Mimetization of the elastic properties of cancellous bone via a parameterized cellular material

Bone tissue mechanical properties and trabecular microarchitecture are the main factors that determine the biomechanical properties of cancellous bone. Artificial cancellous microstructures, typically described by a reduced number of geometrical parameters, can be designed to obtain a mechanical behavior mimicking that of natural bone. In this work, we assess the ability of the parameterized microstructure introduced by Kowalczyk (Comput Methods Biomech Biomed Eng 9:135–147, 2006. doi: 10.1080/10255840600751473) to mimic the elastic response of cancellous bone. Artificial microstructures are compared with actual bone samples in terms of elasticity matrices and their symmetry classes. The capability of the parameterized microstructure to combine the dominant isotropic, hexagonal, tetragonal and orthorhombic symmetry classes in the proportions present in the cancellous bone is shown. Based on this finding, two optimization approaches are devised to find the geometrical parameters of the artificial microstructure that better mimics the elastic response of a target natural bone specimen: a Sequential Quadratic Programming algorithm that minimizes the norm of the difference between the elasticity matrices, and a Pattern Search algorithm that minimizes the difference between the symmetry class decompositions. The pattern search approach is found to produce the best results. The performance of the method is demonstrated via analyses for 146 bone samples.     

The role of fabric in the quasi-static compressive mechanical properties of human trabecular bone from various anatomical locations

Osteoporosis leads to an increased risk of bone fracture. While bone density and architecture can be assessed in vivo with increasing accuracy using CT and MRI, their relationship with the critical mechanical properties at various anatomical sites remain unclear. The objective of this study was to quantify the quasi-static compressive mechanical properties of human trabecular bone among different skeletal sites and compare their relationships with bone volume fraction and a measure of microstructural anisotropy called fabric. Over 600 trabecular bone samples from six skeletal sites were assessed by and tested in uniaxial compression. Bone volume fraction correlated positively with elastic modulus, yield stress, ultimate stress, and the relationships depended strongly on skeletal site. The account of fabric improved these correlations substantially, especially when the data of all sites were pooled together, but the fabric–mechanical property relationships remained somewhat distinct among the anatomical sites. The study confirms that, beyond volume fraction, fabric plays an important role in determining the mechanical properties of trabecular bone and should be exploited in mechanical analysis of clinically relevant sites of the human skeleton.     

Estimation of the effective yield properties of human trabecular bone using nonlinear micro-finite element analyses

Micro-finite element (\(\upmu \)FE) analyses are often used to determine the apparent mechanical properties of trabecular bone volumes. Yet, these apparent properties depend strongly on the applied boundary conditions (BCs) for the limited size of volumes that can be obtained from human bones. To attenuate the influence of the BCs, we computed the yield properties of samples loaded via a surrounding layer of trabecular bone (“embedded configuration”). Thirteen cubic volumes (10.6 mm side length) were collected from \(\upmu \)CT reconstructions of human vertebrae and femora and converted into \(\upmu \)FE models. An isotropic elasto-plastic material model was chosen for bone tissue, and nonlinear \(\upmu \)FE analyses of six uniaxial, shear, and multi-axial load cases were simulated to determine the yield properties of a subregion (5.3 mm side length) of each volume. Three BCs were tested. Kinematic uniform BCs (KUBCs: each boundary node is constrained with uniform displacements) and periodicity-compatible mixed uniform BCs (PMUBCs: each boundary node is constrained with a uniform combination of displacements and tractions mimicking the periodic BCs for an orthotropic material) were directly applied to the subregions, while the embedded configuration was achieved by applying PMUBCs on the larger volumes instead. Yield stresses and strains, and element damage at yield were finally compared across BCs. Our findings indicate that yield strains do not depend on the BCs. However, KUBCs significantly overestimate yield stresses obtained in the embedded configuration (+43.1 ± 27.9%). PMUBCs underestimate (?10.0 ± 11.2%), but not significantly, yield stresses in the embedded situation. Similarly, KUBCs lead to higher damage levels than PMUBCs (+51.0 ± 16.9%) and embedded configurations (+48.4 ± 15.0%). PMUBCs are better suited for reproducing the loading conditions in subregions of the trabecular bone and deliver a fair estimation of their effective (asymptotic) yield properties.     

