . However, the vascular pore system and LCS are intertwined, rendering the permeability of the much smaller-dimensioned LCS challenging to measure. In this study, we report a combined experimental and analytical approach that allowed the accurate determination of the \(K_\mathrm{LC}\) to be on the order of \(10^{-22}\; \hbox {m}^{2}\) for human osteonal bone. It was found that the \(K_\mathrm{LC}\) has a linear dependence on loading frequency, decreasing at a rate of \(2 \times 10^{-24}\; \hbox {m}^{2}\) /Hz from 1 to 100 Hz, and using the proposed model, the porosity alone was able to explain 86 % of the \(K_\mathrm{LC}\) variability.     

Search for critical loading condition of the spine–A meta analysis of a nonlinear viscoelastic finite element model

The relative vulnerability of spinal motion segments to different loading combinations remains unknown. The meta-analysis described here using the results of a validated L2–L3 nonlinear viscoelastic finite element model was designed to investigate the critical loading and its effect on the internal mechanics of the human lumbar spine. A Box-Behnken experimental design was used to design the magnitude of seven independent variables associated with loads, rotations and velocity of motion. Subsequently, an optimization method was used to find the primary and secondary variables that influence spine mechanical output related to facet forces, disc pressure, ligament forces, annulus matrix compressive/shear stresses and anulus fibers strain. The mechanical responses with respect to the two most-relevant variables were then regressed linearly using the response surface quadratic model. Axial force and sagittal rotation were identified as the most-relevant variables for mechanical responses. The procedure developed can be used to find the critical loading for finite element models with multi input variables. The derived meta-models can be used to predict the risk associated with various loading parameters and in setting safer load limits.     

A model for the blood–brain barrier permeability to water and small solutes

The blood–brain barrier (BBB) has unique structures in order to protect the central nervous system. In addition to the tight junction of the microvessel endothelium, there is a uniform and narrow matrix-like basement membrane (BM) sandwiched between the vessel wall and the astrocyte foot processes ensheathing the cerebral microvessel. To understand the mechanism by which these structural components modulate permeability of the BBB, we developed a mathematical model for water and solute transport across the BBB. The fluid flow in the cleft regions of the BBB were approximated by the Poiseuille flow while those in the endothelial surface glycocalyx layer (SGL) and BM were approximated by the Darcy and Brinkman flows, respectively. Diffusion equations in each region were solved for the solute transport. The anatomical parameters were obtained from electron microscopy studies in the literature. Our model predicts that compared to the peripheral microvessels with endothelium only, the BM and the wrapping astrocytes can reduce hydraulic conductivity (L_p) of the BBB and the permeability to sodium fluorescein (P^NaF) by up to 6-fold when the fiber density in the BM is the same as that in the SGL. Even when the SGL and the tight junctions of the endothelium are compromised, the BM and astrocyte foot processes can still maintain the low L_p and P^NaF of the BBB. Our model predictions indicate that the BM and astrocytes of the BBB provide a great protection to the CNS under both physiological and pathological conditions.     

Influence of single-level lumbar degenerative disc disease on the behavior of the adjacent segments—A finite element model study

The current study investigated mechanical predictors for the development of adjacent disc degeneration. A 3-D finite element model of a lumbar spine was modified to simulate two grades of degeneration at the L4–L5 disc. Degeneration was modeled by changes in geometry and material properties. All models were subjected to follower preloads of 800 N and moment loads in the three principal directions of motion using a hybrid protocol. Degeneration caused changes in the loading and motion patterns of the segments above and below the degenerated disc. At the level (L3–L4) above the degenerated disc, the motion increased due to moderate degeneration by 21% under lateral bending, 26% under axial rotation and 28% under flexion/extension. At the level (L5-S1) below the degenerated disc, motion increased only during lateral bending by 20% due to moderate degeneration. Both the L3–L4 and L5-S1 segment showed a monotonic increase in both the maximum von Mises stress and shear stress in the annulus as degeneration progressed for all loading directions, expect extension at L3–L4. The most significant increase in stress was observed at the L5-S1 level during axial rotation with nearly a ten-fold increase in the maximum shear stress and 103% increase in the maximum von Mises stress. The L5-S1 segment also showed a progressive increase in facet contact force for all loading directions with degeneration. Nucleus pressure did not increase significantly for any loading direction at either the caudal or cephalic adjacent segment. Results suggest that single-level degeneration can increase the risk for injury at the adjacent levels.     

