Subject-specific finite element models can accurately predict strain levels in long bones

The prediction of the stress-state and fracture risk induced in bones by various loading conditions in individual patients using subject-specific finite element models still represents a challenge in orthopaedic biomechanics. The accuracy of the strain predictions reported in the literature is variable and generally not satisfactory. The aim of the present study was to evaluate if a proper choice of the density-elasticity relationship can lead to accurate strain predictions in the frame of an automatic subject-specific model generation strategy. To this aim, a combined numerical-experimental study was performed comparing finite element predicted strains with strain-gauges measurements obtained on eight cadaver proximal femurs, each instrumented with 15 rosettes mostly concentrated in the bone metaphyses, tested non-destructively in vitro under six different loading scenarios. Three different density-elasticity power relationships were selected from the literature and implemented in the finite element models derived from computed tomography data. The results of the present study confirm the great influence of the density-elasticity relationship used on the accuracy of numerical predictions. One of the tested constitutive laws provided a very good agreement (R(2)=0.91, RMSE lower than 10% of the maximum measured value) between numerical calculations and experimental measurements. The presented results show, in addition, that the adoption of a single density-elasticity relationship over the whole bone density range is adequate to obtain an accuracy that is already suitable for many applications.     

Structural consequences of transcortical holes in long bones loaded in torsion

Finite element models were used to predict the structural consequences of transcortical holes through long bones loaded in torsion. Several parameters were investigated including hole size, anelastic behavior of the bone, cortical wall thickness, cortical wall symmetry, curvature along the bone's long axis and the axial length of the defect. Finite element model predictions of percent intact bone strength were compared to experimental data for sheep femora with transcortical drill holes loaded to failure in torsion. Hole size was expressed as hole diameter divided by the outer bone diameter. Linear finite element model predictions were in conservative agreement with the experimental data for large hole sizes. A transcortical hole with a diameter 50% of the outer bone diameter reduced the torsional strength by 60%. However, the linear models predict a 40% drop in strength for small holes whereas in vitro data suggest that small holes have no significant effect on strength. Models which represent non-linear anelastic behavior in bone over-predicted torsional strengths. Asymmetric cortical wall thickness and long bone bowing have minor effects, while the length of an elongated defect strongly influences the torsional strength. Strength reductions are greatest for bones with thin cortical walls.     

High strain rate response of rabbit femur bones

Strain rate dependence of the mechanical response of hard tissues has led to a keen interest in their dynamic properties. The current study attempts to understand the high strain rate characteristics of rabbit femur bones. The testing was conducted using a split-Hopkinson pressure bar equipped with a high speed imaging system to capture the fracture patterns. The bones were also characterized under quasi-static compression to enable comparison with the high strain rate results. The quasi-static compressive moduli of the epiphyseal and diaphyseal regions were measured to be in the range of 2–3 and 5–7 GPa, respectively. Under high strain rate loading conditions the modulus is observed to increase with strain rate and attains values as high as 15 GPa for epiphyseal and 30 GPa for diaphyseal regions of the femur. The strength at high strain rate was measured to be about twice the quasi-static strength value. A large number of small cracks initiated on the specimen surface close to the incident bar. Coalescence of crack branches leading to fewer large cracks resulted in specimen fragmentation. In comparison, the quasi-static failure was due to shear cracking.     

Sampling strategy matters to accurately estimate response curves' parameters in species distribution models

Aim

Assessing how different sampling strategies affect the accuracy and precision of species response curves estimated by parametric species distribution models.

Major Taxa Studied

Virtual plant species.

Location

Abruzzo (Italy).

Time Period

Timeless (simulated data).

Methods

We simulated the occurrence of two virtual species with different ecology (generalist vs specialist) and distribution extent. We sampled their occurrence following different sampling strategies: random, stratified, systematic, topographic, uniform within the environmental space (hereafter, uniform) and close to roads. For each sampling design and species, we ran 500 simulations at increasing sampling efforts (total: 42,000 replicates). For each replicate, we fitted a binomial generalised linear model, extracted model coefficients for precipitation and temperature, and compared them with true coefficients from the known species' equation. We evaluated the quality of the estimated response curves by computing bias, variance and root mean squared error (RMSE). Additionally, we (i) assessed the impact of missing covariates on the performance of the sampling approaches and (ii) evaluated the effect of incompletely sampling the environmental space on the uniform approach.

