Acta Biomaterialia 8 (2012) Contents lists available at SciVerse ScienceDirect. Acta Biomaterialia

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1 Acta Biomaterialia 8 (2012) Contents lists available at SciVerse ScienceDirect Acta Biomaterialia journal omepage: Elastic moduli of untreated, demineralized and deproteinized cortical bone: Validation of a teoretical model of bone as an interpenetrating composite material E. Hamed a, E. Novitskaya b,j.li a, P.-Y. Cen b,c, I. Jasiuk a,, J. McKittrick b a University of Illinois at Urbana Campaign, Department of Mecanical Science and Engineering, 1206 West Green Street, Urbana, IL 61801, USA b University of California, San Diego, Department of Mecanical and Aerospace Engineering and Materials Science and Engineering Program, 9500 Gilman Dr., La Jolla, CA 92093, USA c National Tsing Hua University, Department of Materials Science and Engineering, 101 Sec. 2 Kuang-Fu Rd., Hsincu 30013, Taiwan, ROC article info abstract Article istory: Received 11 July 2011 Received in revised form 6 November 2011 Accepted 7 November 2011 Available online 15 November 2011 Keywords: Cortical bone Elastic moduli Multi-scale modeling Demineralization Deproteinization A teoretical experimentally based multi-scale model of te elastic response of cortical bone is presented. It portrays te ierarcical structure of bone as a composite wit interpenetrating biopolymers (collagen and non-collagenous proteins) and minerals (ydroxyapatite), togeter wit void spaces (porosity). Te model involves a bottom-up approac and employs micromecanics and classical lamination teories of composite materials. Experiments on cortical bone samples from bovine femur include completely demineralized and deproteinized bones as well as untreated bone samples. Porosity and microstructure are caracterized using optical and scanning electron microscopy, and micro-computed tomograpy. Compression testing is used to measure longitudinal and transverse elastic moduli of all tree bone types. Te caracterization of structure and properties of tese tree bone states provides a deeper understanding of te contributions of te individual components of bone to its elastic response and allows fine tuning of modeling assumptions. Very good agreement is found between teoretical modeling and compression testing results, confirming te validity of te interpretation of bone as an interpenetrating composite material. Ó 2011 Acta Materialia Inc. Publised by Elsevier Ltd. All rigts reserved. 1. Introduction Bone tissue is a natural composite material consisting of an organic pase (90% type-i collagen and 10% non-collagenous proteins (NCP)), an inorganic pase (ydroxyapatite-like minerals) and water. On a volumetric basis, mammalian skeletal bone is made up of 32 44% organics, 33 43% minerals and 15 25% water [1]. Tese constituents assemble into a complex ierarcical structure, wic gives bone its excellent mecanical properties [2 4]. In tis paper, te ierarcical structure of cortical bone is described in terms of four structural levels (Fig. 1), spanning from nanoscale to mesoscale levels. Nanoscale (Level I), wic ranges from a few to several undred nanometers, represents a mineralized collagen fibril level. A mineralized fibril as a composite structure made of organic and inorganic pases and water. Type I collagen, wic is te major constituent of te organic pase, consists of triple elical tropocollagen molecules wic are 300 nm long [5,6] and 1.5 nm in diameter [6,7]. Tese molecules assemble into a staggered arrangement wit a periodicity of 67 nm [8,9], wic includes gap and overlap regions. Te inorganic pase consists of non-stoiciometric ydroxyapatite crystals (Ca 10 (PO 4 ) 6 (OH) 2 ), wit 4 6% of te Corresponding autor. Tel.: ; fax: address: ijasiuk@illinois.edu (I. Jasiuk). pospate groups replaced by carbonate groups. Te mineral crystals are in te form of platelets nm long, nm wide and 2 4 nm tick [10 14]. Te remaining pase is water, wic plays an important role in bio-mineralization. Tese constituents are combined into mineralized collagen fibrils ( nm in diameter [1,15]), wic are te primary building blocks of bone. It is generally believed tat crystals initially form witin te gap regions of te collagen fibrils, furter proceed into te overlap regions, and subsequently grow into te extrafibrillar space [16,17]. Consequently, minerals are found bot witin and outside te collagen fibrils, but te exact amount in eac location is still a matter of contention [18 22]. Recent studies [23 25] on completely deproteinized and completely demineralized bones sow tat bot te minerals and protein form continuous pases. Sub-microscale (Level II), wic spans from one to tens of microns, represents a single lamella level. A lamella, wit tickness 3 7 lm [10], is made of preferentially oriented mineralized collagen fibrils. At tis lengt scale, te elliptical cavities called lacunae (typically 5 10 lm wide and lm long [26,27]) can be observed. Connecting te lacuna are small cannels ( nm in diameter [28]), called canaliculi. Microscale (Level III), ranging from tens to undreds of microns, denotes lamellar structures, wic are made of lamellae stacked togeter at different orientations, i.e., te fibrils in eac lamella are oriented at a different angle wit respect to te adjacent one /$ - see front matter Ó 2011 Acta Materialia Inc. Publised by Elsevier Ltd. All rigts reserved. doi: /j.actbio

