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).     

Density of fresh and embalmed human compact and cancellous bone

This investigation was undertaken to determine the densities of fresh human compact and cancellous bone, with its marrow constituents intact, and compact and cancellous bone taken from embalmed cadavers. The density of 107 bone specimens was determined on the basis of the weight and volume of each specimen. Mean density results were: (1) fresh compact bone, 1.85; (2) fresh cancellous bone, 1.08; (3) embalmed compact bone, 1.85; and (4) embalmed cancellous bone, 1.09. Embalming did not appear to have altered the density of either compact or cancellous bone. It was observed also that the densities of several bone specimens taken from the same area of one bone varied greatly.     

The mechanical behaviour of cancellous bone

Cancellous bone has a cellular structure: it is made up of a connected network of rods and plates. Because of this, its mechanical behaviour is similar to that of other cellular materials such as polymeric foams. A recent study on the mechanisms of deformation in such materials has led to an understanding of how their mechanical properties depend on their relative density, cell wall properties and cell geometry. In this paper, the results of this previous study are applied to cancellous bone in an attempt to further understand its mechanical behaviour. The results of the analysis agree reasonably well with experimental data available in the literature.     

The fracture toughness of cancellous bone

The mechanical capacity and integrity of cancellous bone is crucial in osteoporosis, a condition which is set to become more prevalent with increasing lifespan and population sizes. The fracture toughness (FT) of cancellous bone has never been examined before and the conditions associated with the growth of a major crack through the lattice of cancellous bone, a cellular solid, may improve our understanding for structural integrity of this material. The aim of this study is to provide (i) basic data on cancellous bone FT and (ii) the experimental support for the hypothesis of Gibson, L.J., Ashby, M.F. [1997a. Chapter 10: Wood. In: Cellular Solids: Structure and Properties, second ed. Cambridge University Press, pp. 387–428; Gibson, L.J., Ashby, M.F., 1997b. Chapter 11: Cancellous Bone. In: Cellular Solids: Structure and Properties, second ed. Cambridge University Press, pp. 429–52] that the FT of cancellous bone tissue is governed by the density of the tissue to a power function of between one and two. 294 SENB and 121 DC(T) specimen were manufactured from 45 human femoral heads, 37 osteoporotic and 8 osteoarthritic, as well as 19 equine thoracic vertebrae. The samples were manufactured in two groups: the first aligned with the trabecular structure (A_∥), the second orientated at 90° to the main trabecular orientation (A_⊥). The samples were tested in either tensile or bending mode to provide values of the stress intensity factor (K). The results which were obtained show a strong and significant link between the density of the cancellous bone tissue and that the critical stress intensity values are governed by the density of the tissue to a power function of between 1 and 2 (K_Q vs. apparent density: A_⊥=1.58, A_∥=1.6). Our results provide some fundamental values for the critical stress intensity factor for cancellous bone and also support the previous hypothesis as set by Gibson, L.J., Ashby, M.F., 1997a. Chapter 10: Wood. In: Cellular Solids: Structure and Properties, second ed. Cambridge University Press, pp. 387–428; Gibson, L.J., Ashby, M.F., (1997b). Chapter 11: Cancellous Bone. In: Cellular Solids: Structure and Properties, second ed. Cambridge University Press, pp. 429–52.     

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.     

The cancellous bone multiscale morphology-elasticity relationship

The cancellous bone effective properties relations are analysed on multiscale across two aspects; properties of representative volume element on micro scale and statistical measure of trabecular trajectory orientation on mesoscale. Anisotropy of the microstructure is described across fabric tensor measure with trajectory orientation tensor as bridging scale connection. The scatter measured data (elastic modulus, trajectory orientation, apparent density) from compression test are fitted by stochastic interpolation procedure. The engineering constants of the elasticity tensor are estimated by last square fitt procedure in multidimensional space by Nelder-Mead simplex. The multiaxial failure surface in strain space is constructed and interpolated by modified super-ellipsoid.     