Results

For the generalist species, we found the lowest RMSE when uniformly sampling the environmental space, while sampling occurrence data close to roads provided the worst performance. For the specialist species, all sampling designs showed comparable outcomes. Excluding important predictors similarly affected all sampling strategies. Sampling limited portions of the environmental space reduced the performance of the uniform approach, regardless of the portion surveyed.

Main Conclusions

Our results suggest that a proper estimate of the species response curve can be obtained when the choice of the sampling strategy is guided by the species' ecology. Overall, uniformly sampling the environmental space seems more efficient for species with wide environmental tolerances. The advantage of seeking the most appropriate sampling strategy vanishes when modelling species with narrow realised niches.     

Xylem vulnerability to cavitation can be accurately characterised in species with long vessels using a centrifuge method

Vulnerability to cavitation curves describe the decrease in xylem hydraulic conductivity as xylem pressure declines. Several techniques for constructing vulnerability curves use centrifugal force to induce negative xylem pressure in stem or root segments. Centrifuge vulnerability curves constructed for long‐vesselled species have been hypothesised to overestimate xylem vulnerability to cavitation due to increased vulnerability of vessels cut open at stem ends that extend to the middle or entirely through segments. We tested two key predictions of this hypothesis: (i) centrifugation induces greater embolism than dehydration in long‐vesselled species, and (ii) the proportion of open vessels changes centrifuge vulnerability curves. Centrifuge and dehydration vulnerability curves were compared for a long‐ and short‐vesselled species. The effect of open vessels was tested in four species by comparing centrifuge vulnerability curves for stems of two lengths. Centrifuge and dehydration vulnerability curves agreed well for the long‐ and short‐vesselled species. Centrifuge vulnerability curves constructed using two stem lengths were similar. Also, the distribution of embolism along the length of centrifuged stems matched the theoretical pressure profile induced by centrifugation. We conclude that vulnerability to cavitation can be accurately characterised with vulnerability curves constructed using a centrifuge technique, even in long‐vesselled species.     

Structural behaviour and strain distribution of the long bones of the human lower limbs

Although stiffness and strength of lower limb bones have been investigated in the past, information is not complete. While the femur has been extensively investigated, little information is available about the strain distribution in the tibia, and the fibula has not been tested in vitro. This study aimed at improving the understanding of the biomechanics of lower limb bones by: (i) measuring the stiffness and strain distributions of the different low limb bones; (ii) assessing the effect of viscoelasticity in whole bones within a physiological range of strain-rates; (iii) assessing the difference in the behaviour in relation to opposite directions of bending and torsion. The structural stiffness and strain distribution of paired femurs, tibias and fibulas from two donors were measured. Each region investigated of each bone was instrumented with 8–16 triaxial strain gauges (over 600 grids in total). Each bone was subjected to 6–12 different loading configurations. Tests were replicated at two different loading speeds covering the physiological range of strain-rates. Viscoelasticity did not have any pronounced effect on the structural stiffness and strain distribution, in the physiological range of loading rates explored in this study. The stiffness and strain distribution varied greatly between bone segments, but also between directions of loading. Different stiffness and strain distributions were observed when opposite directions of torque or opposite directions of bending (in the same plane) were applied. To our knowledge, this study represents the most extensive collection of whole-bone biomechanical properties of lower limb bones.     

Combining specimen-specific finite-element models and optimization in cortical-bone material characterization improves prediction accuracy in three-point bending tests

Although the beam theory is widely used for calculating material parameters in three-point bending test, it cannot accurately describe the biomechanical properties of specimens after the yield. Hence, we propose a finite element (FE) based optimization method to obtain accurate bone material parameters from three-point bending test. We tested 80 machined bovine cortical bone specimens at both longitudinal and transverse directions using three-point bending. We then adopted the beam theory and the FE-based optimization method combined with specimen-specific FE models to derive the material parameters of cortical bone. We compared data obtained using these two methods and further evaluated two groups of parameters with three-point bending simulations. Our data indicated that the FE models with material properties from the FE-based optimization method showed best agreements with experimental data for the entire force-displacement responses, including the post-yield region. Using the beam theory, the yield stresses derived from 0.0058% strain offset for the longitudinal specimen and 0.0052% strain offset for the transverse specimen are closer to those derived from the FE-based optimization method, compared to yield stresses calculated without strain offset. In brief, we conclude that the optimization FE method is more appropriate than the traditional beam theory in identifying the material parameters of cortical bone for improving prediction accuracy in three-point bending mode. Given that the beam theory remains as a popular method because of its efficiency, we further provided correction functions to adjust parameters calculated from the beam theory for accurate FE simulation.     

