Materials Science and Engineering C 54 (2015) 207–216

Contents lists available at ScienceDirect

Materials Science and Engineering C journal homepage: www.elsevier.com/locate/msec

Experimentally-based multiscale model of the elastic moduli of bovine trabecular bone and its constituents Elham Hamed a,1, Ekaterina Novitskaya b,⁎, Jun Li a,2, Iwona Jasiuk a, Joanna McKittrick b a b

University of Illinois at Urbana-Champaign, Department of Mechanical Science and Engineering, 1206 West Green Street, Urbana, IL 61801, USA University of California, San Diego, Department of Mechanical and Aerospace Engineering, Materials Science and Engineering Program, 9500 Gilman Dr., La Jolla, CA 92093, USA

a r t i c l e

i n f o

Article history: Received 21 October 2014 Received in revised form 14 January 2015 Accepted 24 February 2015 Available online 11 March 2015 Keywords: Trabecular bone Elastic moduli Multiscale modeling Demineralization Deproteinization Micro-computed tomography

a b s t r a c t The elastic moduli of trabecular bone were modeled using an analytical multiscale approach. Trabecular bone was represented as a porous nanocomposite material with a hierarchical structure spanning from the collagen–mineral level to the trabecular architecture level. In parallel, compression testing was done on bovine femoral trabecular bone samples in two anatomical directions, parallel to the femoral neck axis and perpendicular to it, and the measured elastic moduli were compared with the corresponding theoretical results. To gain insights on the interaction of collagen and minerals at the nanoscale, bone samples were deproteinized or demineralized. After such processing, the treated samples remained as self-standing structures and were tested in compression. Micro-computed tomography was used to characterize the hierarchical structure of these three bone types and to quantify the amount of bone porosity. The obtained experimental data served as inputs to the multiscale model and guided us to represent bone as an interpenetrating composite material. Good agreement was found between the theory and experiments for the elastic moduli of the untreated, deproteinized, and demineralized trabecular bone. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Bone is a hierarchically structured biological material composed of an organic phase, inorganic phase, and water. The organic phase (32– 44 vol.%) contains a type-I collagen and a small amount of noncollagenous proteins (NCPs). Hydroxyapatite minerals form the inorganic phase (33–43 vol.%) [1]. Water occupies about 15–25 vol.% of the bone material. There are two bone types: cortical (compact, dense) and trabecular (cancellous, porous). The cortical bone is a dense tissue (porosity b10 vol.%), which forms a protective shield around the more porous (porosity ~70–90 vol.%) trabecular bone. Different levels of hierarchy of trabecular bone, including nanoscale, submicroscale, microscale, and mesoscale, were discussed in detail in our previous study [2]. Mechanical properties of trabecular bone have been extensively studied using various theoretical and experimental approaches. At the nanoscale, the mineralized collagen fibril was modeled either as a matrix-inclusion composite material consisting of collagen and hydroxyapatite crystals [3–6] or as an interpenetrating composite built

⁎ Corresponding author. E-mail address: [email protected] (E. Novitskaya). 1 Now at Northwestern University, Civil and Environmental Engineering Department, 2145 Sheridan Road, Evanston, IL 60208, USA. 2 Now at California Institute of Technology, Division of Engineering and Applied Science, 1200 E California Blvd, MC 301-46, Pasadena, CA 91125, USA.

http://dx.doi.org/10.1016/j.msec.2015.02.044 0928-4931/© 2015 Elsevier B.V. All rights reserved.

up by the two phases [2,5,7]. The collagen–mineral interactions were also investigated using computational methods, including a finite element method (FEM) [8–12] and molecular dynamics simulations [13–15]. Modeling of bone at the nanoscale is reviewed in [12]. At the sub-microscale, a single lamella was modeled analytically [2,7,16,17] or computationally [12,18] as a network of mineralized collagen fibrils including lacunar cavities. Elastic properties of a single lamella were also measured by a nanoindentation technique [19–24]. At the microscale, various experimental techniques including a microtensile test [25–27], bending test [28,29], and ultrasound [24,26,30] were used to estimate the elastic properties of a single trabecula. Some studies measured the Young's modulus of trabecular bone experimentally and used that data to back-calculate the elastic constants of trabecular bone tissue [31,32]. At the mesoscale, the majority of analytical studies modeled trabecular bone as a cellular foam and expressed its Young's modulus by power law relations in terms of bone density [33–38]. Other researchers took into consideration the density as well as the fabric tensor [39–41], which characterizes the structural anisotropy of bone. Since the trabecular bone architecture, including thickness, number, separation distance, and connectivity of trabeculae, affects bone's mechanical response, high-resolution imaging techniques such as a micro-computed tomography (μ-CT) have been widely used to characterize the architecture of trabecular bone. Accordingly, many researchers used μ-CT based FEM to model the mechanical behavior of trabecular bone [21,42–47]. Advances in theoretical calculations and experimental measurements of the elastic moduli of trabecular bone were recently summarized in

208

E. Hamed et al. / Materials Science and Engineering C 54 (2015) 207–216

2. Materials and methods 2.1. Sample preparation

Fig. 1. Schematic illustration of physiological and experimental loadings on a bovine femur head, and the sample orientation for two directions: A and B. The samples are not shown to scale. Image adapted from [81].

[48]. It was illustrated that the complex hierarchical structure of trabecular bone plays a crucial role in the mechanical response and makes the modeling of elastic properties of bone very challenging. Several recent studies [49–51] demonstrated that the two main constituents of bone (collagen and hydroxyapatite) can be separated by aging bone in hydrochloric acid (demineralization) or sodium hypochlorite (deproteinization) solutions, and their mechanical properties can be examined separately. These studies proposed that bone tissue could be considered as an interpenetrating composite of collagen and minerals. Despite advances in characterization of bone's mechanical properties, a comprehensive multiscale model that incorporates experimental observations of trabecular bone as an interpenetrating composite material and elucidates the structure–property relations in bone is still absent. As a step toward developing such a model, here, we introduce a multiscale modeling approach, which spans from the nanoscale to the mesoscale, to predict the elastic moduli of trabecular bone, representing the minerals and collagen as interpenetrating phases. The developed model is employed to investigate the elastic behaviors of the untreated (UT), demineralized (DM), and deproteinized (DP) bovine trabecular bone. This is a follow up study to a similar analysis performed on bovine cortical bone, where we demonstrated that the predicted elastic moduli, using a multiscale model of bone as an interpenetrating composite of minerals and collagen, compared well with those obtained experimentally [49]. We further developed ideas generated in a previous study, where preliminary results presented were based on an isotropic model of bone at the mesoscale [52]. In particular, the model outlined here retains the features of our previous study at the nano and submicro scales [49], while different modeling approaches are employed at the micro and meso scales, due to inherent differences in the structure of cortical and trabecular bones at the two latter scales. The obtained modeling results are compared to the experimentally measured elastic moduli of trabecular bone samples tested in compression. Volume fraction, shape, and distribution of porosity at different levels of hierarchy estimated by μ-CT are the inputs to the model. Experimentallybased theoretical models of these three types of bone (UT, DM, and DP) can provide additional insights into the structure and structure–property relations of trabecular bone and allow to fine tune modeling approaches.

