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Nanogranular Origins of the Strength of
Bone
Kuangshin Tai, †^ Franz-Josef Ulm, ‡^ and Christine Ortiz* ,†
Department of Materials Science and Engineering and Department of Ci V il Engineering, Massachusetts Institute of Technology, 77 Massachusetts A V enue, Cambridge, Massachusetts 02139
Received August 10, 2006; Revised Manuscript Received September 13, 2006
ABSTRACT
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 studies^1 -^3 have primarily probed tensile modes of deformation, which are relevant, for example, to avulsion fractures at tendinous and ligamen- tous 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 signifi- cant physiological loading condition^5 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 microdamage^7 and fracture^8 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 indicative of pressure sensitive plasticity. 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, 3-5 nm in thickness^10 ) that permeate in and around type I collagen
fibrils in an overlapping manner.^1 In this paper, we explore the possibility of nanogranular friction from 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 inor- ganic component is known to be a critical determinant of the macroscopic compressive mechanical properties of bone; the yield stress,^11 maximal stress,^12 and failure strength^13 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 com- bined 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, deformation away from their unstressed configurations likely involve mineral-mineral displacement. Hence, we hypoth- esize that the ultrastructure of bone is a cohesive-frictional material, 18 following a Mohr-Coulomb pressure dependent strength criterion 19 (i.e., which arises from the pressure dependence of the density of interparticle contacts).
- To whom correspondence should be addressed. Phone: 617-452-3084. Fax: 617-258-6936. E-mail: [email protected]. † (^) Department of Materials Science and Engineering. ‡ (^) Department of Civil Engineering.
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To explore this hypothesis, the nanoindentation of cortical bone was predicted using an elastic-plastic three-dimensional finite element analysis (FEA) model for two independent triaxial stress states^20 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 dependent strength criterion. In this case, the strength domain in the principal stress space, σI g σII g σIII, is defined by
where c is the interparticle cohesion, which is the finite value
of cohesive shear strength required to cause sliding when the normal stress is zero, and Ê 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 formula- tion.^21 A Poisson’s ratio of 0.3 was also fixed.^22 Two material properties were reduced from the experimental data: c and Ê, which were free fitting parameters. The difference between the experimental indentation response and the theoretical predictions were minimized by the quadratic error 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, characteriza- tion, and experimental protocols have been reported previ- ously. 16 The best fit Ê and c values for both indenter geometries were found to be 15° and 100 MPa, respectively ( R^2 ) 0.99). The indentation simulations on loading exhibited a small elastic region (indentation depths <∼10 nm) fol- lowed 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 compressive strength-to-tensile-strength ratio ( Y c /Y t ) (1 + sin Ê)/ (1 - sin Ê)) of 1.7, which is remarkably close to the known macroscopic ratio ∼2. 9 Theoretical fits were also carried out on nanoindentation data taken on fully deminer- alized bone (treated for 5 days in 0.5 M ethylenediamine- tetraacetic acid). The modulus was fixed to that reduced from the unloading slope (∼2.3 GPa) and Poisson’s ratio was set to 0.3 as previously, yielding best fit Ê and c values for both indenter geometries of 5° and 100 MPa, respectively (Figure
- 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 nano- scale porosity (∼20 nm in size) 23 which is a characteristic of a nanogranular material. 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 derive^24 ) 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 μm) immediately after loading to 7000 μN followed by unloading ( Hysitron Tri- boindenter, loading/unloading rate of 50 μN/second) using (a) Berkovich (∼850 nm depth) and (b) Cube Corner (∼1.5 μm 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 protocols were reported previously.^16 The undeformed regions away from the residual indent area are composed of nanogranular topographical features in contact with each other which have a heterogeneous shape and size distribution (maximum lateral dimen- sion ) 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
f (σ ij ) ) σI (1 + sin Ê) - σIII (1 - sin Ê) - 2 c cos Ê e 0 (1)
B Nano Lett.
capacity compared to the cohesive shear strength (primarily carried by the organic component) due to frictional interac- tions.
It is interesting to note that c (intact bone) ≈ c (deminer- alized 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 hydroxy- lysine aldehyde residues 27,28) or noncovalent “sacrificial” bonding in noncollagenous proteins (e.g., proteoglycans, osteopontin, and bone sialoprotein 29,30).
