Multilevel finite element modeling for the prediction of local cellular deformation in bone

The underlying mechanisms by which bone cells respond to mechanical stimuli or how mechanical loads act on osteocytes housed in lacunae in bone are not well understood. In this study, a multilevel finite element (FE) approach is applied to predict local cell deformations in bone tissue. The local structure of the matrix dictates the local mechanical environment of an osteocyte. Cell deformations are predicted from detailed linear FE analysis of the microstructure, consisting of an arrangement of cells embedded in bone matrix material. This work has related the loads applied to a whole femur during the stance phase of the gait cycle to the strain of a single lacuna and of canaliculi. The predicted bone matrix strains around osteocyte lacunae and canaliculi were nonuniform and differed significantly from the macroscopically measured strains. Peak stresses and strains in the walls of the lacuna were up to six times those in the bulk extracellular matrix. Significant strain concentrations were observed at sites where the process meets the cell body.     

Experimental and finite element analysis for prediction of kidney injury under blunt impact

Kidneys are third most injured organs in abdominal trauma after liver and spleen; this study therefore is an attempt to understand the behaviour of kidneys under blunt trauma. Dynamic impact tests were performed on 20 fresh porcine kidneys to study the injury propagation in the organ, and the acceleration of the impactor was measured. A kidney model was developed with structural details like capsule and cortex. The kidney cortex was modelled with solid hexahedral elements and the capsule was modelled with quadratic shell elements. The material models for the capsule and cortex were used from the experimental data reported in our previous study. The developed model was calibrated using previous and current experimental results to reproduce the injuries of the organ in terms of acceleration of the impactor, and the injuries sustained by the organ during the experiments. The developed kidney model is observed to be robust and can be integrated with the available human body finite element models to simulate accidents and to predict or simulate injuries.     

Stability analysis and finite element simulation of bone remodeling model

Bone remodeling is widely viewed as a dynamic process--maintaining bone structure through a balance between the opposed activities of osteoblast and osteoclast cells--in which the stability problem is often pointed out. By an analytical approach, we present a bone remodeling model applied to n unit-elements in order to analyze the stationary states and the condition of their stability. In addition, this theory has been simulated in a computer model using the Finite Element Method (FEM) to show a relationship between the bone remodeling process and the stability analysis.     

Experimental and finite element analysis of the mouse caudal vertebrae loading model: prediction of cortical and trabecular bone adaptation

In this study, we attempt to predict cortical and trabecular bone adaptation in the mouse caudal vertebrae loading model using knowledge of bone’s local mechanical environment at the onset of loading. In a previous study, we demonstrated appreciable 25.9 and 11% increases in both trabecular and cortical bone volume density, respectively, when subjecting the fifth caudal vertebrae (C5) of C57BL/6 (B6) mice to an acute loading regime (amplitude of 8N, 3000 cycles, 10 Hz, 3 times a week for 4 weeks). We have also established a validated finite element (FE) model of the C5 vertebra using micro-computed tomography (micro-CT), which characterizes, in 3D, the micro-mechanical strains present in both cortical and trabecular compartments due to the applied loads. To investigate the relationship between load-induced bone adaptation and mechanical strains in-vivo and in-silico data sets were compared. Using data from the previous cross-sectional study, we divided cortical and trabecular compartments into 15 subregions and determined, for each region, a bone formation parameter ΔBV/BS (a cross-sectional measure of the bone volume added to cortical and trabecular surfaces following the described loading regime). Linear regression was then used to correlate mean regional values of ΔBV/BS with mean values of mechanical strains derived from the FE models which were similarly regionalized. The mechanical parameters investigated were strain energy density (SED), the orthogonal strains (e _x , e _y , e _z ) and the three shear strains (e _xy , e _yz , e _zx ). For cortical regions, regression analysis showed SED to correlate extremely well with ΔBV/BS (R ² =?0.82) and e _z (R ²?=?0.89). Furthermore, SED was found to predict expansion of the cortical shell correlating significantly with the regional percentage increases in cortical tissue volume (R ² = 0.92), cortical marrow volume (R ² =?0.91) and cortical thickness (R ² = 0.56). For trabecular regions, FE parameters were found not to correlate with load-induced trabecular bone morphology. These results indicate that load-induced cortical morphology can be predicted from population data, whereas the prediction of trabecular morphology requires subject-specific micro- architecture.     