Effect of age on bone mineral density and micro architecture in the radius and tibia of horses: An Xtreme computed tomographic study

Background

The effect of age on the bone mineral density and microarchitecture of the equine radius and tibia was investigated. Fifty-six bones from 15 horses aged four to 21 years were used. There were nine geldings and six mares, and none of the horses had any disease influencing bone properties. Xtreme computed tomography was used to evaluate a 9-mm segment of the diaphysis and metaphysis of each bone. The following variables were determined: length of the bone, circumference and diameter in the frontal and sagittal planes in the middle of the bone.Diaphysis: total volume, bone volume, bone volume ratio, slice area, bone area, marrow area, cortical and marrow thickness, bone mineral density, polar moment of inertia of the cortex.Metaphysis: total area, bone area, cortical bone area, cortical thickness, bone mineral density, bone mineral density in the cortex, bone mineral density in the trabecular region, trabecular number, trabecular thickness, trabecular separation, polar moment of inertia of the metaphysis, polar moment of inertia of the cortex of the metaphysis.

Results

Bone density and microarchitecture were not affected by breed or gender. However, the microarchitecture varied with the age of the horse; the number of trabeculae decreased significantly and the distance between trabeculae increased significantly with increasing age. There were no significant differences between bones of the left and right limbs or between the radius and tibia.

Conclusion

The variables investigated did not differ between geldings and mares. However, there were age-related changes in the microstructure of the bones. Further experimental studies are necessary to determine whether these changes reduce bone strength. Age-related changes in the bones were seen and may explain the higher incidence of fractures and fissures in older horses.

     

Dietary l-carnitine supplementation improves bone mineral density by suppressing bone turnover in aged ovariectomized rats

Postmenopausal bone loss is a major public health concern. Although drug therapies are available, women are interested in alternative/adjunct therapies to slow down the bone loss associated with ovarian hormone deficiency. The purpose of this study was to determine whether dietary supplementation of l-carnitine can influence bone density and slow the rate of bone turnover in an aging ovariectomized rat model. Eighteen-month-old Fisher-344 female rats were ovariectomized and assigned to two groups: (1) a control group in which rats were fed ad libitum a carnitine-free (−CN) diet (AIN-93M) and (2) another fed the same diet but supplemented with l-carnitine (+CN). At the end of 8 weeks of feeding, animals were sacrificed and bone specimens were collected for measuring bone mineral content (BMC) and density (BMD) using dual energy X-ray absorptiometry. Femoral microarchitectural properties were assessed by microcomputed tomography. Femoral mRNA levels of selected bone matrix proteins were determined by northern blot analysis. Data showed that tibial BMD was significantly higher in the rat fed the +CN diet than those fed the −CN (control) diet. Dietary carnitine significantly decreased the mRNA level of tartrate-resistant acid phosphatase (TRAP), an indicator of bone resorption by 72.8%, and decreased the mRNA abundance of alkaline phosphatase (ALP) and collagen type-1 (COL), measures of bone formation by 63.6% and 61.2%, respectively. The findings suggest that carnitine supplementation slows bone loss and improves bone microstructural properties by decreasing bone turnover.     

Mechanical properties of cancellous bone in the human mandibular condyle are anisotropic

The objective of the present study was (1) to test the hypothesis that the elastic and failure properties of the cancellous bone of the mandibular condyle depend on the loading direction, and (2) to relate these properties to bone density parameters. Uniaxial compression tests were performed on cylindrical specimens (n=47) obtained from the condyles of 24 embalmed cadavers. Two loading directions were examined, i.e., a direction coinciding with the predominant orientation of the plate-like trabeculae (axial loading) and a direction perpendicular to the plate-like trabeculae (transverse loading). Archimedes' principle was applied to determine bone density parameters. The cancellous bone was in axial loading 3.4 times stiffer and 2.8 times stronger upon failure than in transverse loading. High coefficients of correlation were found among the various mechanical properties and between them and the apparent density and volume fraction. The anisotropic mechanical properties can possibly be considered as a mechanical adaptation to the loading of the condyle in vivo.     

