Zonal changes in the three-dimensional morphology of the chondron under compression: the relationship among cellular, pericellular, and extracellular deformation in articular cartilage

The pericellular matrix (PCM) is a narrow region of tissue that completely surrounds chondrocytes in articular cartilage. Previous theoretical models of the "chondron" (the PCM with enclosed cells) suggest that the structure and properties of the PCM may significantly influence the mechanical environment of the chondrocyte. The objective of this study was to quantify changes in the three-dimensional (3D) morphology of the chondron in situ at different magnitudes of compression applied to the cartilage extracellular matrix. Fluorescence immunolabeling for type-VI collagen was used to identify the boundaries of the cell and PCM, and confocal microscopy was used to form 3D images of chondrons from superficial, middle, and deep zone cartilage in explants compressed to 0%, 10%, 30%, and 50% surface-to-surface strain. Lagrangian tissue strain, determined locally using texture correlation, was highly inhomogeneous and revealed depth-dependent compressive stiffness and Poisson's ratio of the extracellular matrix. Compression significantly decreased cell and chondron height and volume, depending on the zone and magnitude of compression. In the superficial zone, cellular-level strains were always lower than tissue-level strains. In the middle and deep zones, however, tissue strains below 25% were amplified at the cellular level, while tissue strains above 25% were decreased at the cellular level. These findings are consistent with previous theoretical models of the chondron, suggesting that the PCM can serve as either a protective layer for the chondrocyte or a transducer that amplifies strain, such that cellular-level strains are more homogenous throughout the tissue depth despite large inhomogeneities in local ECM strains.     

Spatial Mapping of the Biomechanical Properties of the Pericellular Matrix of Articular Cartilage Measured In Situ via Atomic Force Microscopy

In articular cartilage, chondrocytes are surrounded by a narrow region called the pericellular matrix (PCM), which is biochemically, structurally, and mechanically distinct from the bulk extracellular matrix (ECM). Although multiple techniques have been used to measure the mechanical properties of the PCM using isolated chondrons (the PCM with enclosed cells), few studies have measured the biomechanical properties of the PCM in situ. The objective of this study was to quantify the in situ mechanical properties of the PCM and ECM of human, porcine, and murine articular cartilage using atomic force microscopy (AFM). Microscale elastic moduli were quantitatively measured for a region of interest using stiffness mapping, or force-volume mapping, via AFM. This technique was first validated by means of elastomeric models (polyacrylamide or polydimethylsiloxane) of a soft inclusion surrounded by a stiff medium. The elastic properties of the PCM were evaluated for regions surrounding cell voids in the middle/deep zone of sectioned articular cartilage samples. ECM elastic properties were evaluated in regions visually devoid of PCM. Stiffness mapping successfully depicted the spatial arrangement of moduli in both model and cartilage surfaces. The modulus of the PCM was significantly lower than that of the ECM in human, porcine, and murine articular cartilage, with a ratio of PCM to ECM properties of ∼0.35 for all species. These findings are consistent with previous studies of mechanically isolated chondrons, and suggest that stiffness mapping via AFM can provide a means of determining microscale inhomogeneities in the mechanical properties of articular cartilage in situ.     

A Mechano-chemical Model for the Passive Swelling Response of an Isolated Chondron under Osmotic Loading

The chondron is a distinct structure in articular cartilage that consists of the chondrocyte and its pericellular matrix (PCM), a narrow tissue region surrounding the cell that is distinguished by type VI collagen and a high glycosaminoglycan concentration relative to the extracellular matrix. We present a theoretical mechano-chemical model for the passive volumetric response of an isolated chondron under osmotic loading in a simple salt solution at equilibrium. The chondrocyte is modeled as an ideal osmometer and the PCM model is formulated using triphasic mixture theory. A mechano-chemical chondron model is obtained assuming that the chondron boundary is permeable to both water and ions, while the chondrocyte membrane is selectively permeable to only water. For the case of a neo-Hookean PCM constitutive law, the model is used to conduct a parametric analysis of cell and chondron deformation under hyper- and hypo-osmotic loading. In combination with osmotic loading experiments on isolated chondrons, model predictions will aid in determination of pericellular fixed charge density and its relative contribution to PCM mechanical properties.     