A voxel-based formulation for contact finite element analysis

To date, voxel-based finite element models have not been feasible for contact problems, owing to the inherent stair-step boundary discontinuities. New preprocessing techniques are reported herein to mesh these boundaries smoothly, for purposes of contact stress analysis. Further, new techniques are reported to concentrate the mesh resolution automatically near the articular surface, thus reducing the problem size to levels compatible with executing nonlinear problems on contemporary engineering workstations. Close approximations to Hertzian analytical solutions were obtained for spherical and cylindrical geometries meshed in this manner, and an illustrative anatomical contact problem of the human hip joint is presented.     

A novel adaptive algorithm for 3D finite element analysis to model extracortical bone growth

Extracortical bone growth with osseointegration of bone onto the shaft of massive bone tumour implants is an important clinical outcome for long-term implant survival. A new computational algorithm combining geometrical shape changes and bone adaptation in 3D Finite Element simulations has been developed, using a soft tissue envelope mesh, a novel concept of osteoconnectivity, and bone remodelling theory. The effects of varying the initial tissue density, spatial influence function and time step were investigated. The methodology demonstrated good correspondence to radiological results for a segmental prosthesis.     

Fracture prediction for the proximal femur using finite element models: Part II--Nonlinear analysis.

In Part I we reported the results of linear finite element models of the proximal femur generated using geometric and constitutive data collected with quantitative computed tomography. These models demonstrated excellent agreement with in vitro studies when used to predict ultimate failure loads. In Part II, we report our extension of those finite element models to include nonlinear behavior of the trabecular and cortical bone. A highly nonlinear material law, originally designed for representing concrete, was used for trabecular bone, while a bilinear material law was used for cortical bone. We found excellent agreement between the model predictions and in vitro fracture data for both the onset of bone yielding and bone fracture. For bone yielding, the model predictions were within 2 percent for a load which simulated one-legged stance and 1 percent for a load which simulated a fall. For bone fracture, the model predictions were within 1 percent and 17 percent, respectively. The models also demonstrated different fracture mechanisms for the two different loading configurations. For one-legged stance, failure within the primary compressive trabeculae at the subcapital region occurred first, leading to load transfer and, ultimately, failure of the surrounding cortical shell. However, for a fall, failure of the cortical and trabecular bone occurred simultaneously within the intertrochanteric region. These results support our previous findings that the strength of the subcapital region is primarily due to trabecular bone whereas the strength of the intertrochanteric region is primarily due to cortical bone.     

Fracture prediction for the proximal femur using finite element models: Part I--Linear analysis.

Over 90 percent of the more than 250,000 hip fractures that occur annually in the United States are the result of falls from standing height. Despite this, the stresses associated with femoral fracture from a fall have not been investigated previously. Our objectives were to use three-dimensional finite element models of the proximal femur (with geometries and material properties based directly on quantitative computed tomography) to compare predicted stress distributions for one-legged stance and for a fall to the lateral greater trochanter. We also wished to test the correspondence between model predictions and in vitro strain gage data and failure loads for cadaveric femora subjected to these loading conditions. An additional goal was to use the model predictions to compare the sensitivity of several imaging sites in the proximal femur which are used for the in vivo prediction of hip fracture risk. In this first of two parts, linear finite element models of two unpaired human cadaveric femora were generated. In Part II, the models were extended to include nonlinear material properties for the cortical and trabecular bone. While there was poor correspondence between strain gage data and model predictions, there was excellent agreement between the in vitro failure data and the linear model, especially using a von Mises effective strain failure criterion. Both the onset of structural yielding (within 22 and 4 percent) and the load at fracture (within 8 and 5 percent) were predicted accurately for the two femora tested. For the simulation of one-legged stance, the peak stresses occurred in the primary compressive trabeculae of the subcapital region.(ABSTRACT TRUNCATED AT 250 WORDS)     

A finite element study of mechanical stimuli in scaffolds for bone tissue engineering