2 E. Hamed et al. / Acta Biomaterialia 8 (2012) Fig. 1. Hierarcical structure of cortical bone constituents and associated porosity. HA = ydroxyapatite, NCP = non-collagenous proteins. Constituents Porosity I At te nanoscale, tropocollagen molecules ( nm) and ydroxyapatite minerals ( nm) combine to form te basic unit of all bone te mineralized collagen fibril (100 nm in diameter). Gaps witin and between te collagen molecules ( nm) configure te first level of porosity, were minerals, water, and non-collagenous proteins are deposited. II Lamellae (3 7 lm tick) are formed from preferentially oriented mineralized collagen fibrils. Embedded in te lamellae are lacuna spaces ( lm) were bone cells reside, connected by small cannels (canaliculi). III Te lamellae form osteons wit a central vascular cannel (Haversian system, lm diameter), te primary feature of cortical bone. Te orientation of te mineralized collagen fibrils is different between te adjacent lamellae, Porosity at tis level includes te Haversian canals (50 lm in diameter), wic are oriented longitudinally, and Volkmann s canals (50 lm in diameter), wic are oriented in transverse direction. wic form a twisted plywood structure. IV Osteonal and interstitial bones wit resorption cavities form te cortical bone. Resorption cavities (300 lm in diameter), sites of bone remodeling, are te main porosity at tis scale. [29,30]. In cortical bone, several layers of te lamellae, arranged in concentric rings around te vascular cannels, form osteons (Haversian system), wile interstitial lamellae, wic are remnants of old osteons, fill spaces between osteons. Mesoscale (Level IV), wic spans several undred microns to several millimeters or more, depending on species, represents te cortical bone level. Te cortical bone consists of osteons embedded in interstitial lamellae wit some resorption cavities. In order to understand te structure and mecanical properties of bone and its protein and mineral constituents, bone can be demineralized or deproteinized by aging in HCl or NaOCl solutions, respectively. Previous studies on te structure of cancellous and

3 1082 E. Hamed et al. / Acta Biomaterialia 8 (2012) cortical bones [23 25] sowed tat, after complete deproteinization and demineralization, te bone samples maintained structural integrity, and te mineral and protein constituents sowed surprisingly similar microstructures and retained features suc as Haversian systems, lacunae and canaliculi. Compression tests on untreated (UT), demineralized (DM) and deproteinized (DP) cancellous [31] and cortical bones [25] sowed tat te mecanical properties of DM and DP bone are muc lower tan tose of UT bone. Tese results indicate tat bone is an interpenetrating composite material wose properties are enanced by te intertwining of te two pases: proteins and minerals. Te structure of bone is more complex tan tat of most engineering composite materials and modeling of its mecanical properties as long been a callenging task. In te literature, a number of teoretical studies ave addressed te mecanical properties of UT bone at different ierarcical levels. At te nanoscale, most of te existing models consider bone as a composite material made up of collagen matrix and ydroxyapatite inclusions [32,33] or, conversely, ydroxyapatite matrix and collagen inclusions [34,35]. Tere are also a few models representing collagen and crystals as two interpenetrating pases [34,36]. Computational models, involving a finite element metod, ave also been used to investigate te collagen mineral interactions in bone [37 40]. However, in suc models te ydroxyapatite pase is treated as disconnected crystals rater tan a continuous pase. At te submicrostructural level, Jasiuk and Ostoja-Starzewski [41] modeled a single lamella as a spatially random network of mineralized collagen fibrils and computed its effective anisotropic stiffness tensor. Fritsc and Hellmic [36] and Yoon and Cowin [42] used micromecanics metods to obtain te effective elastic properties of a single lamella. At te microstructural level, Dong and Guo [43] modeled a single osteon as a two-pase composite wit osteonal lamellae being a matrix and a Haversian canal being an inclusion in te form of an elongated pore. Tey also extended teir micromecanical model to assess te elastic properties of cortical bone by modeling te interstitial lamellae as a matrix, and osteons and resorption cavities as inclusions [43]. Tere are also oter models tat capture te elastic beavior of UT cortical bone. However, tere are no multi-scale models of DM and DP cortical bones available in te literature. In tis study, te elastic moduli of te UT, DM and DP cortical bone are investigated. A step-by-step modeling approac, involving four different ierarcical levels (Levels I IV), is proposed, and teoretical results at te cortical bone level (mesoscale) are compared wit compression test data. Te experimental observations on te structure and composition of tese tree bone types serve as inputs for te teoretical model. To te autors knowledge, tis is te first report on te development of te multi-scale model tat incorporates experimental observations of bone as an interpenetrating composite material composed of contiguous biopolymer and mineral pases. 2. Materials and experimental metods 2.1. Sample preparation A single fres bovine femur bone from an animal of unknown breed was obtained from a local butcer (La Jolla, CA). Te slaugter age was 18 monts. Samples, 3 cm tick, were cut from te mid-diapysis region. Te bone was torougly cleaned wit water, and soft tissue was removed wit a scalpel. Sixty cortical bone samples ( mm) were prepared for compression testing of UT, DM and DP bones (20 samples for eac bone type). Te samples were first rougly cut wit a andsaw and ten precisely wit a diamond blade under constant water irrigation, wit te compression surfaces cut as parallel as possible. An aspect ratio of 1.5 was cosen in order to reduce te influence of sample surface traction forces, according to Ref. [44], wic suggested using an aspect ratio between one and two for compression samples. Samples were prepared for testing in two directions: longitudinal and transverse. Te longitudinal direction was cosen along te long axis of te bone, wile te transverse direction was perpendicular to te long axis in te circumferential (angular) direction. Ten samples of eac bone type were prepared