Multiscale modeling of human bone
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Abstract
A multiscale modeling technique was developed to predict mechanical properties of human bone, which utilizes the hierarchies of human bone in different length scales from nanoscale to macroscale. Bone has a unique structure displaying high stiffness with minimal weight. This is achieved through a hierarchy of complex geometries composed of three major materials: hydroxyapatite, collagen and water. The identifiable hierarchical structures of bone are hydroxyapatite, tropocollagen, fibril, fiber, lamellar layer, trabecular bone, cancellous bone and cortical bone. A helical spring model was used to represent the stiffness of collagen. A unit cell-based micromechanics model computed both the normal and shear stiffness of fibrils, fibers, and lamellar layers. A laminated composite model was applied to cortical and trabecular bone, while a simplified finite element model for the tetrakaidecahedral shape was used to evaluate cancellous bone. Modeling bone from nanoscale components to macroscale structures allows the influence of each structure to be assessed. It was found that the distribution of hydroxyapatite within the tropocollagen matrix at the fibril level influences the macroscale properties significantly. Additionally, the multiscale analysis model can vary any parameter of any hierarchical level to determine its effect on the bone property. With so little known about the detailed structure of nanoscale and microscale bone, a model encompassing the complete hierarchy of bone can be used to help validate assumptions or hypotheses about those structures.
Keywords
Multiscale model Biomaterial Bone1 Introduction
Biomaterials are living tissues that have developed through evolutionary processes. They are distinct for their complex hierarchies built by simple materials (Vaughan et al. 2012). There are a limited number of structural materials in the human body, but living organisms rely on composite hierarchical structures to achieve macroscale forms and functions (Cui et al. 2007). Biomaterials such as skin, ligaments, tendons, muscles and bones exhibit this hierarchical organization. This study focuses on the structure of bone.
Bone is the main structural component of the body. Unlike tendons and ligaments, bone structures support both tensile and compressive loads, as well as bending, torsional and shearing loads (Hench and Jones 2005). To understand and predict the structural properties of bone, a multiscale model is presented. This model begins at the nanoscale level and continues up the hierarchies to the macroscale level. The hierarchical structure of the bone is sketched in Fig. 1. To the authors’ best knowledge, no attempt has been undertaken to link the material properties in the nanoscale to those in the macroscale. Almost all previous studies investigated material properties in a single length scale, especially at the macroscale. Reviews on previous studies are discussed later when appropriate to be introduced.
The multiscale analysis model is presented in the following section, adhering to the hierarchical order as shown in Fig. 1. Then, the predicted results at each length scale model are compared to available data, if any, to validate the model in the subsequent section. Finally, conclusions are provided.
2 Multiscale analysis
2.1 Nanoscale model
Human bone consists of hydroxyapatite (HA), tropocollagen (TC), water and others. Among them, hydroxyapatite and tropocollagen are the major load-bearing components. Hydroxyapatite has a large surface area to volume ratio, allowing for rapid absorption and dissolution when ions are needed (Buschow et al. 2001; Cui et al. 2007). Furthermore, the HA structures are organized into small plates ranging from 1.5 to 5 nm thick (Rho et al. 1998). Although the size of HA crystals varies based on location, mineral density and time allowed for growth; the average crystal size is considered as 50 nm \(\times \) 25 nm \(\times \) 3 nm (Nikolaeva et al. 2007; Buehler 2006). The crystal lattice of HA has the hexagonal close packed structure. From the nanoindentation testing completed on single HA crystals, the stiffness in the [0001] direction was 150 GPa, while the stiffness in the [1010] direction was 143 GPa (Zamiri and De 2011).
To determine the quantitative values for each tropocollagen triple helix, a helical spring model is used. It is assumed that each repeating subunit of the polyproline helices is represented by Gly-Pro-Hyp. A helical spring is used to assess the stiffness of a single molecule of tropocollagen, because each left-handed polyproline helix of the TC molecule can be independently treated as a spring. Buckling is prevented through stabilizing hydrogen bonds, van der Waals attraction and the close packing of the TC molecules within collagen fibrils. The stiffness of a single TC molecule is equal to the stiffness of the combined alpha-1 type-1 and alpha-1 type-2 strands.