Expression profiles of genes involved in apoptosis and selenium metabolism in articular cartilage of patients with Kashin–Beck osteoarthritis

Kashin–Beck disease (KBD) is a special type of endemic osteoarthritis. It has been suggested that alterations in selenium metabolism and apoptosis play a role in KBD. However, the underlying molecular mechanism remains largely unclear. We performed a microarray analysis using RNA isolated from cartilages of KBD patients and healthy controls, through Significance Analysis of Microarray (SAM) software. Functional gene networks and crucial molecules associated with differentially expressed genes were investigated via Ingenuity Pathway Analysis (IPA) and hub gene analysis. Quantitative real-time PCR was used to check the validation of chip test. We identified 52 up-regulated apoptosis-related genes and 26 down-regulated selenium-related genes between KBD and controls, and these genes associated with the “MYC-mediated apoptosis signaling pathway”. We confirmed the results from array studies with quantitative real-time PCR analysis. Our results suggest that abnormal regulation of selenium metabolism and apoptosis through the MYC mediated signaling pathway contributes to the pathogenesis of KBD, but the relationship between apoptosis gene and selenium gene was not found.     

Characterization of the effects of inter-tree competition on source–sink balance in Chinese pine trees with the GreenLab model

We investigate the characteristics of individual tree response to competition on source–sink balance through the functional–structural plant model GreenLab. Four Chinese pine (Pinus tabulaeformis Carr.) trees were destructively sampled and were divided into two groups: high-density group and low-density group. First, the effects of density on organ dimensions and on organ relative mass were analysed based on experimental measurements. These were primary indicators of the plant response to competition. Second, the hidden parameters of the GreenLab model, as well as a tree-specific characteristic surface, were estimated using the data of total tree biomass for needle and wood compartments, for each of the four trees in parallel. The quality of the fitting is finally validated using data of individual organ mass at shoot level for the sampled branches. The Mann–Whitney Student’s t test showed that there were significant differences between the shoot attributes of the two groups for shoot diameter, shoot biomass and needle biomass. No significant difference was found for current year shoot lengths of the two groups. The parametric identification of the model allowed estimating and comparing the amount of biomass that was allocated to primary growth and to secondary growth in the two density conditions. It showed that biomass allocated to secondary growth (ring compartment) was the most strongly affected by density, and that the organ demand satisfaction ratio profiles of each of these trees were a relevant, integrated indicator of the tree state.     

Validation of density–elasticity relationships for finite element modeling of human pelvic bone by modal analysis

In total hip arthroplasty and particularly in revision surgery, computer assisted pre-operative prediction of the best possible anchorage strategy for implant fixation would be a great help to the surgeon. Computer simulation relies on validated numerical models. In the current study, three density–elasticity relationships (No. 1–3) from the literature for inhomogeneous material parameter assignment from CT data in automated finite element (FE) modeling of long bones were evaluated for their suitability for FE modeling of human pelvic bone. Numerical modal analysis was conducted on 10 FE models of hemipelvic bone specimens and compared to the gold standard provided by experimental modal analysis results from a previous in-vitro study on the same specimens. Overall, calculated resonance frequencies came out lower than measured values. Magnitude of mean relative deviation of numerical resonance frequencies with regard to measured values is lowest for the density–elasticity relationship No. 3 (−15.9%) and considerably higher for both density–elasticity relationships No. 1 (−41.1%) and No. 2 (−45.0%). Mean MAC values over all specimens amount to 77.8% (No. 1), 78.5% (No. 2), and 83.0% (No. 3). MAC results show, that mode shapes are only slightly influenced by material distribution. Calculated resonance frequencies are generally lower than measured values, which indicates, that numerical models lack stiffness. Even when using the best suited (No. 3) out of three investigated density–elasticity relationships, in FE modeling of pelvic bone a considerable underestimation of model stiffness has to be taken into account.     

Sensitivity analysis of the Poisson Nernst–Planck equations: a finite element approximation for the sensitive analysis of an electrodiffusion model

Biological structures exhibiting electric potential fluctuations such as neuron and neural structures with complex geometries are modelled using an electrodiffusion or Poisson Nernst–Planck system of equations. These structures typically depend upon several parameters displaying a large degree of variation or that cannot be precisely inferred experimentally. It is crucial to understand how the mathematical model (and resulting simulations) depend on specific values of these parameters. Here we develop a rigorous approach based on the sensitivity equation for the electrodiffusion model. To illustrate the proposed methodology, we investigate the sensitivity of the electrical response of a node of Ranvier with respect to ionic diffusion coefficients and the membrane dielectric permittivity.