for eac testing direction. Samples were wrapped in a wet paper towel, placed in plastic zipped bags and stored in a refrigerator (T =4 C) for 1 2 days until cemical treatment and mecanical testing. Te paper towels were moistened wit Hank s balanced saline solution (HBSS). HBSS was cosen because it is a standard saline solution recommended for storage and ydration of bone and oter mineralized tissues [45] Demineralization and deproteinization process Cortical bone samples were demineralized by aging in 0.6 N ydrocloric acid (HCl) at room temperature, following te procedures described in Refs. [24,46]. Acid solutions were canged daily in order to avoid saturation, wic can affect te demineralization process. Te complete demineralization process took 7 days. All solutions were quantitatively analyzed by inductively coupled plasma optical emission spectrometry, to evaluate te Ca concentration. Te completeness of demineralization was verified by te absence of Ca in te solutions. Bone samples were deproteinized by aging in a 5.6 wt.% sodium ypocloride (NaOCl) solution at 37 C, following te procedures in Ref. [24]. Te solutions were canged every 12. Te wole process took about 2 weeks. Full deproteinization was verified by subsequent demineralization, wic resulted in te disappearance of te sample (deproteinization followed by demineralization). Previous work on bone demineralization and deproteinization sowed tat te amount of proteins left in te solution after subsequent demineralization of previously DP samples is <0.001 wt.% [24] Structural caracterization In order to estimate te area fractions of osteonal and interstitial lamellae, ten representative images of five different UT bone samples, all obtained from te same femur, were used for analysis by optical microscopy (OM) using a Zeiss Axio imager equipped wit a CCD camera (Zeiss Microimaging Inc., Tornwood, NY). Optical images were taken at 100 and 200 in order to view significant bone features clearly. Fracture surfaces of te specimens from all tree groups were analyzed by scanning electron microscopy (SEM) equipped wit energy dispersive spectroscopy (FEI-XL30, FEI Company, Hillsboro, OR). DM bone samples were subjected to critical point drying procedure wit a purge time equal to 20 min. using te fully automatic critical point drier (Tousimis Autosamdri-815, Rockville, MD) before SEM imaging in order to avoid excessive srinkage and deformation. Samples from all groups were mounted on aluminum sample olders, air dried and sputter-coated (Emitec K575X, Quorum Tecnologies Ltd., West Sussex, UK) wit cromium for 30 s before imaging. Samples were observed at a 20 kv accelerating voltage wit a working distance of 10 mm. OM and SEM images gave two-dimensional (2-D) information about bone microstructure. To investigate te microstructure in tree-dimensions (3-D), te micro-computed tomograpy (l-ct) imaging was performed on te tree bone groups at a nominal isotropic resolution of 1 lm. Tis allowed for ig-resolution 3-D imaging witout destruction of te specimen. Te scan produced 1024 slices ( image pixels per slice) resulting in a

4 E. Hamed et al. / Acta Biomaterialia 8 (2012) field of view (FOV) of rougly 1 mm 3 cube. Te l-ct measurements were conducted in air using Xradia MicroXCT-200 (for UT and DP samples) and MicroXCT-400 (for DM samples) (Xradia Inc., Pleasanton, CA) instruments. Te two instruments were cosen based on te range of X-ray electric potential tey could acquire. Samples were scanned at various X-ray poton energies, depending on teir pysical properties, to obtain optimum imaging. Scanning of te DM group was performed at low energies (30 kv, 200 la; acquirable only on MicroXCT-400) owing to low X-ray absorption of proteins, wile te DP group required ig energies (80 kv, 100 la). For UT samples, te parameter was set between te two treated cases (40 kv, 200 la). For all measurements 729 projections were acquired over a range of 182 wit 12 s exposure time for eac projection. No frame averaging was taken at eac tomograpy projection. Te data were reconstructed using Xradia TXMReconstructor. Ring artifacts and beam ardening effects (BHE) were corrected in te reconstruction software. For te DP group (ydroxyapatite only), te BHE were pronounced and, terefore, a beam-flattening filter was placed in te X-ray pat during te scan. Te reconstructed l-ct tomograms were post-processed using Amira (Visage Imaging, Inc., Berlin, Germany) to analyze 3-D microstructures. No filtering was applied on images of UT samples, wile for treated cases (DM and DP) te median filter was applied to suppress noises and enance image contrast. Te gray image slices were ten segmented to binarized data sets, separating voids from bone regions. Te tresold value was critically judged around te middle point between two peaks corresponding to bone and voids in te gray level istogram [47]. After image segmentation, te 3-D microstructures were reconstructed, and quantitative analysis was implemented to obtain te 3-D morpometric information Compression testing Specimens from te tree groups (UT, DM and DP) were submerged in HBSS for 24 before testing and were tested in te ydrated condition (te time between taking te samples out from te solution and testing tem was 1 min). Novitskaya et al. [25] sowed tat te saline solution fills all pores generated by removal of protein or mineral pases, wic significantly affects te corresponding mecanical properties. Compression testing was performed wit a 30 kn load cell universal testing macine (Instron 3367 Dual Column Testing System, Norwood, MA) on UT samples, wile a 500 N load cell testing macine (Instron 3342 Single Column Testing System) was used on DM and DP samples. Specimens were tested at a 0.5 mm min 1 crossead speed, wic translated to a strain rate of 10 3 s 1. An external deflectometer SATEC model I3540 (Epsilon Tecnology Corp., Jackson, WY) was used to measure small displacements wit a precision linearity reading of 0.25% of full measuring range. Compression tests were performed in te unconstrained conditions. 