Elastic properties of cancellous bone derived from finite element models of parameterized microstructure cells

Evaluation of constitutive properties of cancellous bone and their relationships to microstructural parameters is a crucial issue in analysis of stresses and strains in bone tissues and simulation of their remodelling. Known limitations of experimental methods as well as of the micro-FE techniques make the analysis and homogenization of 'equivalent' trabecular microstructures an advantageous tool for this task. In this study, parameterized orthotropic constitutive models of cancellous bone are derived from finite element analysis of repeatable microstructure cells. Two cell types are analysed: cube- and prism-based. The models are fully three-dimensional, have realistic curvilinear shapes and are parameterized with three shape parameters. Variation of the parameters allows to imitate most of the typical microstructure patterns observed in real bones, along with variety of intermediate geometries. Finite element models of cells are generated by a special-purpose structured mesh generator for any arbitrary set of shape parameter values. Six static numerical tests are performed for an exhaustive number of parameter value sets (microstructure instances). Multi-point boundary conditions imposed on the models ensure mutual fitting of deformed neighbouring cells. Values of computed stresses allow to determine all coefficients of elastic orthotropic stiffness matrix. Results have a form of tabularized functions of elastic constants versus the shape parameters. Comparison of the results with micro-FE data obtained for a large set of cancellous bone specimens proves a good agreement, though evidently better in the case of the prism-based cell model.     

Tensile and compressive properties of cancellous bone

The relationship between the mechanical properties of trabecular bone in tension and compression was investigated by non-destructive testing of the same specimens in tension and compression, followed by random allocation to a destructive test in either tension or compression. There was no difference between Young's modulus in tension and compression, and there was a strong positive correlation between the values (R = 0.97). Strength, ultimate strain and work to failure was significantly higher in tensile testing than in compressive testing.     

Elastic constants of composites formed from PMMA bone cement and anisotropic bovine tibial cancellous bone

An ultrasonic pulse-transit time technique is used to determine the nine orthotropic engineering constants of 32 cement-cancellous bone composites as a function of volume fractions of bone ranging from 0.0 to 0.4. The composites are manufactured using well-aligned bovine cancellous bone from the proximal end of the tibia and low viscosity bone cement. Selected composites are also subjected to mechanical compression tests to compare with the ultrasonic results. There is excellent correlation between the dynamic or ultrasonically determined moduli and the static or mechanically determined moduli; the dynamic moduli are approximately twice the static moduli and this difference is thought to be due to the effect of strain rate. An orthotropic model is assumed requiring nine independent elastic constants to be determined. The dynamic Young's modulus in the direction of major trabecular alignment, E1, increases linearly from 4.9 to 10.4 GPa as bone volume fraction increases from 0 to 0.4; dynamic E2 and E3 values increase from 4.9 to 7 GPa as bone volume fractions increase from 0 to 0.4, with E2 being slightly higher than E3. The dynamic shear modulus, G12, increases from 1.8 to 3.0 GPa, and G31 and G23 increase slightly from 1.8 to 2.2 GPa as bone volume fractions increase from 0 to 0.4. The Poisson's ratios are more sensitive than the Young's moduli and shear moduli to experimental error in the velocity measurements. The mechanically tested modulus (static modulus) in the direction of major trabecular alignment, E1, increases with volume fraction of bone from 2.4 to 4.4 GPa as the bone volume fraction increases from 0 to 0.25; static E2 and E3 values are either equal to or lower than that of pure PMMA.     

Orthotropic properties of cancellous bone modelled as parameterized cellular material

Analysis of stresses and strains in bone tissues and simulation of their adaptive remodelling require exhaustive information about distribution of constitutive properties of cancellous bone and their relationships to microstructural parameters. Homogenization of "equivalent" trabecular microstructures appears to be an advantageous tool for this task. In this study, parameterized orthotropic constitutive models of cancellous bone are derived from finite element analysis of repeatable microstructure cells. The models, based on a space-filling dodecahedron, are fully three-dimensional and are parameterized with four shape parameters. Variation of the parameters allows to imitate most of typical microstructure patterns observed in real bones, along with a variety of intermediate geometries. Finite element models of cells are generated by a special-purpose structured mesh generator for any arbitrary set of shape parameter values. Static numerical tests are performed for an exhaustive number of parameter value sets (microstructure instances). Coefficients of elastic orthotropic stiffness matrix are determined as tabularized functions of elastic constants versus the shape parameters. Additionally, they are correlated to apparent density and principal fabric tensor values. Comparison of the results with micro-FE data obtained for a large set of cancellous bone specimens proves a good agreement.     