The effect of charge density on the velocity and attenuation of ultrasound waves in human cancellous bone

Cancellous bone is a highly porous material, and two types of waves, fast and slow, are observed when ultrasound is used for detecting bone diseases. There are several possible stimuli for bone remodelling processes, including bone fluid flow, streaming potential, and piezoelectricity. Poroelasticity has been widely used for elucidating the bone fluid flow phenomenon, but the combination of poroelasticity with charge density has not been introduced. Theoretically, general poroelasticity with a varying charge density is employed for determining the relationship between wave velocity and attenuation with charge density. Fast wave velocity and attenuation are affected by porosity as well as charge density; however, for a slow wave, both slow wave velocity and attenuation are not as sensitive to the effect of charge density as they are for a fast wave. Thus, employing human femoral data, we conclude that charged ions gather on trabecular struts, and the fast wave, which moves along the trabecular struts, is significantly affected by charge density.     

The effects of tensile-compressive loading mode and microarchitecture on microdamage in human vertebral cancellous bone

The amount of microdamage in bone tissue impairs mechanical performance and may act as a stimulus for bone remodeling. Here we determine how loading mode (tension vs. compression) and microstructure (trabecular microarchitecture, local trabecular thickness, and presence of resorption cavities) influence the number and volume of microdamage sites generated in cancellous bone following a single overload. Twenty paired cylindrical specimens of human vertebral cancellous bone from 10 donors (47–78 years) were mechanically loaded to apparent yield in either compression or tension, and imaged in three dimensions for microarchitecture and microdamage (voxel size 0.7×0.7×5.0 μm³). We found that the overall proportion of damaged tissue was greater (p=0.01) for apparent tension loading (3.9±2.4%, mean±SD) than for apparent compression loading (1.9±1.3%). Individual microdamage sites generated in tension were larger in volume (p<0.001) but not more numerous (p=0.64) than sites in compression. For both loading modes, the proportion of damaged tissue varied more across donors than with bone volume fraction, traditional measures of microarchitecture (trabecular thickness, trabecular separation, etc.), apparent Young?s modulus, or strength. Microdamage tended to occur in regions of greater trabecular thickness but not near observable resorption cavities. Taken together, these findings indicate that, regardless of loading mode, accumulation of microdamage in cancellous bone after monotonic loading to yield is influenced by donor characteristics other than traditional measures of microarchitecture, suggesting a possible role for tissue material properties.     

The contribution of cancellous bone to long bone strength and rigidity

A recent article (Burr and Piotrowski, 1982) suggested that structural analyses of long bone cross-sectional geometry will be inaccurate and should be considered inappropriate when cancellous bone accounts for 10-15% or more of the cross-sectional area. Consideration of material property differences between compact and cancellous bone, however, indicates that even significant proportions of cancellous bone (10-40% of total cross-sectional area) will very likely have negligible effects on bone strength and rigidity, and can be effectively ignored in geometrical analyses of diaphyseal sections. In metaphyseal and epiphyseal regions, however, geometric analyses of section properties such as area moments of inertia are inappropriate, both because of significant trabecular bone effects, and because of the inherent constraints of mechanical beam models.     

A theoretical model to predict distribution of the fabric tensor and apparent density in cancellous bone

 The adaptation of cancellous bone to mechanical forces is well recognized. Theoretical models for predicting cancellous bone architecture have been developed and have mainly focused on the distribution of trabecular mass or the apparent density. The purpose of this study was to develop a theoretical model which can simultaneously predict the distribution of trabecular orthotropy/orientation, as represented by the fabric tensor, along with apparent density. Two sets of equations were derived under the assumption that cancellous bone is a biological self-optimizing material which tends to minimize strain energy. The first set of equations provide the relationship between the fabric tensor and stress tensor, and have been verified to be consistent with Wolff’s law of trabecular architecture, that is, the principal directions of the fabric tensor coincide with the principal stress trajectories. The second set of equations yield the apparent density from the stress tensor, which was shown to be identical to those obtained based on local optimization with strain energy density of true bone tissue as the objective function. These two sets of equations, together with elasticity field equations, provide a complete mathematical formulation for the adaptation of cancellous bone. Received: 25 February 1997/Revised version: 23 September 1997     