3. Modeling metods In tis section, te multi-scale approac for modeling of cortical bone is introduced wic consists of successive omogenization steps from nano to mesoscale levels (Levels I IV). Te effective elastic properties of UT, DM and DP cortical bones at eac structural level were found in a bottom-up fasion, using te results from a lower level as te inputs for a iger level. Continuum micromecanics metods and classical lamination teory of composite materials were employed to account for te microstructure of bone at different scales. Te elastic properties and volume fractions of te constituents (collagen, ydroxyapatite, water and NCP) were te main inputs to te model. A wide range of values for te elastic moduli of collagen and ydroxyapatite as been reported in te literature (see Table 1 in Ref. [48]). Te coice of properties for bone constituents is tabulated in Table 1. For simplicity, in te present model, all components were assumed to be linear elastic and isotropic. Te model is well suited to account for anisotropy. However, tere are limited data in te literature on a complete set of anisotropic constants for collagen and ydroxyapatite crystals. Table 2 lists definitions of all te symbols and abbreviations used trougout te modeling sections Modeling of UT cortical bone Nanoscale Level I At te nanoscale, water and NCP fill te spaces between collagen and ydroxyapatite, and solid pases interact wit water [42]. Terefore, te properties of collagen sould be considered in its wet state. Following Ref. [36], a Mori Tanaka sceme [49,50], wit te cross-linked collagen molecules being a matrix and te intermolecular voids (filled wit water and NCP) being te inclusions, was used to obtain te effective elastic properties of collagen water composite C colw as i 1 C colw ¼ C col þ U w ðc w C col Þ : I þ S sp col : C 1 col : ðc w C col Þ i 1 1 : U col I þ U w I þ S sp col : C 1 col : ðc w C col Þ ; ð1þ were subscripts colw, col and w refer, respectively, to collagen water composite, dry collagen, and water and NCP. Te derivation of elastic stiffness tensors of collagen C col and water wit NCP C w is given in Appendix A. S r 0 is te fourt-order Eselby tensor [51] accounting for te sape of pase r in a matrix wit stiffness tensor C 0, were 0 is a generic subscript. Here, superscript sp denotes te sape of sperical void inclusions. Moreover, mineral crystals ave some internal defects (porosities). Tus, a mineral interacting wit water can be represented as a porous solid wit some intercrystalline voids witin, filled wit water and NCP. Te Mori Tanaka metod was applied to predict te stiffness of ydroxyapatite water mixture C HAw, as follows i 1 C HAw ¼ C HA þ U w ðc w C HA Þ : I þ S sp HA : C 1 HA : ðc w C HA Þ i 1 1 : U HA I þ U w I þ S sp HA : C 1 HA : ðc w C HA Þ : ð2þ In Eq. (2), subscripts HAw and HA denote, respectively, ydroxyapatite water composite and ydroxyapatite crystals. Te elastic stiffness of ydroxyapatite C HA is given by Eq. (A.1) in Appendix A. A mineralized collagen fibril was modeled as a nanocomposite material using a micromecanics continuum approac involving a self-consistent metod [52,53] wit two interpenetrating pases: collagen water mixture and interfibrillar ydroxyapatite water composite. Te self-consistent formulation developed for polycrystalline materials was used suc tat tere was no distinct matrix, and bot pases were represented as ellipsoidal inclusions. Te collagen water pase was assumed to be cylindrical in sape wit an aspect ratio of 1000:1:1 following te 100 lm lengt [54] and 100 nm diameter of collagen fibrils [1,15], wile ydroxyapatite water minerals were represented as ellipsoids wit an aspect ratio of 50:25:3 following Ref. [55]. Te effective stiffness tensor of a mineralized collagen fibril C fib, given in terms of stiffness tensors of collagen water C colw and interfibrillar ydroxyapatite water C HAw, was predicted as

5 1084 E. Hamed et al. / Acta Biomaterialia 8 (2012) Table 1 Elastic properties and volume fractions of bone constituents employed in te modeling. Material Young s modulus (GPa) Poisson s ratio Volume fraction (%) Collagen 1.5 [32,84] 0.28 [20] 41 Hydroxyapatite 114 [85,86] 0.23 [87] 42 Non-collagenous proteins 1.0 [20] 0.45 [20] 4 Bulk modulus (GPa) Poisson s ratio Volume fraction (%) Water Table 2 Symbols used in te modeling and teir definitions. C fib ¼ U colw C colw : I þ S cyl : fib C 1 fib : ðc colw C fib Þ i 1 þu HAw C HAw : I þ S ellipse fib : C 1 fib : ðc HAw C fib Þ : U colw I þ S cyl : fib C 1 fib : ðc colw C fib Þ þu HAw Symbol C cfib col colw cyl DOM Efoam ellipse fib HA HAw I Ilam lac NCP os p U S sp w Definition Elastic stiffness tensor Hydroxyapatite coated fibril Dry collagen Collagen water composite Cylinder Degree of mineralization Extrafibrillar ydroxyapatite foam Ellipse Mineralized collagen fibril Hydroxyapatite crystal Hydroxyapatite water composite Identity tensor Interstitial lamellae Lacuna Non-collagenous protein Osteon Pores (due to resorption cavities) Volume fraction Eselby tensor Spere Water and non-collagenous proteins i 1 1 I þ S ellipse fib : C 1 fib : ðc HAw C fib Þ ; ð3þ were subscript fib denotes te mineralized collagen fibril. Here, superscripts cyl and ellipse refer, respectively, to te cylindrical and ellipsoidal sapes of collagen and ydroxyapatite. Since te effective properties of a mineralized collagen fibril C fib were not isotropic, te components of te Eselby tensor were evaluated numerically for an ellipsoidal inclusion embedded in a general anisotropic matrix [56]. Eq. (3) was solved iteratively, wit Eselby tensors S cyl fib and Sellipse fib being updated at eac iteration, to obtain te implicit unknown C fib. Te elastic constants of a mineralized collagen fibril C fib obtained