Role of pericellular matrix in development of a mechanically functional neocartilage

The role of the chondrocyte pericellular matrix (PCM) was examined in a three-dimensional chondrocyte culture system to determine whether retention of the native pericellular matrix could stimulate collagen and proteoglycan accumulation and also promote the formation of a mechanically functional hyaline-like neocartilage. Porcine chondrocytes and chondrons, consisting of the chondrocyte with its intact pericellular matrix, were maintained in pellet culture for up to 12 weeks. Sulfated glycosaminoclycans and type II collagen were measured biochemically. Immunocytochemistry was used to examine collagen localization as well as cell distribution within the pellets. In addition, the equilibrium compressive moduli of developing pellets were measured to determine whether matrix deposition contributed to the mechanical stiffness of the cartilage constructs. Pellets increased in size and weight over a 6-week period without apparent cell proliferation. Although chondrocytes quickly rebuilt a PCM rich in type VI collagen, chondron pellets accumulated significantly more proteoglycan and type II collagen than did chondrocyte pellets, indicating a greater positive effect of the native PCM. After 5 weeks in chondron pellets, matrix remodeling was evident by microscopy. Cells that had been uniformly distributed throughout the pellets began to cluster between large areas of interterritorial matrix rich in type II collagen. After 12 weeks, clusters were stacked in columns. A rapid increase in compressive strength was observed between 1 and 3 weeks in culture for both chondron and chondrocyte pellets and, by 6 weeks, both had achieved 25% of the equilibrium compressive stiffness of cartilage explants. Retention of the in vivo PCM during chondrocyte isolation promotes the formation of a mechanically functional neocartilage construct, suitable for modeling the responses of articular cartilage to chemical stimuli or mechanical compression.     

A biomechanical role for perlecan in the pericellular matrix of articular cartilage

Chondrocytes are surrounded by a narrow pericellular matrix (PCM) that is biochemically, structurally, and biomechanically distinct from the bulk extracellular matrix (ECM) of articular cartilage. While the PCM is often defined by the presence of type VI collagen, other macromolecules such as perlecan, a heparan sulfate (HS) proteoglycan, are also exclusively localized to the PCM in normal cartilage and likely contribute to PCM structural integrity and biomechanical properties. Though perlecan is essential for normal cartilage development, its exact role in the PCM is unknown. The objective of this study was to determine the biomechanical role of perlecan in the articular cartilage PCM in situ and its potential as a defining factor of the PCM. To this end, atomic force microscopy (AFM) stiffness mapping was combined with dual immunofluorescence labeling of cryosectioned porcine cartilage samples for type VI collagen and perlecan. While there was no difference in overall PCM mechanical properties between type VI collagen- and perlecan-based definitions of the PCM, within the PCM, interior regions containing both type VI collagen and perlecan exhibited lower elastic moduli than more peripheral regions rich in type VI collagen alone. Enzymatic removal of HS chains from perlecan with heparinase III increased PCM elastic moduli both overall and locally in interior regions rich in both perlecan and type VI collagen. Heparinase III digestion had no effect on ECM elastic moduli. Our findings provide new evidence for perlecan as a defining factor in both the biochemical and biomechanical properties of the PCM.     

Osteoarthritic changes in the biphasic mechanical properties of the chondrocyte pericellular matrix in articular cartilage

The pericellular matrix (PCM) is a narrow region of cartilaginous tissue that surrounds chondrocytes in articular cartilage. Previous modeling studies indicate that the mechanical properties of the PCM relative to those of the extracellular matrix (ECM) can significantly affect the stress-strain, fluid flow, and physicochemical environments of the chondrocyte, suggesting that the PCM plays a biomechanical role in articular cartilage. The goals of this study were to measure the mechanical properties of the PCM using micropipette aspiration coupled with a linear biphasic finite element model, and to determine the alterations in the mechanical properties of the PCM with osteoarthritis (OA). Using a recently developed isolation technique, chondrons (the chondrocyte and its PCM) were mechanically extracted from non-degenerate and osteoarthritic human cartilage. The transient mechanical behavior of the PCM was well-described by a biphasic model, suggesting that the viscoelastic response of the PCM is attributable to flow-dependent effects, similar to that of the ECM. With OA, the mean Young's modulus of the PCM was significantly decreased (38.7+/-16.2 kPa vs. 23.5+/-12.9 kPa, p < 0.001), and the permeability was significantly elevated (4.19+/-3.78 x10(-17) m(4)/Ns vs. 10.2+/-9.38 x 10(-17) m(4)/Ns, p < 0.01). The Poisson's ratio was similar for both non-degenerate and OA PCM (0.044+/-0.063 vs. 0.030+/-0.068, p > 0.6). These findings suggest that the PCM may undergo degenerative processes with OA, similar to those occurring in the ECM. In combination with previous theoretical models of cell-matrix interactions in cartilage, our findings suggest that changes in the properties of the PCM with OA may have an important influence on the biomechanical environment of the chondrocyte.     