One trabecular bovine femoral bone from an approximately 18month-old animal was obtained from a local butcher. Bovine bone was chosen for the current study since it is readily available and is commonly used as a model material for the investigation of bone mechanical properties [53–56]. The bone marrow was carefully removed with water using a water pick. Altogether fifty-six samples (6 mm × 6 mm × 8 mm) were prepared for compression testing. Sample sizes were chosen in accordance with [57], who suggested to use an aspect ratio between 1 and 1.5 for compression tests on trabecular bone samples. The samples were first roughly cut by a handsaw and then precisely with a diamond blade under a constant water irrigation with the surfaces as parallel as possible. Samples were cut in two directions. The direction oriented along the femur neck axis was called as the Adirection, while the direction normal to the A-direction was labeled as the B-direction (Fig. 1). Seventeen UT samples in the A-direction and eleven UT samples in the B-direction were prepared. Samples were stored in closed zip lock bags filled with Hank's balanced saline solution in refrigerator (T = 4 °C) for 1–2 days until chemical procedure and mechanical testing. Five DM samples in the A-direction and seven DM samples in the Bdirection were prepared following the procedures given in [51]. In parallel, four DP samples in the A-direction and four DP samples in the Bdirections were prepared following the procedures given in [51]. To avoid additional uncertainties associated with samples from different animals, only one bovine femur bone was used for the current study. Therefore, there was a limitation to the number of samples that could be prepared in the A- and B-directions. The DM and DP processes brought additional challenges to the sample preparation (since DP and DM samples are extremely fragile). Therefore, several samples were excluded from the investigation due to their failure during sample preparation. As a result, there is a difference in the number of samples in the A- and B-directions. All samples were stored in closed zip lock bags filled with Hank's balanced saline solution in refrigerator (T = 4 °C) until mechanical testing. 2.2. Micro-computed tomography characterization All samples cut in the two directions (A and B) were scanned using Xradia MicroXCT-200 (− 400 for DM, due to low X-ray absorption of protein) (Carl Zeiss X-ray Microscopy, Inc., Pleasanton, CA). For each sample the scan generated 729 radiographic images over a range of 182° with a 12 s exposure time for each image and no frame averaging. The raw images were then reconstructed using the Xradia TXMReconstructor software, where ring artifacts and beam hardening effects (BHE) were corrected. For the DP group (hydroxyapatite only), due to the high X-ray attenuation of minerals [58], the BHE were so pronounced that a 3.7-mm-Al beam-flattening filter was placed in the Xray path during the scan. The reconstructed tomographic images consisted of 1024 slices (1024 × 1024 image pixels per slice) with a resolution of 10 μm and a field of view of roughly 1 cm3 cube. The stacks of two-dimensional (2D) slices were post-processed in the software Amira 5.2 (FEI Visualization Sciences Group, Burlington, MA) for image visualizations and quantifications of 3D microstructures. No image filtering was used on images of UT samples, while for treated cases (DM and DP) a median image filter with kernel size 3 was applied for noise reduction. Initially, the 2D tomographic slices were 16 bits gray scale images (0 ~ 65536 from black to white) and, then, they were segmented to black-white binary images for the separation of voids and bone tissues. A global threshold was selected around the middle point between the two peaks of black and white in gray level histograms [59] and verified by an external method of porosity measurements [60].

E. Hamed et al. / Materials Science and Engineering C 54 (2015) 207–216

209

Table 1 Elastic properties and volume fractions of bone constituents employed in the modeling procedure. Material

Young's modulus (GPa)

Poisson's ratio

Volume fraction (%)

Collagen Hydroxyapatite Non-collagenous proteins

1.5 [3,82] 114 [84,85] 1 [83]

0.28 [83] 0.23 [86] 0.45 [83]

41 42 4

Bulk modulus (GPa) Water

2.3 [17]

2.3. Compression testing Compression testing was performed using a testing machine equipped with a 500 N load cell (Instron 3342 Single Column Testing System, Instron, Norwood, MA). Specimens were tested in the hydrated

Poisson's ratio

Volume fraction (%)

0.49 [17]

13

condition at a strain rate of 10−3 s−1. An external deflectometer SATEC model I3540 (Epsilon Technology Corp., Jackson, WY) was used to measure displacements with precision linearity, reading of 0.25% of full measuring range. Compression tests were performed in an unconstrained condition.

3. Modeling methods The elastic moduli of UT, DM, and DP trabecular bones were predicted using a multiscale modeling approach, which involved successive steps from the nano to mesoscale levels. Micromechanics approaches, laminated composite materials theories, and strength of materials cellular solid methods were employed to account for bone structures at different scales. The elastic properties, volume fractions, and structural geometries of the constituents (collagen, hydroxyapatite, water, and NCPs) as well as the porosities at different length scales were the inputs to the model. Table 1 lists the properties of the bone constituents used in the modeling. For simplicity, all bone components were assumed to be linear elastic and isotropic. Throughout the following sections, Cr and Φr denote, respectively, the elastic stiffness tensor and volume fraction of phase r. I is the identity tensor, while Sr0 is the Eshelby tensor [61] accounting for the shape of phase r in a matrix with stiffness tensor C0, where 0 is a generic subscript. 3.1. Modeling of untreated trabecular bone 3.1.1. Nanoscale The mineralized collagen fibril at the nanoscale was modeled as a composite material with two interpenetrating phases, collagen and interfibrillar hydroxyapatite crystals. Both phases were represented as inclusions and there was no matrix in the model. Collagen molecules were assumed to be cylindrical in shape with an aspect ratio of 1000:1:1 following the ~ 100 μm length [62] and ~ 100 nm diameter of collagen fibrils [1,63], while hydroxyapatite crystals were represented as ellipsoidal inclusions with an aspect ratio of 50:25:3 following [64]. Using a self-consistent method [65, 66], the effective stiffness tensor of a mineralized collagen fibril, Cfib, in terms of stiffness tensors of collagen, Cwcol, and interfibrillar hydroxyapatite, CIHA, was predicted as  h
i−1 h
i−1  wcol −1 IHA −1 Cfib ¼ Φwcol Cwcol : I þ Sfib : Cfib : Cwcol −Cfib þ ΦIHA CIHA : I þ Sfib : Cfib : CIHA −Cfib  h
i−1 h
i−1 −1 wcol −1 IHA −1 þ ΦIHA I þ Sfib : Cfib : CIHA −Cfib ; : Φwcol I þ Sfib : Cfib : Cwcol −Cfib

Fig. 2. A representative idealized cubic cell of the foam-like structure of trabecular bone.