in tis level were ten used as inputs for te next level Sub-microscale Level II At te sub-microstructural level, two different modeling steps were defined as in Ref. [48]: (1) mineralized collagen fibrils interacting wit an extrafibrillar ydroxyapatite foam (coated fibrils), and (2) combining te matrix of step 1 wit lacunar cavities to form a single lamella. Different experimental tecniques confirmed te existence of extrafibrillar ydroxyapatite crystals on te outer surface of te fibrils [19,22,57,58]. Tese crystals are dispersed and randomly oriented [59 61]. Terefore, te extrafibrillar ydroxyapatites can be treated as a porous foam consisting of ydroxyapatite minerals wit intercrystalline pores in-between, filled wit water and NCP [34,36,62,63]. Te self-consistent sceme wit two interpenetrating pases, ydroxyapatite water minerals and pores, was used to evaluate te overall elastic constants of te extrafibrillar foam C Efoam as C Efoam ¼ U w C w : I þ S sp : Efoam C 1 Efoam : ðc w C Efoam Þ i 1 þu HAw C HAw : I þ S sp : Efoam C 1 Efoam : ðc HAw C Efoam Þ : U w I þ S sp : Efoam C 1 Efoam : ðc w C Efoam Þ þu HAw i 1 1 I þ S sp : Efoam C 1 Efoam : ðc HAw C Efoam Þ ; ð4þ were te subscript Efoam refers to te extrafibrillar ydroxyapatite foam. Te random arrangement of extrafibrillar ydroxyapatite minerals leads to isotropy of te omogenized material. Terefore, for te sake of simplicity, bot pases were assumed to be sperical in sape, following Ref. [64]. Furtermore, it was assumed tat 75% of te total ydroxyapatite crystals were interfibrillar, and te remaining 25% were extrafibrillar [48]. Mineralized collagen fibrils, wit te elastic properties obtained in Eq. (3), and te extrafibrillar ydroxyapatite foam, wit te elastic properties obtained in Eq. (4), interpenetrate eac oter to form coated fibrils. Te effective stiffness tensor of coated fibrils C cfib was predicted by using te self-consistent sceme as C cfib ¼ U fib C fib : I þ S cyl : cfib C 1 cfib : ðc fib C cfib Þ þ UEfoam C Efoam i 1 : I þ S sp : cfib C 1 cfib : ðc Efoam C cfib Þ : U fib I þ S cyl : cfib C 1 cfib : ðc fib C cfib Þ þu Efoam i 1 1 I þ S sp : cfib C 1 cfib : ðc Efoam C cfib Þ : ð5þ In Eq. (5), subscript cfib refers to te mineralized collagen fibrils coated wit extrafibrillar ydroxyapatite foam (coated fibrils). Also, superscripts cyl and sp denote, respectively, te cylindrical and sperical sapes of fibrils and extrafibrillar ydroxyapatite foam. Here again, two interpenetrating pases were considered, wic were modeled as two different types of inclusions, wit no matrix. A single lamella was modeled as aving coated fibrils as a matrix, wit properties given in Eq. (5), perforated by osteocyte-filled ellipsoidal cavities called lacunae. Subscript lac denotes te ellipsoidal lacunae, wit an aspect ratio of 5:2:1 following teir approximate lm 3 dimensions [26,42]. Te major axes of lacunae were assumed to be oriented along te longitudinal direction of a single lamella. Te effective elastic constants of a single lamella C lamella were obtained using te Mori Tanaka sceme as i 1 C lamella ¼ C cfib þ U lac ðc lac C cfib Þ : I þ S ellipse cfib : C 1 cfib : ðc lac C cfib Þ i 1 1 : U cfib I þ U lac I þ S ellipse cfib : C 1 cfib : ðc lac C cfib Þ : ð6þ

6 E. Hamed et al. / Acta Biomaterialia 8 (2012) Table 3 3-D bone morpometry results from l-ct image analysis. Porosity Canals (UT samples only) Lacunae (UT samples only) UT sample (%) DM sample (%) DP sample (%) Volume fraction (%) Mean widt (lm) Volume fraction (%) Mean lengt (lm) Mean widt (lm) 7.9 ± ± ± ± ± ± ± ± 0.3 Fig. 2. Optical microscopy image of a cross section of bovine femoral cortical bone. In te present model, te effect of canaliculi on elastic properties of te single lamella was neglected. Te elastic properties of a single lamella C lamella were used as inputs for te next level Microscale Level III At te microscale, te lamellae are oriented in a twisted pattern caracterized by a continuous rotation of lamellae. Te properties of lamellae were obtained using Eq. (6), and te starting angle was cosen to be 0 degree for te innermost layer. To te autors knowledge, tere is no consensus in te literature on te number of lamellar layers in an osteon or te orientation of te outermost layer. Here, it was assumed tat te fibrils complete a 180 turn from te innermost to te outermost layer. As long as te layers are not ortogonal to eac oter, te angle cange between successive layers as a negligible effect on te results [65]. Te elastic properties of an osteonal lamella were obtained following te omogenization sceme of Sun and Li [66] developed for laminated composite materials. More details on te modeling procedure of an osteonal lamella are given in Ref. [48]. Te properties of an interstitial lamella were evaluated following te same omogenization procedure as for te osteonal lamella. Te osteons are generally less stiff and less mineralized tan te interstitial lamellae [67,68]. In order to capture suc beavior, one could use a iger degree of mineralization (DOM) for an interstitial lamella compared wit an osteonal lamella [48]. However, for simplicity ere, bot te osteonal and interstitial lamellae were assumed to ave te same DOM and, terefore, te same elastic properties. Having found te elastic properties of an osteonal lamella, a generalized self-consistent metod [69] was used to calculate te effective elastic constants of an osteon C os, following te approac of Dong and Guo [43]. To tis end, te osteon was modeled as a two-pase composite wit te osteonal lamella being a matrix and te Haversian canal being a cylindrical inclusion. An osteon Fig. 3. SEM images of (a) untreated (UT), (b) demineralized (DM) (continuous protein network) and (c) deproteinized (DP) (continuous mineral network) cortical bones sowing microstructural features: osteons (Os), lacuna spaces (Lac), Haversian cannels (HC) and Volkmann s canals (VC).

