Elastic Properties Of Trabecular Bone Research Articles

Probability-based approach for characterization of microarchitecture and its effect on elastic properties of trabecular bone

Trabecular bone is a random cellular solid with an interconnected network of plate-like and rod-like components. However, the structural randomness and complexity have hindered rigorous mathematical modeling of trabecular bone microarchitecture. Recent advancements in imaging processing techniques have enabled us to define the size, orientation, and spatial location of individual trabecular plates and rods in trabecular bone. Based on the essential information, this study proposed a probability-based approach to define the size, orientation, and spatial distributions of trabecular plates and rods for trabecular bone cubes (N = 547) acquired from six human cadaver proximal femurs. Using two groups of probability-based parameters, it was attempted to capture microarchitectural details, which could not be captured by the existing histomorphometric parameters, but crucial to the elastic properties of trabecular bone. The elastic properties of the trabecular bone cubes in three principal axes were estimated using microCT based finite element (FE) simulations. Based on the results of multivariate multiple regression modeling, the efficacy of the two groups of probability-based parameters in prediction of the elastic properties was verified in comparison with that of the existing histomorphometric parameters (BV/TV, Tb.Th, Tb.Sp, DA, EF.Med, and Conn.D). The results indicated that the regression models trained using the probability-based parameters had a comparable and even better accuracy (rMSE = 0.621 and 0.548) than that of the histomorphometric parameters (rMSE = 0.647). More importantly, the probability-based parameters could provide more insights into some unexplored microarchitectural features, such as individual trabecular size, orientation, and spatial distributions, which are also critical to the elastic properties of trabecular bone.

Post-yield mechanical properties of bovine trabecular bone – Relationships with bone volume fraction and strain rate

Bone finite element models, which are increasingly used to assess fracture risk in patients with degenerative bone diseases, require extensive experimental data to adequately model the material properties being assigned. Previous studies have demonstrated that bone elastic mechanical properties are dependent on density as well as strain rate, however, the effect these same variables on the post-yield behaviour of trabecular bone has not been thoroughly studied. For this study, bovine femoral samples were tested at several strain rates using three different tests devices: a conventional constant strain rate apparatus for low strain rates, a Drop Weight Impact Test device for the intermediate regime and a Split Hopkinson Pressure Bar System with magnesium bar and a PVCA pulse shaper for high strain rates. Ultimate and plateau stress, as well as absorbed energy were recorded. The results of this experimentation show that post-yield mechanical properties of trabecular bone are strain rate and volume fraction dependant. Two constitutive material models describing the Ultimate strength, Plateau stress and absorbed energy are proposed based on published models valid for elastic properties of trabecular bone.

Effect of SR-microCT radiation on the mechanical integrity of trabecular bone using in situ mechanical testing and digital volume correlation

The use of synchrotron radiation micro-computed tomography (SR-microCT) is becoming increasingly popular for studying the relationship between microstructure and bone mechanics subjected to in situ mechanical testing. However, it is well known that the effect of SR X-ray radiation can considerably alter the mechanical properties of bone tissue. Digital volume correlation (DVC) has been extensively used to compute full-field strain distributions in bone specimens subjected to step-wise mechanical loading, but tissue damage from sequential SR-microCT scans has not been previously addressed. Therefore, the aim of this study is to examine the influence of SR irradiation-induced microdamage on the apparent elastic properties of trabecular bone using DVC applied to in situ SR-microCT tomograms obtained with different exposure times. Results showed how DVC was able to identify high local strain levels (> 10,000 µε) corresponding to visible microcracks at high irradiation doses (~ 230 kGy), despite the apparent elastic properties remained unaltered. Microcracks were not detected and bone plasticity was preserved for low irradiation doses (~ 33 kGy), although image quality and consequently, DVC performance were reduced. DVC results suggested some local deterioration of tissue that might have resulted from mechanical strain concentration further enhanced by some level of local irradiation even for low accumulated dose.