Orthotropic properties of cancellous bone modelled as parameterized cellular material

Analysis of stresses and strains in bone tissues and simulation of their adaptive remodelling require exhaustive information about distribution of constitutive properties of cancellous bone and their relationships to microstructural parameters. Homogenization of “equivalent” trabecular microstructures appears to be an advantageous tool for this task. In this study, parameterized orthotropic constitutive models of cancellous bone are derived from finite element analysis of repeatable microstructure cells. The models, based on a space-filling dodecahedron, are fully three-dimensional and are parameterized with four shape parameters. Variation of the parameters allows to imitate most of typical microstructure patterns observed in real bones, along with a variety of intermediate geometries. Finite element models of cells are generated by a special-purpose structured mesh generator for any arbitrary set of shape parameter values. Static numerical tests are performed for an exhaustive number of parameter value sets (microstructure instances). Coefficients of elastic orthotropic stiffness matrix are determined as tabularized functions of elastic constants versus the shape parameters. Additionally, they are correlated to apparent density and principal fabric tensor values. Comparison of the results with micro-FE data obtained for a large set of cancellous bone specimens proves a good agreement.     

Vascularized periosteum associated with cancellous bone graft: an experimental study

The association of a vascularized periosteal flap with a cancellous bone graft was studied on a group of 20 Wistar rats. Ten rats were sacrificed at 6 weeks and seven at 12 weeks (three died prematurely). The behavior of the cancellous bone graft buried in striated muscle and the osteogenic capacity of a simple vascularized periosteal flap also were observed on the same animals. Results of the study are as follows: In 14 of 17 animals, a vascularized periosteal flap wrapped around a cancellous bone graft resulted in new cortical bone formation with little resorption of the initial cancellous graft. A vascularized musculoperiosteal flap has produced a small amount of new compact bone only in 4 of 17 animals. A cancellous bone graft buried into well-vascularized muscle tissue was resorbed (15 cases) or necrotic (2 cases) at 12 weeks. In conclusion, the association of a vascularized periosteal flap and cancellous bone is a better means to produce compact bone than a vascularized periosteal flap alone or an isolated cancellous bone graft.     

The role of an effective isotropic tissue modulus in the elastic properties of cancellous bone.

Conceptually, the elastic characteristics of cancellous bone could be predicted directly from the trabecular morphology--or architecture--and by the elastic properties of the tissue itself. Although hardly any experimental evidence exists, it is often implicitly assumed that tissue anisotropy has a negligible effect on the apparent elastic properties of cancellous bone. The question addressed in this paper is whether this is actually true. If it is, then micromechanical finite element analysis (micro-FEA) models, representing trabecular architecture, using an 'effective isotropic tissue modulus' should be able to predict apparent elastic properties of cancellous bone. To test this, accurate multi-axial compressive mechanical tests of 29 whale bone specimens were simulated with specimen-specific micro-FEA computer models built from true three-dimensional reconstructions. By scaling the micro-FEA predictions by a constant tissue modulus, 92% of the variation of Young's moduli determined experimentally could be explained. The correlation even increased to 95% when the micro-FEA moduli were scaled to the isotropic tissue moduli of individual specimens. Excellent agreement was also found in the elastic symmetry axes and anisotropy ratios. The prediction of Poisson's ratios was somewhat less precise at 85% correlation. The results support the hypothesis; for practical purposes, the concept of an 'effective isotropic tissue modulus' concept is a viable one. They also suggest that the value of such a modulus for individual cases might be inferred from the average tissue density, hence the degree of mineralization. Future studies must clarify how specific the tissue modulus should be for different types of bone if adequate predictions of elastic behavior are to be made in this way.     

Linear poroelastic cancellous bone anisotropy: trabecular solid elastic and fluid transport properties

