A new material mapping procedure for quantitative computed tomography-based, continuum finite element analyses of the vertebra

Inaccuracies in the estimation of material properties and errors in the assignment of these properties into finite element models limit the reliability, accuracy, and precision of quantitative computed tomography (QCT)-based finite element analyses of the vertebra. In this work, a new mesh-independent, material mapping procedure was developed to improve the quality of predictions of vertebral mechanical behavior from QCT-based finite element models. In this procedure, an intermediate step, called the material block model, was introduced to determine the distribution of material properties based on bone mineral density, and these properties were then mapped onto the finite element mesh. A sensitivity study was first conducted on a calibration phantom to understand the influence of the size of the material blocks on the computed bone mineral density. It was observed that varying the material block size produced only marginal changes in the predictions of mineral density. Finite element (FE) analyses were then conducted on a square column-shaped region of the vertebra and also on the entire vertebra in order to study the effect of material block size on the FE-derived outcomes. The predicted values of stiffness for the column and the vertebra decreased with decreasing block size. When these results were compared to those of a mesh convergence analysis, it was found that the influence of element size on vertebral stiffness was less than that of the material block size. This mapping procedure allows the material properties in a finite element study to be determined based on the block size required for an accurate representation of the material field, while the size of the finite elements can be selected independently and based on the required numerical accuracy of the finite element solution. The mesh-independent, material mapping procedure developed in this study could be particularly helpful in improving the accuracy of finite element analyses of vertebroplasty and spine metastases, as these analyses typically require mesh refinement at the interfaces between distinct materials. Moreover, the mapping procedure is not specific to the vertebra and could thus be applied to many other anatomic sites.     

Confocal-based cell-specific finite element modeling extended to study variable cell shapes and intracellular structures: The example of the adipocyte

This communication extends the recently reported cell-specific finite element (FE) method in Slomka and Gefen (2010) in which geometrically realistic FE cell models are created from confocal microscopy scans for large deformation analyses. The cell-specific FE method is extended here in the following aspects: (i) we demonstrate that cell-specific FE is versatile enough to deal with cells of substantially different geometrical shapes. The examples of an “elongated” pre-adipocyte and a “round” mature adipocyte are used to demonstrate this feature. (ii) We demonstrate that cell-specific FE can be used to analyze the mechanical behavior of cells that incorporate complex intracellular structures and are subjected to large deformations—again through the example of an adipocyte which contains a multitude of lipid droplets, each having a different size and shape. By demonstrating feasibility of inclusion of such inhomogeneities in the cytoplasm, the present work paves the way for modeling cellular organelles such as Golgi bodies, lysosomes and mitochondria in mechanically loaded cells using cell-specific FE.     

Validation performance comparison for finite element models of the human brain

The objective of this study was to compare the performance of six validated brain finite element (FE) models to localized brain motion validation data in five experimental configurations. Model performance was measured using the objective metric CORA (CORrelation and Analysis), where higher ratings represent better correlation. The KTH model achieved the highest average CORA rating, and the ABM received the highest average rating among models robustly validated against five cadaver impacts in three directions. This technique can be more frequently employed to build better models and, when associated limitations are well understood, to compare inter-model performance under similar conditions.     

Morphing methods to parameterize specimen-specific finite element model geometries

Shape plays an important role in determining the biomechanical response of a structure. Specimen-specific finite element (FE) models have been developed to capture the details of the shape of biological structures and predict their biomechanics. Shape, however, can vary considerably across individuals or change due to aging or disease, and analysis of the sensitivity of specimen-specific models to these variations has proven challenging. An alternative to specimen-specific representation has been to develop generic models with simplified geometries whose shape is relatively easy to parameterize, and can therefore be readily used in sensitivity studies. Despite many successful applications, generic models are limited in that they cannot make predictions for individual specimens.We propose that it is possible to harness the detail available in specimen-specific models while leveraging the power of the parameterization techniques common in generic models. In this work we show that this can be accomplished by using morphing techniques to parameterize the geometry of specimen-specific FE models such that the model shape can be varied in a controlled and systematic way suitable for sensitivity analysis. We demonstrate three morphing techniques by using them on a model of the load-bearing tissues of the posterior pole of the eye. We show that using relatively straightforward procedures these morphing techniques can be combined, which allows the study of factor interactions. Finally, we illustrate that the techniques can be used in other systems by applying them to morph a femur. Morphing techniques provide an exciting new possibility for the analysis of the biomechanical role of shape, independently or in interaction with loading and material properties.     

