Statistical methods in finite element analysis

Finite element analysis (FEA) is a commonly used tool within many areas of engineering and can provide useful information in structural analysis of mechanical systems. However, most analyses within the field of biomechanics usually take no account either of the wide variation in material properties and geometry that may occur in natural tissues or manufacturing imperfections in synthetic materials. This paper discusses two different methods of incorporating uncertainty in FE models. The first, Taguchi's robust parameter design, uses orthogonal matrices to determine how to vary the parameters in a series of FE models, and provides information on the sensitivity of a model to input parameters. The second, probabilistic analysis, enables the distribution of a response variable to be determined from the distributions of the input variables. The methods are demonstrated using a simple example of an FE model of a beam that is assigned material properties and geometry over a range similar to an orthopaedic fixation plate. In addition to showing how each method may be used on its own, we also show how computational effort may be minimised by first identifying the most important input variables before determining the effects of imprecision.     

Pressfit of equatorially roughened cementless acetabular components--a finite element analysis]

AIM: Does the pressfit anchorage of cementless acetabular cups depend on the roughness of the pole? To answer this question the primary pressfit of two cementless acetabular cups which differ only with regard to the roughness of their poles were compared by means of finite elements analysis. MATERIALS AND METHODS: It was assumed that the material properties of bone are homogeneous, isotropic and linearly elastic. Material-specific values of cancellous bone with three different bone densities were used. Assumption of isotropy represents an approximation. RESULTS: Comparison of the two prosthesis designs revealed that both designs/shapes cause similar patterns of bone deformation and tension. CONCLUSIONS: It can therefore be concluded that with regard to pressfit anchorage the prosthesis with milled polar surface is according to FEA mechanically equivalent to the prosthesis with non-milled polar surface.     

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.     

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.     

Considerations for reporting finite element analysis studies in biomechanics

Simulation-based medicine and the development of complex computer models of biological structures is becoming ubiquitous for advancing biomedical engineering and clinical research. Finite element analysis (FEA) has been widely used in the last few decades to understand and predict biomechanical phenomena. Modeling and simulation approaches in biomechanics are highly interdisciplinary, involving novice and skilled developers in all areas of biomedical engineering and biology. While recent advances in model development and simulation platforms offer a wide range of tools to investigators, the decision making process during modeling and simulation has become more opaque. Hence, reliability of such models used for medical decision making and for driving multiscale analysis comes into question. Establishing guidelines for model development and dissemination is a daunting task, particularly with the complex and convoluted models used in FEA. Nonetheless, if better reporting can be established, researchers will have a better understanding of a model's value and the potential for reusability through sharing will be bolstered. Thus, the goal of this document is to identify resources and considerate reporting parameters for FEA studies in biomechanics. These entail various levels of reporting parameters for model identification, model structure, simulation structure, verification, validation, and availability. While we recognize that it may not be possible to provide and detail all of the reporting considerations presented, it is possible to establish a level of confidence with selective use of these parameters. More detailed reporting, however, can establish an explicit outline of the decision-making process in simulation-based analysis for enhanced reproducibility, reusability, and sharing.     

Parametric finite element analysis and closed-form solutions in orthodontics

The goal and clinical relevance of this work was the development of closed formulas that are correct and simple enough for a fast decision making by the orthodontist in the daily praxis. This paper performs a parametric three-dimensional finite element linear analysis on a maxillary central incisor with a root of paraboloidal shape, which is subjected to typical orthodontic force-systems. Parameters of most importance, such as the tooth mobility in translation and in pure moment rotation including orthodontic centers, as well as the stresses inside the periodontal ligament are calculated for a large variety of over four hundred different couples of root lengths and root diameters around a nominal value. Regression analysis is afterwards performed and establishes closed-form solutions, which are also explained in terms of analytical strain energy and hydrostatic stress considerations within the periodontal ligament characterised by a small compressibility. The obtained expressions include both the root length as well as the root diameter.     