ð1Þ

210

E. Hamed et al. / Materials Science and Engineering C 54 (2015) 207–216

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 3. Micro-computed tomography 3D isosurface images of (a,b) untreated bone, (c,d) demineralized bone and (e,f) deproteinized bone in the (a,c,e) A-direction and (b,d,f) B-direction. Scale bar = 700 μm. Taken from [52].

where subscript “fib”, “wcol”, and “IHA” denote, respectively, the mineralized collagen fibril, collagen, and interfibrillar hydroxyapatite. It should be noted that water and NCPs fill all the pores at the nanoscale, and both collagen and hydroxyapatite phases interact with water. Therefore, in Eq. (1), the elastic properties of collagen, Cwcol, and hydroxyapatite, CIHA, are those corresponding to the mixture of each phase with water, whose formulation is provided in Appendix A (Eqs. (A.1) and (A.2)).

Table 2 Porosity estimation from micro-computed tomography (μ-CT) images and physical density measurements. Bone type

Untreated

Orientation

A

B

A

Demineralized B

A

Deproteinized B

Porosity, μ-CT (%) Porosity, physical measurements (%)

83.9 ± 3.3 82.3 ± 6.1

83.2 ± 4.6 75.4 ± 6.4

91.6 ± 1.4 91 ± 2

90.6 ± 1.6 91 ± 2

89.1 ± 1.1 88.5 ± 1.9

86.9 ± 2.2 84.6 ± 3.8

E. Hamed et al. / Materials Science and Engineering C 54 (2015) 207–216

(a)

(b)

211

(c)

Fig. 4. Representative stress–strain curves for trabecular bone: (a) untreated, (b) deproteinized and (c) demineralized.

3.1.2. Sub-microscale Following our previous studies [16,49], two modeling steps are introduced at the sub-microscale to capture the structure and properties of bone: (1) mineralized collagen fibrils interacting with extrafibrillar hydroxyapatite (coated fibrils), and (2) a single lamella formed by coated fibrils comprising some lacunar cavities. Various experimental studies showed that some hydroxyapatite crystals exist on the outer surface of fibrils [67–70], which contribute to the stiffness of bone at the sub-microscale. In our model, it is assumed that this extrafibrillar hydroxyapatite and the mineralized collagen fibrils interpenetrate each other to form coated fibrils. The effective stiffness tensor of coated fibrils, Ccfib, was predicted by using the self-consistent scheme as  h
i−1 h
i−1  fib −1 EHA −1 þ ΦEHA CEHA : I þ Scfib : Ccfib : CEHA −Ccfib Ccfib ¼ Φfib Cfib : I þ Scfib : Ccfib : Cfib −Ccfib  h
i−1 h
i−1 −1 fib −1 EHA −1 : Φfib I þ Scfib : Ccfib : Cfib −Ccfib þ ΦEHA I þ Scfib : Ccfib : CEHA −Ccfib ;

ð2Þ

where subscript “cfib” refers to the mineralized collagen fibrils coated with extrafibrillar hydroxyapatite (coated fibrils) and subscript “EHA” refers to the extrafibrillar hydroxyapatite foam built up by hydroxyapatite crystals with intercrystalline pores in-between, filled with water and NCPs. Mineralized collagen fibrils were assumed to be cylindrical with an aspect ratio of 1000:1:1 having the effective elastic properties as obtained in Eq. (1). Furthermore, the extrafibrillar hydroxyapatite was represented as a spherical inclusion whose elastic properties are given in Appendix B

Fig. 5. Relative elastic modulus versus relative density for untreated, demineralized, and deproteinized trabecular bone for two anatomical directions (UT_A = untreated bone, A-direction; UT_B = untreated bone, B-direction; DM_A = demineralized bone, A-direction; DM_B = demineralized bone, B-direction; DP_A = deproteinized bone, A-direction; DP_B = deproteinized bone, B-direction).

212

E. Hamed et al. / Materials Science and Engineering C 54 (2015) 207–216

Table 3 Normalizing parameters for untreated, demineralized, and deproteinized trabecular bone obtained experimentally and used in Fig. 5. Bone type

ρtrabecula, g/cm3

Etrabecula, GPa

Untreated (UT) Demineralized (DM) Deproteinized (DP)

2.05 [50] 1.18 [50] 1.98 [50]

19.4 [50] 0.182 [50] 5.9 [50]

(Eq. (B.1)). In our model, it was assumed that 75% of the total hydroxyapatite minerals were interfibrillar and the remaining 25% were extrafibrillar [16]. A single lamella was represented by a continuous matrix made of coated fibrils comprising some lacunar cavities as inclusions. In our model, it was assumed that lacunae have a volume fraction of ~ 4% [49] and an aspect ratio of 5:2:1 following their approximate 25 × 10 × 5 μm3 dimensions [17, 71]. The effective elastic constants of a single lamella, Clamella, were obtained using the Mori–Tanaka scheme [72,73] as 
 h
i−1  lacuna −1 Clamella ¼ Ccfib þ Φlacuna Clacuna −Ccfib : I þ Scfib : Ccfib : Clacuna −Ccfib :  h
i−1 −1 lacuna −1 : Φcfib I þ Φlacuna I þ Scfib : Ccfib : Clacuna −Ccfib

ð3Þ

3.1.3. Microscale The elastic behavior of trabecular bone tissue at the microscale was modeled following the homogenization scheme of Sun and Li [74] developed for laminated composite materials that are composed of several lamellae with different stackings. Similarly, in our problem, the single trabecula consists of a number of single lamellae each having a preferential orientation of collagen fibrils. The elastic properties for each single lamella, Clamella, were obtained from the previous level (Eq. (3)). The lamellae are generally arranged into orthogonal, rotated, or twisted motifs in trabecular bone [75]. Due to lack of experimental information on the actual arrangement of lamellae in each trabecula, here the lamellae are assumed, for simplicity, to be oriented randomly, which gave rise to an isotropic elastic response for the trabecular bone tissue. In reality, however, the trabeculae may be anisotropic. The detailed formulation for the modeling of a single trabecula is given in Appendix C. The prediction of Young's modulus of trabecular bone tissue, Etrabecula, from this level served as the modeling input for the next structural level. 3.1.4. Mesoscale At the mesoscale, the trabecular bone was modeled as an idealized open-cell foam built up by arrays of cubic cells. Fig. 2 shows a representative anisotropic cubic cell, which has a length of l in x1 and x2 directions and a height of h in the x3 direction. The choice of an anisotropic modeling cell was motivated by the μ-CT images of trabecular bone samples showing elongated pores as opposed to round ones (Fig. 3). The degree of anisotropy is defined as D = h/l. The Young's modulus of trabecular bone in the x3 direction, E3, was obtained using the analytical model proposed by Huber and Gibson [76] as: E3 Etrabecula