Computational study of the elastic properties of Rheum rhabarbarum tissues via surrogate models of tissue geometry

Plant petioles and stems are hierarchical cellular structures, displaying geometrical features defined at multiple length scales. One or more of the intermediate hierarchical levels consists of tissues in which the cellular distribution is quasi-random, a factor that affects the elastic properties of the tissues. The current work focuses on the finite element analysis (FEA) of the constituent tissues of the plant Rheum rhabarbarum (rhubarb). The geometric model is generated via a recently introduced method: the finite edge centroidal Voronoi tessellation (FECVT), which is capable to capture the gradients of cellularity and diversified pattern of cellular materials, as opposed to current approaches in literature. The effective stiffness of the tissues is obtained by using an accurate numerical homogenization technique via detailed finite element analysis of the models of sub-regions of the tissues. As opposed to a large-scale representative volume element (RVE), statistical volume elements (SVE) are considered in this work to model tissue microstructures that are highly random. 2D finite element analyses demonstrate that the distribution of cells in collenchyma and parenchyma tissue make them stiffer in two different directions, while the overall effect of the combined tissues results in approximately equal stiffness in both directions. The rhubarb tissues, on the other hand, are more compliant than periodic and quasi-uniform random cellular materials by a factor of up to 47% and 44%, respectively. The variations of the stiffness shows the stiffening role that cell shape, size, and graded cellular distribution play in the mechanics of the rhubarb tissue.     

The Elastic Behaviour of Sintered Metallic Fibre Networks: A Finite Element Study by Beam Theory

Background

The finite element method has complimented research in the field of network mechanics in the past years in numerous studies about various materials. Numerical predictions and the planning efficiency of experimental procedures are two of the motivational aspects for these numerical studies. The widespread availability of high performance computing facilities has been the enabler for the simulation of sufficiently large systems.

Objectives and Motivation

In the present study, finite element models were built for sintered, metallic fibre networks and validated by previously published experimental stiffness measurements. The validated models were the basis for predictions about so far unknown properties.

Materials and Methods

The finite element models were built by transferring previously published skeletons of fibre networks into finite element models. Beam theory was applied as simplification method.

Results and Conclusions

The obtained material stiffness isn’t a constant but rather a function of variables such as sample size and boundary conditions. Beam theory offers an efficient finite element method for the simulated fibre networks. The experimental results can be approximated by the simulated systems. Two worthwhile aspects for future work will be the influence of size and shape and the mechanical interaction with matrix materials.     

Experimental observation, theoretical models, and biomechanical inference in the study of mandibular form

Experimental studies and mathematical models are disparate approaches for inferring the stress and strain environment in mammalian jaws. Experimental designs offer accurate, although limited, characterization of biomechanical behavior, while mathematical approaches (finite element modeling in particular) offer unparalleled precision in depiction of strain magnitudes, directions, and gradients throughout the mandible. Because the empirical (experimental) and theoretical (mathematical) perspectives differ in their initial assumptions and their proximate goals, the two methods can yield divergent conclusions about how masticatory stresses are distributed in the dentary. These different sources of inference may, therefore, tangibly influence subsequent biological interpretation. In vitro observation of bone strain in primate mandibles under controlled loading conditions offers a test of finite element model predictions. Two issues which have been addressed by both finite element models and experimental approaches are: (1) the distribution of torsional shear strains in anthropoid jaws and (2) the dissipation of bite forces in the human alveolar process. Not surprisingly, the experimental data and mathematical models agree on some issues, but on others exhibit discordance. Achieving congruence between these methods is critical if the nature of the relationship of masticatory stress to mandibular form is to be intelligently assessed. A case study of functional/mechanical significance of gnathic morphology in the hominid genus Paranthropus offers insight into the potential benefit of combining theoretical and experimental approaches. Certain finite element analyses claim to have identified a biomechanical problem unrecognized in previous comparative work, which, in essence, is that the enlarged transverse dimensions of the postcanine corpus may have a less important role in resisting torsional stresses than previously thought. Experimental data have identified subperiosteal cortical thinning as a culprit in diminishing the role of cross-sectional geometry in conditioning the strain environment. These observations raise questions concerning the biomechanical significance of mandibular form in early hominids, fueling persistent arguments over whether gnathic morphology can be related to dietary specialization in the "robust" australopithecines. Nonmechanical explanations (e.g., tooth size or body size) for Paranthropus mandibular dimensions, however, are not compelling as competing hypotheses. Both theoretical and experimental models are in need of refinement before it is possible to conclude that the jaws of the "robust" australopithecines are not functionally linked to elevated masticatory loads.     