Investigating the mechanical response of paediatric bone under bending and torsion using finite element analysis

Fractures of bone account 25% of all paediatric injuries (Cooper et al. in J Bone Miner Res 19:1976–1981, 2004.  https://doi.org/10.1359/JBMR.040902). These can be broadly categorised into accidental or inflicted injuries. The current clinical approach to distinguish between these two is based on the clinician’s judgment, which can be subjective. Furthermore, there is a lack of studies on paediatric bone to provide evidence-based information on bone strength, mainly due to the difficulties of obtaining paediatric bone samples. There is a need to investigate the behaviour of children’s bones under external loading. Such data will critically enhance our understanding of injury tolerance of paediatric bones under various loading conditions, related to injuries, such as bending and torsional loads. The aim of this study is therefore to investigate the response of paediatric femora under two types of loading conditions, bending and torsion, using a CT-based finite element approach, and to determine a relationship between bone strength and age/body mass of the child. Thirty post-mortem CT scans of children aged between 0 and 3 years old were used in this study. Two different boundary conditions were defined to represent four-point bending and pure torsional loads. The principal strain criterion was used to estimate the failure moment for both loading conditions. The results showed that failure moment of the bone increases with the age and mass of the child. The predicted failure moment for bending, external and internal torsions were 0.8–27.9, 1.0–31.4 and 1.0–30.7 Nm, respectively. To the authors’ knowledge, this is the first report on infant bone strength in relation to age/mass using models developed from modern medical images. This technology may in future help advance the design of child, car restrain system, and more accurate computer models of children.     

Firing-rate models capture essential response dynamics of LGN relay cells

Firing-rate models provide a practical tool for studying signal processing in the early visual system, permitting more thorough mathematical analysis than spike-based models. We show here that essential response properties of relay cells in the lateral geniculate nucleus (LGN) can be captured by surprisingly simple firing-rate models consisting of a low-pass filter and a nonlinear activation function. The starting point for our analysis are two spiking neuron models based on experimental data: a spike-response model fitted to data from macaque (Carandini et al. J. Vis., 20(14), 1–2011, 2007), and a model with conductance-based synapses and afterhyperpolarizing currents fitted to data from cat (Casti et al. J. Comput. Neurosci., 24(2), 235–252, 2008). We obtained the nonlinear activation function by stimulating the model neurons with stationary stochastic spike trains, while we characterized the linear filter by fitting a low-pass filter to responses to sinusoidally modulated stochastic spike trains. To account for the non-Poisson nature of retinal spike trains, we performed all analyses with spike trains with higher-order gamma statistics in addition to Poissonian spike trains. Interestingly, the properties of the low-pass filter depend only on the average input rate, but not on the modulation depth of sinusoidally modulated input. Thus, the response properties of our model are fully specified by just three parameters (low-frequency gain, cutoff frequency, and delay) for a given mean input rate and input regularity. This simple firing-rate model reproduces the response of spiking neurons to a step in input rate very well for Poissonian as well as for non-Poissonian input. We also found that the cutoff frequencies, and thus the filter time constants, of the rate-based model are unrelated to the membrane time constants of the underlying spiking models, in agreement with similar observations for simpler models.     

Marginalized models for moderate to long series of longitudinal binary response data

Marginalized models (Heagerty, 1999, Biometrics 55, 688-698) permit likelihood-based inference when interest lies in marginal regression models for longitudinal binary response data. Two such models are the marginalized transition and marginalized latent variable models. The former captures within-subject serial dependence among repeated measurements with transition model terms while the latter assumes exchangeable or nondiminishing response dependence using random intercepts. In this article, we extend the class of marginalized models by proposing a single unifying model that describes both serial and long-range dependence. This model will be particularly useful in longitudinal analyses with a moderate to large number of repeated measurements per subject, where both serial and exchangeable forms of response correlation can be identified. We describe maximum likelihood and Bayesian approaches toward parameter estimation and inference, and we study the large sample operating characteristics under two types of dependence model misspecification. Data from the Madras Longitudinal Schizophrenia Study (Thara et al., 1994, Acta Psychiatrica Scandinavica 90, 329-336) are analyzed.     