  ρbone n ¼ CD ; ρtrabecula

ð4Þ

where Etrabecula is Young's modulus of a single trabecula as obtained theoretically in the previous level, ρbone and ρtrabecula are, respectively, the density of trabecular bone and the density of solid trabeculae, and C is a constant of proportionality. Gibson [36] proposed two types of structures for trabecular bone: an open cell (a network of rod-like elements) at relative densities smaller than 0.2 and a closed cell (a network of plate-like elements) at relative densities greater than 0.2. The power n was determined to be equal to 2 for an open cell and 3 for a closed cell [36]. In our model, the relative density, ρbone/ρtrabecula, was equal to the bone volume fraction as determined by μ-CT. Here, since the experimentally measured relative densities of most bone samples were smaller than 0.2, we assumed n = 2 and C = 1 in the model. Using the μ-CT data, the degree of anisotropy was quantified as the length of longest divided by shortest mean intercept length vector. The Young's modulus of trabecular bone in the direction x1 or x2, E1 = E2, was determined as [76]: E1 1 þ 1=D3 ¼ : E3 2D2

ð5Þ

The modeling results obtained at this scale, namely E1 and E3, represent the elastic moduli of trabecular bone and thus are comparable to the results obtained via compression testing of bone samples. 3.2. Modeling of demineralized and deproteinized trabecular bone Modeling of DM and DP bones involved the same procedure as for the UT bone with the following modification: the removed phases in the treated bones, namely the collagen in the case of DP bone and the hydroxyapatite in the case of DM bone, were replaced with voids in all the modeling steps. In other words, in case of DP bone, Cwcol → 0 in Eq. (1), whereas in case of DM bone, CIHA → 0 in Eq. (1) and CEHA → 0 in Eq. (2). 4. Results and discussion Fig. 3 illustrates representative μ-CT 3D isosurface images of UT, DM, and DP bone samples for both A- and B-directions. These images confirm that trabecular bone is indeed a porous network and this sustained

architecture is preserved in DM and DP samples as well. Also, in all cases, the trabecular pores are mostly elongated, as opposed to equiaxed, with different dimensions in the two orthogonal directions. Analysis of μ-CT images, using Amira 5.2, for one UT sample yielded the degree of anisotropy D = 1.94 for the A-direction and D = 1.56

E. Hamed et al. / Materials Science and Engineering C 54 (2015) 207–216

for the B-direction, implying that trabecular bone behaves as an anisotropic material with non-uniform structures in both A- and Bdirections. Porosity values from μ-CT analysis are given in Table 2, which were used in Eq. (4) for the modeling analysis. These values are close to density measurements made through weight and dimensional analysis. The porosities of the DM and DP samples are close to each other (within experimental error): ~ 90%. The slightly higher values for the DM samples could be due to water absorption, as the measurements were made in the wet condition. Fig. 4 shows the representative stress–strain curves for UT, DP, and DM trabecular bones. The elastic moduli of trabecular bone samples (Ebone) were calculated from the initial elastic regions of stress–strain curves according to [37,38]. Fig. 5 shows the experimentally determined relative elastic modulus (Ebone/Etrabecula) plotted against the relative density (ρbone/ρtrabecula) for UT, DM and DP samples in the A- and B-directions. The measured density, ρbone, was obtained from weight and sample dimensions measurements from UT, DM and DP samples. The normalizing parameters ρtrabecula and Etrabecula were chosen to be the density and the elastic modulus of the cell wall material (bone tissue) in agreement with [37,38]. These parameters were different for UT, DM, and DP samples (see Table 3). Etrabecula was the average between three anatomical directions (longitudinal, radial, and transverse) taken from [50]. For UT samples, the normalizing parameters were selected as the Young's modulus and density of cortical bone. For DM and DP samples, the parameters were chosen as the Young's modulus and density of DM and DP cortical bone [50], respectively. Trabecular bone forms perforated plates, shown in Fig. 3b, d, f, that are mostly oriented in the B-direction (direction of the larger stress), whereas thinner rods are formed in the A-direction. According to [36,37], the stress applied normal to these plates (in the A-direction) made them bend, and thus data points for this kind of

(a)

213

loading should fall closer to the line with slope 3. In the case of stress applied parallel to those plates (B-direction), data points should be close to the line with slope equal to 1 (see Fig. 5). It can be concluded that not only UT trabecular bone behaves as a cellular solid material, as found by Gibson and Ashby [37,38], but DP and DM trabecular bone also demonstrate this behavior. Fig. 6 illustrates the experimental results for the elastic moduli of UT, DP, and DM trabecular bone obtained by compression testing of bone samples in the A- and B-directions and compares them with the corresponding theoretical results predicted by the multiscale modeling approach. The reported mean values for the modeling results were calculated using the average values of porosity (Table 2). Bars represent the standard deviation for the experimental data and the range (due to the scatter in porosity) for the theoretical results. Two sets of modeling results are predicted for the elastic modulus of each bone type: the upper bound which is equivalent to E3 given by Eq. (4) and the lower bound which is equivalent to E1 as in Eq. (5). It is not known whether the experimental elastic moduli are measured in the x1 direction or x3 direction (Fig. 2), or possibly other directions, since these axes may not be aligned in the A- or B-directions. In our model, for simplicity, we considered only two characteristic lengths for the trabecular cubic cell; the longest length along x3 and the shortest length along x1, and, accordingly, we defined the degree of anisotropy between these two directions. However, the actual trabecular bone has an orthotropic response along its three principal axes. In spite of this simplifying assumption in the model, the experimental data in all cases falls in between the calculated lower and upper modeling bounds. Alternately, we could have used more precise methods involving a fabric tensor [40] or a μ-CT based FEM [2,20]. We chose the bounds approach using cell models for simplicity. A better correlation between modeling and experimental results could be obtained by first identifying the orientation of the trabeculae using µ-CT and then cutting the samples in that orientation. The differences in porosities for different parts of the femur head (the porosity is higher in the middle part of femoral head compared to its peripheral region), and orientation of trabeculae were the main reasons for the scatter in experimental results. The discrepancies between experimental and modeling results are mainly due to simplifying assumptions and selections made at different stages of modeling and a large variety of trabecular bone density values, even in the limit case of one femoral head studied here. The largest difference in the elastic modulus appeared between the two anatomical directions for UT samples. Our choice of sample orientations (Fig. 1) shows that the Bdirection is more optimized for physiological loading conditions, having an elastic modulus over twice that measured in the A-direction. Somewhat noticeable difference in the elastic modulus was found between the A- and B-directions for the DM and DP samples. This supports the concept that the orientation of minerals as well as collagen fibers contributes significantly to the bone stiffness, in agreement with [1,50]. 5. Conclusions

(b) Fig. 6. Comparison of the experimental and modeling results for elastic moduli of untreated (UT), demineralized (DM, magnified by 100× for clarity), and deproteinized (DP) trabecular bone in the (a) A-direction and (b) B-direction.