Static and free-vibration analyses of dental prosthesis and atherosclerotic human artery by refined finite element models

Static and modal responses of representative biomechanical structures are investigated in this paper by employing higher-order theories of structures and finite element approximations. Refined models are implemented in the domain of the Carrera unified formulation (CUF), according to which low- to high-order kinematics can be postulated as arbitrary and, eventually, hierarchical expansions of the generalized displacement unknowns. By using CUF along with the principle of virtual work, the governing equations are expressed in terms of fundamental nuclei of finite element arrays. The fundamental nuclei are invariant of the theory approximation order and can be opportunely employed to implement variable kinematics theories of bio-structures. In this work, static and free-vibration analyses of an atherosclerotic plaque of a human artery and a dental prosthesis are discussed. The results from the proposed methodologies highlight a number of advantages of CUF models with respect to already established theories and commercial software tools. Namely, (i) CUF models can represent correctly the higher-order phenomena related to complex stress/strain field distributions and coupled mode shapes; (ii) bio-structures can be modeled in a component-wise sense by only employing the physical boundaries of the problem domain and without making any geometrical simplification. This latter aspect, in particular, can be currently accomplished only by using three-dimensional analysis, which may be computationally unbearable as complex bio-systems are considered.     

A new cell-based FE model for the mechanics of embryonic epithelia

In order to overcome a significant stiffening artefact associated with current finite element (FE) models for the mechanics of embryonic epithelia, two new FE formulations were developed. Cell–cell interfacial tensions γ are represented by constant-force rod elements as in previous models. However, the viscosity of the cytoplasm with its embedded organelles and filament networks is modeled using viscous triangular elements, it is modeled using either radial and circumferential dashpots or an orthogonal dashpot system rather than the viscous triangular elements typical of previous two-dimensional FE models. The models are tested against tissue (epithelium) stretching because it gives rise to significant changes in cell shape and against cell sorting because it involves high rates of cell rearrangement. The orthogonal dashpot system is found to capture cell size and shape effects well, give the model cells characteristics that are consistent with those of real cells, provide high computational efficiency and hold promise for future three-dimensional analyses.     

A new cell-based FE model for the mechanics of embryonic epithelia

In order to overcome a significant stiffening artefact associated with current finite element (FE) models for the mechanics of embryonic epithelia, two new FE formulations were developed. Cell-cell interfacial tensions gamma are represented by constant-force rod elements as in previous models. However, the viscosity of the cytoplasm with its embedded organelles and filament networks is modeled using viscous triangular elements, it is modeled using either radial and circumferential dashpots or an orthogonal dashpot system rather than the viscous triangular elements typical of previous two-dimensional FE models. The models are tested against tissue (epithelium) stretching because it gives rise to significant changes in cell shape and against cell sorting because it involves high rates of cell rearrangement. The orthogonal dashpot system is found to capture cell size and shape effects well, give the model cells characteristics that are consistent with those of real cells, provide high computational efficiency and hold promise for future three-dimensional analyses.     

The use of strain energy as a convergence criterion in the finite element modelling of bone and the effect of model geometry on stress convergence

Keyak and Skinner¹ state that the adequacy of a finite element mesh may be verified by confirming that convergence of strain energy has been achieved. The paper concludes by stating that convergence of strain energy does not ensure that a particular mesh is adequate for producing accurate stress/strain results, and that only qualitative results can be achieved from 3D models of bone. This conclusion is challenged and the effect of element size on stress convergence is clarified. The influence of element size on stress convergence in regions of surface irregularity is studied and current methods are discussed that facilitate convergence.     

An exploratory study on the combined effects of external and internal morphology on load dissipation in primate capitates: its potential for an understanding of the positional and locomotor repertoire of early hominins

This pilot study explored whether the redirection of stress through trabeculae within morphologically constrained capitates provides information about habitual/positional behaviours unavailable from the study of external morphology alone. To assess this possibility, an experimental finite element approach was taken, whereby no attempt was made to reconstruct the actual magnitudes and loading conditions experienced by the capitates in vivo. Rather, this work addressed fundamental biological questions relating to bone plasticity, i.e. internal versus external bone morphology. The capitates of 7 species with different and - in the case of fossils - inferred locomotor behaviours were selected. Virtual models of capitates were created, scaled to the same size and subjected to the same theoretical load. In the first set of analyses, models were assigned the material properties of bone throughout, whereas in the second set, models were assigned 11 different material properties representing the trabecular architecture derived from high-resolution CT. Species with arboreal behaviours consistently redirected loads towards the ulnar aspect of the capitate when trabeculae were introduced, while terrestrial species, and the bipedal Homo, redirected stress towards the radial side. From these preliminary analyses, it is tentatively concluded that Australopithecus anamensis habitually engaged in arboreal behaviours, whereas Australopithecus afarensis did not.     