Locomotor variation and bending regimes of capuchin limb bones

Primates are very versatile in their modes of progression, yet laboratory studies typically capture only a small segment of this variation. In vivo bone strain studies in particular have been commonly constrained to linear locomotion on flat substrates, conveying the potentially biased impression of stereotypic long bone loading patterns. We here present substrate reaction forces (SRF) and limb postures for capuchin monkeys moving on a flat substrate (“terrestrial”), on an elevated pole (“arboreal”), and performing turns. The angle between the SRF vector and longitudinal axes of the forearm or leg is taken as a proxy for the bending moment experienced by these limb segments. In both frontal and sagittal planes, SRF vectors and distal limb segments are not aligned, but form discrepant angles; that is, forces act on lever arms and exert bending moments. The positions of the SRF vectors suggest bending around oblique axes of these limb segments. Overall, the leg is exposed to greater moments than the forearm. Simulated arboreal locomotion and turns introduce variation in the discrepancy angles, thus confirming that expanding the range of locomotor behaviors studied will reveal variation in long bone loading patterns that is likely characteristic of natural locomotor repertoires. “Arboreal” locomotion, even on a linear noncompliant branch, is characterized by greater variability of force directions and discrepancy angles than “terrestrial” locomotion (significant for the forearm only), partially confirming the notion that life in trees is associated with greater variation in long bone loading. Directional changes broaden the range of external bending moments even further. Am J Phys Anthropol, 2009. © 2009 Wiley‐Liss, Inc.     

The role of strain in the response of rapidly growing young male rat bones to parathyroid hormone

Human parathyroid hormone (hPTH 1-34) stimulates an anabolic response in human and animal skeletons; however, it is unclear if the effect is strain dependent. To determine if the anabolic response to hPTH (1-34) was dependent upon strain in rats we used 2 outbred strains (Sprague Dawley, Wistar), 2 inbred strains (Fischer 344, Wistar spontaneously hypertensive:SHR), and 2 mutant strains (Zucker obese, Zucker lean) of rats. Male rats, 5 weeks of age, from each strain were treated subcutaneously with 80 microg/kg body weight hPTH (1-34) or vehicle for 12 days. The response to PTH was similar in all strains whereby PTH exerted an anabolic effect on femoral bone mass and cancellous bone histology that was independent of strain differences. Histomorphometric indices of bone volume, mineralized surface and bone formation in lumbar vertebrae increased in all PTH-treated rats. Additionally, femur bone mineral content and bone mineral density measured by dual energy X-ray absorptiometry (DEXA), and ash weight increased in all PTH-treated rats. These increases occurred regardless of strain. In summary, PTH exerted comparable anabolic effects on bone mass, bone mineral density and bone formation in all rat models tested demonstrating that the skeletal responsiveness to PTH was not dependent upon strain.     

Scanner influence on the mechanical response of QCT-based finite element analysis of long bones

Patient-specific QCT-based finite element (QCTFE) analyses enable highly accurate quantification of bone strength. We evaluated CT scanner influence on QCTFE models of long bones.A femur, humerus, and proximal femur without the head were scanned with K₂HPO₄ phantoms by seven CT scanners (four models) using typical clinical protocols. QCTFE models were constructed. The geometrical dimensions, as well as the QCT-values expressed in Hounsfield unit (HU) distribution was compared. Principal strains at representative regions of interest (ROIs), and maximum principal strains (associated with fracture risk) were compared. Intraclass correlation coefficients (ICCs) were calculated to evaluate strain prediction reliability for different scanners. Repeatability was examined by scanning the femur twice and comparing resulting QCTFE models.Maximum difference in geometry was 2.3%. HU histograms before phantom calibration showed wide variation between QCT scans; however, bone density histogram variability was reduced after calibration and algorithmic manipulation. Relative standard deviation (RSD) in principal strains at ROIs was <10.7%. ICC estimates between scanners were >0.9. Fracture-associated strain had 6.7%, 8.1%, and 13.3% maximum RSD for the femur, humerus, and proximal femur, respectively. The difference in maximum strain location was <2 mm. The average difference with repeat scans was 2.7%.Quantification of strain differences showed mean RSD bounded by ∼6% in ROIs. Fracture-associated strains in “regular” bones showed a mean RSD bounded by ∼8%. Strains were obtained within a ±10% difference relative to the mean; thus, in a longitudinal study only changes larger than 20% in the principal strains may be significant. ICCs indicated high reliability of QCTFE models derived from different scanners.     