A multiscale theoretical model was extended to predict the elastic moduli of trabecular bone at different length scales. The model predictions at the trabecular bone level were verified by the experimentally measured elastic moduli using compression test of the bovine femoral trabecular bone samples loaded in two directions: parallel to the femoral neck axis and perpendicular to that. The model portrays trabecular bone as a porous composite material built up by proteins and hydroxyapatite minerals. In order to gain more insights on the structure and interaction of these main bone constituents, trabecular bone samples were either demineralized or deproteinized. The micro-computed tomography (µ-CT) images clearly illustrate that trabecular bone is a porous network of plate-like and rod-like elements and this structure is preserved upon demineralization or deproteinization process. Furthermore, the degree of anisotropy, D, calculated from

214

E. Hamed et al. / Materials Science and Engineering C 54 (2015) 207–216

analysis of μ-CT images revealed that trabecular bone is an anisotropic material with non-equiaxed trabecular cells. This finding motivated the use of an anisotropic cubic cell in the modeling procedure to better capture the actual structure of trabecular bone. The developed multiscale model using experimental inputs for values of porosity under statistical variance demonstrated that the elastic moduli of the untreated, demineralized and deproteinized bones fell between theoretical bounds. Our findings on the structure and interactions of collagen and hydroxyapatite crystals are significantly helpful for understanding the structure of human bone at the nanoscale. Our study reveals two important lessons about bone: (1) the constituent materials, collagen and hydroxyapatite, have nanometer dimensions and interpenetrate each other, and (2) the structure has a hierarchical organization spanning multiple length scales. The remarkable material properties of bone are mainly due to these two features. This knowledge can be employed to design novel interpenetrating nanocomposite materials using basic material building blocks, such as proteins and crystals, with applications as bone implants and synthetic bone substitutes. Acknowledgments The authors gratefully acknowledge support from the National Science Foundation, Ceramics Program Grant 1006931 and a MultiUniversity Research Initiative through the Air Force Office of Scientific Research (AFOSR-FA9550-15-1-0009) (JM) and the CMMI Program Grant 09-27909 (IJ). They thank Professor Marc A. Meyers for his enthusiastic support and Leilei Yin (Beckman Institute, UIUC) for the assistance in μ-CT scanning.

hydroxyapatite crystals with some intercrystalline pores filled with water and NCPs [5,7,77,78]. The self-consistent scheme with two interpenetrating phases, namely hydroxyapatite crystals and pores, was used to estimate the elastic constants of the extrafibrillar foam, CEHA, as  h i−1 h i−1  w −1 HA −1 CEHA ¼ Φw Cw : I þ SEHA : CIHA : ðCw −CEHA Þ þ ΦHA CHA : I þ SEHA : CEHA : ðCHA −CEHA Þ  h  i−1 h i−1 −1 w −1 HA −1 : Φw I þ SEHA : CEHA : ðCw −CEHA Þ þ ΦHA I þ SEHA : CEHA : ðCHA −CEHA Þ ; ðB:1Þ

where subscripts “EHA” and “HA” denote, respectively, extrafibrillar hydroxyapatite foam and hydroxyapatite crystals. The disorder of extrafibrillar crystals leads to isotropy of the homogenized material. Therefore, for the sake of simplicity, both phases were assumed to be spherical in shape, following [79]. Appendix C Modeling a single trabecula at the microscale follows the approach given in our previous study [2]. Three vectors V1, V2, and V3 are used to define a preferential orientation of the kth lamella. The vector V1 is obtained through the following equations ðkÞ

V1 ¼ φ

ðkÞ

 h i−1  w −1 Cwcol ¼ Ccol þ Φw ðCw −Ccol Þ : I þ Scol : Ccol : ðCw −Ccol Þ :  ðA:1Þ h i−1 −1 w −1 Φcol I þ Φw I þ Scol : Ccol : ðCw −Ccol Þ ; where subscripts “wcol”, “col”, and “w” refer, respectively, to wet collagen, dry collagen, and water and NCPs. The elastic properties of the wet collagen phase as well as water and NCPs are given in Table 1. Additionally, the interfibrillar hydroxyapatite crystals interacting with water can be represented as a porous foam made of hydroxyapatite crystals along with some intercrystalline voids, filled with water and NCPs. The self-consistent method was applied to predict the stiffness of an interfibrillar hydroxyapatite foam, CIHA, as follows  h i−1 h i−1  w −1 HA −1 CIHA ¼ Φw Cw : I þ SIHA : CIHA : ðCw −CIHA Þ þ ΦHA CHA : I þ SIHA : CIHA : ðCHA −CIHA Þ  h i−1 h i−1 −1 w −1 HA −1 þ ΦHA I þ SIHA : CIHA : ðCHA −CIHA Þ : : Φw I þ SIHA : CIHA : ðCw −CIHA Þ ðA:2Þ

In Eq. (A.2), subscripts “IHA” and “HA” denote, respectively, interfibrillar hydroxyapatite foam and hydroxyapatite crystals. Hydroxyapatite crystals were represented as ellipsoidal inclusions with an aspect ratio of 50:25:3 following [64], having isotropic elastic constants as listed in Table 1. Water-filled voids were assumed to be spherical.