2.2 Microscale model
2.2.1 Fibril model
For many years, a linear fibril orientation has been the accepted model (Nikolaeva et al. 2007). This model was viewed as a long thin filament with alternating bands of mineral rich phase and mineral deficient phase. Although the linear model presents a viable solution, there has not been any evidence proving lateral growth is a purely linear process. Fibrils grow laterally in 4 nm steps by electrostatic attraction during the formation of the microscale fibril (Gibson 1994). A twisting crystalline structure was more recently considered for fibril model. The 67 nm periodicity, which determined the two-dimensional stacking, was assumed to direct the three-dimensional pattern. For this reason, the lateral periodicity is also 67 nm. A simplified twisted model is shown in Fig. 3b.
The twisting fibril takes into account a 67 nm periodicity in the 2 direction as well as the 3 direction. Additionally, because each repeated stack in the 2 direction contains the same 67 nm periodicity in the 3 direction, the subunit is shortened in the 3 direction. The repeated unit cell for the twisting fibril model is shown in Fig. 3b. The dimensions associated with the twisting model are \(a_1 = 50\) nm, \(a_2 = 17\) nm, \(b_1 = 25\) nm, \(b_2 = 75\) nm, \(c_1 = 3\) nm, and \(c_2 = 6\) nm. For both fibril models, subcell #1 in Fig. 2 is assigned HA properties; the remaining subcells are assigned TC properties.
2.2.2 Fiber model
The fibers found in the micro level are single bundles of fibrils or a combination of many bundles: each bundle is an organized and repeating arrangement of fibrils and HA. Accordingly, the unit cell micromechanics model is aptly suited for modeling bone fiber, as the fiber unit cell relies on volume fractions of each material.
Dimension for fiber unit cell with extrafibrillar mineralization
%EFM | \(a_1 \) | \(a_2 \) |
---|---|---|
50 | 0.50 | 0.50 |
70 | 0.70 | 0.30 |
90 | 0.90 | 0.10 |
95 | 0.95 | 0.05 |
2.3 Macroscale model
2.3.1 Lamellar bone
To model the lamellar layer as a continuous fiber, the material properties assigned to subcells #1, #3, #5 and #7 in Fig. 2 are mirrored for subcells #2, #4, #6 and #8. This produces a continuous fiber reinforced composite. The dimensionless values to be used for this model are \(a_1 = 0.5\), \(a_2 = 0.5\), \(b_1 = 0.62\), \(b_2 = 0.38\), \(c_1 = 0.62\), and \(c_2 = 0.38\).
The material properties of a single lamellar layer in a different fiber orientation can be found by rotating the layer with respect to the 3-axis. This rotation can be completed by rotating the stiffness matrix of the material (Gibson 1994).
2.3.2 Cortical bone
Cortical bone is composed of concentric layers of lamellar bone, known as osteons. Osteons can be made of primary or secondary bone. Secondary osteons are called Haversian systems, which are analyzed in this study. The concentric layers composing Haversian systems form a cylindrical structure approximately 200 \(\upmu \)m in diameter (Reznikov et al. 2014) as shown in Fig. 6. Each concentric layer associated with the osteon is composed of lamellar bone. The fibers of each individual layer are unidirectional in alignment. The center cylinder is representative of a Haversian canal and will be represented as un-bound water. Multiple Haversian systems are packed together in cortical bone. Due to their circular shape, there are incomplete layers at the interface of each Haversian system where the boundaries intersect. These boundaries are defined by a cement line, which is an identifiable region where osteon growth direction has transitioned. The properties of cement lines are similar to those of the surrounding bone, despite the misnomer of ‘cement’ (Skedros et al. 2005).