Mechanical stimuli are one of the factors that affect cell proliferation and differentiation in the process of bone tissue regeneration. Knowledge on the specific deformation sensed by cells at a microscopic level when mechanical loads are applied is still missing in the development of biomaterials for bone tissue engineering. The objective of this study was to analyze the behavior of the mechanical stimuli within some calcium phosphate-based scaffolds in terms of stress and strain distributions in the solid material phase and fluid velocity, fluid pressure and fluid shear stress distributions in the pores filled of fluid, by means of micro computed tomographed (CT)-based finite element (FE) models. Two samples of porous materials, one of calcium phosphate-based cement and another of biodegradable glass, were used. Compressive loads equivalent to 0.5% of compression applied to the solid material phase and interstitial fluid flows with inlet velocities of 1, 10 and 100 microm/s applied to the interconnected pores were simulated, changing also the inlet side and the viscosity of the medium. Similar strain distributions for both materials were found, with compressive and tensile strain maximal values of 1.6% and 0.6%, respectively. Mean values were consistent with the applied deformation. When 10 microm/s of inlet fluid velocity and 1.45 Pas viscosity, maximal values of fluid velocity were 12.76 mm/s for CaP cement and 14.87 mm/s for glass. Mean values were consistent with the inlet ones applied, and mean values of shear stress were around 5 x 10(-5)Pa. Variations on inlet fluid velocity and fluid viscosity produce proportional and independent changes in fluid velocity, fluid shear stress and fluid pressure. This study has shown how mechanical loads and fluid flow applied on the scaffolds cause different levels of mechanical stimuli within the samples according to the morphology of the materials.     

Cement flow during impaction allografting: a finite element analysis

Cement intrusion into cancellous or impacted bone is not well understood. We adopted an engineering mechanics approach to predict the effect of surgical variables on the cement intrusion into impacted cancellous bone, used for the revision of failed total hip replacement with the impaction allografting technique. Specifically, a three-dimensional finite element model was used to determine the effects of cement viscosity, the magnitude and duration of pressurization, and the distribution of the porosity along the femur on cement intrusion. The overall averaged mean intrusion depth difference between the finite element model prediction and the cadaveric measurements was 1.1mm. The depth of penetration increased with higher pressurization pressure, duration of pressurization, and earlier stem insertion (lower viscosity), but maintained a similar profile. The distribution of the porosity along the femur determined the intrusion profile. Cement viscosity, the applied pressure or the duration of the pressurization can be adjusted to limit the cement volume injected into the medullary canal and therefore prevent the cement from reaching the endosteal surface.     

Intravoxel bone micromechanics for microCT-based finite element simulations

While micro-FE simulations have become a standard tool in computational biomechanics, the choice of appropriate material properties is still a relevant topic, typically involving empirical grey value-to-elastic modulus relations. We here derive the voxel-specific volume fractions of mineral, collagen, and water, from tissue-independent bilinear relations between mineral and collagen content in extracellular bone tissue (J. Theor. Biol. 287: 115, 2011), and from the measured X-ray attenuation information quantified in terms of grey values. The aforementioned volume fractions enter a micromechanics representation of bone tissue, as to deliver voxel-specific stiffness tensors. In order to check the relevance of this strategy, we convert a micro Computer Tomograph of a mouse femur into a regular Finite Element mesh, apply forces related to the dead load of a standing mouse, and then compare simulation results based on voxel-specific heterogeneous elastic properties to results based on homogeneous elastic properties related to the spatial average over the solid bone matrix compartment, of the X-ray attenuation coefficients. The element-specific strain energy density illustrates that use of homogeneous elastic properties implies overestimation of the organ stiffness. Moreover, the simulation reveals large tensile normal stresses throughout the femur neck, which may explain the mouse femur neck's trabecular morphology being quite different from the human case, where the femur neck bears compressive forces and bending moments.     

Visualization and finite element analysis of pulsatile flow in models of the abdominal aortic aneurysm

Pulsatile flows in glass models simulating fusiform and lateral saccular aneurysms were investigated by a flow visualization method. When resting fluid starts to flow, the initial fluid motion is practically irrotational. After a short period of time, the flow began to separate from the proximal wall of the aneurysm. Then the separation bubble or vortex grew rapidly in size and filled the whole area of the aneurysm circumferentially. During this period of time, the center of the vortex moved from the proximal end to the distal point of the aneurysm. The transient reversal flow, for instance, which may occur at the end of the ejection period, passed between the wall of the aneurysm and the centrally located vortex. When the rate and pulsatile frequency of flow were high, the vortex broke down into highly disturbed flow (or turbulence) at the distal portion of the aneurysm. The same effect was observed when the length of the aneurysm was increased. A reduction in pulsatile amplitude made the flow pattern close to that in steady flow. A finite element analysis was made to obtain velocity and pressure fields in pulsatile flow through a tube with an axisymmetric expansion. Calculations were performed with the pulsatile flows used in the visualization experiment in order to study the effects of change in the pulsatile wave form by keeping the time-mean Reynolds number and Womersley's parameter unchanged. Calculated instantaneous patterns of velocity field and stream lines agreed well with the experimental results. The appearance and disappearance of the vortex in the dilated portion and its development resulted in complex distributions of pressure and shear fields. Locally minimum and maximum values of wall shear stress occurred at points just upstream and downstream of the distal end of the expansion when the flow rate reached its peak.     

