Discretization error when using finite element models: Analysis and evaluation of an underestimated problem

Mesh convergence tests are often insufficiently performed in finite element analyses. There are many parameters which may have an effect on the mesh convergence behavior. The aim of this study was to identify the influence of different parameters on the mesh convergence behavior.For this purpose we used a simplified axis-symmetrical model of a single pedicle screw flank with surrounding bone to simulate a pull-out test. In parameter studies, the flank radii and the contact conditions at the bone–screw interface were varied. These parameter studies were carried out using an implicit and explicit solver. Thereby, the convergence criteria and the number of the substeps for the implicit nonlinear iteration process as well as the velocity and the material density for the explicit approach were considered.The mesh convergence behavior was influenced by varying the flank radii and the contact conditions. The implicit calculations led to a reaction force, which converged rapidly to a certain value with increasing mesh density, whereas the maximum von-Mises stress showed substantial convergence problems. The number of substeps and the convergence criteria of the iteration process strongly influenced the implicit solutions. In contrast, the maximum von-Mises stresses resulting from explicit calculations converged to a certain value after only a few refinement steps. Different pull-out velocities substantially affected the mesh convergence behavior, while the material density showed only a negligible influence.The results indicated the need to perform an appropriate mesh convergence test when using finite element methods. We were able to show that different parameters strongly influence the mesh convergence behavior and we demonstrated that convergence tests do not always lead to a satisfactory or acceptable solution.     

An automated method to morph finite element whole-body human models with a wide range of stature and body shape for both men and women

Field data analyses have shown that small female, obese, and/or older occupants are at increased risks of death and serious injury in motor-vehicle crashes compared with mid-size young men. The current adult finite element (FE) human models represent occupants in the same three body sizes (large male, mid-size male, and small female) as those for the contemporary adult crash dummies. Further, the time needed to develop an FE human model using the traditional method is measured in months or even years. In the current study, an improved regional mesh morphing method based on landmark-based radial basis function (RBF) interpolation was developed to rapidly morph a mid-size male FE human model into different geometry targets. A total of 100 human models with a wide range of human attributes were generated. A pendulum chest impact condition was applied to each model as an initial assessment of the resulting variability in response. The morphed models demonstrated mesh quality similar to the baseline model. The peak impact forces and chest deflections in the chest pendulum impacts varied substantially with different models, supportive of consideration of population variation in evaluating the occupant injury risks. The method developed in this study will enable future safety design optimizations targeting at various vulnerable populations that cannot be considered with the current models.     

Cervical spine morphology and ligament property variations: A finite element study of their influence on sagittal bending characteristics

Cervical spine finite element models reported in biomechanical literature usually represent a static morphology. Not considering morphology as a model parameter limits the predictive capabilities for applications in personalized medicine, a growing trend in modern clinical practice. The objective of the study was to investigate the influence of variations in spinal morphology on the flexion-extension responses, utilizing mesh-morphing-based parametrization and metamodel-based sensitivity analysis. A C5-C6 segment was used as the baseline model. Variations of intervertebral disc height, facet joint slope, facet joint articular processes height, vertebral body anterior-posterior depth, and segment size were parametrized. In addition, material property variations of ligaments were considered for sensitivity analysis. The influence of these variations on vertebral rotation and forces in the ligaments were analyzed. The disc height, segmental size, and body depth were found to be the most influential (in the cited order) morphology variations; while among the ligament material property variations, capsular ligament and ligamentum flavum influenced vertebral rotation the most. Changes in disc height influenced forces in the posterior ligaments, indicating that changes in the anterior load-bearing column of the spine could have consequences on the posterior column. A method to identify influential morphology variations is presented in this work, which will help automation efforts in modeling to focus on variations that matter. This study underscores the importance of incorporating influential morphology parameters, easily obtained through computed tomography/magnetic resonance images, to better predict subject-specific biomechanical responses for applications in personalized medicine.     