Preferential orientation of layered composite model
Layer | Fiber orientation (\(^{\circ }\)) | Volume fraction |
---|---|---|
Transverse | 72.5 | 0.45 |
Longitudinal | 22.5 | 0.35 |
Intermediate | 0 | 0.05 |
Intermediate | 40 | 0.05 |
Intermediate | 55 | 0.05 |
Intermediate | 90 | 0.05 |
2.3.3 Trabecular bone
Fiber orientation of trabecular bone layer model
Fiber direction | Volume fraction |
---|---|
\(-\) 10\(^{\circ }\) | 0.20 |
\(-\) 5\(^{\circ }\) | 0.20 |
0\(^{\circ }\) | 0.20 |
5\(^{\circ }\) | 0.20 |
10\(^{\circ }\) | 0.20 |
2.3.4 Cancellous bone
While a laminated composite material represents the material properties of a single trabecula, it does not quantify the properties of cancellous bone. The structure of cancellous bone is a uniquely three-dimensional problem, that cannot be solved through a two dimensional approximation (Odgaard 1997). Additionally, there is much heterogeneity in cancellous bone at different anatomical locations (Parkinson and Fazzalari 2013).
Early models of cancellous bone utilized a model of rods and plates. Four early models were an asymmetric rod-like cubic model, a plate-like cubic model, a rod-like hexagonal columnar model, and a plate-like hexagonal columnar model (Gibson 1985). These early models helped to shape simple models of cancellous bone. However, these models provided asymmetrical properties for cancellous bone. Additionally, the early models included Euler buckling as a failure mechanism. Later experiments have shown that failure of cancellous bone is most commonly due to microscopic cracking, which removes buckling as a failure mode for trabeculae (Fyhrie and Schaffler 1994).
Complex finite element models have been used to model cancellous bone. Three-dimensional models formed from micro-computed tomography can replicate small sections of bone (Kadir et al. 2010). Furthermore, two unit cells have been proposed that are able to accurately model cancellous bone. Kadir et al. (2010) compared the results of prismatic unit cells and tetrakaidecahedral unit cells to those of a micro-computed tomography model. The authors found that both unit cells accurately represent the mechanical properties. Additionally, Guo and Kim (2002) have shown that a complex finite element model of several tetrakaidecahedron cells can accurately represent different levels of bone loss due to aging.
The TKDH has eight hexagonal faces, six square faces, 36 edges and 24 vertices. For the open cell structures present in cancellous bone, the faces are treated as voids and the edges are treated as trabeculae. A simple finite element model was developed by (Kwon et al. 2003) to analyze the TKDH frame. Each ligament in the TKDH geometry is modeled as a 3-D beam which has the property of the trabecula. The details of the 3-D beam element are shown in (Kwon et al. 2003) and it is omitted here to save space.
3 Results and discussion
3.1 Nanoscale result
To compute the elastic modulus of the tropocollagen, the geometric and material properties as discussed in the previous section are used for Eq. (1). In addition, the length and diameter of the helix are considered as 300 and 0.7 nm, respectively, for Eq. (2). Then, a single helix has the elastic modulus of 1.18 GPa and TC has three helices with the modulus of 3.54 GPa. The experimental data in (Carter and Caler 1981; Zysset et al. 1999) showed 3.0 and 5.1 GPa. The present value agrees well with the data.
3.2 Microscale result
3.2.1 Fibril
Material properties of fibril model component
Material | \(E_L \) (GPa) | \(E_T \) (GPa) | \(\nu \) | G (GPa) |
---|---|---|---|---|
Hydroxyapatite | 150 | 143 | 0.23 | 59.7 |
Tropocollagen under compression | 3.43 | 3.43 | 0.35 | 1.27 |
Tropocollagen under tension | 3.26 | 3.26 | 0.35 | 1.21 |
In addition to using two different models to explore the crystal arrays within fibrils, the presence of water is taken into account with the calculation of material properties. The presence of water produces a bi-modulus composite material, as water is only considered in compression. While the volume percent water is much higher in tendons and ligaments, bone is known to have approximately 10–25% water. Some of this is thought be found within the nanoscale tropocollagen and hydroxyapatite. Water serves as a binding and stabilizing agent within the triple helix of tropocollagen. Additionally, small amounts of water are tightly bound within the HA crystal. However, the rest is assumed be held within the various hierarchies. For the purpose of the fibrillar model, each unit cell is assumed to have 8% volume of water. This is calculated by adjusting the stiffness of collagen to assume an 8% volume of water in compression and an 8% volume of void space in tension. Tropocollagen in compression and tension exhibits the different properties as shown in Table 4.