Finite element implementation of mechanochemical phenomena in neutral deformable porous media under finite deformation

Biological soft tissues and cells may be subjected to mechanical as well as chemical (osmotic) loading under their natural physiological environment or various experimental conditions. The interaction of mechanical and chemical effects may be very significant under some of these conditions, yet the highly nonlinear nature of the set of governing equations describing these mechanisms poses a challenge for the modeling of such phenomena. This study formulated and implemented a finite element algorithm for analyzing mechanochemical events in neutral deformable porous media under finite deformation. The algorithm employed the framework of mixture theory to model the porous permeable solid matrix and interstitial fluid, where the fluid consists of a mixture of solvent and solute. A special emphasis was placed on solute-solid matrix interactions, such as solute exclusion from a fraction of the matrix pore space (solubility) and frictional momentum exchange that produces solute hindrance and pumping under certain dynamic loading conditions. The finite element formulation implemented full coupling of mechanical and chemical effects, providing a framework where material properties and response functions may depend on solid matrix strain as well as solute concentration. The implementation was validated using selected canonical problems for which analytical or alternative numerical solutions exist. This finite element code includes a number of unique features that enhance the modeling of mechanochemical phenomena in biological tissues. The code is available in the public domain, open source finite element program FEBio (http:∕∕mrl.sci.utah.edu∕software).     

Poroelastic finite element analysis of a bone specimen under cyclic loading

It had been suggested that the fluid embodied in bone lacunar-canalicular porosity may play an important role in bone remodelling [Weinbaum et al., 1994. Journal of Biomechanics 27, 339-360]. In this paper a finite element model of a poroelastic prismatic solid of rectangular cross-section is considered to simulate bone behaviour, precisely as in the previous work by Zhang and Cowin [Zhang and Cowin, 1994. Journal of Mechanical Physics of Solids 42, 1575-1599]. This solid is subject to combined cyclic axial and bending loads at its end. The objectives of the study are: (1) to verify the accuracy of the simplifying hypotheses underlying the analytical solutions established by the above authors; (2) to provide further insight into the behaviour of that solid; (3) to test the advantages in generality and versatility and the computing costs of general-purpose finite element codes in poroelastic analysis. The study is parametric with respect to the fluid leakage coefficient, to the ratio of the bending moment and axial load, and to the ratio of the characteristic relaxation time of the pore pressure over the excitation period. Results show that, for all the cases considered, the pore pressure distribution along the section height of the poroelastic beam exhibits a very good matching with previous analytical results. Stresses transversal with respect to the beam axis (assumed as constant or zero in previous analytical solutions) are evaluated. The analysis pointed out that: (1) the effects due to end-loads with zero resultants practically extinguish within a distance from the beam end almost equal to a typical length of the cross-section; (2) cross-sections remain plane above that distance; (3) the transversal total stresses are three orders of magnitude lower than axial stress.     

Simulation of vertebral trabecular bone loss using voxel finite element analysis

Trabecular bone loss in human vertebral bone is characterised by thinning and eventual perforation of the horizontal trabeculae. Concurrently, vertical trabeculae are completely lost with no histological evidence of significant thinning. Such bone loss results in deterioration in apparent modulus and strength of the trabecular core. In this study, a voxel-based finite element program was used to model bone loss in three specimens of human vertebral trabecular bone. Three sets of analyses were completed. In Set 1, strain adaptive resorption was modelled, whereby elements which were subject to the lowest mechanical stimulus (principal strain) were removed. In Set 2, both strain adaptive and microdamage mechanisms of bone resorption were included. Perforation of vertical trabeculae occurred due to microdamage resorption of elements with strains that exceeded a damage threshold. This resulted in collapse of the trabecular network under compression loading for two of the specimens tested. In Set 3, the damage threshold strain was gradually increased as bone loss progressed, resulting in reduced levels of microdamage resorption. This mechanism resulted in trabecular architectures in which vertical trabeculae had been perforated and which exhibited similar apparent modulus properties compared to experimental values reported in the literature. Our results indicate that strain adaptive remodelling alone does not explain the deterioration in mechanical properties that have been observed experimentally. Our results also support the hypothesis that horizontal trabeculae are lost principally by strain adaptive resorption, while vertical trabeculae may be lost due to perforation from microdamage resorption followed by rapid strain adaptive resorption of the remaining unloaded trabeculae.     