The mechanical performance of cancellous bone is characterized using experiments which apply linear poroelasticity theory. It is hypothesized that the anisotropic organization of the solid and pore volumes of cancellous bone can be physically characterized separately (no deformable boundary interactive effects) within the same bone sample. Due to its spongy construction, the in vivo mechanical function of cancellous or trabecular bone is dependent upon fluid and solid materials which may interact in a hydraulic, convective fashion during functional loading. This project provides insight into the organization of the tissue, ie., the trabecular connectivity, by defining the separate nature of this biphasic performance. Previous fluid flow experiments [Kohles et al., 2001, Journal of Biomechanics, 34(11), pp. 1197-1202] describe the pore space via orthotropic permeability. Ultrasonic wave propagation through the trabecular network is used to describe the solid component via orthotropic elastic moduli and material stiffness coefficients. The linear poroelastic nature of the tissue is further described by relating transport (fluid flow) and elasticity (trabecular load transmission) during regression analysis. In addition, an empirical relationship between permeability and porosity is applied to the collected data. Mean parameters in the superior-inferior (SI) orientation of cubic samples (n=20) harvested from a single bovine distal femur were the largest (p<0.05) in comparison to medial-lateral (ML) and anterior-posterior (AP) orientations: Apparent elastic modulus (2,139 MPa), permeability (4.65x10(-10) m2), and material stiffness coefficient (13.6 GPa). A negative correlation between permeability as a predictor of structural elastic modulus supported a parametric relationship in the ML (R2=0.4793), AP (R2=0.3018), and SI (R2=0.6445) directions (p<0.05).     

A new bone banking technique to maintain osteoblast viability in frozen human iliac cancellous bone

The aim of this study was to develop a new cryopreservation technique to maintain the osteoblast viability in frozen iliac bone and to prove cell viability using cell culture techniques.Human iliac cancellous bones were frozen with and without 10% Me(2)SO at -80 degrees C. The tubes were kept in a -80 degrees C freezer for at least 2 days. After the storage period, the frozen bone was thawed by placing the tube in a 37 degrees C water bath. A serial enzymatic digestion technique using 0.2% collagenase was employed to isolate osteoblast-like cells from the bone. The cells that were released were inoculated into tissue culture flasks containing DMEM supplemented with 10% FCS. They were incubated at 37 degrees C in a humidified atmosphere of 95% air and 5% CO(2). Cells of the second passage were plated at a density of 5 x 10(3)cells/cm(2) in a 24-well plate and used for characterization. For characterization, WST-1 assay, determination of alkaline phosphatase, Type I collagen assay, osteocalcin assay, and von Kossa staining were used. The assays were performed at 3, 6, 9, and 12 days after plating the cells. Based on the results of this study, we conclude that the osteoblast-like cells in the frozen bone can survive, only when the bone is frozen with cryoprotectants to prevent injury during freezing and thawing.     

Elastic properties of human supraorbital and mandibular bone

Elastic constants, including the elastic modulus, the shear modulus, and Poisson's ratio, were measured on human craniofacial bone specimens obtained from the supraorbital region and the buccal surfaces of the mandibles of unembalmed cadavers. Constants were determined using an ultrasonic wave technique in three directions relative to the surface of each sample: 1) normal, 2) tangential, and 3) longitudinal. Statistical analysis of these elastic constants indicated that significant differences in the relative proportions of elastic properties existed between the regions. Bone from the mandible along its longitudinal axis was stiffer than bone from the supraorbital region. Directional differences in both locations demonstrated that cranial bone was not elastically isotropic. It is suggested that differences in elastic properties correspond to regional differences in function. © 1993 Wiley-Liss, Inc.     

A continuous wave technique for the measurement of the elastic properties of cortical bone

A continuous wave technique is described for measuring the nine independent orthotropic elastic coefficients from a single cubic specimen. The side dimensions of this cubic specimen are on the order of 5 mm. Because of the small size of the specimen, the spatial resolution of material inhomogeneity using this technique is quite good. Although it is possible to apply this technique to any elastic material such as woods or metals, the elastic properties of human and canine cortical femora are presented here. The orthotropic elastic coefficients and the variation of these coefficients are presented as a function of anatomical position.     

Mechanical properties of compacted morselized cancellous bone graft using one-dimensional consolidation testing

Failures of orthopaedic procedures that use morselized cancellous bone (MCB) graft for load bearing are often due to gross displacement within the graft material. For this reason the mechanical behavior of MCB must be better understood. Our purpose is to present a detailed testing methodology for the mechanical characterization of MCB, and to illustrate how this methodology can be used to study the influence of water and fat content. Complete one-dimensional consolidation testing was performed on bovine cancellous bone processed to represent MCB typically used in surgery (52% water, 31% fat). The one-dimensional consolidation strain under a stress of 1.09 MPa was 30.9% and the confined modulus was 8.0 MPa. The coefficient of consolidation (rate of consolidation) was 2.2×10⁻⁵ cm²/s and the coefficient of secondary strain (steady-state creep rate) was 1.9%. While reducing the water content alone had some influence on properties, reducing the fat content improved both the static and dynamic behavior. A sample of MCB which had fat intentionally minimized and a lower overall moisture content (56% water, 5% fat) demonstrated 23.1% strain, a confined modulus of 9.6 MPa, a coefficient of consolidation of 3.4×10⁻³ cm²/s, and a coefficient of secondary strain of 0.9%. The test methods described in this technical note can be used to evaluate the influence of fluid content on the mechanical behavior of MCB.     