7 1086 E. Hamed et al. / Acta Biomaterialia 8 (2012) Fig. 4. l-ct 3-D isosurface images of untreated (UT) cortical bone sowing (a) side view of canal network (red: Haversian, vertical; Volkmann s, orizontal) and osteocyte lacunae (yellow); te lacunae are preferentially oriented in vertical direction, indicating te long axis of bone. (b) Side view of canal network only. (c) Top view of canal network and osteocyte lacunae. (d) Top view of canal network only. A volumetric filter ( lm 3 ) was applied to separate te canal network from te lacuna space. is typically a cylinder 250 lm in diameter and 1 cm long, wile te diameter of te Haversian canal is 50 lm [70]. Consequently, te volume fraction of te Haversian canals is 4%. Tis coice for te volume fraction of longitudinal Haversian canals was also in good agreement wit te present l-ct findings (Table 3). Te outputs of tis level, wic were te elastic stiffness tensors of an osteon C os and an interstitial lamella C Ilam, served as te inputs for te next level Mesoscale Level IV Te ybrid Mori Tanaka sceme [71], wit te interstitial lamella being te matrix and te osteons and resorption cavities being inclusions, was applied to evaluate te elastic constants of cortical bone at te mesoscale (tissue level). Bot te osteons and te resorption sites were assumed to be cylindrical, wit aspect ratios of 4:1:1, following te 1 cm lengt and 250 lm diameter of osteons [70], and aligned along te long axis of bone. Te subscripts Ilam, os and p denote te interstitial lamellae, osteons and pores, respectively. Ten, te transversely isotropic effective stiffness tensor of te cortical bone C bone was computed as C bone ¼ U Ilam C Ilam þ U os C os : I þ S cyl : Ilam C 1 Ilam : ðc os C Ilam Þ i 1 þu p C p : I þ S cyl : Ilam C 1 Ilam : ðc p C Ilam Þ : U Ilam þ U os I þ S cyl : Ilam C 1 Ilam : ðc os C Ilam Þ þu p i 1 1 I þ S cyl : Ilam C 1 Ilam : ðc p C Ilam Þ : ð7þ Te volume fraction of osteons was taken as 68% based on te present optical images Modeling of treated cortical bone Modeling of treated (DP and DM) bones followed te same modeling procedure as for te UT cortical bone wit te following exceptions. (i) In te treated bones, one pase was removed: te collagen in te case of DP bone, and te ydroxyapatite in te case of DM bone. Te removed pase was replaced wit voids in all te modeling steps. (ii) In te modeling of UT bone, it was assumed tat alf te volume fraction of water at nanoscale interacted wit collagen, wile te oter alf interacted wit ydroxyapatite. However, in te case of treated bones, were one pase is no longer available, all te water content at nanoscale went to te remaining pase. In oter words, in te case of DP bone, U w ¼ 0 in Eq. (1), meaning tat C colw ¼ C col! 0, wereas in te case of DM bone, U w ¼ 0 in Eqs. (2) and (4), meaning tat C HAw ¼ C Efoam ¼ C HA! Results and discussion Estimation of te amount of osteonal and interstitial bone area fractions was done by analyzing ten representative optical images of five different UT bone samples, all taken from te same femur (Fig. 2 is a representative optical image). Te osteonal area fraction was found to be 0.68 ± 0.05, wit te balance being te interstitial bone fraction; bot area fractions incorporate teir respective porosities. Tese numbers were later used in te present model for Level IV (mesoscale) as volume fractions. SEM images of UT, DM and DP samples are sown in Fig. 3. Fig. 3a illustrates te fracture surface of an UT bone sample; vascular cannels are visible and osteon structure is well defined. Fig. 3b and c demonstrate tat DM and DP bones are contiguous, standalone structures (continuous protein network and continuous mineral network). Microscopic features, suc as te Haversian

8 E. Hamed et al. / Acta Biomaterialia 8 (2012) Fig. 5. l-ct 3-D detailed surface view images of: (a) voids in untreated (UT) bone; te lacunae are preferentially oriented along te long axis of bone; (b) protein pase in demineralized (DM) bone; (c) mineral pase in deproteinized (DP) bone. cannels (20 40 lm in diameter) and Volkmann s canals, are preserved in DM and DP samples in agreement wit Ref. [24]. Moreover, a well-defined osteon structure is clearly seen in bot types of treated bone. Fig. 4 sows te 3-D isosurface overview of all voids in te UT sample. Te two types of porosity in sub-microscale to microscale levels, te osteocyte lacunar system and canal network, are clearly observed. Te quantification was ten performed on four samples per eac group, and te results are presented in Table 3, wic lists means and standard deviations. To illustrate 3-D microstructures in detail, a partial region inside ( lm 3 cube) was cosen, and a complicated surface view was generated terein (Fig. 5). In UT bones, te lacunae ave ellipsoid sapes tat are oriented in te longitudinal direction of te bone. Te collagen fibril (Fig. 5b) and mineral-grain (Fig. 5c) like patterns are well observed in DM and DP bones, respectively. Fig. 6 sows experimental stress strain curves for UT, DM and DP cortical bones measured in longitudinal and transverse directions. UT and DP samples sowed a well-defined initial linear elastic region, wile DM samples sowed te beavior typical for biopolymers wit long toe in region in te initial part of te stress strain curve. For UT bone, te longitudinal direction (wit elastic modulus ± 1.79 GPa) was stiffer and stronger tan te transverse direction (wit elastic modulus of ± 1.44 GPa), in good agreement wit te results of oter researcers [3]. Removal of te mineral or protein pase resulted in a drastic cange in te stress strain curves. First, for te DM samples (Fig. 6b), te elastic modulus was calculated using te steepest portions of te curves, and te modulus in te longitudinal direction (0.23 ± 0.01 GPa) was found to be larger tan in te transverse direction (0.13 ± 0.02 GPa). As pointed out by Gibson and Asby [72], te elastic modulus of a polymer foam is igly