Mechanical properties of nacre and highly mineralized bone

We compared the mechanical properties of 'ordinary' bovine bone, the highly mineralized bone of the rostrum of the whale Mesoplodon densirostris, and mother of pearl (nacre) of the pearl oyster Pinctada margaritifera. The rostrum and the nacre are similar in having very little organic material. However, the rostral bone is much weaker and more brittle than nacre, which in these properties is close to ordinary bone. The ability of nacre to outperform rostral bone is the result of its extremely well-ordered microstructure, with organic material forming a nearly continuous jacket round all the tiny aragonite plates, a design well adapted to produce toughness. In contrast, in the rostrum the organic material, mainly collagen, is poorly organized and discontinuous, allowing the mineral to join up to form, in effect, a brittle stony material.     

Porous surface modified bioactive bone cement for enhanced bone bonding

Background

Polymethylmethacrylate bone cement cannot provide an adhesive chemical bonding to form a stable cement-bone interface. Bioactive bone cements show bone bonding ability, but their clinical application is limited because bone resorption is observed after implantation. Porous polymethylmethacrylate can be achieved with the addition of carboxymethylcellulose, alginate and gelatin microparticles to promote bone ingrowth, but the mechanical properties are too low to be used in orthopedic applications. Bone ingrowth into cement could decrease the possibility of bone resorption and promote the formation of a stable interface. However, scarce literature is reported on bioactive bone cements that allow bone ingrowth. In this paper, we reported a porous surface modified bioactive bone cement with desired mechanical properties, which could allow for bone ingrowth.

Materials and Methods

The porous surface modified bioactive bone cement was evaluated to determine its handling characteristics, mechanical properties and behavior in a simulated body fluid. The in vitro cellular responses of the samples were also investigated in terms of cell attachment, proliferation, and osteoblastic differentiation. Furthermore, bone ingrowth was examined in a rabbit femoral condyle defect model by using micro-CT imaging and histological analysis. The strength of the implant–bone interface was also investigated by push-out tests.

Results

The modified bone cement with a low content of bioactive fillers resulted in proper handling characteristics and adequate mechanical properties, but slightly affected its bioactivity. Moreover, the degree of attachment, proliferation and osteogenic differentiation of preosteoblast cells was also increased. The results of the push-out test revealed that higher interfacial bonding strength was achieved with the modified bone cement because of the formation of the apatite layer and the osseointegration after implantation in the bony defect.

Conclusions

Our findings suggested a new bioactive bone cement for prosthetic fixation in total joint replacement.     

The effects of organic nitrates on osteoporosis: a randomized controlled trial [ISRCTN94484747]

Background

Osteoporotic fractures are common and are associated with increased morbidity, mortality and health care costs. The most effective way to moderate increases in health care costs and the sickness and premature death associated with osteoporotic fractures, is to prevent osteoporosis. Several lines of evidence suggest that nitrates, drugs typically prescribed for the treatment of angina, may be effective in preventing postmenopausal osteoporosis.

Methods

We have designed a multicentre randomized controlled trial to determine the effects of nitrates on bone. The trial consists of two studies. The objective of the first study is to determine whether isosorbide mononitrate at 20 mg/day or nitroglycerin ointment at 15 mg/day leads to fewer headaches. The nitrate that is best tolerated will be used in a second study with one main objective: To determine if postmenopausal women with a T-score at the lumbar spine (L1 to L4) between 0 and -2.0 randomized to two years of treatment with intermittent nitrates have a greater increase in spine bone mineral density as compared to women randomized to placebo. We hypothesize that: 1. Women will report fewer headaches when they are randomized to intermittent nitroglycerin ointment at 15 mg/day compared to intermittent oral isosorbide mononitrate at 20 mg/day, and, 2. After two years, women randomized to intermittent nitrates will have a greater percent increase in lumbar spine bone mineral density compared with women randomized to placebo.

Discussion

We have completed our pilot study and found that transdermal nitroglycerin was associated with fewer headaches than oral isosorbide mononitrate. We are currently recruiting patients for our second main study.     