Molecular motors: how to make models that can be used to convey the concept of molecular ratchets and thermal capture

A wide variety of cellular processes use molecular motors, including processive motors that move along some form of track (e.g., myosin with actin, kinesin or dynein with tubulin) and polymerases that move along a template (e.g., DNA and RNA polymerases, ribosomes). In trying to understand how these molecular motors actually move, many apply their understanding of how man-made motors work: the latter use some form of energy to exert a force or torque on its load. However, quite a different mechanism has been proposed to possibly account for the movement of molecular motors. Rather than hydrolyzing ATP to push or pull their load, they might use their own thermal vibrational energy as well as that of their load and their environment to move the load, capturing those movements that occur along a desired vector or axis and resisting others; ATP hydrolysis is required to make backward movements impossible. This intriguing thermal capture or Brownian ratchet model is relatively more difficult to convey to students. In this report, we describe several teaching aids that are very easily constructed using widely available household materials to convey the concept of a molecular ratchet.     

Proportionality of the long bones and their relation to stature among Mesoamericans

A total of 235 (176 ♂ and 59 ♀) cadavers were measured. Of these blood samples of 132 (103 ♂ and 29 ♀) are drawn. Only those cadavers of which the long bones could be measured afterwards were used in the investigation, and of these only those whose blood was of group O and Rh +. A sample of 98 (69 ♂ and 29 ♀) was thus arrived at and divided morphoscopically into seven categories going from “pure” Indian to “pure” white. With electronic computers, means and standard deviations were calculated for all six unpaired long bones and for stature, as well as the coefficient of multiple correlation among the seven variables, for both males and females. As categories leading from “pure” white towards the other extreme were withdrawn, the sample size was reduced but the coefficient of multiple correlation increased from 0.71 to 0.90 in males, and from 0.74 to 0.94 in females, indicating that a more homogeneous population was being dealt with. Mean stature for this population was 161.50 cm and 149.80 cm for males and females, after 2.5 cm are deducted from cadaveral measurements. As the sample is morphoscopically and serologically as close as one could get to pre-hispanic conditions and as the statures arrived at are representative of what is known, tables were drawn giving the corresponding values of statures of males and females going from 180 cm to 130 cm at steps of 0.5 cm. It is submitted that until a larger sample is obtained, the newly drawn tables and formulae are more appropriate to calculate stature from long bones of American pre-hispanic populations than any other hitherto used.     

Subject-specific finite element models of long bones: An in vitro evaluation of the overall accuracy

The determination of the mechanical stresses induced in human bones is of great importance in both research and clinical practice. Since the stresses in bones cannot be measured non-invasively in vivo, the only way to estimate them is through subject-specific finite element modelling. Several methods exist for the automatic generation of these models from CT data, but before bringing them in the clinical practice it is necessary to assess their accuracy in the predictions of the bone stresses. Particular attention should be paid to those regions, like the epiphyseal and metaphyseal parts of long bones, where the automatic methods are typically less accurate. Aim of the present study was to implement a general procedure to automatically generate subject-specific finite element models of bones from CT data and estimate the accuracy of this general procedure by applying it to one real femur. This femur was tested in vitro under five different loading scenarios and the results of these tests were used to verify how the adoption of a simplified two-material homogeneous model would change the accuracy with respect to the density-based inhomogeneous one, with special attention paid to the epiphyseal and metaphyseal proximal regions of the bone. The results showed that the density-based inhomogeneous model predicts with a very good accuracy the measured stresses (R(2)=0.91, RMSE=8.6%, peak error=27%), while the two-material model was less accurate (R(2)=0.89, RMSE=9.6%, peak error=35%). The results showed that it is possible to automatically generate accurate finite element models of bones from CT data and that the strategy of material properties mapping has a significant influence on its accuracy.