Nanogranular Origins of the Strength of bone

Nanogranular Origins of the Strength of bone

(Parte 1 de 2)

Nanogranular Origins of the Strength of Bone

Kuangshin Tai,† Franz-Josef Ulm,‡ and Christine Ortiz*,†

Department of Materials Science and Engineering and Department of CiVil Engineering, Massachusetts Institute of Technology, 7 Massachusetts AVenue, Cambridge, Massachusetts 02139

Received August 10, 2006; Revised Manuscript Received September 13, 2006

Here, we investigate the ultrastructural origins of the strength of bone, which is critical for proper physiological function. A combination of dual nanoindentation, three-dimensional elastic-plastic finite element analysis using a Mohr-Coulomb cohesive-frictional strength criterion, and angle of repose measurements was employed. Our results suggest that nanogranular friction between mineral particles is responsible for increased yield resistance in compression relative to tension and that cohesion originates from within the organic matrix itself, rather than organic−mineral bonding.

The ultrastructural origins of the plasticity of bone and its complex relationship to damage accumulation and fracture risk are poorly understood. Recent studies1-3 have primarily probed tensile modes of deformation, which are relevant, for example, to avulsion fractures at tendinous and ligamentous insertions and bending fractures in the diaphyseal regions of long bones.4 In this study, we focus on the nanoscale compressive strength of bone, which is a significant physiological loading condition5 and a key requirement in vivo. While day-to-day deformation of bone generally takes place in the linear elastic regime,6 excessive injurious loads, fatigue, and degradation of biomechanical properties due to age or disease can lead to microdamage7 and fracture8 in vivo at compressive locations such as metaphyseal areas, vertebral bodies, and the calcaneus.4 Hence, a fundamental mechanistic understanding of how the structural design of bone is able to achieve optimal resistance to compressive yield is critically important for predicting tissue-level fracture, simulating remodeling processes, and developing clinical approaches to treat biomechanical degradation.

It is known that cortical bone exhibits a macroscopic yield strength in compression that is greater ( 2 ) than in tension or torsion,9 which is indicativeof pressuresensitiveplasticity. The strength of bone must begin at the ultrastructural level. At this length scale, plate-like carbonated apatite mineralites exist ( 10 s of nm in length and width, 3-5n mi n thickness10) that permeate in and around type I collagen fibrils in an overlapping manner.1 In this paper, we explore the possibilityof nanogranularfrictionfrom mineral-mineral interparticle interactions as a contributing source of the compressive yield strength of bone. This hypothesis is based on a number of experimental observations. First, the inorganic component is known to be a critical determinant of the macroscopiccompressivemechanicalpropertiesof bone; the yield stress,1 maximal stress,12 and failure strength13 are all known to increase with increasing mineral content. Second, the fact that mineral content of human bone is typically above the percolation threshold of 50% packing density (corresponding to 43% mineral content).14,15 Last, previously reported data of the direct visualization of the ultrastructural plasticity of bone via nanoindentation combined with high resolution atomic force microscopy (AFM) imaging of residual impressions show the nanogranular structure of contacting mineralites flattened, but still visible, within the plastically deformed indented region (Figure 1).16 The appearance of the undeformed mineralites outside of the indent region compared to within suggest mineral displacement and the possibility of interparticle frictional interactions. These data are consistent with recent scanning electron microscopy images of collagen fibrils bridging a crack within a compressed trabeculae.17 Given that many mineralized fibrils are in direct contact with each other, deformationaway from their unstressedconfigurationslikely involve mineral-mineral displacement. Hence, we hypothesize that the ultrastructure of bone is a cohesive-frictional material,18 following a Mohr-Coulomb pressure dependent strength criterion19 (i.e., which arises from the pressure dependence of the density of interparticle contacts).

* To whom correspondenceshould be addressed. Phone: 617-452-3084.

Fax: 617-258-6936. E-mail: cortiz@mit.edu

† Department of Materials Science and Engineering. ‡ Department of Civil Engineering.