Micro-finite-element method to assess elastic properties of trabecular bone at micro- and macroscopic level

The objective of the present study is to assess the mechanical behavior of trabecular bone based on microCT imaging and micro-finite-element analysis. In this way two methods are detailed: (i) direct determination of macroscopic elastic property of trabecular bone; (ii) inverse approach to assess mechanical properties of trabecular bone tissue. Thirty-five females and seven males (forty-two subjects) mean aged (±SD) 80±11.7 years from hospitals of Assistance publique-Hôpitaux de Paris (AP-HP) diagnosed with osteoporosis following a femoral neck fracture due to a fall from standing were included in this study. Fractured heads were collected during hip replacement surgery. Standardized bone cores were removed from the femoral head's equator by a trephine in a water bath. MicroCT images acquisition and analysis were performed with CTan® software and bone volume fraction was then determined. Micro-finite-element simulations were per-formed using Abaqus 6.9-2® software in order to determine the macroscopic mechanical behaviour of the trabecular bone. After microCT acquisition, a longitudinal compression test was performed and the experimental macroscopic Young's Modulus was extracted. An inverse approach based on the whole trabecular bone's mechanical response and micro-finite-element analysis was performed to determine microscopic mechanical properties of trabecular bone. In the present study, elasticity of the tissue was shown to be similar to that of healthy tissue but with a lower yield stress. Classical histomorphometric analysis form microCT imaging associated with an inverse micro-finite-element method allowed to assess microscopic mechanical trabecular bone parameters.

Nonlinear homogenisation of trabecular bone: Effect of solid phase constitutive model.

Micro-finite element models have been extensively employed to evaluate the elastic properties of trabecular bone and, to a limited extent, its yield behaviour. The macroscopic stiffness tensor and yield surface are of special interest since they are essential in the prediction of bone strength and stability of implants at the whole bone level. While macroscopic elastic properties are now well understood, yield and post-yield properties are not. The aim of this study is to shed some light on what the effect of the solid phase yield criterion is on the macroscopic yield of trabecular bone for samples with different microstructure. Three samples with very different density were subjected to a large set of apparent load cases (which is important since physiological loading is complex and can have multiple components in stress or strain space) with two different solid phase yield criteria: Drucker-Prager and eccentric-ellipsoid. The study found that these two criteria led to small differences in the macroscopic yield strains for most load cases except for those that were compression-dominated; in these load cases, the yield strains for the Drucker-Prager criterion were significantly higher. Higher density samples resulted in higher differences between the two criteria. This work provides a comprehensive assessment of the effect of two different solid phase yield criteria on the macroscopic yield strains of trabecular bone, for a wide range of load cases, and for samples with different morphology.

Nonlinear viscoelastic characterization of bovine trabecular bone

The time-independent elastic properties of trabecular bone have been extensively investigated, and several stiffness–density relations have been proposed. Although it is recognized that trabecular bone exhibits time-dependent mechanical behaviour, a property of viscoelastic materials, the characterization of this behaviour has received limited attention. The objective of the present study was to investigate the time-dependent behaviour of bovine trabecular bone through a series of compressive creep–recovery experiments and to identify its nonlinear constitutive viscoelastic material parameters. Uniaxial compressive creep and recovery experiments at multiple loads were performed on cylindrical bovine trabecular bone samples (n = 19). Creep response was found to be significant and always comprised of recoverable and irrecoverable strains, even at low stress/strain levels. This response was also found to vary nonlinearly with applied stress. A systematic methodology was developed to separate recoverable (nonlinear viscoelastic) and irrecoverable (permanent) strains from the total experimental strain response. We found that Schapery’s nonlinear viscoelastic constitutive model describes the viscoelastic response of the trabecular bone, and parameters associated with this model were estimated from the multiple load creep–recovery (MLCR) experiments. Nonlinear viscoelastic recovery compliance was found to have a decreasing and then increasing trend with increasing stress level, indicating possible stiffening and softening behaviour of trabecular bone due to creep. The obtained parameters from MLCR tests, expressed as second-order polynomial functions of stress, showed a similar trend for all the samples, and also demonstrate stiffening–softening behaviour with increasing stress.