Principal trabecular structural orientation predicted by quantitative ultrasound is strongly correlated with \upmu FEA determined anisotropic apparent stiffness

The microarchitecture and alignment of trabecular bone adapts to the particular mechanical milieu applied to it. Due to this anisotropic mechanical property, measurement orientation has to be taken into consideration when assessing trabecular bone quality and fracture risk prediction. Quantitative ultrasound (QUS) has demonstrated the ability in predicting the principal structural orientation (PSO) of trabecular bone. Although the QUS prediction for PSO is very close to that of \(\upmu \) CT, certain angle differences still exist. It remains unknown whether this angle difference can induce significant differences in mechanical properties or not. The objective of this study was to evaluate the mechanical properties in different PSOs predicted using different methods, QUS and \(\upmu \) CT, thus to investigate the ability of QUS as a means to predict the PSO of trabecular bone noninvasively. By validating the ability of QUS to predict the PSO of trabecular bone, it is beneficial for future QUS applications because QUS measurements in the PSO can provide information more correlated with the mechanical properties than with other orientations. In this study, seven trabecular bone balls from distal bovine femurs were used to generate finite element models based on the 3-dimensional \(\upmu \) CT images. Uniaxial compressive loading was performed on the bone ball models in the finite element analysis (FEA) in six different orientations (three anatomical orientations, two PSOs predicted by QUS and the longest vector of mean intercept length (MIL) tensor calculated by \(\upmu \) CT). The stiffness was calculated based on the reaction force of the bone balls under loading, and the von Mises stress results showed that both the mechanical properties in the PSOs predicted by QUS are significantly higher than the anatomical orientations and comparatively close to the longest vector of MIL tensor. The stiffness in the PSOs predicted by QUS is also highly correlated with the stiffness in the MIL tensor orientation (ATTmax vs. MIL, \(R^{2}\)  = 0.98, \(p<001\) ; UVmax vs. MIL, \(R^{2}\)  = 0.92, \(p<001\) ). These results were validated by in vitro mechanical testing on the bone ball samples. This study demonstrates that the PSO of trabecular bone predicted by QUS has an equally strong apparent stiffness with the orientation predicted by \(\upmu \) CT.     

Comparison between quantitative X-ray imaging,dual energy X-ray absorptiometry and microCT in the assessment of bone mineral density in disuse-induced bone loss

Objectives:

We recently introduced a new methodology called quantitative X-ray imaging (qXRI) to investigate bone mineral density in isolated rodent bones. The aims of the present study were to compare DXA and microCT with qXRI in a rat model of disuse osteoporosis.

Methods:

Fourteen Copenhagen rats were injected with a single dose of botulinum toxin (BTX - 2 UI) in the right Mus quadriceps femoris. The left hindlimb serves as control. Areal BMD and vBMD were determined with a Hologic Discovery-W device and a Skyscan 1172 microcomputed tomograph (microCT). Absorbing material density (AMD) was determined on digitized X-ray images obtained with a Faxitron M020 device.

Results:

All three methods highlighted significant lower values for aBMD, vBMD and AMD in trabecular and cortical bone in the BTX-injected side. In trabecular bone, aBMD, vBMD and AMD were significantly correlated with BV/TV. In cortical bone, only aBMD and vBMD were significantly correlated with cortical bone mass On the other hand, only AMD was significantly correlated with the mechanical parameters bending strength and bending modulus.

Conclusions:

qXRI is a rapid and cheap method to assess trabecular bone mass in isolated rodent bones and can be used as a surrogate for the densitometry of small animals.     

Least significant changes and monitoring time intervals for high-resolution pQCT-derived bone outcomes in postmenopausal women

Background:

Least Significant Change (LSC) assists in determining whether observed bone change is beyond measurement precision. Monitoring Time Interval (MTI) estimates time required to reliably detect skeletal changes. MTIs have not been defined for bone outcomes provided by high resolution peripheral quantitative computed tomography (HR-pQCT). The purpose of this study was to determine the LSCs and MTIs for HR-pQCT derived bone area, density and micro-architecture with postmenopausal women.