dependent on te density, and large deformations can be accommodated. Te difference between te beavior in te longitudinal and transverse directions can be explained by te preferential orientation of te osteons, among oter factors (see Refs. [73,74]). Secondly, te DP samples (Fig. 6c) appear to beave as a classic cellular solid, demonstrating a linear elastic region up to a peak stress, after wic a plateau region is sustained. Te longitudinal elastic modulus (9.23 ± 2.82 GPa) is larger tan te transverse one (2.45 ± 0.78 GPa). Tis again can be attributed to te alignment of osteons in te longitudinal direction. As a side note, it is clear tat a rule of mixtures law (Voigt average) does not apply ere: te volume fraction averaged elastic modulus of te mineral and protein constituents (4.6 GPa in longitudinal direction) is not close to tat of te UT bone. Fig. 7a illustrates te modeling results for te longitudinal and transverse elastic moduli of UT cortical bone as a function of bone porosity. Te same parameters are sown in Fig. 7b and c for DP and DM cortical bones, respectively. Te range of porosities selected in te modeling for eac bone type was based on te l-ct measurements as listed in Table 3 (5 10% for UT bone, and 45 60% for DP and DM bones). Clearly, te elastic moduli of all tree bone groups decrease as te porosity increases. Suc a trend was also reported by oter researcers for UT cortical bone [75,76] and for cellular structures [72,77], suc as DP and DM bones.

9 1088 E. Hamed et al. / Acta Biomaterialia 8 (2012) Fig. 6. Stress strain curves for (a) untreated (UT), (b) demineralized (DM) and (c) deproteinized (DP) cortical bone for two anatomical directions. N = 10 for eac curve. Fig. 8 illustrates te experimental results obtained from compression testing for te longitudinal and transverse (in te circumferential direction) elastic moduli of UT, DP and DM cortical bones and compares tem wit modeling results. Te mean values reported for teoretical results (Fig. 8) were calculated by averaging over te different values of porosity (Fig. 7). Te bars in Fig. 8 represent te standard deviation and te range, respectively, for te experimental and modeling data. Experimental and modeling results are in very good agreement and, in most cases, teir discrepancies, wenever present, are mainly due to simplifying assumptions and selections made at different stages of modeling. Te main discrepancy between experiments and modeling occurs for te transverse elastic modulus of UT cortical bone. One possible reason may be tat, in te present model, for simplicity, all osteons were assumed to be aligned along te long axis of bone. However, tere are some drifting osteons in bone wit off-axis, rater tan te longitudinal, alignment [78]. Te transverse elastic modulus of UT bone is underestimated by neglecting te presence of tose misaligned osteons in te model. In addition, as mentioned in Section 3.1.3, in te present model te osteonal and interstitial lamellae were assumed to ave te same DOM (42% mineral volume fraction [79]). Te modeling results reported in Figs. 7 and 8 are based on tat assumption. In reality, owever, tis is not te case, and te interstitial lamellae are more mineralized tan te osteonal lamellae. Te present model can easily andle different mineral contents for interstitial and osteonal lamellae. In order to address tis issue, first, te average mineral volume fraction was assumed to be 37% for osteon and 43% for interstitial lamella, following Gupta et al. [80], and te modeling steps were repeated for UT bone. In tis case, te longitudinal and transverse elastic moduli were found to be, respectively, and 8.91 GPa. Clearly, te values were lower compared wit te previous results, since te overall mineral content became lower. Next, te case of 42% mineral volume fraction for osteons and 48% mineral volume fraction for interstitial bone was considered. Te longitudinal elastic modulus of UT bone increased to GPa, wile te transverse modulus increased to GPa. However, no experimental references support inputs of suc iger mineral content. Ideally, te actual mineral content specific to te present bone type sould be used in te model, but suc measurements are not available for te present samples. Te oter somewat large discrepancy occurs between te teoretical and experimental longitudinal elastic modulus of DM bone. Tis can be explained by te fact tat only te presence of longitudinal Haversian vascular cannels was incorporated in te present model, but te existence of Volkmann s canals, wic are

10 E. Hamed et al. / Acta Biomaterialia 8 (2012) Fig. 7. Teoretical prediction of te elastic modulus as a function of porosity for longitudinal and transverse directions of (a) untreated (UT), (b) demineralized (DM) and (c) deproteinized (DP) cortical bone. Fig. 8. Comparison of te experimental and modeling results for (a) longitudinal, and (b) transverse elastic moduli of untreated (UT), demineralized (DM) (magnified 100 for clarity) and deproteinized (DP) cortical bone. Te capped lines sow te standard deviation for experimental data and te range for modeling results. oriented perpendicular to te main Haversian system (see Fig. 4), was neglected. Including some voids in te transverse direction (Volkmann s canals) would decrease te computed elastic moduli of DM bone, along wit considering te off-axis alignment of te Haversian canals. Anoter reason for te difference of experimental and modeling results for DM bone is a possible degradation of collagen structure during te demineralization process due to enzymatic autolysis [25]. In te case of DP bone, te results of modeling and experiments could be closer if more information about te type and magnitude of forces tat old te ydroxyapatite crystals togeter were available. In te present model, it was assumed tat, after removing te protein pase, ydroxyapatite crystals were perfectly bonded to eac oter. Based on SEM images sown in Fig. 9, te ydroxyapatite crystals in DP bone form an aligned nanocrystalline network, coinciding wit te collagen alignment. Tese nanocrystals are most likely eld togeter by weak electrostatic forces and/or troug mecanical interlocking, as demonstrated by te low fracture strengt sown in Fig. 6c. Incorporating suc geometry tat could allow some slip in te model could give rise to a better matc