Evaluating the macroscopic yield behaviour of trabecular bone using a nonlinear homogenisation approach

Computational homogenisation approaches using high resolution images and finite element (FE) modelling have been extensively employed to evaluate the anisotropic elastic properties of trabecular bone. The aim of this study was to extend its application to characterise the macroscopic yield behaviour of trabecular bone. Twenty trabecular bone samples were scanned using a micro-computed tomography device, converted to voxelised FE meshes and subjected to 160 load cases each (to define a homogenised multiaxial yield surface which represents several possible strain combinations). Simulations were carried out using a parallel code developed in-house. The nonlinear algorithms included both geometrical and material nonlinearities. The study found that for tension-tension and compression-compression regimes in normal strain space, the yield strains have an isotropic behaviour. However, in the tension-compression quadrants, pure shear and combined normal-shear planes, the macroscopic strain norms at yield have a relatively large variation. Also, our treatment of clockwise and counter-clockwise shears as separate loading cases showed that the differences in these two directions cannot be ignored. A quadric yield surface, used to evaluate the goodness of fit, showed that an isotropic criterion adequately represents yield in strain space though errors with orthotropic and anisotropic criteria are slightly smaller. Consequently, although the isotropic yield surface presents itself as the most suitable assumption, it may not work well for all load cases. This work provides a comprehensive assessment of material symmetries of trabecular bone at the macroscale and describes in detail its macroscopic yield and its underlying microscopic mechanics.

A three-scale identification of orthotropic properties of trabecular bone

The bone is a hierarchically structured material with mechanical properties depending on its architecture at all scales. In human bodies, the bone consists of three types: Cortical bone, trabecular bone and marrow. It has been shown that the mechanical properties of the bone vary at different structural scales ( Rho et al. 1998 ). Therefore, in order to find the macroscopic properties of the bone, it is important to consider all of these scales. Many researchers have been investigated in this field, but the behaviour of the bone materials are modeled at different scales separately. Structurally, bone is considered like a composite material with a complex structure. For simplicity, many studies consider an isotropic linear elastic behaviour ( Sansalone et al. 2010; Hambli et al. 2010 ). In other research, several authors simulate bone as anisotropic behaviour ( Martinez-Reina et al. 2009; Doblare et al. 2002 ). To predict the orthotropic properties at different scales, Hamed et al. (2012) used homogenization approach. Elastic properties of trabecular bone are calculated at each structural level, from nanoscale to mesoscale. In the analysis, results from a lower level are used as inputs for a higher level. In the same way, Vaughan et al. (2012) proposed a three scale homogenisation scheme to estimate the effective properties of trabecular and cortical bone, based on finite element models. The objective of this study is to develop a multiscale approach that can incorporate multiscale variability in both bone composition and organization to predict the orthotropic behaviour.

Stochastic multi-scale prediction on the apparent elastic moduli of trabecular bone considering uncertainties of biological apatite (BAp) crystallite orientation and image-based modelling

An assessment of the mechanical properties of trabecular bone is important in determining the fracture risk of human bones. Many uncertainty factors contribute to the dispersion of the estimated mechanical properties of trabecular bone. This study was undertaken in order to propose a computational scheme that will be able to predict the effective apparent elastic moduli of trabecular bone considering the uncertainties that are primarily caused by image-based modelling and trabecular stiffness orientation. The effect of image-based modelling which focused on the connectivity was also investigated. A stochastic multi-scale method using a first-order perturbation-based and asymptotic homogenisation theory was applied to formulate the stochastically apparent elastic properties of trabecular bone. The effective apparent elastic modulus was predicted with the introduction of a coefficient factor to represent the variation of bone characteristics due to inter-individual differences. The mean value of the predicted effective apparent Young's modulus in principal axis was found at approximately 460 MPa for respective 15.24% of bone volume fraction, and this is in good agreement with other experimental results. The proposed method may provide a reference for the reliable evaluation of the prediction of the apparent elastic properties of trabecular bone.

Predicting trabecular bone elastic properties from measures of bone volume fraction and fabric on the basis of micromagnetic resonance images

The relationship between fabric (a measure of structural anisotropy) and elastic properties of trabecular bone was examined by invoking morphology and homogenization theory on the basis of micromagnetic resonance images from the distal tibia in specimens (N = 30) and human subjects (N = 16) acquired at a 160 × 160 × 160 μm(3) voxel size. The fabric tensor was mapped in 7.5 × 7.5 × 7.5 mm(3) cubic subvolumes by a three-dimensional mean-intercept-length method. Elastic constants (three Young's and three shear moduli) were derived from linear microfinite element simulations of three-dimensional grayscale bone volume fraction-mapped images. In the specimen data, moduli fit power laws of bone volume fraction (bone volume/total volume) for all three test directions and subvolumes (R(2) = 0.92-0.98) with exponents ranging from 1.3 to 1.8. Weaker linear relationships were found for the in vivo data because of a narrower range in bone volume/total volume. When pooling the data for all test directions and subvolumes, bone volume/total volume predicted elastic moduli less well in the specimens (mean R(2) = 0.74) and not at all in vivo. A model of bone volume/total volume and fabric was highly predictive of microfinite element-derived Young's moduli: mean R(2) s of 0.98 and 0.82 (in vivo). The results show that fabric, an important predictor of bone mechanical properties, can be assessed in the limited resolution and signal-to-noise ratio regime of micromagnetic resonance images.