10.1021/nl061877k C: $3.50 © x American Chemical Society PAGE EST: 5.7 Published on Web 10/03/2006

To explore this hypothesis, the nanoindentation of cortical bone was predictedusing an elastic-plasticthree-dimensional finite element analysis (FEA) model for two independent triaxial stress states20 achieved with two different indenter geometries, Berkovich (included angle 142.3°, half angle 65.3°) and Cube Corner (included angle 90°, half angle 54.6°), which incorporated the Mohr-Coloumb pressure dependentstrengthcriterion.In this case, the strengthdomain in the principal stress space, óI g óII g óIII, is defined by

where c is the interparticlecohesion, which is the finite value of cohesive shear strength required to cause sliding when the normal stress is zero, and æ is the internal friction angle, which provides the failure envelope given by the relationship of the linear slope between shear and normal stress. This internal friction approaches the “dry” (zero cohesion) angle of repose, which is measured when a bulk quantity of particles is poured onto a horizontal surface and is defined as the angle formed by the inclined edge of the pile and the horizontal plane. The FEA model utilized large deformation theory and incorporated a rigid indenter with frictionless contacts between the tip and sample. A modulus of 18 GPa was fixed in the simulations and approximated from the unloading slope of the nanoindentation data using an isotropic, elastic continuum mechanical half-space formulation.21 A Poisson’sratio of 0.3 was also fixed.2 Two material properties were reduced from the experimental data: c and æ, which were free fitting parameters. The difference between the experimental indentation response and the theoreticalpredictionswere minimizedby the quadraticerror for values that best fit both indenter geometries.

The predictions of the theoretical fits were compared to nanoindentation force versus indentation depth data taken on loading of adult bovine cortical bone perpendicular to the long bone axis for both the Berkovich and Cube Corner geometries (Figure 2). The sample preparation, characterization, and experimental protocols have been reported previously.16 The best fit æ and c values for both indenter geometries were found to be 15° and 100 MPa, respectively (R2 ) 0.9).The indentationsimulationson loadingexhibited a small elastic region (indentation depths < 10 nm) followed by yield (as assessed by monitoring the plastic equivalent strain of each element) due to local stress concentrations at the tip apex. The friction angle that best fits both indenter geometries translates into a uniaxial known macroscopic ratio 2.9 Theoretical fits were also carried out on nanoindentation data taken on fully demineralized bone (treated for 5 days in 0.5 M ethylenediaminetetraaceticacid). The moduluswas fixed to that reducedfrom the unloading slope ( 2.3 GPa) and Poisson’s ratio was set to 0.3 as previously, yielding best fit æ and c values for both indenter geometries of 5° and 100 MPa, respectively (Figure 3) translating to a compressive strength-to-tensile-strength ratio of 1.2. Figure 4 compares all four experimental nanoindentation datasets on a single plot and shows an increased force for a given depth (i.e., resistance to yield) for the intact compared to demineralized bone.

We note in Figure 1 the absence of pileup and rather the presence of sink-in which is explained as follows. The ultrastructure of bone has been suggested to possess nanoscale porosity ( 20 nm in size)23 which is a characteristic of a nanogranularmaterial. Hence, when bone is compressed during nanoindentation, there is most likely a plastic contracting behavior until the material reaches a state (called the critical state, a concept introduced for granular soils from which all Cam-Clay models derive24) at which it behaves like a cohesive-frictional material in the Mohr-Coulomb

Figure 1. Tapping mode atomic force microscopy amplitude images (Quesant) of a residual nanoindentation impression in adult bovine cortical bone ( 65 wt % measured through back-scattered electron imaging, which probes a depth of 1 ím) immediately after loading to 7000 íN followed by unloading (Hysitron Triboindenter, loading/unloading rate of 50 íN/second) using (a) Berkovich ( 850 nm depth) and (b) Cube Corner ( 1.5 ím depth) geometry. Experiments were conducted with the loading direction perpendicular to the long bone axis in ambient conditions. Details of the sample preparation and characterization and experimental protocolswere reportedpreviously.16 The undeformedregionsaway from the residual indent area are composed of nanogranular topographical features in contact with each other which have a heterogeneousshape and size distribution(maximum lateral dimension ) 51.0 ( 30.7 nm) which is consistent with the known dimensions of mineral particles, as measured by scanning electron microscopy,41 transmission electron microscopy,42 and small-angle X-ray scattering.43

sense (i.e., the state which we model with FEA). In return, the plastic contracting behavior can explain why we observe sink-in rather than pile-up. This is observed in the AFM images (Figure 1): rather than being squeezed out, the particles in the imprint area are more highly compacted than far away. Refined models, currently in development, aim at taking this contracting behavior into account, by considering a critical-state Cam-Clay type plasticity model for bone nanoindentation. This refined model, which considers nanoscale porosity, is expected to shed light on the observable contracting behavior in nanoindentation, whereas it is not expectedto changethe overallresultthat bone’sultrastructure is a cohesive-frictional material.