Predicted material property of fibril
Model | \(E_L \) (GPa) | \(E_T \) (GPa) | \(G_{TT} \) (GPa) | \(G_{LT} \) (GPa) | \(\nu _{TL} \) | \(\nu _{TT} \) |
---|---|---|---|---|---|---|
Linear | ||||||
Compression | 6.05 | 6.60 | 1.53 | 1.68 | 0.30 | 0.30 |
Tension | 5.76 | 6.30 | 1.46 | 1.59 | 0.30 | 0.30 |
Twisting | ||||||
Compression | 4.26 | 3.81 | 1.35 | 1.37 | 0.31 | 0.37 |
Tension | 4.05 | 3.62 | 1.29 | 1.30 | 0.31 | 0.37 |
Predicted fiber compressive material property
50% EFM | 70% EFM | 90% EFM | 95% EFM | |||||
---|---|---|---|---|---|---|---|---|
Linear | Twisting | Linear | Twisting | Linear | Twisting | Linear | Twisting | |
\(E_L \) (GPa) | 6.51 | 5.45 | 7.57 | 6.51 | 11.6 | 10.5 | 16.2 | 15.0 |
\(E_T \) (GPa) | 16.7 | 14.6 | 21.4 | 19.2 | 26.4 | 23.9 | 27.7 | 25.2 |
\(G_{TT} \) (GPa) | 1.05 | 0.93 | 1.47 | 1.30 | 1.89 | 1.67 | 2.00 | 1.77 |
\(G_{LT} \) (GPa) | 0.01 | 0.01 | 0.02 | 0.02 | 0.06 | 0.06 | 0.11 | 0.11 |
\(\nu _{TL} \) | 0.37 | 0.41 | 0.36 | 0.40 | 0.30 | 0.31 | 0.25 | 0.25 |
\(\nu _{TT} \) | 0.14 | 0.13 | 0.13 | 0.12 | 0.13 | 0.12 | 0.13 | 0.12 |
Estimated mineral volume percent of fiber model
%EFM | Mineral volume fraction (%) | |
---|---|---|
Linear | Twisting | |
50 | 25.9 | 18.4 |
70 | 31.5 | 23.9 |
90 | 37.0 | 29.5 |
95 | 38.4 | 30.9 |
Predicted lamellar compressive material property
50% EFM | 70% EFM | 90% EFM | 95% EFM | |||||
---|---|---|---|---|---|---|---|---|
Linear | Twisting | Linear | Twisting | Linear | Twisting | Linear | Twisting | |
\(E_L \) (GPa) | 6.43 | 4.61 | 6.84 | 5.02 | 8.39 | 6.55 | 10.1 | 8.27 |
\(E_T \) (GPa) | 9.00 | 6.43 | 9.72 | 6.93 | 10.4 | 7.41 | 10.6 | 7.58 |
\(G_{TT} \) (GPa) | 1.34 | 1.16 | 1.55 | 1.34 | 1.71 | 1.47 | 1.74 | 1.49 |
\(G_{LT} \) (GPa) | 0.62 | 0.53 | 0.630 | 0.54 | 0.67 | 0.57 | 0.72 | 0.62 |
\(\nu _{TL} \) | 0.33 | 0.37 | 0.32 | 0.35 | 0.28 | 0.29 | 0.24 | 0.24 |
\(\nu _{TT} \) | 0.24 | 0.27 | 0.24 | 0.27 | 0.24 | 0.29 | 0.25 | 0.30 |
3.2.2 Fiber
The results for the fiber model are calculated both in compression and tension using the geometric data. As expected, due to the applied symmetry, the fiber exhibits transverse isotropic material properties. Table 6 shows the results for compression.