ðkÞ

sin φ

¼ Rπ; θ

ðkÞ

cos θ

ðkÞ

sin φ

ðkÞ

sin θ

ðkÞ

cos φ

ðkÞ

E ;

¼ 2Rπ;

ðC:1Þ

where R is a random number, with values 0 ≤ R ≤ 1, generating a randomly-oriented pattern of lamellae. Given that vectors V2 and V3 are contingent on vector V1, they can be obtained as follows D

E ðkÞ ðkÞ ðkÞ X3 Y3 Z3 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi


 ffi; ðkÞ 2 ðkÞ 2 ðkÞ 2 þ Y3 þ Z3 X3

Appendix A To consider the interaction of collagen with bound water at the nanoscale, we assumed that cross-linked collagen molecules form a continuous matrix perforated by spherical intermolecular voids filled with water and NCPs. Using the Mori–Tanaka homogenization scheme, the elastic stiffness tensor of wet collagen, Cwcol, was obtained as

D

ðkÞ V3

ðkÞ

X 3 ¼ 2R−1;

ðkÞ Y3

¼ 2R−1; ðkÞ ðkÞ X sin φðkÞ cos θðkÞ þ Y 3 sin φðkÞ sin θðkÞ ðkÞ Z3 ¼ − 3 ; ðkÞ cos φ ðkÞ ðkÞ ðkÞ V2 ¼ V3  V1 :

ðC:2Þ

Now that the orientation of each single lamella is defined, a transformation matrix, Tij, is used to account for different fibril orientations in different lamellae [80] 2

2

m1 6 6 m22 h i 6 2 6 T i j ¼ 6 m3 6m m 6 2 3 4m m 3 1 m1 m2

2

n1 2 n2 2 n3 n2 n3 n3 n1 n1 n2

2

p1 2 p1 2 p3 p2 p3 p3 p1 p1 p2

2n1 p1 2n2 p2 2n3 p3 n2 p3 þ n3 p2 n3 p1 þ n1 p3 n1 p2 þ n2 p1

2p1 m1 2p2 m2 2p3 m3 p2 m3 þ p3 m2 p3 m1 þ p1 m3 p1 m2 þ p2 m1

3 2m1 n1 7 7 2m2 n2 7 7 2m3 n3 7; m2 n3 þ m3 n2 7 7 m3 n1 þ m1 n3 5 m1 n2 þ m2 n1

ðC:3Þ where mi, ni, and pi are direction cosines of axis i (i = 1, 2, 3), that is 2

m1 4 m2 m3

n1 n2 n3

3 2 ðkÞ 3 V1 p1 6 7 p2 5 ¼ 4 V ð2kÞ 5: ðkÞ p3 V

ðC:4Þ

3

After transformation, the stiffness tensor of the kth lamella, C(k), is obtained as ðkÞ

C

¼T

−1ðkÞ

ðkÞ

Clamella T ;

ðC:5Þ

Appendix B The dispersed and randomly oriented extrafibrillar hydroxyapatite crystals can be represented as a porous foam consisting of

where Clamella is the stiffness tensor of a single lamella as obtained in Eq. (3). These k lamellae are then stacked together, according to Sun and Li's formulation [74], to build a single strut. A nearly isotropic

E. Hamed et al. / Materials Science and Engineering C 54 (2015) 207–216

response will be obtained if a large number of randomly oriented lamellae are used.

References [1] M.J. Olszta, X.G. Cheng, S.S. Jee, R. Kumar, Y.Y. Kim, M.J. Kaufman, E.P. Douglas, L.B. Gower, Bone structure and formation: a new perspective, Mater. Sci. Eng. R 58 (2007) 77–116. [2] E. Hamed, I. Jasiuk, A. Yoo, Y. Lee, T. Liszka, Multi-scale modelling of elastic moduli of trabecular bone, J. R. Soc. Interface (2012). [3] J.D. Currey, Relationship between stiffness and mineral content of bone, J. Biomech. 2 (1969) 477–480. [4] J.L. Katz, Hard tissue as a composite material. I. Bounds on elastic behavior, J. Biomech. 4 (1971) 455–473. [5] C. Hellmich, J.F. Barthelemy, L. Dormieux, Mineral–collagen interactions in elasticity of bone ultrastructure — a continuum micromechanics approach, Eur. J. Mech. A 23 (2004) 783–810. [6] C. Hellmich, F.J. Ulm, Are mineralized tissues open crystal foams reinforced by crosslinked collagen? — some energy arguments, J. Biomech. 35 (2002) 1199–1212. [7] A. Fritsch, C. Hellmich, ‘Universal’ microstructural patterns in cortical and trabecular, extracellular and extravascular bone materials: micromechanics-based prediction of anisotropic elasticity, J. Theor. Biol. 244 (2007) 597–620. [8] B.H. Ji, H.J. Gao, Elastic properties of nanocomposite structure of bone, Compos. Sci. Technol. 66 (2006) 1212–1218. [9] S.P. Kotha, N. Guzelsu, The effects of interphase and bonding on the elastic modulus of bone: changes with age-related osteoporosis, Med. Eng. Phys. 22 (2000) 575–585. [10] T. Siegmund, M.R. Allen, D.B. Burr, Failure of mineralized collagen fibrils: modeling the role of collagen cross-linking, J. Biomech. 41 (2008) 1427–1435. [11] Q. Luo, R. Nakade, X. Dong, Q. Rong, X. Wang, Effect of mineral–collagen interfacila behavior on the microdamage progression in bone using probabilistic cohesive finite element model, J. Mech. Behav. Biomed. Mater. 4 (2011) 943–952. [12] E. Hamed, I. Jasiuk, Multiscale damage and strength of lamellar bone modeled by cohesive finite elements, J. Mech. Behav. Biomed. Mater. 28 (2013) 94–110. [13] R. Bhowmik, K.S. Katti, D.R. Katti, Mechanics of molecular collagen is influenced by hydroxyapatite in natural bone, J. Mater. Sci. 42 (2007) 8795–8803. [14] M.J. Buehler, Molecular nanomechanics of nascent bone: fibrillar toughening by mineralization, Nanotechnology 18 (2007). [15] D.K. Dubey, V. Tomar, Microstructure dependent dynamic fracture analyses of trabecular bone based on nascent bone atomistic simulations, Mech. Res. Commun. 35 (2008) 24–31. [16] E. Hamed, Y. Lee, I. Jasiuk, Multiscale modeling of elastic properties of cortical bone, Acta Mech. 213 (2010) 131–154. [17] Y.J. Yoon, S.C. Cowin, The estimated elastic constants for a single bone osteonal lamella, Biomech. Model. Mechanobiol. 7 (2008) 1–11. [18] I. Jasiuk, M. Ostoja-Starzewski, Modeling of bone at a single lamella level, Biomech. Model. Mechanobiol. 3 (2004) 67–74. [19] O. Brennan, O.D. Kennedy, T.C. Lee, S.M. Rackard, F.J. O'Brien, Biomechanical properties across trabeculae from the proximal femur of normal and ovariectomised sheep, J. Biomech. 42 (2009) 498–503. [20] Y. Chevalier, D.H. Pahr, H. Allmer, M. Charlebois, P.K. Zysset, Validation of a voxelbased FE method for prediction of the uniaxial apparent modulus of human trabecular bone using macroscopic mechanical tests and nanoindentation, J. Biomech. 40 (2007) 3333–3340. [21] N.M. Harrison, P.F. McDonnell, D.C. O'Mahoney, O.D. Kennedy, F.J. O'Brien, P.E. McHugh, Heterogeneous linear elastic trabecular bone modelling using micro-CT attenuation data and experimentally measured heterogeneous tissue properties, J. Biomech. 41 (2008) 2589–2596. [22] J.Y. Rho, T.Y. Tsui, G.M. Pharr, Elastic properties of human cortical and trabecular lamellar bone measured by nanoindentation, Biomaterials 18 (1997) 1325–1330. [23] P.K. Zysset, X.E. Guo, C.E. Hoffler, K.E. Moore, S.A. Goldstein, Elastic modulus and hardness of cortical and trabecular bone lamellae measured by nanoindentation in the human femur, J. Biomech. 32 (1999) 1005–1012. [24] C.H. Turner, J.Y. Rho, Y. Takano, T.Y. Tsui, G.M. Pharr, The elastic properties of trabecular and cortical bone tissues are similar: results from two microscopic measurement techniques, J. Biomech. 32 (1999) 437–441. [25] F. Bini, A. Marinozzi, F. Marinozzi, F. Patane, Microtensile measurements of single trabeculae stiffness in human femur, J. Biomech. 35 (2002) 1515–1519. [26] J.Y. Rho, R.B. Ashman, C.H. Turner, Young's modulus of trabecular and cortical bone material-ultrasonic and microtensile measurements, J. Biomech. 26 (1993) 111–119. [27] S.D. Ryan, J.L. Williams, Tensile testing of rodlike trabeculae excised from bovine femoral bone, J. Biomech. 22 (1989) 351–355. [28] K. Choi, J.L. Kuhn, M.J. Ciarelli, S.A. Goldstein, The elastic moduli of human subchondral, trabecular, and cortical bone tissue and the size-dependency of cortical bone modulus, J. Biomech. 23 (1990) 1103–1113. [29] K. Tanaka, Y. Kita, T. Katayama, M. Matsukawa, Mechanical properties of a single trabecula in bovine femur by the three point bending test, IFMBE Proceedings, LNCSE, 27, 2010, pp. 235–238. [30] C.S. Jorgensen, T. Kundu, Measurement of material elastic constants of trabecular bone: a micromechanical study using a 1 GHz acoustic microscope, J. Orthop. Res. 20 (2002) 151–158. [31] J.C. Rice, S.C. Cowin, J.A. Bowman, On the dependence of the elasticity and strength of cancellous bone on apparent density, J. Biomech. 21 (1988) 155–168.