TRANSVERSE ISOTROPIC ELASTIC PROPERTIES OF VERTEBRAL TRABECULAR BONE MATRIX MEASURED USING MICROINDENTATION UNDER DRY CONDITIONS (EFFECTS OF AGE, GENDER, AND VERTEBRAL LEVEL)

The mechanical properties of bone extracellular matrix have become of increasing interest for the understanding of vertebral fracture risk. Depth-sensing indentation techniques allow the measurement of directional elastic properties of trabecular bone ex vivo with a high spatial resolution. Transverse isotropic elastic properties of vertebral trabecular bone obtained from two orthogonal directions were investigated using microindentation under dry conditions focusing on the influence of microanatomical location, age, gender, vertebral level, and anatomic direction on these properties. Biopsies were obtained from 104 human vertebrae (T1–L3) with a median age of 65 (21–94) years. Significantly, higher indentation moduli were found for indentations on axial than on transverse cross-sections of trabeculae (p < 0.01). Indentation moduli in the core were 1.05 to 1.12 times higher than in the periphery (p < 0.01). No difference in stiffness could be detected between males and females (p > 0.05) and different ages (p > 0.5). Vertebral level showed a weak correlation (p = 0.073, r2 ≈ 0.17). These results provide insights in the transverse isotropic properties of trabecular bone matrix related to age, gender, microanatomical location, and anatomic direction for a broad spectrum of human vertebrae.

Micromechanical Modeling of the Anisotropy of Elastic Biological Composites

In this paper, a Mori–Tanaka micromechanical modeling of effective elastic properties of trabecular bone is coupled with experimental measurements of morphology obtained from X-ray microtomography. The principle of modeling is based on the Eshelby and Hill polarization tensors of isolated ellipsoidal inclusions embedded in an infinite matrix. The problem is entirely geometrized and is treated in terms of averages of Walpole's components of the fourth-order tensors describing the problem. The structural anisotropy of trabecular bone was determined in three-dimensional space by means of the mean intercept length method and expressed by a fabric tensor, which is a second rank tensor describing the orientation distribution of inclusions volume fractions and degree of anisotropy of trabecular bone. Modeling results showed the high effects of the aspect ratio and orientation distribution of pores on the effective elastic properties of trabecular bone.

Complete Volumetric Decomposition of Individual Trabecular Plates and Rods and Its Morphological Correlations With Anisotropic Elastic Moduli in Human Trabecular Bone

Trabecular plates and rods are important microarchitectural features in determining mechanical properties of trabecular bone. A complete volumetric decomposition of individual trabecular plates and rods was used to assess the orientation and morphology of 71 human trabecular bone samples. The ITS-based morphological analyses better characterize microarchitecture and help predict anisotropic mechanical properties of trabecular bone. Standard morphological analyses of trabecular architecture lack explicit segmentations of individual trabecular plates and rods. In this study, a complete volumetric decomposition technique was developed to segment trabecular bone microstructure into individual plates and rods. Contributions of trabecular type-associated morphological parameters to the anisotropic elastic moduli of trabecular bone were studied. Seventy-one human trabecular bone samples from the femoral neck (FN), tibia, and vertebral body (VB) were imaged using muCT or serial milling. Complete volumetric decomposition was applied to segment trabecular bone microstructure into individual plates and rods. The orientation of each individual trabecula was determined, and the axial bone volume fractions (aBV/TV), axially aligned bone volume fraction along each orthotropic axis, were correlated with the elastic moduli. The microstructural type-associated morphological parameters were derived and compared with standard morphological parameters. Their contributions to the anisotropic elastic moduli, calculated by finite element analysis (FEA), were evaluated and compared. The distribution of trabecular orientation suggested that longitudinal plates and transverse rods dominate at all three anatomic sites. aBV/TV along each axis, in general, showed a better correlation with the axial elastic modulus (r(2) = 0.95 approximately 0.99) compared with BV/TV (r(2) = 0.93 approximately 0.94). The plate-associated morphological parameters generally showed higher correlations with the corresponding standard morphological parameters than the rod-associated parameters. Multiple linear regression models of six elastic moduli with individual trabeculae segmentation (ITS)-based morphological parameters (adjusted r(2) = 0.95 approximately 0.98) performed equally well as those with standard morphological parameters (adjusted r(2) = 0.94 approximately 0.97) but revealed specific contributions from individual trabecular plates or rods. The ITS-based morphological analyses provide a better characterization of the morphology and trabecular orientation of trabecular bone. The axial loading of trabecular bone is mainly sustained by the axially aligned trabecular bone volume. Results suggest that trabecular plates dominate the overall elastic properties of trabecular bone.