Methods:

Distal radius and tibia of 33 postmenopausal women (mean age: 77, SD: ±7 years), from the Saskatoon cohort of the Canadian Multicentre Osteoporosis Study (CaMos), were measured using HR-pQCT at baseline and 1-year later. We determined LSC from precision errors and divided them by the median annual percent changes to define MTIs for bone area, density, and micro-architecture.

Results:

Distal radius: HR-pQCT LSCs indicated a 1-8% observed change was needed for reliable monitoring of bone area and density while a 3-18% change was needed for micro-architectural measures. The longest MTIs (>3 years) pertained to cortical and trabecular area and density measures, cortical thickness and bone volume fraction; the shortest MTIs (~2 years) pertained to bone micro-architectural measures (trabecular number, thickness, separation and heterogeneity). Distal tibia: LSCs indicated a <1-5% observed change was needed for reliable monitoring of bone area and density, while a 3-19% change was needed for micro-architectural measures. The longest MTIs (>3 years) pertained to trabecular density, bone volume fraction, number, separation and heterogeneity; the shortest MTIs (~1 year) pertained to cortical and trabecular area, cortical density and thickness.

Conclusion:

MTIs suggest that performing HR-pQCT follow-up measures in postmenopausal women every 2 years at the distal radius and every 1 year at the distal tibia to monitor true skeletal changes as indicated by the LSCs.     

Influences of Endplate Removal and Bone Mineral Density on the Biomechanical Properties of Lumbar Spine

Purpose

To investigate (1) effects of endplate removal and bone mineral density (BMD) on biomechanical properties of lumbar vertebrae (2) whether the distributions of mechanical strength and stiffness of endplate are affected by BMD.

Methods

A total of thirty-one lumbar spines (L1-L5) collected from fresh cadavers were used in this study. Bone density was measured using lateral DEXA scans and parts of samples were performed with partial or entire endplate removal. All the specimens were divided into three BMD groups. According to endplate integrity of the lumbar vertebrae, each BMD group was then divided into three subgroups: subgroup A: intact endplate; subgroup B: central region of endplate removal; subgroup C: entire endplate removal. The axial compression test was conducted with material testing system at a speed of 2mm/min. The experimental results were statistically analyzed using SPSS 17.0.

Results

(1) Significant differences of biomechanical properties occurred among normal BMD, osteoporotic and serious osteoporotic group (P<0.05). (2) Spearman analysis showed that BMD was positively correlated with the failure load and stiffness of lumbar vertebrae. (3) For each BMD group, significant differences of biomechanical properties were found between subgroup A and C, and between subgroup B and C (P<0.05). (4) For each BMD group, there was no statistical difference of biomechanical properties between subgroup A and B (P>0.05).

Conclusions

Entire endplate removal can significantly decrease the structural properties of lumbar vertebrae with little change in biomechanical properties by preservation of peripheral region of the endplate. BMD is positively correlated to the structural properties of the lumbar vertebrae.     