In order to further investigate the nanogranular friction in bone, angle of reposeexperimentswere performedon ground and ultrasonicated powders of deorganified adult bovine cortical bone mineral (Figure 5) in a sealed chamber in ambient (æambient) and vacuum environments (ævacuum). The smallest particle size distribution obtained was 914 ( 4 nm (as measured by dynamic light scattering) and the corresponding angles of repose were found to be æambient ) 32.8 ( 3.1° and ævacuum ) 18.2 ( 2.5° (number of measurements, n ) 4). Since ævacuum < æambient, this suggests that the vacuum was effective in minimizing interparticle cohesion likely due to hydration layers. ævacuum was found to be statistically independent (p < 0.05) of particle size up to 25 ím. It was observed that the internal friction angle æ obtained from fitting nanoindentation data to the Mohr- Coloumb formulation in intact bone ( 15°) was slightly lower than the measured angle of repose in vacuum ævacuum ( 18.2°) for the deorganified bone. Hence, rather than introducing additional friction due to macromolecular shear (as might be expected since æ ) 5° for the demineralized bone), the interfacial organic has a slight lubricating effect, which is consistentwith nanomechanicalstudies which show that certain interfacial organics, in particular polyelectrolyte macromolecules, can enhance lubrication.25 This may be a tradeoff effect, in order to counteract the reduction of tensile

Figure 2. Comparison of the predictions of a three-dimensional elastic-perfectly plastic finite element analyses (FEA) model (ABAQUS) incorporating a Mohr-Coulomb cohesive-frictional yield strength criterion to averaged nanoindentation data on loading of intact adult bovine cortical bone perpendicular to the long bone axis in ambient conditions (Hysitron Triboindenter, loading rate 50 íN/s) using; (a) Berkovich and (b) Cube Corner probe tips. The sample preparation, characterization, and experimental protocols were reported previously.16 The two fitting parameters used were the internal friction angle (æ) and cohesion (c). The best-fit parameters (R2 ) 0.9) produced the same value for the two different indenter geometries. (a) Berkovich and (b) Cube Corner FEA meshes are also shown. Eight-node linear elastic brick hybrid elements (C3D8H) were used. To reduce computational cost and due to symmetry, 1/6 of the tip and the surface were modeled and the corresponding boundary conditions were applied to ensure the symmetry; that is, the nodes on the sidewalls were fixed in the direction normal to the sidewall surface (the nodal displacement in this direction is set to be zero). The probe tip end radius and truncate height were approximated from control experiments on a fused quartz sample. Each averaged force versus depth curve represents 80 individual nanoindentation experiments; horizontal bars are one standard deviation.

Nano Lett. C

capacity compared to the cohesive shear strength (primarily carried by the organic component) due to frictional interactions.

It is interesting to note that c (intact bone) c (demineralized bone) which may well suggest that cohesion values quantified by nanoindentation are attributable to the organic itself, rather than to interfacial mineral-organic bonding.26 Indeed, cohesion may arise from collagen crosslinking (which includes intermolecular pyridinoline and pyrrole linkages arising from linkages between lysine and hydroxylysine aldehyde residues27,28) or noncovalent “sacrificial” bonding in noncollagenous proteins (e.g., proteoglycans, osteopontin, and bone sialoprotein29,30).

The fact that æ (intact bone) . æ (demineralized bone)

(Figures 2 and 3) and æ (intact bone) ævacuum (deorganified bone)(Figures2 and 5) suggeststhat the ultrastructuralorigins of the friction angle arise primarilyfrom mineral interparticle interactions and that organic frictional contribution (e.g., internal friction arising from, for example, molecular rotations, stick-slip sliding, and/or barrier-hop fluctuations31) is minimal.Potentialinterparticlefrictionalmechanismscited in the literature that may be relevant include mechanical interlockingand deformationof surfacenanoasperities,which has been observed directly by scanning electron microscopy for adjacent aragonite-based nacre tablets (from the inner layer of a gastropod mollusk).32 In this paper,32 it was postulated that inter-tablet shear resistance was enhanced as nanoasperities were required to “climb” over one another in order for intertablet sliding. Another potential mechanism at even smaller length scales has been studied by atomistic molecular dynamics modeling and involves mechanical locking due to surface roughness between atoms, as well as dynamic frictioncharacterizedby translationalkinetic energy that dissipates during sliding into internal energy motions.3 Stick-slip and smooth sliding between atoms and the transition between the two at atomic length scales could be relevant to bone mineral interactions as well.34

(Parte 1 de 2)

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