215

[32] B. Van Rietbergen, H. Weinans, R. Huiskes, A. Odgaard, A new method to determine trabecular bone elastic properties and loading using micromechanical finiteelement models, J. Biomech. 28 (1995) 69–81. [33] L.J. Gibson, M.F. Ashby, The mechanics of 3-dimensional cellular materials, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 382 (1982) 43–59. [34] L.J. Gibson, M.F. Ashby, G.S. Schajer, C.I. Robertson, The mechanics of twodimensional cellular materials, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 382 (1982) 25–42. [35] K. Rajan, Linear elastic properties of trabecular bone — a cellular solid approach, J. Mater. Sci. Lett. 4 (1985) 609–611. [36] L.J. Gibson, The mechanical behavior of cancellous bone, J. Biomech. 18 (1985) 317–328. [37] L.J. Gibson, M.F. Ashby, Cellular Solids: Structure and Properties, 2nd ed. University Press, Cambridge, 1997. [38] L.J. Gibson, M.F. Ashby, B.A. Harley, Foam-like Materials in Nature, Cambridge University Press, 2010. [39] J. Kabel, B. Van Rietbergen, A. Odgaard, R. Huiskes, Constitutive relationships of fabric, density, and elastic properties in cancellous bone architecture, Bone 25 (1999) 481–486. [40] C.H. Turner, S.C. Cowin, J.Y. Rho, R.B. Ashman, J.C. Rice, The fabric dependence of the orthotropic elastic constants of cancellous bone, J. Biomech. 23 (1990) 549–561. [41] P.K. Zysset, A review of morphology–elasticity relationships in human trabecular bone: theories and experiments, J. Biomech. 36 (2003) 1469–1485. [42] R. Muller, P. Ruegsegger, Three-dimensional finite element modelling of noninvasively assessed trabecular bone structures, Med. Eng. Phys. 17 (1995) 126–133. [43] D. Ulrich, B. van Rietbergen, H. Weinans, P. Ruegsegger, Finite element analysis of trabecular bone structure: a comparison of image-based meshing techniques, J. Biomech. 31 (1998) 1187–1192. [44] B.C. Bourne, M.C.H. van der Meulen, Finite element models predict cancellous apparent modulus when tissue modulus is scaled from specimen CT-attenuation, J. Biomech. 37 (2004) 613–621. [45] C.A. Dobson, G. Sisias, R. Phillips, M.J. Fagan, C.M. Langton, Three dimensional stereolithography models of cancellous bone structures from mu CT data: testing and validation of finite element results, Proc. Inst. Mech. Eng. Part H J. Eng. Med. 220 (2006) 481–484. [46] H. Follet, F. Peyrin, E. Vidal-Salle, A. Bonnassie, C. Rumelhart, P.J. Meunier, Intrinsic mechanical properties of trabecular calcaneus determined by finite-element models using 3D synchrotron microtomography, J. Biomech. 40 (2007) 2174–2183. [47] D.H. Pahr, P.K. Zysset, Influence of boundary conditions on computed apparent elastic properties of cancellous bone, Biomech. Model. Mechanobiol. 7 (2008) 463–476. [48] E. Novitskaya, P.-Y. Chen, E. Hamed, J. Li, V.A. Lubarda, I. Jasiuk, J. McKittrick, Recent advances on the measurement and calculation of the elastic moduli of cortical and trabecular bone: a review, Theor. Appl. Mech. 38 (2011) 209–297. [49] E. Hamed, E. Novitskaya, J. Li, P.-Y. Chen, I. Jasiuk, J. McKittrick, Elastic moduli of untreated, demineralized, and deproteinized cortical bone: validation of a theoretical model of bone as an interpenetrating composite material, Acta Biomater. 8 (2012) 1080–1092. [50] E. Novitskaya, P.Y. Chen, S. Lee, A. Castro-Ceseña, G. Hirata, V.A. Lubarda, J. McKittrick, Anisotropy in the compressive mechanical properties of bovine cortical bone and the mineral and protein constituents, Acta Biomater. 7 (2011) 3170–3177. [51] P.-Y. Chen, D. Toroian, P.A. Price, J. McKittrick, Minerals form a continuum phase in mature cancellous bone, Calcif. Tissue Int. 88 (2011) 351–361. [52] E. Hamed, E. Novitskaya, J. Li, A. Setters, W. Lee, J. McKittrick, I. Jasiuk, Correlation of multiscale modeling and experimental results for the elastic modulus of trabecular bone, in: F. Barthelat, P. Zavattieri, C.S. Korach, B.C. Prorok, K.J. Grande-Allen (Eds.), Mechanics of Biological Systems and Materials, Springer, Heidelberg, 2013, pp. 59–65. [53] J.D. Currey, Bones: Structure and Mechanics, Cambridge University Press, 2002. [54] J.D. Currey, What determines the bending strength of bone? J. Exp. Biol. 202 (1999) 2495–2503. [55] P. Zioupos, J.D. Currey, A. Casinos, Exploring the effects of hypermineralisation in bone tissue by using an extreme biological example, Connect. Tissue Res. 41 (2000) 229–248. [56] G.H. van Lenthe, J.P.W. van den Bergh, A.R.M.M. Hermus, R. Huiskes, The prospects of estimating trabecular bone tissue properties from the combination of ultrasound, dual-energy X-ray absorptiometry, microcomputed tomography, and microfinite element analysis, J. Bone Miner. Res. 16 (2001) 550–555. [57] C.H. Turner, D.B. Burr, Experimental techniques for bone mechanics, in: S.C. Cowin (Ed.), Bone Mechanics Handbook, CRC Press, Boca Raton, FL, 2001 (p. 7–1–7–35). [58] F.E. Boas, D. Fleischmann, CT artifacts: causes and reduction techniques, Imaging Med. 4 (2) (2012) 229–240. [59] A. Basillais, S. Bensamoun, C. Chappard, B. Brunet-Imbault, G. Lemineur, B. Ilharreborde, M.C. Ho Ba Tho, C.L. Benhamou, Three-dimensional characterization of cortical bone microstructure by microcomputed tomography: validation with ultrasonic and microscopic measurements, J. Orthop. Sci. 12 (2007) 141–148. [60] E. Perilli, F. Baruffaldi, M. Visentin, B. Bordini, F. Traina, A. Cappello, M. Viceconti, MicroCT examination of human bone specimens: effects of polymethylmethacrylate embedding on structural parameters, J. Microsc. 225 (2007) 192–200. [61] J.D. Eshelby, The elastic field outside an ellipsoidal inclusion, Proc. R. Soc. Lond. A 252 (1959) 561–569. [62] M. Minary-Jolandan, M.F. Yu, Nanoscale characterization of isolated individual type I collagen fibrils: polarization and piezoelectricity, Nanotechnology 20 (2009). [63] F. Hang, A.H. Barber, Nano-mechanical properties of individual mineralized collagen fibrils from bone tissue, J. R. Soc. Interface 8 (2011) 500–505. [64] R. Robinson, An electron microscopic study of the crystalline inorganic component of bone and its relationship to the organic matrix, J. Bone Joint Surg. 344 (1952) 389–435.