FEBio finite element models of the human lumbar spine

Finite element analysis has proven to be a viable method for assessing many structure-function relationships in the human lumbar spine. Several validated models of the spine have been published, but they typically rely on commercial packages and are difficult to share between labs. The goal of this study is to present the development of the first open-access models of the human lumbar spine in FEBio. This modeling framework currently targets three deficient areas in the field of lumbar spine modeling: 1) open-access models, 2) accessibility for multiple meshing schemes, and 3) options to include advanced hyperelastic and biphasic constitutive models.     

Harmonizing finite element modelling for non-invasive strength estimation by high-resolution peripheral quantitative computed tomography

The finite element (FE) method based on high-resolution peripheral quantitative computed tomography (HR-pQCT) use a variety of tissue constitutive properties and boundary conditions at different laboratories making comparison of mechanical properties difficult. Furthermore, the advent of a second-generation HR-pQCT poses challenges due to improved resolution and a larger region of interest (ROI). This study addresses the need to harmonize results across FE models. The aims are to establish the relationship between FE results as a function of boundary conditions and a range of tissue properties for the first-generation HR-pQCT system, and to determine appropriate model parameters for the second-generation HR-pQCT system. We implemented common boundary conditions and tissue properties on a large cohort (N = 1371), and showed the relationships were highly linear (R² > 0.99) for yield strength and reaction force between FE models. Cadaver radii measured on both generation HR-pQCT with matched ROIs were used to back-calculate a tissue modulus that accounts for the increased resolution (61 µm versus 82 µm), resulting in a modulus of 8748 MPa for second-generation HR-pQCT to produce bone yield strength and reaction force equivalent to using 6829 MPa for first-generation HR-pQCT. Finally, in vivo scans (N = 61) conducted on both generations demonstrated that the larger ROI in the second-generation system results in stronger bone outcome measures, suggesting it is not advisable to convert FE results across HR-pQCT generations without matching ROIs. Together, these findings harmonize FE results by providing a means to compare findings with different boundary conditions and tissue properties, and across scanner generations.     

Finite element analysis and computed tomography based structural rigidity analysis of rat tibia with simulated lytic defects

Finite element analysis (FEA), CT based structural rigidity analysis (CTRA) and mechanical testing is performed to assess the compressive failure load of rat tibia with simulated lytic defects.     

Reliable simulations of the human proximal femur by high-order finite element analysis validated by experimental observations

Background: The mechanical response of patient-specific bone to various load conditions is of major clinical importance in orthopedics. Herein we enhance the methods presented in Yosibash et al. [2007. A CT-based high-order finite element analysis of the human proximal femur compared to in-vitro experiments. ASME Journal of Biomechanical Engineering 129(3), 297–309.] for the reliable simulations of the human proximal femur by high-order finite elements (FEs) and validate the simulations by experimental observations.

Method of approach: A fresh-frozen human femur was scanned by quantitative computed tomography (QCT) and thereafter loaded (in vitro experiments) by a quasi-static force of up to 1250 N. QCT scans were manipulated to generate a high-order FE bone model with distinct cortical and trabecular regions having inhomogeneous isotropic elastic properties with Young's modulus represented by continuous spatial functions. Sensitivity analyses were performed to quantify parameters that mostly influence the mechanical response. FE results were compared to displacements and strains measured in the experiments.

Results: Young moduli correlated to QCT Hounsfield Units by relations in Keyak and Falkinstein [2003. Comparison of in situ and in vitro CT scan-based finite element model predictions of proximal femoral fracture load. Medical Engineering and Physics 25, 781–787.] were found to provide predictions that match the experimental results closely. Excellent agreement was found for both the displacements and strains. The presented study demonstrates that reliable and validated high-order patient-specific FE simulations of human femurs based on QCT data are achievable for clinical computer-aided decision making.     

3D finite element model of the diabetic neuropathic foot: A gait analysis driven approach

Diabetic foot is an invalidating complication of diabetes that can lead to foot ulcers. Three-dimensional (3D) finite element analysis (FEA) allows characterizing the loads developed in the different anatomical structures of the foot in dynamic conditions. The aim of this study was to develop a subject specific 3D foot FE model (FEM) of a diabetic neuropathic (DNS) and a healthy (HS) subject, whose subject specificity can be found in term of foot geometry and boundary conditions. Kinematics, kinetics and plantar pressure (PP) data were extracted from the gait analysis trials of the two subjects with this purpose. The FEM were developed segmenting bones, cartilage and skin from MRI and drawing a horizontal plate as ground support. Materials properties were adopted from previous literature. FE simulations were run with the kinematics and kinetics data of four different phases of the stance phase of gait (heel strike, loading response, midstance and push off). FEMs were then driven by group gait data of 10 neuropathic and 10 healthy subjects. Model validation focused on agreement between FEM-simulated and experimental PP.     