Virtual Functional Morphology: Novel Approaches to the Study of Craniofacial Form and Function

Recent developments in simulating musculoskeletal functioning in the craniofacial complex using multibody dynamic analysis and finite elements analysis enable comprehensive virtual investigations into musculoskeletal form and function. Because the growth of the craniofacial skeleton is strongly influenced by mechanical functioning, these methods have potential in investigating the normal and abnormal development of the skull: loading history during development can be predicted and bony adaptations to these loads simulated. Thus these methods can be used to predict the impact of altered loading or modifications of skull form early in ontogeny on the subsequent development of structures. Combining functional models with geometric morphometric methods (GMM), which are principally concerned with the study of variations of form, offers the opportunity to examine variations in form during development and the covariations between form and factors such as functional performance. Such a combination of functional models and GMM can potentially be applied in many useful ways, for example: to build and modify functional models, to assess the outcomes of remodelling studies by comparing the results with morphological changes during ontogeny, and to compare the outcomes of finite element analyses within a multivariate framework. Studies using these tools can not only investigate the development of the skull but also the mechanical processes and thus to some degree, behaviours underlying the development of variation among extant and fossil skeletal elements. By bringing together these tools from quite different comparative traditions, a novel and potentially powerful framework for simulation and statistical biomechanical analyses of form and function emerges. This paper reviews these recent developments in the context of the evolutionary and functional influences on skull development.     

Scaling of form and function in the xenarthran femur: a 100-fold increase in body mass is mitigated by repositioning of the third trochanter

How animals cope with increases in body size is a key issue in biology. Here, we consider scaling of xenarthrans, particularly how femoral form and function varies to accommodate the size range between the 3 kg armadillo and its giant relative the 300 kg glyptodont. It has already been noted that femoral morphology differs between these animals and suggested that this reflects a novel adaptation to size increase in glyptodont. We test this idea by applying a finite element analysis of coronal plane forces to femoral models of these animals, simulating the stance phase in the hind limb; where the femur is subject to bending owing to longitudinal compressive as well as abduction loads on the greater trochanter. We use these models to examine the hypothesis that muscles attaching on the third trochanter (T3) can reduce this bending in the loaded femur and that the T3 forces are more effective at reducing bending in glyptodont where the T3 is situated at the level of the knee. The analysis uses traditional finite element methods to produce strain maps and examine strains at 200 points on the femur. The coordinates of these points before and after loading are also used to carry out geometric morphometric (GM) analyses of the gross deformation of the model in different loading scenarios. The results show that longitudinal compressive and abductor muscle loading increases bending in the coronal plane, and that loads applied to the T3 reduce that bending. In the glyptodont model, the T3 loads are more effective and can more readily compensate for the bending owing to longitudinal and abductor loads. This study also demonstrates the usefulness of GM methods in interpreting the results of finite element analyses.     

Stochastic finite element analysis of biological systems: comparison of a simple intervertebral disc model with experimental results

Statistical methods allow the effects of uncertainty to be incorporated into finite element models. This has potential benefits for the analysis of biological systems where natural variability can give rise to substantial uncertainty in both material and geometrical properties. In this study, a simple model of the intervertebral disc under compression was created and analysed as both a deterministic and a stochastic system. Factorial analysis was used to determine the important parameters to be included in the stochastic analysis. The predictions from the model were compared to experimental results from 21 sheep discs. The size and shape of the distribution of the axial deformations predicted by the model was consistent with the experimental results given that the number of model solutions far exceeded the number of experimental results. Stochastic models could be valuable in determining the range and most likely value of stress in a tissue or implant.     