Cardiovascular stent design and vessel stresses: a finite element analysis

Intravascular stents of various designs are currently in use to restore patency in atherosclerotic coronary arteries and it has been found that different stents have different in-stent restenosis rates. It has been hypothesized that the level of vascular injury caused to a vessel by a stent determines the level of restenosis. Computational studies may be used to investigate the mechanical behaviour of stents and to determine the biomechanical interaction between the stent and the artery in a stenting procedure. In this paper, we test the hypothesis that two different stent designs will provoke different levels of stress within an atherosclerotic artery and hence cause different levels of vascular injury. The stents analysed using the finite-element method were the S7 (Medtronic AVE) and the NIR (Boston Scientific) stent designs. An analysis of the arterial wall stresses in the stented arteries indicates that the modular S7 stent design causes lower stress to an atherosclerotic vessel with a localized stenotic lesion compared to the slotted tube NIR design. These results correlate with observed clinical restenosis rates, which have found higher restenosis rates in the NIR compared with the S7 stent design. Therefore, the testing methodology outlined here is proposed as a pre-clinical testing tool, which could be used to compare and contrast existing stent designs and to develop novel stent designs.     

Three-dimensional finite element analysis of glass-ceramic dental crowns

Because of the improved esthetic potential of glass-ceramic crowns as dental restorations, they are sometimes preferred over metal-ceramic crowns for restoration of anterior teeth. Because of their relatively high strength, these ceramic crowns are also frequently used for restoration of posterior teeth. However, due to the larger magnitude of biting forces on posterior teeth, intraoral fracture of all-ceramic crowns tends to occur more frequently in posterior crowns (Moffa, 1988). The objective of this study was to determine the relative influence of load orientation and the occlusal thickness of posterior ceramic crowns on the stress distribution which develops under these loading and design conditions. Three-dimensional finite element models for a molar crown were developed to determine the stress distribution under simulated applied loads. Glass-ceramic crowns with occlusal thicknesses of 0.5, 1.5, and 3.0 mm were considered. The largest principal tensile stresses induced in ceramic due to a distributed load of 600 N applied in a cuspal region were approximately 12 and 182 MPa for vertical and horizontal loading orientations, respectively. Stresses which developed in the facial and lingual marginal regions were primarily compressive under vertical loads. However, tensile stresses developed when the load was applied horizontally. Differences in stress distribution within crowns with the three occlusal thicknesses occurred only near the site of loading. Because of the relatively large failure rates of ceramic crowns in the posterior regions, these restorations should be strengthened by improvement in design, composition, and thermal processing conditions. Before any significant progress is made in these areas, these restorations should be used for the anterior teeth. The results of this study suggest that orientation of the applied load has a more important effect on development of large tensile stresses than the occlusal thickness of ceramic.     

Tissue-fluid interface analysis using biphasic finite element method

Numerical studies on fluid-structure interaction have primarily relied on decoupling the solid and fluid sub-domains with the interactions treated as external boundary conditions on the individual sub-domains. The finite element applications for the fluid-structure interactions can be divided into iterative algorithms and sequential algorithms. In this paper, a new computational methodology for the analysis of tissue-fluid interaction problems is presented. The whole computational domain is treated as a single biphasic continuum, and the same space and time discretisation is carried out for the sub-domains using a penalty-based finite element model. This procedure does not require the explicit modelling of additional boundary conditions or interface elements. The developed biphasic interface finite element model is used in analysing blood flow through normal and stenotic arteries. The increase in fluid flow velocity when passing through a stenosed artery and the drop in pressure at the region are captured using this method.     

Potential of finite element analysis for cephalometric investigation

In view of concerns relating to the validity of traditional cephalometric appraisals, we undertook this study to use a potentially powerful method, finite element analysis, to compare cephalometric changes between two samples (subjects with and without orthodontic treatment). The derived data show varying sample contrasts depending on the particular finite element array included in the analysis. Thus, although finite element analysis facilitates rigorous morphometric analysis, further investigation is required before it can be applied universally in cephalometric studies.     

A contact-coupled finite element analysis of the natural adult hip

A non-linear two-dimensional finite element model was used to study phenomena of stress redistribution in the natural adult hip resulting from parametric material property variations in the juxtarticular regions of the femoral head. Despite the geometrical simplifications employed, the intra-articular contact stresses (computed using the FEAP program) were found to be in reasonable qualitative agreement with previous in vitro data for the case of a normal hip. Generalized sclerotic changes in the subchondral plate, as reflected either in apparent modulus increases or in plate thickening, were found to have only minor effects on the computed contact stress distribution, although stress levels within the plate itself were markedly influenced. Localized subchondral plate sclerosis, by contrast, led to marked stress elevations in the cartilage immediately overlying the stiffened bone. Cartilage modulus increases caused increased load uptake for a given imposed deformation, but involved stress distribution increases which were very nearly linearly proportional to the increases in resultant load magnitude. Friction coefficient elevations had no noticeable effects on normal contact stress or upon overall load transmission, but involved complex, possibly slip-related, changes in intra-articular and cartilaginous shear stresses.     