The elastic anisotropy of bone

In modeling the anisotropic properties of hydroxyapatite (HAp), Katz found that two kinds of phenomenological relationships held among the elastic stiffness coefficients. Firstly, there are three linear combinations--(c11 + c22 + c33), (c44 + c55 + c66), (c12 + c13 + c23)--which arise naturally when computing the isotropic averages of anisotropic crystal systems over all possible spatial orientations. Secondly, the degree of elastic anisotropy in such crystal systems is characterized by two specific factors: (a) the ratio of the linear compressibility along the unique axis to that perpendicular to it, (c11 + c12 - 2c23)(c33 - c13); and (b) the ratio of the two shear moduli, c44/c66. There have been a number of experiments in recent years which have used either mechanical methods or ultrasonic techniques to measure the anisotropic elastic properties of bovine and human cortical bone. Analyses of data from these experiments show that the above relationships also play a significant role in characterizing the elastic anisotropy in bone.     

Limitations of the continuum assumption in cancellous bone

Most existing stress analyses of the skeleton which consider cancellous bone assume that it can be modelled as a continuum. In this paper we develop a criterion for the validity of this assumption. The limitations of the continuum assumption appear in two areas: near biologic interfaces, and in areas of large stress gradients. These limitations are explored using a probabilistic line scanning model for density measurement, resulting in an estimate of density accuracy as a function of line length which is experimentally verified. Within three to five trabeculae of an interface, a continuum model is suspect. When results as predicted using continuum analyses vary by more than 20-30% over a distance spanning three to five trabeculae, the results are suspect.     

Trabecular shear stress in human vertebral cancellous bone: intra- and inter-individual variations

Correlation of the mean and standard deviation of trabecular stresses has been proposed as a mechanism by which a strong relationship between the apparent strength and stiffness of cancellous bone can be achieved. The current study examined whether the relationship between the mean and standard deviation of trabecular von Mises stresses can be generalized for any group of cancellous bone. Cylindrical human vertebral cancellous bone specimens were cut in the infero-superior direction from T12 of 23 individuals (inter-individual group). Thirty nine additional specimens were prepared similarly from the T4-T12 and L2-L5 vertebrae of a 63 year old male (intra-individual group). The specimens were scanned by micro-computed tomography (microCT) and trabecular von Mises stresses were calculated using finite element modeling. The expected value, standard deviation and coefficient of variation of the von Mises stress were calculated form a three-parameter Weibull function fitted to von Mises stress data from each specimen. It was found that the average and standard deviation of trabecular von Mises shear stress were: (i) correlated with each other, supporting the idea that high correlation between the apparent strength and stiffness of cancellous bone can be achieved through controlling the trabecular level shear stress variations, (ii) dependent on anatomical site and sample group, suggesting that the variation of stresses are correlated to the mean stress to different degrees between vertebrae and individuals, and (iii) dependent on bone volume fraction, consistent with the idea that shear stress is less well controlled in bones with low BV/TV. The conversion of infero-superior loading into trabecular von Mises stresses was maximum for the tissue at the junction of the thoracic and lumbar spine (T12-L1) consistent with this junction being a common site of vertebral fracture.     

Finite element calculated uniaxial apparent stiffness is a consistent predictor of uniaxial apparent strength in human vertebral cancellous bone tested with different boundary conditions

Strong correspondence between the uniaxial apparent strength and stiffness of cancellous bone allows the use of stiffness as a predictor of bone strength. Measured values of mechanical properties in cancellous bone can be different between experiments due to different experimental conditions. In the current study, bone volume fraction, experimentally determined and finite element (FE) predicted stiffness were examined as predictors of cancellous bone ultimate strength in two different groups each of which was tested using a different end constraint. It is demonstrated that, although always significant, the relationships of strength with bone volume fraction and experimentally determined stiffness are different between test groups. Apparent stiffness, estimated by FE modeling, predicts the ultimate strength of human cancellous bone consistently for all examined experimental protocols.