A New Computational Efficient Approach for Trabecular Bone Analysis using Beam Models Generated with Skeletonized Graph Technique

Micro-finite element (FE) analysis is a well established technique for the evaluation of the elastic properties of trabecular bone, but is limited in its application due to the large number of elements that it requires to represent the complex internal structure of the bone. In this paper, we present an alternative FE approach that makes use of a recently developed 3D-Line Skeleton Graph Analysis (LSGA) technique to represent the complex internal structure of trabecular bone as a network of simple straight beam elements in which the beams are assigned geometrical properties of the trabeculae that they represent. Since an enormous reduction of cputime can be obtained with this beam modeling approach, ranging from approximately 1,200 to 3,600 for the problems investigated here, we think that the FE modeling technique that we introduced could potentially constitute an interesting alternative for the evaluation of the elastic mechanical properties of trabecular bone.

On the importance of geometric nonlinearity in finite-element simulations of trabecular bone failure

The finite element method, which has been successfully applied to studies of the elastic properties of trabecular bone, is now being used to simulate its failure. These simulations have used a geometrically linear (linear kinematic) approximation to the total stiffness matrix; nonlinear terms in the total stiffness matrix have been excluded from the computation in order to achieve efficiency. Because trabecular bone appears to be a slender (i.e., geometrically nonlinear) structure, we studied the validity of the linear kinematic approximation for simulating its failure. Two cases, designed to bracket the extremes of stability behavior, were explored: a single representative spicule of trabecular bone (case 1) and a volume of trabecular bone consisting of relatively low aspect ratio members (case 2). For case 1, geometrically linear (GL) and nonlinear (GNL) analyses were performed with two different materials models: a plastic damage model and a brittle damage model. When GNL terms were included in the total stiffness matrix, we found that load-path bifurcation preceded tissue failure regardless of the form of the damage model. This bifurcation was the result of a complex coupling between material yield and structural instability. The nature of this coupling was highly sensitive to the form of the damage model. None of these behaviors was observed in the linear analyses, where failure was insensitive to the form of the damage model and where structural instabilities were prevented from occurring. For case 2, compressive loading of a volume of trabecular bone, geometric nonlinear effects were pronounced. There was a bifurcation in load response that resulted in large apparent strain to failure. The GL simulations, on the other hand, precluded this bifurcation. We hypothesize that trabecular bone is a geometric nonlinear structure; nonlinear terms must be included in the total stiffness matrix to accurately simulate its failure.

Micro-FE analyses of bone: state of the art.

The ability to provide a complete characterization of elastic properties of bone has vastly improved our understanding of trabecular bone mechanical properties. Based on this information, it was possible to validate several mechanical concepts related to the elastic behavior of trabecular bone that could not be validated earlier. With recently developed micro-CT scanners and the availability of large parallel computer systems, this technique has also enabled the determination of physiological bone tissue loading conditions from very large microFE models that can represent whole human bones in detail. Such analyses can provide the data needed for a better understanding of bone failure processes or cell mediated load adaptive remodeling processes. Computational demands for whole bone analyses, however, are still excessive. Unlike linear stress and strain analyses, the application of PFE to study non-linear processes, in particular bone failure mechanisms, is still in an early phase Results of recent studies, however, are promising and indicate that an accurate prediction of bone failure with these techniques is possible. Compelling features of such analyses are that they enable multi-axial failure criteria at the apparent level to be developed using primarily computational methods as well as that they can provide a basis for detailed analysis of micro-mechanics associated with trabecular failure at the apparent level. The application of microFE techniques to analyze bone in vivo is in an early stage as well. First results have indicated that, although the resolution of presently available in vivo imaging techniques (i.e. pQCT and MR) is much less than that of images used so far for uFE analyses, the technique can provide meaningful elastic properties of trabecular bone in vivo in most cases. It is expected that the remaining uncertainties in the microFE results can be eliminated as soon as the resolution of in vivo images is improved. With the fast developments in pQCT(47) and MR scanning, this will probably be in the near future.