Assessment of biomechanical properties of the extracellular and pericellular matrix and their interconnection throughout the course of osteoarthritis

During osteoarthritis (OA)-triggered cartilage degeneration, the chondrocytes spatially rearrange from single to double strings, and then to small and finally big clusters. Both the extracellular matrix (ECM) and the pericellular matrix (PCM) progressively degrade in osteoarthritis, changing the overall mechanical properties of the cartilage. We investigated the mechanical properties particularly elasticity of the ECM and PCM and their interconnection as a function of chondrocyte spatial organisation.Human articular cartilage samples from 30 patients were categorised according to their cellular pattern. Elasticity of the ECM and PCM was assessed by means of atomic force microscopy (AFM). Significant decreases were observed in the elasticity of both the ECM and the PCM with each change of cellular pattern, except from single to double strings in the ECM (p = 0.072). Spatial reorganisation strongly correlated with the elasticity of the ECM (r = −0.768, p < 0.001) and of the PCM (r = −0.729, p < 0.001). The ECM/PCM ratio remained unchanged (r = −0.099, p = 0.281).This study is the first to describe and quantify the differences in the elastic moduli of the ECM in relation to the PCM on the basis of chondrocyte spatial arrangement. This study shows that the elastic changes of the ECM and the PCM occur simultaneously, unidirectionally, and to a comparable degree.     

Rheological properties of cross-linked hyaluronan-gelatin hydrogels for tissue engineering

Hydrogels that mimic the natural extracellular matrix (ECM) are used in three-dimensional cell culture, cell therapy, and tissue engineering. A semi-synthetic ECM based on cross-linked hyaluronana offers experimental control of both composition and gel stiffness. The mechanical properties of the ECM in part determine the ultimate cell phenotype. We now describe a rheological study of synthetic ECM hydrogels with storage shear moduli that span three orders of magnitude, from 11 to 3 500 Pa, a range important for engineering of soft tissues. The concentration of the chemically modified HA and the cross-linking density were the main determinants of gel stiffness. Increase in the ratio of thiol-modified gelatin reduced gel stiffness by diluting the effective concentration of the HA component.     

Mechanical and failure properties of extracellular matrix sheets as a function of structural protein composition

The goal of this study was to determine how alterations in protein composition of the extracellular matrix (ECM) affect its functional properties. To achieve this, we investigated the changes in the mechanical and failure properties of ECM sheets generated by neonatal rat aortic smooth muscle cells engineered to contain varying amounts of collagen and elastin. Samples underwent static and dynamic mechanical measurements before, during, and after 30 min of elastase digestion followed by a failure test. Microscopic imaging was used to measure thickness at two strain levels to estimate the true stress and moduli in the ECM sheets. We found that adding collagen to the ECM increased the stiffness. However, further increasing collagen content altered matrix organization with a subsequent decrease in the failure strain. We also introduced collagen-related percolation in a nonlinear elastic network model to interpret these results. Additionally, linear elastic moduli correlated with failure stress which may allow the in vivo estimation of the stress tolerance of ECM. We conclude that, in engineered replacement tissues, there is a tradeoff between improved mechanical properties and decreased extensibility, which can impact their effectiveness and how well they match the mechanical properties of native tissue.     