The results in compression show that the linear fibril model produces a stiffer fiber in all normal and shear cases and in all directions. For all %EFM, the fiber shear modulus \(G_{LT} \) is much less than \(G_{TT} \). As %EFM increases from 50 to 95%, the shear modulus \(G_{LT} \) increases by an order of magnitude and \(G_{TT} \) increases approximately double. This is exhibited for both the linear and the twisting models. The Poisson’s ratio in the transverse plane remains relatively constant, but as %EFM increases, \(\nu _{TL} \) decreases. The results in tension show similar results to those in compression.
The difficulty with evaluating the results of the fiber model is that there is an absence of experimental testing available for comparison. Fibers exist within the macrostructures of bone and are not easily isolated for testing. Additionally, almost all theoretical calculations assume that macrostructure bone is composed solely of layered fibers. However, the macrostructures of bone are fiber reinforced composites with fibrils acting as the matrix and bone fibers as the fibers.
Estimated mineral volume percent of lamellar model
%EFM | Mineral volume fraction (%) | |
---|---|---|
Linear | Twisting | |
50 | 20.2 | 10.9 |
70 | 22.4 | 13.0 |
90 | 24.5 | 15.2 |
95 | 25.0 | 15.7 |
Predicted cortical bone material property under compression
50% EFM | 70% EFM | 90% EFM | 95% EFM | |||||
---|---|---|---|---|---|---|---|---|
Pref. | Smooth | Pref. | Smooth | Pref. | Smooth | Pref. | Smooth | |
\(E_L \) [GPa] | 5.07 | 4.64 | 5.39 | 4.91 | 5.94 | 5.45 | 6.42 | 5.92 |
\(E_T \) [GPa] | 5.84 | 5.68 | 6.22 | 6.05 | 6.82 | 6.59 | 7.28 | 6.99 |
\(G_{TT} \)[GPa] | 0.70 | 0.74 | 0.77 | 0.82 | 0.84 | 0.89 | 0.87 | 0.92 |
\(G_{LT} \) [GPa] | 0.94 | 0.96 | 1.04 | 1.05 | 1.14 | 1.16 | 1.21 | 1.23 |
\(\nu _{TL} \) | 0.32 | 0.36 | 0.32 | 0.36 | 0.31 | 0.35 | 0.31 | 0.34 |
\(\nu _{TT} \) | 0.23 | 0.23 | 0.23 | 0.22 | 0.22 | 0.22 | 0.22 | 0.22 |
Published cortical elastic modulus
Elastic modulus (GPa) | Method | Source |
---|---|---|
6 | FEM, 20% mineral volume fraction | Buehler (2006) |
9 | FEM, 30% mineral volume fraction | |
15 | FEM, 40% mineral volume fraction | |
21 | FEM, 50% mineral volume fraction |
3.3 Macroscale result
The results of the lamellar micromechanics model were calculated in both compression and tension for four different %EFM. The resulting stiffness and Poisson’s ratios are shown in Table 8 for compression. Both linear and twisting fibril models were used in the prediction. Through the bone hierarchies, the mineral content of the lamellar layers was calculated. The mineral volume fraction of each model is displayed in Table 9. The mineral volume fraction within lamellar layers is representative of macroscale bone mineral content and comparisons can be made to theoretical and experimental values. Early studies of fibril organization suggested total bone mineral volume content at 50% (Guo and Kim 2002). Calculation of the bone ash content of adult cows was found to be close to 70%. The volume percent of bone ash does not directly correlate to bone mineral content. This method results in a large estimate of mineral content, as additional residues remain from sources other than hydroxyapatite. More recent studies estimate a lower bone mineral volume. Kotha and Guzelsu (2007) postulate that mineral volume content is 40% for bone, while (Ashman and Rho 1988) approximates 33–43% hydroxyapatite mineral volume. These estimates are based on a more complete understanding of the hierarchical structure of bone, but are still estimates as bone mineral content varies with bone type, anatomical position, age, and gender.