High-resolution three-dimensional-pQCT images can be an adequate basis for in-vivo microFE analysis of bone.

Micro-finite element (microFE) models based on high-resolution images have enabled the calculation of elastic properties of trabecular bone in vitro. Recently, techniques have been developed to image trabecular bone structure in vivo, albeit at a lesser resolution. The present work studies the usefulness of such in-vivo images for microFE analyses, by comparing their microFE results to those of models based on high-resolution micro-CT (microCT) images. Fifteen specimens obtained from human femoral heads were imaged first with a 3D-pQCT scanner at 165 microns resolution and a second time with a microCT scanner at 56 microns resolution. A third set of images with a resolution of 165 microns was created by downscaling the microCT measurements. The microFE models were created directly from these images. Orthotropic elastic properties and the average tissue von Mises stress of the specimens were calculated from six FE-analyses per specimen. The results of the 165 microns models were compared to those of the 56 microns model, which was taken as the reference model. The results calculated from the pQCT-based models, correlated excellent with those calculated from the reference model for both moduli (R2 > 0.95) and for the average tissue von Mises stress (R2 > 0.83). Results calculated from the downscaled micro-CT models correlated even better with those of the reference models (R2 > 0.99 for the moduli and R2 > 0.96 for the average von Mises stress). In the case of the 3D-pQCT based models, however, the slopes of the regression lines were less than one and had to be corrected. The prediction of the Poisson's ratios was less accurate (R2 > 0.45 and R2 > 0.67) for the models based on 3D-pQCT and downscaled microCT images respectively). The fact that the results from the downscaled and original microCT images were nearly identical indicates that the need for a correction in the case of the 3D-pQCT measurements was not due to the voxel size of the images but due to a higher noise level and a lower contrast in these images, in combination with the application of a filtering procedure at 165 micron images. In summary: the results of microFE models based on in-vivo images of the 3D-pQCT can closely resemble those obtained from microFE models based on higher resolution microCT system.

A global relationship between trabecular bone morphology and homogenized elastic properties.

An alternative concept of the relationship between morphological and elastic properties of trabecular bone is presented and applied to human tissue from several anatomical locations using a digital approach. The three-dimensional morphology of trabecular bone was assessed with a microcomputed tomography system and the method of directed secants as well as the star volume procedure were used to compute mean intercept length (MIL) and average bone length (ABL) of 4 mm cubic specimens. Assuming isotropic elastic properties for the trabecular tissue, the general elastic tensors of the bone specimens were determined using the homogenization method and the closest orthotropic tensors were calculated with an optimization algorithm. The assumption of orthotropy for trabecular bone was found to improve with specimen size and hold within 6.1 percent for a 4 mm cube size. A strong global relationship (r2 = 0.95) was obtained between fabric and the orthotropic elastic tensor with a minimal set of five constants. Mean intercept length and average bone length provided an equivalent power of prediction. These results support the hypothesis that the elastic properties of human trabecular bone from an arbitrary anatomical location can be estimated from an approximation of the anisotropic morphology and a prior knowledge of tissue properties.

The relationship between three-dimensional connectivity and the elastic properties of trabecular bone.

A finite-element model was used to explore the relationship between connectivity density and the elastic modulus of trabecular bone. Six cubic specimens of trabecular bone, three prepared from human distal radii and three from L1 vertebrae, were imaged with synchrotron microtomography. The three-dimensional images were reconstructed into binary volumes of mineralized bone and soft tissue, and incorporated into the finite-element model. The relationship between three-dimensional connectivity and elastic modulus was explored by uniform thinning (atrophy) and thickening (recovery) of the trabecular bone. Though no functional relationship was found between connectivity and elastic modulus, there was a linear relationship, after a full cycle of atrophy and recovery, between the loss of elastic modulus and the overall loss of connectivity. The results indicate that recovery of mechanical function depends on preserving or restoring trabecular connectivity.

Linear elastic properties of trabecular bone: a cellular solid approach

Linear elastic properties of trabecular bone: a cellular solid approach