Interrelationship of cartilage composition and chondrocyte mechanics after a partial meniscectomy in the rabbit knee joint – Experimental and numerical analysis

Site-specific and depth-dependent properties of cartilage were implemented within a finite element (FE) model to determine if compositional or structural changes in the tissue could explain site-specific alterations of chondrocyte deformations due to cartilage loading in rabbit knee joints 3 days after a partial meniscectomy (PM). Depth-dependent proteoglycan (PG) content, collagen content and collagen orientation in the cartilage extracellular matrix (ECM), and PG content in the pericellular matrix (PCM) were assessed with microscopic and spectroscopic methods. Patellar, femoral groove and samples from both the lateral and medial compartments of the femoral condyle and tibial plateau were extracted from healthy controls and from the partial meniscectomy group. For both groups and each knee joint site, axisymmetric FE models with measured properties were generated. Experimental cartilage loading was applied in the simulations and chondrocyte volumes were compared to the experimental values. ECM and PCM PG loss occurred within the superficial cartilage layer in the PM group at all locations, except in the lateral tibial plateau. Collagen content and orientation were not significantly altered due to the PM. The FE simulations predicted similar chondrocyte volume changes and group differences as obtained experimentally. Loss of PCM fixed charge density (FCD) decreased cell volume loss, as observed in the medial femur and medial tibia, whereas loss of ECM FCD increased cell volume loss, as seen in the patella, femoral groove and lateral femur. The model outcome, cell volume change, was also sensitive to applied tissue geometry, collagen fibril orientation and loading conditions.     

A depth-dependent model of the pericellular microenvironment of chondrocytes in articular cartilage

Experimental studies suggest that the magnitude of chondrocyte deformation is much smaller than expected based on the material properties of extracellular matrix (ECM) and cells, and that this result could be explained by a structural unit, the chondron, that is thought to protect chondrocytes from large deformations in situ. We extended an existing numerical model of chondrocyte, ECM and pericellular matrix (PCM) to include depth-dependent structural information. Our results suggest that superficial zone chondrocytes, which lack a pericellular capsule (PC), are relatively stiff, and therefore are protected from excessive deformations, whereas middle and deep zone chondrocytes are softer but are protected by the PC that limits cell deformations in these regions. We conclude that cell deformations sensitively depend on the immediate structural environment of the PCM in a depth-dependent manner, and that the functional stiffness of chondrocytes in situ is much larger than experiments on isolated cells would suggest.     

Retention of the native chondrocyte pericellular matrix results in significantly improved matrix production.

The interaction of the cell with its surrounding extracellular matrix (ECM) has a major effect on cell metabolism. We have previously shown that chondrons, chondrocytes with their in vivo-formed pericellular matrix, can be enzymatically isolated from articular cartilage. To study the effect of the native chondrocyte pericellular matrix on ECM production and assembly, chondrons were compared with chondrocytes isolated without any pericellular matrix. Immediately after isolation from human cartilage, chondrons and chondrocytes were centrifuged into pellets and cultured. Chondron pellets had a greater increase in weight over 8 weeks, were more hyaline appearing, and had more type II collagen deposition and assembly than chondrocyte pellets. Minimal type I procollagen immunofluorescence was detected for both chondron and chondrocyte pellets. Chondron pellets had a 10-fold increase in proteoglycan content compared with a six-fold increase for chondrocyte pellets over 8 weeks (P<0.0001). There was no significant cell division for either chondron or chondrocyte pellets. The majority of cells within both chondron and chondrocyte pellets maintained their polygonal or rounded shape except for a thin, superficial edging of flattened cells. This edging was similar to a perichondrium with abundant type I collagen and fibronectin, and decreased type II collagen and proteoglycan content compared with the remainder of the pellet. This study demonstrates that the native pericellular matrix promotes matrix production and assembly in vitro. Further, the continued matrix production and assembly throughout the 8-week culture period make chondron pellet cultures valuable as a hyaline-like cartilage model in vitro.     

Strain-dependent viscoelastic behaviour and rupture force of single chondrocytes and chondrons under compression

The chondron in articular cartilage includes the chondrocyte and its surrounding pericellular matrix (PCM). Single chondrocytes and chondrons were compressed between two parallel surfaces by a micromanipulation technique to investigate their biomechanical properties and to discover the mechanical significance of the PCM. The force imposed on the cells was measured directly during deformation at various compression speeds and deformations up to cell rupture. When the deformation at the end of compression was 50%, relaxation showed that the cells were viscoelastic, but this viscoelasticity was generally insignificant at 30% deformation or lower. When the deformation was 70%, the cells had deformed plastically. Chondrons ruptured at a mean deformation of 85 ± 1%, whilst chondrocytes ruptured at a mean deformation of 78 ± 1%. Chondrons were generally stiffer than chondrocytes and showed less viscoelastic behaviour than chondrocytes. Thus, the PCM significantly influences the mechanical properties of the cells.     