When comparing the bone mineral content from this study to those found through experimental and theoretical methods, the linear crystal pattern emerges as the more viable model. The mineral volume fractions of the twisting model are too low to validate its use. Even with slight perturbations to the constraints applied to the fibril and fiber models, the mineral content of the twisting model does not match the current estimates.
For the subsequent analyses, only the linear fibril model is used. Additionally, the preferential and smooth layered models were calculated in both tension and compression. The compressive results are shown in Table 10. The results of the cortical model can be compared to results from (Buehler 2006) as shown in Table 11.
When compared based on the similar mineral volume content, the present model computes comparable moduli with the finite element results with 20% mineral volume percent as shown in Table 11. Increase in mineral content for different hierarchical models would increase the stiffness of the cortical bone as shown in Table 11.
Predicted trabecular bone property in compression and tension
50% EFM | 70% EFM | 90% EFM | 95% EFM | |||||
---|---|---|---|---|---|---|---|---|
Comp. | Tens. | Comp. | Tens. | Comp. | Tens. | Comp. | Tens. | |
\(E_L \) (GPa) | 6.05 | 5.10 | 6.43 | 5.16 | 7.85 | 5.23 | 9.45 | 5.25 |
\(E_T \) (GPa) | 8.52 | 7.76 | 9.19 | 8.57 | 9.79 | 9.13 | 9.98 | 9.25 |
\(G_{TT} \) (GPa) | 1.26 | 1.20 | 1.46 | 1.39 | 1.61 | 1.53 | 1.64 | 1.56 |
\(G_{LT} \) (GPa) | 0.66 | 0.62 | 0.67 | 0.64 | 0.72 | 0.67 | 0.77 | 0.72 |
\(\nu _{TL} \) | 0.24 | 0.25 | 0.24 | 0.26 | 0.24 | 0.32 | 0.24 | 0.38 |
\(\nu _{TT} \) | 0.23 | 0.25 | 0.23 | 0.24 | 0.24 | 0.24 | 0.25 | 0.24 |
Published values for trabecular property
Predicted cancellous elastic modulus (all expressed in MPa)
50% EFM | 70% EFM | 90% EFM | 95% EFM | |||||
---|---|---|---|---|---|---|---|---|
Comp. | Tens. | Comp. | Tens. | Comp. | Tens. | Comp. | Tens. | |
Femoral neck | 292 | 246 | 311 | 249 | 379 | 252 | 457 | 253 |
Femoral head | 196 | 165 | 209 | 167 | 255 | 170 | 307 | 170 |
Distal/proximal | 300 | 253 | 319 | 256 | 390 | 259 | 469 | 260 |
4 Conclusions
The structures of biomaterials are highly dependent on complex geometries. This study has shown that material properties of hierarchical structures can be found by analyzing each level independently. By linking all hierarchies and adjusting parameters, the influence of each level can then be analyzed. Additionally, changes to the geometry at each level can be completed to test assumptions about the structure of bone.
The nanoscale constituents of bone were described, and a spring model was utilized to calculate the longitudinal stiffness of tropocollagen. This simple model produced accurate results. Additionally, a micromechanical unit cell model was used to analyze the microscale components of bone. Bone fibrils represent a particle reinforced matrix, while bone fibers are a fiber reinforced composite. The fibrillar model produced accurate results as compared to accepted values. The fiber model could not be compared to experimental results, so different variations were carried over to the next hierarchy.
The first macroscale structures of bone are the lamellar layers. These were modeled as a fiber reinforced composite and compared to accepted values of mineral fraction volume. The results disproved the twisting hydroxyapatite fibrillar model. The linear fibril model was utilized for both cortical and trabecular bone. These macroscale structures utilized a layered composite model to calculate their transverse isotropic material properties.
A simple finite element model of a tetrakaidecahedron was used to model cancellous bone. From experimental testing, the volume fractions of bone and trabecular thicknesses were defined for different anatomical locations. The model was shown to accurately predict the properties of cancellous bone.