Integration of pecking,filter feeding and drinking mechanisms in waterfowl

This paper is one of several contributions in a series, illustrating the application of a specific deductive methodology to explain diversity of form. The methodology facilitates the explanation of feeding morphologies in various ducks as a transformation of the mallard's feeding design maximized for specific proportions of performance that are contributed by pecking and filter feeding mechanisms.The earlier described anatomy and formal analyses of the three mechanisms in the mallard served as the initial conditions used in simulation models. Four elements of the feeding system were chosen that play a major role in all three mechanisms. For each element, the main parameter was selected: storage capacity of the rostral mouth cavity, transport capacity of the rostral mouth tube, storage capacity of the caudal mouth cavity and transport capacity of the caudal mouth tube. The boundary conditions for the simulation were determined from internal organismic constraints. The total food uptake of the mallard was regarded as the function to be maximized. This object function is the summation of the food uptake by one second of pecking and one second of filter feeding. The drinking mechanism was shown not to interfere, since that mechanism operates sufficiently whenever the pumping mechanism works properly.The object function, made up by the pecking and filter feeding performances was graphed. From these graphs a morphospace was developed: the region within which modifications of the mouth design are feasible. This procedure allowed examination of the general hypothesis that different modifications of one design for a complex multi-role system are explainable from differences in proportions of the functional performance contributed by each of the roles.Two predictions were evaluated more specifically: 1) If filter feeding performance must increase for a specific change in total food uptake, the volume of the rostral mouth cavity must increase; this requires widening and lengthening of the rostral maxillar portion and also a phase shift in jaw and lingual motion patterns, increasing the stroke volume. 2) If pecking performance must increase, the transport capacity of the rostral mouth tube must increase; this requires shortening of the maxillar mid portion. These two predictions regarding change in mouth morphology were borne out by shovelers and tufted ducks, respectively.The deductive analysis of the feeding system's morphospace allowed the explanation of the wide diversity of forms in the duck mouth designs as emerging from the great radiation capacity of the ancestral duck's mouth design: it is a versatile, potentially multi-role system in which pecking, drinking and filter feeding mechanisms are mutually very tolerant and, by epiphenomena, includes the elements of a grazing mechanism. In addition, the deduced morphospace was shown to explain phenotypic plasticity, since it explains the occurrence of two morphotypes that develop due to habitat discrimination: one type that has been forced to filter feed, and one type that feeds by pecking.Address reprint requests to Dr. G.A. Zweers     

Preparation and anti-osteoporotic activities in vivo of phosphorylated peptides from Antarctic krill (Euphausia superba)

Antarctic krill (Euphausia superba) protein serves as a novel sustainable protein source for human. Krill protein isolate was phosphorylated by the dry-heating method with sodium pyrophosphate. Phosphorylated peptides from Antarctic krill (PP-AKP) were obtained from phosphorylated protein through tryptic hydrolysis. Two types of phosphate bonds were introduced by phosphorylation, i.e. PO and PO bonds. The anti-osteoporotic activities of PP-AKP at two doses (400 and 800 mg/kg body weight) were investigated with an osteoporotic rat model, which was established with bilateral ovariectomy surgery. Different doses of PP-AKP were given intraperitoneal injections to rats once a day with alendronate as a positive control. Phosphorylated peptides from Antarctic krill dose-dependently preserved bone mineral density in osteoporotic rats by increasing the degree of bone mineralization. Both trabecular and cortical bone strength in osteoporotic rats was significantly improved with PP-AKP treatment. The mechanism by which PP-AKP augmented bone mineral density and bone strength was relation to the reduction in osteoclast-mediated bone remodeling, as was supported by the decrease in bone resorption markers. Phosphorylated peptides from Antarctic krill could be developed as functional food or nutritional supplements.     

Cellular organization of the lateral and postinfundibular regions of the median eminence in the rat

Summary The structural organization of the rostral, lateral and postinfundibular regions of the median eminence (ME) of 5-day cyclic diestrous rats was studied with light and electron microscopic methods. The ependymal cells lining (i) the floor of the infundibular recess (IR) at rostral levels, (ii) the lateral extensions of the IR, and (iii) the floor of the premammillary recess appear to represent the same type of tanycyte ependyma (1 tanycytes). In the entire width of the rostral and postinfundibular palisade regions, as well as in the lateral palisade region of the preinfundibular ME, the processes of the 1 tanycytes form a continuous cuff. This cuff separates the nerve endings from the blood vessels and the pars tuberalis. At this level, synaptoid contacts between neurosecretory axons and the ependymal cuff can be observed. The ultrastructural characteristics of the 1 tanycytes are described and their ependymal endings tentatively classified into three types. In the lateral regions of the ME, the Golgi study revealed the presence of two fiber systems: (i) one possessing a latero-medial trajectory and distributed in the subependymal region; (ii) the other formed by a loose longitudinal tract originating from neurons of the arcuate nucleus. Some functional implications of the cellular organization of the rat ME are discussed.Supported by Grants from PLAMIRH (92.171.2.77) and from the Dirección de Investigaciones, Universidad Austral (S-77-28)The authors wish to thank Miss Rosario Andrade, Mrs. Elizabeth Santibáñez and Mr. Armando Bilbao for their assistance     