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.     

Influence of boundary conditions on computed apparent elastic properties of cancellous bone

High-resolution finite element models of trabecular bone can be used to study trabecular structure–function relationships, elasticity, multiaxial strength, and tissue remodelling in more detail than experiments. Beside effects of the model size, scan/analysis resolution, segmentation process, etc., the type of the applied boundary conditions (BCs) have a strong influence on the predicted elastic properties. Appropriate BCs have to be applied on hexahedral digital finite element models in order to obtain effective elastic properties. Homogeneous displacement BCs as proposed by Van Rietbergen et al. (J Biomech 29(12):1653–1657, 1996) lead to “apparent” rather than to “effective” elastic properties. This study provides some answers concerning such differences by comparing various BC types (uniform displacement, mixed BCs, periodic BCs), different volume element definitions (original and mirrored models), and several bone volume fractions (BVTV ranging from 6.5 to 37.6%). First, the mixed BCs formulated by Hazanov (Arch Appl Mech 68(6):385–394, 1998) are theoretically extended to shear loading of a porous media. Second, six human bone samples are analyzed, their orthotropic Young’s moduli, shear moduli, and Poisson’s ratios computed and compared. It is found that the proposed mixed BCs give exactly the same effective elastic properties as periodic BCs if a periodic and orthotropic micro-structured material is used and thus denoted as “periodicity compatible” mixed uniform BCs (PMUBCs). As bone samples were shown to be nearly orthotropic for volume element side lengths ≥5 mm the proposed mixed BCs turn out to be the best choice because they give again essentially the same overall elastic properties as periodic BCs. For bone samples of smaller dimensions ( < 5 mm) with a strong anisotropy (beyond orthotropy) uniform displacement BCs remain applicable but they can significantly overestimate the effective stiffness. In Memoriam, Prof. Christian Huet.     

Elastic properties of an intact and ACL-ruptured knee joint: measurement, mathematical modelling, and haptic rendering

An analytical, dynamic model of the human knee joint has been developed to simulate the unloaded knee joint behaviour in 6 degrees of freedom. It is based on extensive robot-based measurements of the elastic properties of a human cadaver knee joint. The measured data are compared with data from the literature to ensure that a proper database for modelling is used. The analytical modelling of the passive elastic joint properties is done with Local Linear Model Trees. The deduced knee joint model incorporates passive elastic properties of the internal knee joint structures, passive elastic muscle forces, damping forces, gravitational forces, and external forces. There are two sets of parameters, one simulating the movement of the intact knee joint, and a second simulating the knee joint with ruptured anterior cruciate ligament. The dynamic model can be easily processed in real-time. It is implemented in the haptic display of the Munich Knee Joint Simulator (MKS), which enables a person to move a plastic leg driven by a robot manipulator and feel the simulated knee joint force. Orthopaedic physicians judged the performance of the dynamic knee joint model by executing physical knee joint tests at the MKS.     

Elastic moduli, yield stress, and ultimate stress of cancellous bone in the canine proximal femur

Elastic moduli, yield stress and ultimate compressive stress were determined for cancellous bone from the femoral head and neck regions of the canine femur. Unconfined compression tests were performed on 5 mm cubic samples which were cut from two femurs. Elastic moduli were measured in three orthogonal directions, and the yield stress and ultimate stress were measured along the proximal-distal axis. The results from this investigation support previous assumptions that the mechanical behavior of canine cancellous bone is qualitatively similar to human cancellous bone. The canine cancellous bone was observed to be anisotropic in elastic modulus. For two thirds of the cubic specimens tested, the elastic modulus was largest in the load-bearing, proximal-distal direction. A linear relationship between yield stress and elastic modulus was observed for canine bone, as is typical of human bone. A similar linear relationship between ultimate stress and elastic modulus was observed. Thus, for canine bone as well as for human bone, failure appears to be governed by a strain level which is position independent. The yield strain of 0.0259 and ultimate strain of 0.0288 for canine bone were both less than the yield strain of 0.0395 reported for human bone.     

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.