Strain accommodation in the zygomatic arch of the pig: A validation study using digital speckle pattern interferometry and finite element analysis

It has been repeatedly suggested that mammalian cranial sutures act not only to allow growth but also to reduce the levels of strain experienced by the skull during feeding. However, because of the added complexity they introduce, sutures are rarely included in finite element (FE) models, despite their potential to influence strain results. Because sutures present different morphologies and with differing degrees of internal fusion, many different methods of modeling may be necessary to accurately measure strain environments. Alternatively, these variables may exert very little influence on the scale of a whole‐skull model. To validate suture modeling methods, four alternative ways of including a suture in 3D FE models of the pig zygomatic arch were considered and compared with ex vivo experimental data from digital speckle pattern interferometry (DSPI). The use of DSPI rather than traditional strain gauge techniques allows strain gradients around the suture as well as the motions of the two bones to be observed. Results show that the introduction of 3D elements assigned more compliant material properties than the surrounding bone, is the most effective way of modeling both morphologies of suture, both in tension and compression. However, models containing no suture are almost indistinguishable from these compliant suture models, beyond the high strain gradient immediately adjacent to the suture. Conversely, modeling the suture as an open break in the mesh, or with spring elements assigned suture properties, fails to reproduce the experiment. Thus, although a solid but flexible model of sutures is preferred, the similarity between these models and those without sutures tentatively suggests that such extra detail may be unnecessary in pigs if the behavior of the whole skull is of interest. J. Morphol., 2011. © 2011 Wiley‐Liss, Inc.     

Using 3D finite element models verified the importance of callus material and microstructure in biomechanics restoration during bone defect repair

Background: There is lack of further observations on the microstructure and material property of callus during bone defect healing and the relationships between callus properties and the mechanical strength. Methods: Femur bone defect model was created in rabbits and harvested CT data to reconstruct finite element models at 1 and 2 months. Three types of assumed finite element models were compared to study the callus properties, which assumed the material elastic property as heterogeneous (R-model), homogenous (H-model) or did not change from 1 to 2 months (U-model). Results: The apparent elastic moduli increased at 2 months (from 355.58 ± 132.67 to 1139.30 ± 967.43 MPa) in R-models. But there was no significant difference in apparent elastic moduli between R-models (355.58 ± 132.67 and 1139.30 ± 967.43 MPa) and H-models (344.79 ± 138.73 and 1001.52 ± 692.12 MPa) in 1 and 2 months. A significant difference of apparent elastic moduli was found between the R-model (1139.30 ± 967.43 MPa) and U-model group (207.15 ± 64.60 MPa) in 2 months. Conclusions: This study showed that the callus structure stability remodeled overtime to achieve a more effective structure, while the material quality of callus tissue is a very important factor for callus strength. At the meantime, this study showed an evidence that the material heterogeneity maybe not as important as it is in bone fracture model.     

Simple finite element models for use in the design of therapeutic footwear

Therapeutic footwear is frequently prescribed in cases of rheumatoid arthritis and diabetes to relieve or redistribute high plantar pressures in the region of the metatarsal heads. Few guidelines exist as to how these interventions should be designed and what effect such interventions actually have on the plantar pressure distribution. Finite element analysis has the potential to assist in the design process by refining a given intervention or identifying an optimal intervention without having to actually build and test each condition. However, complete and detailed foot models based on medical image segmentation have proven time consuming to build and computationally expensive to solve, hindering their utility in practice. Therefore, the goal of the current work was to determine if a simplified patient-specific model could be used to assist in the design of foot orthoses to reduce the plantar pressure in the metatarsal head region. The approach is illustrated by a case study of a diabetic patient experiencing high pressures and pain over the fifth metatarsal head. The simple foot model was initially calibrated by adjusting the individual loads on the metatarsals to approximate measured peak plantar pressure distributions in the barefoot condition to within 3%. This loading was used in various shod conditions to identify an effective orthosis. Model results for metatarsal pads were considerably higher than measured values but predictions for uniform surfaces were generally within 16% of measured values. The approach enabled virtual prototyping of the orthoses, identifying the most favorable approach to redistribute the patient’s plantar pressures.     