A three-dimensional inverse finite element analysis of the heel pad

Quantification of plantar tissue behavior of the heel pad is essential in developing computational models for predictive analysis of preventive treatment options such as footwear for patients with diabetes. Simulation based studies in the past have generally adopted heel pad properties from the literature, in return using heel-specific geometry with material properties of a different heel. In exceptional cases, patient-specific material characterization was performed with simplified two-dimensional models, without further evaluation of a heel-specific response under different loading conditions. The aim of this study was to conduct an inverse finite element analysis of the heel in order to calculate heel-specific material properties in situ. Multidimensional experimental data available from a previous cadaver study by Erdemir et al. ("An Elaborate Data Set Characterizing the Mechanical Response of the Foot," ASME J. Biomech. Eng., 131(9), pp. 094502) was used for model development, optimization, and evaluation of material properties. A specimen-specific three-dimensional finite element representation was developed. Heel pad material properties were determined using inverse finite element analysis by fitting the model behavior to the experimental data. Compression dominant loading, applied using a spherical indenter, was used for optimization of the material properties. The optimized material properties were evaluated through simulations representative of a combined loading scenario (compression and anterior-posterior shear) with a spherical indenter and also of a compression dominant loading applied using an elevated platform. Optimized heel pad material coefficients were 0.001084 MPa (μ), 9.780 (α) (with an effective Poisson's ratio (ν) of 0.475), for a first-order nearly incompressible Ogden material model. The model predicted structural response of the heel pad was in good agreement for both the optimization (<1.05% maximum tool force, 0.9% maximum tool displacement) and validation cases (6.5% maximum tool force, 15% maximum tool displacement). The inverse analysis successfully predicted the material properties for the given specimen-specific heel pad using the experimental data for the specimen. The modeling framework and results can be used for accurate predictions of the three-dimensional interaction of the heel pad with its surroundings.     

Failure locus of the anterior cruciate ligament: 3D finite element analysis

Anterior cruciate ligament (ACL) disruption is a common injury that is detrimental to an athlete's quality of life. Determining the mechanisms that cause ACL injury is important in order to develop proper interventions. A failure locus defined as various combinations of loadings and movements, internal/external rotation of femur and valgus and varus moments at a 25(o) knee flexion angle leading to ACL failure was obtained. The results indicated that varus and valgus movements were more dominant to the ACL injury than femoral rotation. Also, Von Mises stress in the lateral tibial cartilage during the valgus ACL injury mechanism was 83% greater than that of the medial cartilage during the varus mechanism of ACL injury. The results of this study could be used to develop training programmes focused on the avoidance of the described combination of movements which may lead to ACL injury.     

Failure locus of the anterior cruciate ligament: 3D finite element analysis

Anterior cruciate ligament (ACL) disruption is a common injury that is detrimental to an athlete's quality of life. Determining the mechanisms that cause ACL injury is important in order to develop proper interventions. A failure locus defined as various combinations of loadings and movements, internal/external rotation of femur and valgus and varus moments at a 25^o knee flexion angle leading to ACL failure was obtained. The results indicated that varus and valgus movements were more dominant to the ACL injury than femoral rotation. Also, Von Mises stress in the lateral tibial cartilage during the valgus ACL injury mechanism was 83% greater than that of the medial cartilage during the varus mechanism of ACL injury. The results of this study could be used to develop training programmes focused on the avoidance of the described combination of movements which may lead to ACL injury.     