Mechanical Cell–Cell Communication in Fibrous Networks: The Importance of Network Geometry

Cells contracting in extracellular matrix (ECM) can transmit stress over long distances, communicating their position and orientation to cells many tens of micrometres away. Such phenomena are not observed when cells are seeded on substrates with linear elastic properties, such as polyacrylamide (PA) gel. The ability for fibrous substrates to support far reaching stress and strain fields has implications for many physiological processes, while the mechanical properties of ECM are central to several pathological processes, including tumour invasion and fibrosis. Theoretical models have investigated the properties of ECM in a variety of network geometries. However, the effects of network architecture on mechanical cell–cell communication have received little attention. This work investigates the effects of geometry on network mechanics, and thus the ability for cells to communicate mechanically through different networks. Cell-derived displacement fields are quantified for various network geometries while controlling for network topology, cross-link density and micromechanical properties. We find that the heterogeneity of response, fibre alignment, and substrate displacement fields are sensitive to network choice. Further, we show that certain geometries support mechanical communication over longer distances than others. As such, we predict that the choice of network geometry is important in fundamental modelling of cell–cell interactions in fibrous substrates, as well as in experimental settings, where mechanical signalling at the cellular scale plays an important role. This work thus informs the construction of theoretical models for substrate mechanics and experimental explorations of mechanical cell–cell communication.     

Three-dimensional finite element modeling of pericellular matrix and cell mechanics in the nucleus pulposus of the intervertebral disk based on in situ morphology

Nucleus pulposus (NP) cells of the intervertebral disk (IVD) have unique morphological characteristics and biologic responses to mechanical stimuli that may regulate maintenance and health of the IVD. NP cells reside as single cell, paired or multiple cells in a contiguous pericellular matrix (PCM), whose structure and properties may significantly influence cell and extracellular matrix mechanics. In this study, a computational model was developed to predict the stress–strain, fluid pressure and flow fields for cells and their surrounding PCM in the NP using three-dimensional (3D) finite element models based on the in situ morphology of cell–PCM regions of the mature rat NP, measured using confocal microscopy. Three-dimensional geometries of the extracellular matrix and representative cell–matrix units were used to construct 3D finite element models of the structures as isotropic and biphasic materials. In response to compressive strain of the extracellular matrix, NP cells and PCM regions were predicted to experience volumetric strains that were 1.9–3.7 and 1.4–2.1 times greater than the extracellular matrix, respectively. Volumetric and deviatoric strain concentrations were generally found at the cell/PCM interface, while von Mises stress concentrations were associated with the PCM/extracellular matrix interface. Cell–matrix units containing greater cell numbers were associated with higher peak cell strains and lower rates of fluid pressurization upon loading. These studies provide new model predictions for micromechanics of NP cells that can contribute to an understanding of mechanotransduction in the IVD and its changes with aging and degeneration.     

A finite element model of cell-matrix interactions to study the differential effect of scaffold composition on chondrogenic response to mechanical stimulation

Mechanically induced cell deformations have been shown to influence chondrocyte response in 3D culture. However, the relationship between the mechanical stimulation and cell response is not yet fully understood. In this study a finite element model was developed to investigate cell-matrix interactions under unconfined compression conditions, using a tissue engineered encapsulating hydrogel seeded with chondrocytes. Model predictions of stress and strain distributions within the cell and on the cell boundary were shown to exhibit space-dependent responses that varied with scaffold mechanical properties, the presence of a pericellular matrix (PCM), and the cell size. The simulations predicted that when the cells were initially encapsulated into the hydrogel scaffolds, the cell size hardly affected the magnitude of the stresses and strains that were reaching the encapsulated cells. However, with the inclusion of a PCM layer, larger cells experienced enhanced stresses and strains resulting from the mechanical stimulation. It was also noted that the PCM had a stress shielding effect on the cells in that the peak stresses experienced within the cells during loading were significantly reduced. On the other hand, the PCM caused the stresses at the cell-matrix interface to increase. Based on the model predictions, the PCM modified the spatial stress distribution within and around the encapsulated cells by redirecting the maximum stresses from the periphery of the cells to the cell nucleus. In a tissue engineered cartilage exposed to mechanical loading, the formation of a neo-PCM by encapsulated chondrocytes appears to protect them from initially excessive mechanical loading. Predictive models can thus shed important insight into how chondrocytes remodel their local environment in order to redistribute mechanical signals in tissue engineered constructs.     