This model can be used to validate future discoveries about the structure of bone. As technology advances, imaging capabilities will allow the nanostructures of bone to be explored in detail. The discoveries can be checked against this model to assess their impact on macroscale properties. Additionally, the properties of synthetic bone materials can be checked against the hierarchical structure of bone. This would allow for more anatomically beneficial bone grafts.
Notes
Compliance with ethical standards
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
References
- Ashman RB, Rho JY (1988) Elastic modulus of trabecular bone material. J Biomech 21(3):177–181CrossRefGoogle Scholar
- Buehler MJ (2006) Atomistic and continuum modeling of mechanical properties of collagen: elasticity, fracture, and self-assembly. J Mater Res 21(8):1947–1961CrossRefGoogle Scholar
- Buschow KH, Cahn J, Flemings RW, Ilschner MC, Kramer B, Mahajan EJ (2001) Bone mineralization. In: Encyclopedia of materials science and technology, pp 787–794Google Scholar
- Carter DR, Caler WE (1981) Uniaxial fatigue of human cortical bone. The influence of tissue physical characteristics. J Biomech 14(7):461–470CrossRefGoogle Scholar
- Cowin SC, Mehrabadi MM (1989) Identification of the elastic symmetry of bone and other materials. J Biomech 22(6–7):503–515CrossRefGoogle Scholar
- Cui FZ, Li Y, Ge J (2007) Self-assembly of mineralized collagen composites. Mater Sci Eng 57:1–27CrossRefGoogle Scholar
- Ding M, Hvid I (2000) Quantification of age-related changes in the structure model type and trabecular thickness of human tibial cancellous bone. Bone 26(3):291–295CrossRefGoogle Scholar
- Fyhrie DP, Schaffler MB (1994) Failure mechanisms in human vertebral cancellous bone. Bone 15(1):105–109CrossRefGoogle Scholar
- Gautieri A, Vesentini S, Redaelli A, Ballarini R (2013) Modeling and measuring visco-elastic properties: From collagen molecules to collagen fibrils. Int J Non-Linear Mech 56:25–33CrossRefGoogle Scholar
- Gibson LJ (1985) The mechanical behavior of cancellous bone. J Biomech 18(5):317–328CrossRefGoogle Scholar
- Gibson RF (1994) Principles of composite material mechanics. McGraw-Hill Inc., New YorkGoogle Scholar
- Gong H, Zhu D, Gao J, Linwei L, Zhang X (2010) Ad adaptation model for trabecular bone at different mechanical levels. BioMed Eng OnLine 9(32):1–17Google Scholar
- Guo XE, Kim CH (2002) Mechanical consequence of trabecular bone loss and its treatment: a three-dimensional model simulation. Bone 30(2):404–411CrossRefGoogle Scholar
- Hamed E, Jasiuk I (2013) Multiscale damage and strength of lamellar bone modeled by cohesive finite elements. J Mech Behav Biomed Mater 28:94–110CrossRefGoogle Scholar
- Harley R, James D, Miller A, White JW (1997) Phonons and the elastic moduli of collagen and muscle. Nature 267:285–287CrossRefGoogle Scholar
- Hench LL, Jones JR (2005) Clinical needs and concepts of repair, biomaterials. In: Hench LL, Jones JR (eds) Artificial organs and tissue eng. Sawston, UK, Woodhead, pp 79–89Google Scholar
- Kadir MR, Syahrom A, Öchsner A (2010) Finite element analysis of idealised unit cell cancellous structure based on morphological indices of cancellous bone. Med Biol Eng Comput 48:497–505CrossRefGoogle Scholar
- Kotha SP, Guzelsu N (2007) Tensile behavior of cortical bone: dependence of organic matrix material properties on bone mineral content. J Biomech 40:36–45CrossRefGoogle Scholar