216

E. Hamed et al. / Materials Science and Engineering C 54 (2015) 207–216

[65] B. Budiansky, On elastic moduli of some heterogeneous materials, J. Mech. Phys. Solids 13 (1965) 223–227. [66] R. Hill, Elastic properties of reinforced solids — some theoretical principles, Mech. Phys. Solids 11 (1963) 357–372. [67] E.P. Katz, S. Li, Structure and function of bone collagen fibrils, J. Mol. Biol. 80 (1973) 1–15. [68] K.S. Prostak, S. Lees, Visualization of crystal-matrix structure. In situ demineralization of mineralized turkey leg tendon and bone, Calcif. Tissue Int. 59 (1996) 474–479. [69] N. Sasaki, Y. Sudoh, X-ray pole figure analysis of apatite crystals and collagen molecules in bone, Calcif. Tissue Int. 60 (1997) 361–367. [70] N. Sasaki, A. Tagami, T. Goto, M. Taniguchi, M. Nakata, K. Hikichi, Atomic force microscopic studies on the structure of bovine femoral cortical bone at the collagen fibril-mineral level, J. Mater. Sci. Mater. Med. 13 (2002) 333–337. [71] F. Remaggi, V. Cane, C. Palumbo, M. Ferretti, Histomorphometric study on the osteocyte lacuno-canalicular network in animals of different species. I. Woven-fibered and parallel fibered bones, Ital. J. Anat. Embryol. 103 (1998) 145–155. [72] Y. Benveniste, A new approach to the application of Mori–Tanaka theory in composite materials, Mech. Mater. 6 (1987) 147–157. [73] T. Mori, K. Tanaka, Average stress in matrix and average elastic energy of materials with misfitting inclusions, Acta Metall. 21 (1973) 571–574. [74] C.T. Sun, S. Li, Three-dimensional effective elastic constants for thick laminates, J. Compos. Mater. 22 (1988) 629–639. [75] M.A. Rubin, I. Jasiuk, The TEM characterization of the lamellar structure of osteoporotic human trabecular bone, Micron 36 (2005) 653–664. [76] A.T. Huber, L.J. Gibson, Anisotropy of foams, J. Mater. Sci. 23 (1988) 3031–3040.

[77] A. Fritsch, C. Hellmich, L. Dormieux, Ductile sliding between mineral crystals followed by rupture of collagen crosslinks: experimentally supported micromechanical explanation of bone strength, J. Theor. Biol. 260 (2009) 230–252. [78] A. Fritsch, L. Dormieux, C. Hellmich, Porous polycrystals built up by uniformly and axisymmetrically oriented needles: homogenization of elastic properties, C.R. Mec. 334 (2006) 151–157. [79] C. Hellmich, F.J. Ulm, Micromechanical model for ultrastructural stiffness of mineralized tissues, J. Eng. Mech. 128 (2002) 898–908. [80] I.M. Daniel, O. Ishai, Engineering Mechanics of Composite Material, Oxford University Press, New Yorks, 2006. [81] State of South Wales DoEaT., Animal femora, http://lrrpublic.cli.det.nsw.edu.au/ lrrSecure/Sites/Web/gondwana/Animal_Fossils_of_Gondwana/lo/fossils_07/ applets/ph_femur_text.htm. [82] R.H. Hall, Variations with pH of the tensile properties of collagen fibres, J. Soc. Leather Trades Chem. 35 (1951) 195–210. [83] S. Nikolov, D. Raabe, Hierarchical modeling of the elastic properties of bone at submicron scales: the role of extrafibrillar mineralization, Biophys. J. 94 (2008) 4220–4232. [84] R.S. Gilmore, J.L. Katz, Elastic properties of apatites, J. Mater. Sci. 17 (1982) 1131–1141. [85] J.L. Katz, K. Ukraincik, On the anisotropic elastic properties of hydroxyapatite, J. Biomech. 4 (1971) 221–227. [86] R. Snyders, D. Music, D. Sigumonrong, B. Schelnberger, J. Jensen, J.M. Schneider, Experimental and ab initio study of the mechanical properties of hydroxyapatite, Appl. Phys. Lett. 90 (2007).