Concentration independent modulation of local micromechanics in a fibrin gel

Methods for tuning extracellular matrix (ECM) mechanics in 3D cell culture that rely on increasing the concentration of either protein or cross-linking molecules fail to control important parameters such as pore size, ligand density, and molecular diffusivity. Alternatively, ECM stiffness can be modulated independently from protein concentration by mechanically loading the ECM. We have developed a novel device for generating stiffness gradients in naturally derived ECMs, where stiffness is tuned by inducing strain, while local mechanical properties are directly determined by laser tweezers based active microrheology (AMR). Hydrogel substrates polymerized within 35 mm diameter Petri dishes are strained non-uniformly by the precise rotation of an embedded cylindrical post, and exhibit a position-dependent stiffness with little to no modulation of local mesh geometry. Here we present the device in the context of fibrin hydrogels. First AMR is used to directly measure local micromechanics in unstrained hydrogels of increasing fibrin concentration. Changes in stiffness are then mapped within our device, where fibrin concentration is held constant. Fluorescence confocal imaging and orbital particle tracking are used to quantify structural changes in fibrin on the micro and nano levels respectively. The micromechanical strain stiffening measured by microrheology is not accompanied by ECM microstructural changes under our applied loads, as measured by confocal microscopy. However, super-resolution orbital tracking reveals nanostructural straightening, lengthening, and reduced movement of fibrin fibers. Furthermore, we show that aortic smooth muscle cells cultured within our device are morphologically sensitive to the induced mechanical gradient. Our results demonstrate a powerful cell culture tool that can be used in the study of mechanical effects on cellular physiology in naturally derived 3D ECM tissues.     

Synthetic matrices reveal contributions of ECM biophysical and biochemical properties to epithelial morphogenesis

Epithelial cells cultured within collagen and laminin gels proliferate to form hollow and polarized spherical structures, recapitulating the formation of a rudimentary epithelial organ. However, the contributions of extracellular matrix (ECM) biochemical and biophysical properties to morphogenesis are poorly understood because of uncontrolled presentation of multiple adhesive ligands, limited control over mechanical properties, and lot-to-lot compositional variability in these natural ECMs. We engineered synthetic ECM-mimetic hydrogels with independent control over adhesive ligand density, mechanical properties, and proteolytic degradation to study the impact of ECM properties on epithelial morphogenesis. Normal cyst growth, polarization, and lumen formation were restricted to a narrow range of ECM elasticity, whereas abnormal morphogenesis was observed at lower and higher elastic moduli. Adhesive ligand density dramatically regulated apicobasal polarity and lumenogenesis independently of cell proliferation. Finally, a threshold level of ECM protease degradability was required for apicobasal polarity and lumen formation. This synthetic ECM technology provides new insights into how cells transduce ECM properties into complex morphogenetic behaviors.     

Analytical methods to determine the effective mesoscopic and macroscopic elastic properties of cortical bone

We compare theoretical predictions of the effective elastic moduli of cortical bone at both the meso- and macroscales. We consider the efficacy of three alternative approaches: the method of asymptotic homogenization, the Mori-Tanaka scheme and the Hashin-Rosen bounds. The methods concur for specific engineering moduli such as the axial Young's modulus but can vary for others. In a past study, the effect of porosity alone on mesoscopic properties of cortical bone was considered, taking the matrix to be isotropic. Here, we consider the additional influence of the transverse isotropy of the matrix. We make the point that micromechanical approaches can be used in two alternative ways to predict either the macroscopic (size of cortical bone sample) or mesoscopic (in between micro- and macroscales) effective moduli, depending upon the choice of representative volume element size. It is widely accepted that the mesoscale behaviour is an important aspect of the mechanical behaviour of bone but models incorporating its effect have started to appear only relatively recently. Before this only macroscopic behaviour was addressed. Comparisons are drawn with experimental data and simulations from the literature for macroscale predictions with particularly good agreement in the case of dry bone. Finally, we show how predictions of the effective mesoscopic elastic moduli can be made which retain dependence on the well-known porosity gradient across the thickness of cortical bone.