Development and validation of a weight-bearing finite element model for total knee replacement

Total knee arthroplasty (TKA) is a successful procedure for osteoarthritis. However, some patients (19%) do have pain after surgery. A finite element model was developed based on boundary conditions of a knee rig. A 3D-model of an anatomical full leg was generated from magnetic resonance image data and a total knee prosthesis was implanted without patella resurfacing. In the finite element model, a restarting procedure was programmed in order to hold the ground reaction force constant with an adapted quadriceps muscle force during a squat from 20° to 105° of flexion. Knee rig experimental data were used to validate the numerical model in the patellofemoral and femorotibial joint. Furthermore, sensitivity analyses of Young’s modulus of the patella cartilage, posterior cruciate ligament (PCL) stiffness, and patella tendon origin were performed. Pearson’s correlations for retropatellar contact area, pressure, patella flexion, and femorotibial ap-movement were near to 1. Lowest root mean square error for retropatellar pressure, patella flexion, and femorotibial ap-movement were found for the baseline model setup with Young’s modulus of 5 MPa for patella cartilage, a downscaled PCL stiffness of 25% compared to the literature given value and an anatomical origin of the patella tendon. The results of the conducted finite element model are comparable with the experimental results. Therefore, the finite element model developed in this study can be used for further clinical investigations and will help to better understand the clinical aspects after TKA with an unresurfaced patella.     

The influence of ligament modelling strategies on the predictive capability of finite element models of the human knee joint

In finite element (FE) models knee ligaments can represented either by a group of one-dimensional springs, or by three-dimensional continuum elements based on segmentations. Continuum models closer approximate the anatomy, and facilitate ligament wrapping, while spring models are computationally less expensive. The mechanical properties of ligaments can be based on literature, or adjusted specifically for the subject. In the current study we investigated the effect of ligament modelling strategy on the predictive capability of FE models of the human knee joint. The effect of literature-based versus specimen-specific optimized material parameters was evaluated. Experiments were performed on three human cadaver knees, which were modelled in FE models with ligaments represented either using springs, or using continuum representations. In spring representation collateral ligaments were each modelled with three and cruciate ligaments with two single-element bundles. Stiffness parameters and pre-strains were optimized based on laxity tests for both approaches. Validation experiments were conducted to evaluate the outcomes of the FE models.Models (both spring and continuum) with subject-specific properties improved the predicted kinematics and contact outcome parameters. Models incorporating literature-based parameters, and particularly the spring models (with the representations implemented in this study), led to relatively high errors in kinematics and contact pressures. Using a continuum modelling approach resulted in more accurate contact outcome variables than the spring representation with two (cruciate ligaments) and three (collateral ligaments) single-element-bundle representations. However, when the prediction of joint kinematics is of main interest, spring ligament models provide a faster option with acceptable outcome.     

A novel methodology for generating 3D finite element models of the hip from 2D radiographs

Finite element (FE) modelling has been proposed as a tool for estimating fracture risk and patient-specific FE models are commonly based on computed tomography (CT). Here, we present a novel method to automatically create personalised 3D models from standard 2D hip radiographs. A set of geometrical parameters of the femur were determined from seven a–p hip radiographs and compared to the 3D femoral shape obtained from CT as training material; the error in reconstructing the 3D model from the 2D radiographs was assessed. Using the geometry parameters as the input, the 3D shape of another 21 femora was built and meshed, separating a cortical and trabecular compartment. The material properties were derived from the homogeneity index assessed by texture analysis of the radiographs, with focus on the principal tensile and compressive trabecular systems. The ability of these FE models to predict failure load as determined by experimental biomechanical testing was evaluated and compared to the predictive ability of DXA. The average reconstruction error of the 3D models was 1.77 mm (±1.17 mm), with the error being smallest in the femoral head and neck, and greatest in the trochanter. The correlation of the FE predicted failure load with the experimental failure load was r²=64% for the reconstruction FE model, which was significantly better (p<0.05) than that for DXA (r²=24%). This novel method for automatically constructing a patient-specific 3D finite element model from standard 2D radiographs shows encouraging results in estimating patient-specific failure loads.     