- Krug R, Carballido-Gamio J, Burhardt AJ, Kazakia G, Hyun BH, Jobke B, Banerjee S, Huber M, Link TM TM, Majumdar S (2008) Assessment of trabecular bone structure comparing magnetic resonance imaging at 3 Tesla with high-resolution peripheral quantitative computed tomography ex vivo and in vivo. Osteoporos Int 19(5):653–661CrossRefGoogle Scholar
- Kwon YW, Cooke RE, Park C (2003) Representative unit-cell models for open-cell metal foams with or without elastic fillers. Mater Sci Eng A 343:63–70CrossRefGoogle Scholar
- Kwon YW, Darcy J (2018) Failure criteria for fibrous composites based on multiscale model. Multiscale Multidiscip Model Exp Des 1(1):3–17CrossRefGoogle Scholar
- Kwon YW, Kim C (1998) Micromechanical model for thermal analysis of particulate and fibrous composites. J Therm Stress 21:21–39CrossRefGoogle Scholar
- Kwon YW, Park MS (2013) Versatile micromechanics model for multiscale analysis of composite structures. Appl Compos Mater 20(4):673–692CrossRefGoogle Scholar
- Nikolaeva TI, Tiktopulo EI, Il’yasova EN, Kuznetsova SM (2007) Collagen type I fibril packing in vivo and in vitro. Biophysics 52(5):489–497CrossRefGoogle Scholar
- Majumar S, Genant HK, Grampp S, Newitt DC, Truong VH, Lin JC, Mathur A (1997) Correlation of trabecular bone structure with age, bone mineral density, and osteoporotic status: in vivo studies in the distal radius using high resolution magnetic resonance imaging. J Bone Miner Res 12(1):111–118CrossRefGoogle Scholar
- Odgaard A (1997) Three-dimensional methods for quantification of cancellous bone architecture. Bone 20(4):315–328CrossRefGoogle Scholar
- Park MS, Kwon YW (2013) Elastoplastic micromechanics model for multiscale analysis of metal matrix composite structures. Comput Struct 123:28–38CrossRefGoogle Scholar
- Parkinson IH, Fazzalari NL (2013) Characterisation of trabecular bone, studies in mechanobiology, tissue engineering and biomaterials, vol 5, pp 31–51Google Scholar
- Reznikov N, Shahar R, Weiner S (2014) Bone hierarchical structure in three dimensions. Acta Biomater 10:3815–3826CrossRefGoogle Scholar
- Rho JY, Kuhn-Spearing L, Zioupos P (1998) Mechanical properties and the hierarchical structure of bone. Med Eng Phys 20:92–102CrossRefGoogle Scholar
- Ricard-Blum S (2011) The collagen family. Cold Spring Harb Perspect Biol 3(1):a004978CrossRefGoogle Scholar
- Skedros JG, Holmes JL, Vajda EG, Bloebaum RD (2005) Cement lines of secondary osteons in human bone are not mineral-deficient: New data in a historical perspective. Anat Rec Part 286A(1):781–803CrossRefGoogle Scholar
- Tzaphlidou M, Berillis P (2005) Collagen fibril diameter in relation to bone site. A quantitative ultrastructural study. Micron 36:703–705CrossRefGoogle Scholar
- Vaughan TJ, McCarthy CT, McNamara LM (2012) A three-scale finite element investigation into the effects of tissue mineralisation and lamellar organisation in human cortical and trabecular bone. J Mech Behav Biomed Mater 12:50–62CrossRefGoogle Scholar
- Yoon HS, Katz JL (1976) Ultrasonic wave propagation in human cortical bone-II. Measurements of elastic properties and microhardness. J Biomech 9(7):459–464CrossRefGoogle Scholar
- Zamiri A, De S (2011) Mechanical properties of hydroxapatite single crystals from nanoindentation data. J Mech Behav Biomed Mater 2(4):146–152CrossRefGoogle Scholar
- Zysset PK, Guo XE, Hoffler CE, Moore KE, Goldstein SA (1999) Elastic modulus and hardness of cortical and trabecular bone lamellae measured by nanoindentation in the human femur. J Biomech 32(10):1005–1012CrossRefGoogle Scholar