Comparison of the linear finite element prediction of deformation and strain of human cancellous bone to 3D digital volume correlation measurements

The mechanical properties of cancellous bone and the biological response of the tissue to mechanical loading are related to deformation and strain in the trabeculae during function. Due to the small size of trabeculae, their motion is difficult to measure. To avoid the need to measure trabecular motions during loading the finite element method has been used to estimate trabecular level mechanical deformation. This analytical approach has been empirically successful in that the analytical models are solvable and their results correlate with the macroscopically measured stiffness and strength of bones. The present work is a direct comparison of finite element predictions to measurements of the deformation and strain at near trabecular level. Using the method of digital volume correlation, we measured the deformation and calculated the strain at a resolution approaching the trabecular level for cancellous bone specimens loaded in uniaxial compression. Smoothed results from linearly elastic finite element models of the same mechanical tests were correlated to the empirical three-dimensional (3D) deformation in the direction of loading with a coefficient of determination as high as 97% and a slope of the prediction near one. However, real deformations in the directions perpendicular to the loading direction were not as well predicted by the analytical models. Our results show, that the finite element modeling of the internal deformation and strain in cancellous bone can be accurate in one direction but that this does not ensure accuracy for all deformations and strains.     

Mechanical behavior of coronary stents investigated through the finite element method

Intravascular stents are small tube-like structures expanded into stenotic arteries to restore blood flow perfusion to the downstream tissues. The stent is mounted on a balloon catheter and delivered to the site of blockage. When the balloon is inflated, the stent expands and is pressed against the inner wall of the coronary artery. After the balloon is deflated and removed, the stent remains in place, keeping the artery open. Hence, the stent expansion defines the effectiveness of the surgical procedure: it depends on the stent geometry, it includes large displacements and deformations and material non-linearity.In this paper, the finite element method is applied (i) to understand the effects of different geometrical parameters (thickness, metal-to-artery surface ratio, longitudinal and radial cut lengths) of a typical diamond-shaped coronary stent on the device mechanical performance, (ii) to compare the response of different actual stent models when loaded by internal pressure and (iii) to collect suggestions for optimizing the device shape and performance.The stent expansion and partial recoil under balloon inflation and deflation were simulated. Results showed the influence of the geometry on the stent behavior: a stent with a low metal-to-artery surface ratio has a higher radial and longitudinal recoil, but a lower dogboning. The thickness influences the stent performance in terms of foreshortening, longitudinal recoil and dogboning.In conclusion, a finite element analysis similar to the one herewith proposed could help in designing new stents or analyzing actual stents to ensure ideal expansion and structural integrity, substituting in vitro experiments often difficult and unpractical.     

Analysis of indentation: implications for measuring mechanical properties with atomic force microscopy.

Indentation using the atomic force microscope (AFM) has potential to measure detailed micromechanical properties of soft biological samples. However, interpretation of the results is complicated by the tapered shape of the AFM probe tip, and its small size relative to the depth of indentation. Finite element models (FEMs) were used to examine effects of indentation depth, tip geometry, and material nonlinearity and heterogeneity on the finite indentation response. Widely applied infinitesimal strain models agreed with FEM results for linear elastic materials, but yielded substantial errors in the estimated properties for nonlinear elastic materials. By accounting for the indenter geometry to compute an apparent elastic modulus as a function of indentation depth, nonlinearity and heterogeneity of material properties may be identified. Furthermore, combined finite indentation and biaxial stretch may reveal the specific functional form of the constitutive law--a requirement for quantitative estimates of material constants to be extracted from AFM indentation data.     

Pullout performance of self-tapping medical screws

This study investigates the effect of the pilot hole size, implant depth, synthetic bone density, and screw size on the pullout strength of the self-tapping screw using analytical, finite element, and experimental methodologies. Stress distribution and failure propagation mode around the implant thread zone are also investigated. Based on the finite element analysis (FEA) results, an analytical model for the pullout strength of the self-tapping screw is constructed in terms of the (synthetic) bone mechanical properties, screw size, and the implant depth. The pullout performance of self-tapping screws is discussed. Results from the analytical and finite element models are experimentally validated.     

New perspectives for estimating body condition from mass/length data: the scaled mass index as an alternative method

Body condition is assumed to influence an animal's health and fitness. Various non‐destructive methods based on body mass and a measure of body length have been used as condition indices (CIs), but the dominant method amongst ecologists is currently the calculation of residuals from an ordinary least squares (OLS) regression of body mass against length. Recent studies of energy reserves in small mammals and starlings claimed to validate this method, although we argue that they did not include the most appropriate tests since they compared the CI with the absolute size of energy reserves. We present a novel CI (the ‘scaled mass index’) based on the central principle of scaling, with important methodological, biological and conceptual advantages. Through a reanalysis of data from small mammals, starlings and snakes, we show that the scaled mass index is a better indicator of the relative size of energy reserves and other body components than OLS residuals, performing better in all seven species and in 19 out of 20 analyses. We also present an empirical and theoretical comparison of the scaled mass index and OLS residuals as CIs. We argue that the scaled mass index is a useful new tool for ecologists.