The fact that Ê (intact bone). Ê (demineralized bone) (Figures 2 and 3) and Ê (intact bone) ≈ Êvacuum (deorganified bone) (Figures 2 and 5) suggests that the ultrastructural origins of the friction angle arise primarily from mineral interparticle interactions and that organic frictional contribution (e.g., internal friction arising from, for example, molecular rota- tions, stick-slip sliding, and/or barrier-hop fluctuations 31 ) is minimal. Potential interparticle frictional mechanisms cited in the literature that may be relevant include mechanical interlocking and deformation of surface nanoasperities, 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 friction characterized by translational kinetic energy that dissipates during sliding into internal energy motions.^33 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 The addition of mineral and corresponding frictional interactions results in an increase in the resistance to plasticity (strength or elasticity domain) in compression at these small lengths (Figure 4), which is the dominating physiological stress state in bone, and correlates with the correct trend macroscopically. 11 Our interpretation is consistent with the known structure-function relationships of other mineralized tissues; for example, human tendon has no mineral volume and is loaded predominantly in tension, 35 whereas ear or whale bones have a considerable amount of mineral (∼ 90
Figure 3. Adult bovine cortical bone was demineralized using 0. M ethylenediaminetetraacetic acid treatment for 5 days. Lack of mineral was verified through X-ray photoelectron spectroscopy (no evidence of Ca or P), heat treatment (400 °C to show complete loss of material), and energy dispersive X-ray analysis (no Ca, P). A three-dimensional elastic-perfectly plastic finite element analyses (FEA) model incorporating the Mohr-Coulomb cohesive-frictional strength criterion (described in Figure 2 caption) was fit to the averaged experimental nanoindentation force versus depth curve for demineralized bone tested perpendicular to the long bone axis in ambient conditions for (a) Berkovich and (b) Cube Corner probe tips ( R^2 ) 0.99). Each averaged force versus depth curve represents 80 individual nanoindentation experiments; horizontal bars are one standard deviation.
Figure 4. Plot comparing experimental nanoindentation data on loading in ambient conditions perpendicular to the long bone axis comparing intact (data from Figure 2) and demineralized (data from Figure 3) bone indented with both Berkovich and Cube Corner probe tips. Each averaged force versus depth curve represents 80 individual nanoindentation experiments; horizontal bars are one standard deviation.
D Nano Lett.
wt %) and are primarily loaded in compression. 14 Human (as well as bovine) bone (∼60 wt % mineral) seems to be an intermediate compromise, providing strength in compres- sion for physiological loading plus some degree of protection for accidental shear/torsion/bending loading. Mineralization has also been shown to vary with anatomical site,^36 leading to correlated intersite variation in compressive stiffness^37 (reflective of mineral content) and yield strength. 38 The macroscopic experimental value of the uniaxial compressive yield strength of these adult bovine cortical bone samples was measured perpendicular to the long bone axis as σ y ∼ 178 ( 47 MPa (Zwick allround-line mechanical tester, 0.01 mm/s load rate), which is 30% less than that calculated from the Mohr-Coloumb friction angle and cohesion reduced from nanoindentation data (∼260 MPa). This comparison suggests that larger length scale structures serve to weaken, rather than strengthen bone, possibility due to heterogeneity, porosity, or other defects present. A number of nanoscale deformation mechanisms have been proposed in the literature, based on various types of experimental data, including void and crack formation,^39 extension of bridging noncollagenous proteins with sacrificial bonds during collagen fibril separation,^1 and mineralized
collagen fibril-matrix shear. 2 Several of these behaviors are expected to be pressure dependent (i.e., consistent with macroscopic yield). These mechanisms were all observed primarily in tensile modes of deformation. No distinct microcracking was observed in the direct imaging of the compressive deformation (Figure 1), but it should be noted that very small loads and lengths scale are being probed, and hence, our results do not preclude this mechanism from taking place at larger deformations. No jumps or disconti- nuities were observed in the individual force versus depth curves as well (data not shown) suggesting the absence of underlying microcracking 40 and a uniform continuum re- sponse. The mineralized collagen fibril-matrix shear mech- anism in tension could in fact activate the mechanism proposed here, mineralite interparticle friction, as well as rupture of sacrificial bonds in the organic. 1 Our results suggest, however, that the former is more dominant for this particular compressive experimental geometry studied and is an important factor in the pressure-dependent yield strength of bone.