Tissue strain amplification at the osteocyte lacuna: a microstructural finite element analysis

A parametric finite element model of an osteocyte lacuna was developed to predict the microstructural response of the lacuna to imposed macroscopic strains. The model is composed of an osteocyte lacuna, a region of perilacunar tissue, canaliculi, and the surrounding bone tissue. A total of 45 different simulations were modeled with varying canalicular diameters, perilacunar tissue material moduli, and perilacunar tissue thicknesses. Maximum strain increased with a decrease in perilacunar tissue modulus and decreased with an increase in perilacunar tissue modulus, regardless of the thickness of the perilacunar region. An increase in the predicted maximum strain was observed with an increase in canalicular diameter from 0.362 to 0.421 microm. In response to the macroscopic application of strain, canalicular diameters increased 0.8% to over 1.0% depending on the perilacunar tissue modulus. Strain magnification factors of over 3 were predicted. However, varying the size of the perilacunar tissue region had no effect on the predicted perilacunar tissue strain. These results indicate that the application of average macroscopic strains similar to strain levels measured in vivo can result in significantly greater perilacunar tissue strains and canaliculi deformations. A decrease in the perilacunar tissue modulus amplifies the perilacunar tissue strain and canaliculi deformation while an increase in the local perilacunar tissue modulus attenuates this effect.     

Testing dental implants with an in vivo finite element model]

The finite element method (FEM) makes it possible to simulate biomechanical situations on a computer. In the present study the so-called voxel method [9, 14, 17, 18, 19] was used for the construction of the mandible model. For this, the relationship between the biological tissue (e.g. bone) and the corresponding attenuation coefficient of CT data (Hounsfield units = HU) were utilized. The CT data were obtained from an edentulous patient provided with a prosthesis borne on two titanium implants. In a parallel study, the bite forces of the same individual were measured. These were recorded digitally in three dimensions (cranio-caudal, anterior-posterior and left-right) The forces determined by a special program were then transferred to the FEM model implants. We were able to show that a bar joining the implants had a far greater effect on maximum equivalent stress than was expected from the measuring data alone. The highest stress at maximum occlusive force was lowered by 704% on using the connecting bar. On chewing, a stress reduction of 59.9% was observed. The reduction in stress achieved by the bar could, we believe, prolong the life of the implant.     

Stent expansion in curved vessel and their interactions: a finite element analysis

Coronary restenosis after angioplasty has been reduced by stenting procedure, but in-stent restenosis (ISR) has not been eliminated yet, especially in tortuous vessels. In this paper, we proposed a finite element method (FEM) to study the expansion of a stent in a curved vessel (the CV model) and their interactions. A model of the same stent in a straight vessel (the SV model) was also studied and mechanical parameters of both models were researched and compared, including final lumen area, tissue prolapse between stent struts and stress distribution. Results show that in the CV model, the vessel was straightened by stenting and a hinge effect can be observed at extremes of the stent. The maximum tissue prolapse of the CV model was more severe (0.079 mm) than the SV model (0.048 mm); and the minimum lumen area of the CV was decreased (6.10 mm(2)), compared to that of the SV model (6.28 mm(2)). Tissue stresses of the highest level were concentrated in the inner curvature of the CV model. The simulations offered some explanations for the clinical results of ISR in curved vessels and gave design suggestions of the stent and balloon for tortuous vessels. This FEM provides a tool to study mechanisms of stents in curved vessels and can improve new stent designs especially for tortuous vessels.     

Fracture risk in the femoral hip region: A finite element analysis supported experimental approach

The decrease of bone mineral density (BMD) is a multifactorial bone pathology, commonly referred to as osteoporosis. The subsequent decline of the bone's micro-structural characteristics renders the human skeletal system, and especially the hip, susceptible to fragility fractures. This study represents a systematic attempt to correlate BMD spectrums to the mechanical strength characteristics of the femoral neck and determine a fracture risk indicator based on non-invasive imaging techniques. The BMD of 30 patients' femurs was measured in vivo by Dual-energy X-ray absorptiometry (DXA). As these patients were subjected to total hip replacement, the mechanical strength properties of their femurs' were determined ex-vivo using uniaxial compression experiments. FEA simulations facilitated the correlation of the DXA measurements to the apparent fracture risk, indicating critical strain values during complex loading scenarios.     

Comparison of plate-screw systems used in mandibular fracture reduction: finite element analysis

A finite element model of the human dentate mandible has been developed to provide a comparison of fixation systems used currently for fracture reduction. Volume domains for cortical bone, cancellous bone, and teeth were created and meshed in ANSYS 8.0 based on IGES curves created from computerized tomography data. A unilateral molar clench was loaded on the model with a fracture gap simulated along the symphysis. Results based on Von Mises stress in cortical and cancellous bone surrounding the screws, and on fracture surface spatial fixation, show some relative differences between different screw-plate systems, yet all were judged to be appropriate in their reduction potential.