Primary cilium mechanotransduction of tensile strain in 3D culture: Finite element analyses of strain amplification caused by tensile strain applied to a primary cilium embedded in a collagen matrix

Human adipose-derived stem cells (hASC) exhibit multilineage differentiation potential with lineage specification that is dictated by both the chemical and mechanical stimuli to which they are exposed. We have previously shown that 10% cyclic tensile strain increases hASC osteogenesis and cell-mediated calcium accretion. We have also recently shown that primary cilia are present on hASC and that chemically-induced lineage specification of hASC concurrently results in length and conformation changes of the primary cilia. Further, we have observed cilia length changes in hASC cultured within a collagen I gel in response to 10% cyclic tensile strain. We therefore hypothesize that primary cilia may play a key mechanotransduction role for hASC exposed to tensile strain. The goal of this study was to use finite element analysis (FEA) to determine strains occurring within the ciliary membrane in response to 10% tensile strain applied parallel, or perpendicular, to cilia orientation. To elucidate the mechanical environment experienced by the cilium, several lengths were modeled and evaluated based on cilia lengths measured on hASC grown under varied culture conditions. Principal tensile strains in both hASC and ciliary membranes were calculated using FEA, and the magnitude and location of maximum principal tensile strain determined. We found that maximum principal tensile strain was concentrated at the base of the cilium. In the linear elastic model, applying strain perpendicular to the cilium resulted in maximum strains within the ciliary membrane from 150% to 200%, while applying strain parallel to the cilium resulted in much higher strains, approximately 400%. In the hyperelastic model, applying strain perpendicular to the cilium resulted in maximum strains within the ciliary membrane around 30%, while applying strain parallel to the cilium resulted in much higher strains ranging from 50% to 70%. Interestingly, FEA results indicated that primary cilium length was not directly related to ciliary membrane strain. Rather, it appears that cilium orientation may be more important than cilium length in determining sensitivity of hASC to tensile strain. This is the first study to model the effects of tensile strain on the primary cilium and provides newfound insight into the potential role of the primary cilium as a mechanosensor, particularly in tensile strain and potentially a multitude of other mechanical stimuli beyond fluid shear.     

Long-term study of bone remodelling after femoral stem: a comparison between dexa and finite element simulation

A hip replacement with a cemented or cementless femoral stem produces an effect on the bone called adaptive remodelling, attributable to mechanical and biological factors. The objective of all of cementless prostheses designs has been to achieve a perfect transfer of loads in order to avoid stress-shielding, which produces an osteopenia. In order to quantify this, the long term and mass-produced study with dual energy X-ray absorptiometry (DEXA) is necessary. Finite element (FE) simulation makes possible the explanation of the biomechanical changes which are produced in the femur after stem implantation. The good correlation obtained between the results of the FE simulation and the densitometric study allow, on one hand, to explain from the point of view of biomechanical performance the changes observed in bone density in the long-term, where it is clear that these are due to a different transfer of load in the implanted model compared to the healthy femur; on the other hand, it validates the simulation model, in a way that it can be used in different conditions and at different time periods, to carry out a sufficiently precise prediction of the evolution of the bone density from the biomechanical behaviour in the interaction between the prosthesis and femur.     

Implementation of reflex loops in a biomechanical finite element model

In the field of biomechanics, the offer of models which are more and more realistic requires to integrate a physiological response, in particular, the controlled muscle bracing and the reflexes. The following work aims to suggest a unique methodology which couples together a sensory and motor loop with a finite element model. Our method is applied to the study of the oscillation of the elbow in the case of a biceps brachial stretch reflex. The results obtained are promising in the purpose of the development of reactive human body models.