Acknowledgment. The MIT Department of Materials Science and Engineering Nanomechanical Testing Facility. The Whitaker Foundation and the U.S. Army through the MIT Institute for Soldier Nanotechnologies (Contract No. DAAD-19-02-D0002) for funding. [The content does not necessarily reflect the position of the government and no official endorsement should be inferred.] References (1) Fantner, G. E.; Hassenkam, T.; Kindt, J. H.; Weaver, G.; Birkedal, H.; Pechenik, L.; Cutroni, J. A.; Cidade, A.; Stucky, G.; Morse, D. E.; Hansma, H. G. Sacrificial bonds and hidden length dissipate energy as mineralized fibrils separate during bone fracture. Nat. Mater. 2005 , 4 , 612-616. (2) Gupta, H. S.; Wagermaier, W.; Zickler, G. A.; Raz-Ben Aroush, D.; Funari, S. S.; Roschger, P.; Wagner, H. D.; Fratzl, P. Nanoscale deformation mechanisms in bone. Nano Lett. 2005 , 5 (10), 2108-
(3) Nalla, R. K.; Kinney, J. H.; Ritchie, R. O. Mechanistic fracture criteria for the failure of human cortical bone. Nat. Mater. 2003 , 2 , 164-
(4) Cochran, G. V. B. A primer of orthopaedic biomechanics ; Churchill Livingstone: New York, 1982. (5) Nordin, M.; Frankel, V. Biomechanics of Bone. In Basic Biome- chanics of the Musculoskeletal System ; Lea & Febiger: Philadelphia, 1989; Number of 3-29. (6) Biewener, A. A. Safety factors in bone strength. Calcif. Tiss. Int. 1993 , 53 Suppl 1 , S68-74. (7) Vashishth, D.; Koontz, J.; Qiu, S. J.; Lundin-Cannon, D.; Yeni, Y. N.; Schaffler, M. B.; Fyhrie, D. P. In vivo diffuse damage in human vertebral trabecular bone. Bone 2000 , 26 (2), 147-152. (8) Fine, K. M.; Vegso, J. J.; Sennett, B.; Torg, J. S. Prevention of cervical spine injuries in football: A model for other sports. Phys. Sportsmed. 1991 , 19 , 54-62. (9) Yeni, Y. N.; Dong, X. N.; Fyhrie, D. P.; Les, C. M. The dependence between the strength and stiffness of cancellous and cortical bone tissue for tension and compression: extension of a unifying principle. Biomed. Mater. Eng. 2004 , 14 (3), 303-310. (10) Weiner, S.; Wagner, H. D. The material bone: Structure-mechanical function relations. Ann. Re V_. Mater. Sci._ 1998 , 28 , 271-298. (11) Wachter, N. J.; Krischak, G. D.; Mentzel, M.; Sarkar, M. R.; Ebinger, T.; Kinzl, L.; Claes, L.; Augat, P. Correlation of bone mineral density with strength and microstructural parameters of cortical bone in vitro. Bone 2002 , 31 (1), 90-95. (12) Louis, O.; Boulpaep, F.; Willnecker, J.; Van den Winkel, P.; Osteaux, M. Cortical mineral content of the radius assessed by peripheral QCT predicts compressive strength on biomechanical testing. Bone 1995 , 16 (3), 375-379.
Figure 5. Angle of repose experiments performed in ambient environmental conditions and in vacuum. The deorganified bone (400 °C heat treatment where no weight change after 5 days was recorded) powder is shown (a) before vacuum, ambient conditions and (b) after vacuum. Angles measured from deorganified bone were compared to one another and found to be ∼ 33 ° in the ambient state and ∼ 18 ° in vacuum. The particle size was varied (∼ 1 - 25 μm) and controlled by the degree of sonication (Branson, Sonifier
- and measured by a Brookhaven 90Plus Particle Size Analyzer. The angle of repose did not change significantly for the range of particle sizes measured.
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