11 1090 E. Hamed et al. / Acta Biomaterialia 8 (2012) wile studying te strengt and fracture of cortical bone, te important role of cement lines sould not be ruled out. In summary, te present multi-scale model assumed tat cortical bone is an interpenetrating composite of continuous pases of collagen and ydroxyapatite. Te model also accounted for te presence of NCP, water and porosity. Te experimental inputs were te volume fractions of porosities at different levels obtained from l-ct scans and te volume fractions of osteonal and interstitial bone obtained from OM. Additionally, te morpology of te collagen fibrils, ydroxyapatite and porosity was incorporated using te Eselby tensor. Given te simplifying assumptions used in te analysis, te model sowed very good agreement wit experimental values. To te autors knowledge, tis is te first multi-scale model tat incorporates experimental observations of bone as an interpenetrating composite material. Also, te study of DM and DP bone, wic provided valuable insigts into te bone structure and its mecanical properties, allowed fine tuning of te teoretical model. 5. Conclusions Fig. 9. SEM micrograp of DP bone (100% minerals) sowing tat minerals align in a preferred orientation. Tis alignment coincides wit te alignment of te collagen fibers. Fig. 10. Backscattered electron image sowing te cross-sectional microstructure of cortical bovine femur bone: osteons (Os); interstitial lamellae (ILam); arrows point to te cement lines. between te actual and te modeled structure and, consequently, elastic properties of DP bone. Anoter limitation of te present model is neglecting te presence of cement lines around te outer boundaries of osteons. Tese tin (<5 lm) lines deflect crack propagation in bone loaded in te transverse direction, wic enances te fracture tougness of te bone in tis direction [81]. A backscattered SEM image of cortical bone is sown in Fig. 10, were te presence of cement lines is clearly observed around te osteons. Tere is no consensus in te literature on te DOM of cement lines. Some researcers found tat cement lines are less mineralized tan te surrounding tissues [82], wile oters described cement lines as igly mineralized tissues [83]. In eiter case, since te volume fraction of tin cement lines is very small compared wit te volume fractions of osteons and interstitial lamella, teir presence does not significantly affect te computed elastic moduli of all tree bone groups. However, A new teoretical model was developed tat accurately predicts te experimentally measured elastic modulus of cortical bovine femur bone. Tis model assumes tat cortical bone as a ierarcical structure, is an interpenetrating composite of biopolymers and ydroxyapatite minerals, and consists of porosity at different ierarcical levels. A bottom-up approac was employed, incorporating outcomes of te previous ierarcical level as te inputs for te next one. Tis model was furter verified by te close agreement found between te model and experimental results taken on DP and DM bone. Te major findings of tis work are as follows. 1. Tis is te first multi-scale model incorporating experimental observations of bone as an interpenetrating composite combined wit interdispersed porosity at different ierarcical levels. Tese results sow te complexity of te bone structure, wic is still not well understood, and te open callenges in modeling it. 2. Te compressive elastic moduli of UT and treated bones sow anisotropy in te elastic modulus between te longitudinal (iger) and transverse (lower) directions. Tis demonstrates tat bot te protein and mineral arcitectures ave preferential structures in te longitudinal direction. 3. Tree-dimensional imaging by l-ct clearly reveals and quantifies te ierarcical structure of te porosity from lacuna spaces to vascular cannels and to resorption cavities. Te lacunae spaces are ellipsoids tat ave te major axis parallel to te long axis of te bone. Te l-ct images illustrate a complex network of canals, including Haversian and Volkmann s canals, perforating te bone structure. 4. Te multi-scale model using experimental values of porosity and volume fractions of constituents demonstrates tat te elastic moduli of te UT, DM and DP bones are in very good agreement wit te experimentally measured values. Conflict of interest statement None of te autors ave any conflicts of interest to report wit respect to te material contained in tis manuscript. Acknowledgements Te autors gratefully acknowledge support from te National Science Foundation, Ceramics Program Grant (J.M.) and

12 E. Hamed et al. / Acta Biomaterialia 8 (2012) te CMMI Program Grant (I.J.). Tey tank Professor Marc A. Meyers (UCSD) for is entusiastic support and Professor Maria-Grazia Ascenzi (UCLA) for saring er expertise on collagen orientations. Tey also tank Ryan Anderson (CalIT2, UCSD) and Aruni Suwarnasarn (SIO, UCSD) for teir elp in SEM and Leilei Yin (Beckman Institute, UIUC) for te assistance in l-ct scanning. Appendix A Given te Young s modulus E r and Poisson s ratio m r of a pase r wit an isotropic elastic beavior, its elastic stiffness tensor C r is represented as 0 1 m r m r m r m r 1 m r m r E r m r m r 1 m r C r ¼ ð1 þ m r Þð1 2m r Þ m r 0 0 : B m r 0 A m r ða:1þ Young s moduli and Poisson s ratios of different pases (collagen, ydroxyapatite, and water and NCP), wic are given in Table 1, were substituted in Eq. (A.1) to obtain teir corresponding elastic stiffness tensors. Appendix B. 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