A three-dimensional finite element analysis of the upper tibia

A three-dimensional finite element model of the proximal tibia has been developed to provide a base line for further modeling of prosthetic resurfaced tibiae. The geometry for the model was developed by digitizing coronal and transverse sections made with the milling machine, from one fresh tibia of average size. The load is equally distributed between the medial and lateral compartments over contact areas that were reported in the literature. An indentation test has been used to measure the stiffness and the ultimate strength of cancellous bone in four cadaver tibiae. These values provided the statistical basis for characterising the inhomogeneous distribution of the cancellous bone properties in the proximal tibia. All materials in the model were assumed to be linearly elastic and isotropic. Mechanical properties for the cortical bone and cartilage have been taken from the literature. Results have been compared with strain gage tests and with a two-dimensional axisymmetric finite element model both from the literature. Qualitative comparison between trabecular alignment, and the direction of the principal compressive stresses in the cancellous bone, showed a good relationship. Maximum stresses in the cancellous bone and cortical bone, under a load which occurs near stance phase during normal gait, show safety factors of approximately eight and twelve, respectively. The load sharing between the cancellous bone and the cortical bone has been plotted for the first 40 mm distally from the tibial eminence.     

Interstitial fluid flow in the osteon with spatial gradients of mechanical properties: a finite element study

Bone remodelling is the process that maintains bone structure and strength through adaptation of bone tissue mechanical properties to applied loads. Bone can be modelled as a porous deformable material whose pores are filled with cells, organic material and interstitial fluid. Fluid flow is believed to play a role in the mechanotransduction of signals for bone remodelling. In this work, an osteon, the elementary unit of cortical bone, is idealized as a hollow cylinder made of a deformable porous matrix saturated with an interstitial fluid. We use Biot’s poroelasticity theory to model the mechanical behaviour of bone tissue taking into account transverse isotropic mechanical properties. A finite element poroelastic model is developed in the COMSOL Multiphysics software. Elasticity equations and Darcy’s law are implemented in this software; they are coupled through the introduction of an interaction term to obtain poroelasticity equations. Using numerical simulations, the investigation of the effect of spatial gradients of permeability or Poisson’s ratio is performed. Results are discussed for their implication on fluid flow in osteons: (i) a permeability gradient affects more the fluid pressure than the velocity profile; (ii) focusing on the fluid flow, the key element of loading is the strain rate; (iii) a Poisson’s ratio gradient affects both fluid pressure and fluid velocity. The influence of textural and mechanical properties of bone on mechanotransduction signals for bone remodelling is also discussed.     

Trabecular bone fracture healing simulation with finite element analysis and fuzzy logic

Trabecular bone fractures heal through intramembraneous ossification. This process differs from diaphyseal fracture healing in that the trabecular marrow provides a rich vascular supply to the healing bone, there is very little callus formation, woven bone forms directly without a cartilage intermediary, and the woven bone is remodelled to form trabecular bone. Previous studies have used numerical methods to simulate diaphyseal fracture healing or bone remodelling, however not trabecular fracture healing, which involves both tissue differentiation and trabecular formation. The objective of this study was to determine if intramembraneous bone formation and remodelling during trabecular bone fracture healing could be simulated using the same mechanobiological principles as those proposed for diaphyseal fracture healing. Using finite element analysis and the fuzzy logic for diaphyseal healing, the model simulated formation of woven bone in the fracture gap and subsequent remodelling of the bone to form trabecular bone. We also demonstrated that the trabecular structure is dependent on the applied loading conditions. A single model that can simulate bone healing and remodelling may prove to be a useful tool in predicting musculoskeletal tissue differentiation in different vascular and mechanical environments.