     

Is the between-population variance negligible in the total variance of heterozygosity? Case of a finite number of loci subject to the infinite-allele model in finite monoecious populations

The decomposition of the variance of the average heterozygosity into variances between and within populations is studied in the general case of a finite number of loci. These loci are assumed randomly distributed over chromosome pairs having a non-interference recombination scheme, and independently subject to mutation according to the infinite-allele model. The equilibrium behavior of that decomposition is discussed in the monoecious mating case with regard to each parameter of the model: mutation rate per gene per generation (u), population size (N), number of loci (n), map length of chomosome pairs (L). It is shown that the proportion Q of the between-population variability in the total variance of the average heterozygosity is decreasing as either the mean heterozygosity (θ = 4Nu/(1 + 4Nu)) or the mean number of mutations per gamete per generation (v = nu) is increasing. Moreover, even if Q is always smaller than for this model, it is not negligible unless θ is close to one or v is much larger than one for L long enough.     

Analytical basis for the determination of the lacunar–canalicular permeability of bone using cyclic loading

An analytical model for the determination of the permeability in the lacunar-canalicular porosity of bone using cyclic loading is described in this contribution. The objective of the analysis presented is to relate the lacunar-canalicular permeability to a particular phase angle that is measurable when the bone is subjected to infinitesimal cyclic strain. The phase angle of interest is the lag angle between the applied strain and the resultant stress. Cyclic strain causes the interstitial fluid to move. This movement is essential for the viability of osteocytes and is believed to play a major role in the bone mechanotransduction mechanism. However, certain bone fluid flow properties, notably the permeability of the lacunar-canalicular porosity, are still not accurately determined. In this paper, formulas for the phase angle as a function of permeability for infinitesimal cyclic strain are presented and mathematical expressions for the storage modulus, loss modulus, and loss tangent are obtained. An accurate determination of the PLC permeability will improve our ability to understand mechanotransduction and mechanosensory mechanisms, which are fundamental to the understanding of how to treat osteoporosis, how to cope with microgravity in long-term manned space flights, and how to increase the longevity of prostheses that are implanted in bone tissue.     

A computational reaction–diffusion model for biosynthesis and linking of cartilage extracellular matrix in cell-seeded scaffolds with varying porosity

Biomechanics and Modeling in Mechanobiology - Cartilage tissue engineering is commonly initiated by seeding cells in porous materials such as hydrogels or scaffolds. Under optimal conditions, the...     

A finite viscoelastic–plastic model for describing the uniaxial ratchetting of soft biological tissues

In this paper, a phenomenological constitutive model is constructed to describe the uniaxial ratchetting (i.e., the cyclic accumulation of inelastic deformation) of soft biological tissues in the framework of finite viscoelastic-plasticity. The model is derived from a polyconvex elastic free energy function and addresses the anisotropy of cyclic deformation of the tissues by means of structural tensors. Ratchetting is considered by the evolution of internal variables, and its time-dependence is described by introducing a pseudo-potential function. Accordingly, all the evolution equations are formulated from the dissipation inequality. In numerical examples, the uniaxial monotonic stress–strain responses and ratchetting of some soft biological tissues, such as porcine skin, coronary artery layers and human knee ligaments and tendons, are predicted by the proposed model in the range of finite deformation. It is seen that the predicted monotonic stress–strain responses and uniaxial ratchetting obtained at various loading rates and in various loading directions are in good agreement with the corresponding experimental results.     

The role of bone sialoprotein in the tendon–bone insertion

Tendons/ligaments insert into bone via a transitional structure, the enthesis, which is susceptible to injury and difficult to repair. Fibrocartilaginous entheses contain fibrocartilage in their transitional zone, part of which is mineralized. Mineral-associated proteins within this zone have not been adequately characterized. Members of the Small Integrin Binding Ligand N-linked Glycoprotein (SIBLING) family are acidic phosphoproteins expressed in mineralized tissues. Here we show that two SIBLING proteins, bone sialoprotein (BSP) and osteopontin (OPN), are present in the mouse enthesis. Histological analyses indicate that the calcified zone of the quadriceps tendon enthesis is longer in Bsp^−/− mice, however no difference is apparent in the supraspinatus tendon enthesis. In an analysis of mineral content within the calcified zone, micro-CT and Raman spectroscopy reveal that the mineral content in the calcified fibrocartilage of the quadriceps tendon enthesis are similar between wild type and Bsp^−/− mice. Mechanical testing of the patellar tendon shows that while the tendons fail under similar loads, the Bsp^−/− patellar tendon is 7.5% larger in cross sectional area than wild type tendons, resulting in a 16.5% reduction in failure stress. However, Picrosirius Red staining shows no difference in collagen organization. Data collected here indicate that BSP is present in the calcified fibrocartilage of murine entheses and suggest that BSP plays a regulatory role in this structure, influencing the growth of the calcified fibrocartilage in addition to the weakening of the tendon mechanical properties. Based on the phenotype of the Bsp^−/− mouse enthesis, and the known in vitro functional properties of the protein, BSP may be a useful therapeutic molecule in the reattachment of tendons and ligaments to bone.