Mechanical properties of the porcine growth plate vary with developmental stage

The objectives of this study were to extract the intrinsic mechanical properties of the growth plate at four different stages of growth and to compare two different methods of extracting these properties. Porcine distal ulnar growth plate samples were obtained from newborn, 4-, 8-, and 18-week (W) pigs and were tested using stress relaxation tests under unconfined compression. A four-parameter curve fitting procedure was developed to extract mechanical properties using the Transversely Isotropic Biphasic Elastic model(TIBPE) (Cohen et?al. in J Biomech Eng Trans Asme 120(4):491–496, 1998) and the Differential Evolution (DE) optimization algorithm (Price et?al. Natural computing series, Springer, Germany 2005). Optimization was done on all experimental curves for the first method and on one average experimental curve per developmental stage in the second. The 4-week stage was studied in two subgroups (a) and (b) due to distinct differences in mechanical properties. Intrinsic mechanical properties of the growth plate varied nonlinearly with developmental stage. Both methods showed that transverse and out-of-plane Young’s moduli (E ₁, E ₃) decrease with developmental stage, whereas transverse permeability (k ₁) increases. The exception is a sharp increase in stiffness and reduction in permeability at the 4W(a) stage, which may be associated with rapid porcine developmental changes at the 3–4 week period. The second method provides a more reliable representation of the average mechanical behavior, whereas the first method allows statistical comparison of optimized mechanical properties. This study characterizes, for the first time, the variation in growth plate mechanical properties for the same animal (porcine) and bone (ulna) model with developmental stage and provides new insight into the progression of musculoskeletal diseases during growth spurts in response to mechanical loading.     

Prediction of the mechanical response of the femur with uncertain elastic properties

A mandatory requirement for any reliable prediction of the mechanical response of bones, based on quantitative computer tomography, is an accurate relationship between material properties (usually Young's modulus E) and bone density ρ. Many such E-ρ relationships are available based on different experiments on femur specimens with a large spread due to uncertainties. The first goal of this study is to pool and analyze the relevant available experimental data and develop a stochasticE-ρ relationship. This analysis highlights that there is no experimental data available to cover the entire density range of the human femur and that some "popular" E-ρ relationships are based on data that contains extreme scatter, while others are based on a very limited amount of information. The second goal is to use the newly developed stochastic E-ρ relationship in high-order finite element analyses (FEAs) for the computation of strains and displacements in two human proximal femurs, mimicking in vitro experiments. When compared with the experimental observations, the FEA predictions using the median of the stochastic E-ρ relationship follow the underlying distribution of the stochastic E-ρ relationship. Thus, most deviations of the FEA predictions from experimental observations can possibly be explained by uncertain elastic properties of the femur.     

Prediction of apparent trabecular bone stiffness through fourth-order fabric tensors

The apparent stiffness tensor is an important mechanical parameter for characterizing trabecular bone. Previous studies have modeled this parameter as a function of mechanical properties of the tissue, bone density, and a second-order fabric tensor, which encodes both anisotropy and orientation of trabecular bone. Although these models yield strong correlations between observed and predicted stiffness tensors, there is still space for reducing accuracy errors. In this paper, we propose a model that uses fourth-order instead of second-order fabric tensors. First, the totally symmetric part of the stiffness tensor is assumed proportional to the fourth-order fabric tensor in the logarithmic scale. Second, the asymmetric part of the stiffness tensor is derived from relationships among components of the harmonic tensor decomposition of the stiffness tensor. The mean intercept length (MIL), generalized MIL (GMIL), and fourth-order global structure tensor were computed from images acquired through microcomputed tomography of 264 specimens of the femur. The predicted tensors were compared to the stiffness tensors computed by using the micro-finite element method (\(\upmu \)FE), which was considered as the gold standard, yielding strong correlations (\(R^2\) above 0.962). The GMIL tensor yielded the best results among the tested fabric tensors. The Frobenius error, geodesic error, and the error of the norm were reduced by applying the proposed model by 3.75, 0.07, and 3.16 %, respectively, compared to the model by Zysset and Curnier (Mech Mater 21(4):243–250, 1995) with the second-order MIL tensor. From the results, fourth-order fabric tensors are a good alternative to the more expensive \(\upmu \)FE stiffness predictions.