Bone mechanics

HomeMUBone mechanics

In this project we model the elastic properties of bone at the level of mineralized collagen fibrils via step-by-step homogenization from the staggered arrangement of collagen molecules up to an array of parallel mineralized fibrils. A new model for extrafibrillar mineralization is proposed, assuming that the extrafibrillar minerals are mechanically equivalent to reinforcing rings coating each individual fibril. Our modeling suggests that no more than 30% of the total mineral content is extrafibrillar and the fraction of extrafibrillar minerals grows linearly with the degree of mineralization. It is shown that the extrafibrillar mineralization considerably reinforces the fibrils properties in the transverse directions and the fibrils shear moduli. The model predictions for the elastic moduli and constants are found to be in a good agreement with the experimental data reported in the literature.

Keywords: modeling, bone, mechanical properties, structure, biominerals

The remarkable mechanical properties of bone related to its low density are essentially due to the bone’s complex, hierarchical microstructure from the macro- down to the nanoscale (Figure 1). At length scales below several microns, the diversity of bone tissues is reduced to different arrangements of mineralized collagen fibrils (Level 4 in Figure 1) formed through self-assembly of soft collagen molecules and hard mineral nanoparticles. At this level of organization, the effective mechanical properties of bone depend on the properties of the fibrils’ constituents, the fibrils’ microstructure and orientation distribution as well as on the mineral content and the shape of the mineral particles. Clearly, the development of reliable structure-properties relations for mineralized collagen fibrils and for fibril arrays incorporating these dependencies is of crucial importance, not only for the evaluation of the mechanical properties of bone, but also for better understanding of how the governing “material design” principles, growth processes or diseased states influence the mechanical properties at submicron length scales. Such relations can also help answering some still unclear questions about the bone structure and serve as a tool for designing new bone implants and bio-inspired nanocomposites.

At the nanoscale (Levels 1 and 2 in Figure 1), bone consists of: i) collagen type I molecules (triple helices  ~ 300 nm long and ~ 1.4 nm in diameter), self-assembled in a staggered fashion to form collagen fibrils with diameter of the order of ~ 100 nm (Weiner and Wagner, 1998); ii) biological hydroxyapatite (HA) minerals with hexagonal unit cell (Weiner and Wagner, 1998); iii) water, part of which provides hydrogen (H) bonds for both the collagen molecules and the collagen-mineral composite; and iv) a relatively limited amount of noncollagenous proteins (NCP) like the extrafibrillar proteins that glue together adjacent collagen fibrils (Fantner et al., 2005).

 

It is now well established that the HA crystals inside the collagen fibrils grow primarily in the gaps between subsequent collagen molecules (Fratzl et al., 2004) and are shaped as platelets with typical average dimensions on the order of 50 x 25 x 3 nm (Weiner and Wagner, 1998), although their length can vary from 15 to 150 nm, their width from 10 to 80 nm and their thickness from 2 to 5 nm (Rubin et al., 2003). The longest dimension of the HA platelets is along the fibril axis. The form, the location and the relative fraction of extrafibrillar minerals is less well understood and is still a matter of debate. Recent study by atomic force microscopy (AFM), (Sasaki et al., 2002), confirmed the existence of mineral-containing “blobs” on the fibrils surface. Hassenkam et al., 2004, Hansma et al., 2005, Kindt et al., 2007, obtained a more detailed picture of the extrafibrillar minerals via High-Resolution AFM imaging. They observed that each collagen fibril is individually coated with extrafibrillar HA minerals with various shapes and sizes and part of these mineral formations are arranged with a period of 67 nm, the same as that of the underlying microstructure of the “naked” collagen fibrils.

 

In the past two decades, mechanical models for mineralized collagen fibrils have been developed by Sasaki et al., 1991, Wagner and Weiner, 1992, Currey et al., 1994, Akiva et al., 1998, Jäger and Fratzl, 2000, Akkus, 2005, Fritsch and Hellmich, 2007. The cooperative collagen-mineral deformation within a single fibril has been studied by Jäger and Fratzl, 2000 in the framework of their shear-lag model with staggered mineral platelets. Akiva et al., 1998, modeled the 3D orthotropic elastic properties of a single collagen fibril as well as of lamellar and fibrolamellar bone tissues taking into account the mineral content and the shape of the HA platelets in the fibrils. Akkus, 2005, and Fritsch and Hellmich, 2007, used continuum micromechanics homogenization to estimate the mechanical properties of mineralized collagen fibrils and bone tissues. Very recently, Molecular Dynamics (MD) simulations have emerged as a promising tool for investigation of the mechanical properties and the deformation of bone and its constituents at the nanoscale. For example, the properties of single collagen molecules have been assessed by Vesentini et al., 2005 while Bhowmik et al., 2007, investigated how the presence of HA crystals influences the mechanical behavior of the collagen molecules. Broedling et al., 2007, showed that the strength and the toughness of platelet-reinforced nanocomposites with bone-like microstructure depend on the size and the arrangement pattern of the platelets. Unfortunately, realistic large-scale MD simulations of mineralized collagen fibrils including the molecular structure of collagen, water and HA crystals require enormous computational power and are still not affordable at present. For this reason, the modeling method used here is based on continuum micromechanics.

 

In this work we model the 3D elastic constants of a single mineralized collagen fibril and of a bundle of fibrils with particular attention to the extrafibrillar mineralization and its influence on the mechanical properties of bone. Different homogenization methods are employed to first find the effective properties at a lower level of hierarchy, starting with the effective properties of the collagen-water composite inside the mineralized collagen fibrils, and then incorporate the obtained results in the modeling of the next higher hierarchy level.

 

Here we propose a new model for extrafibrillar mineralization considering that each individual collagen fibril is reinforced with HA coating “rings” strongly adhering to the fibrils’ surface. Different mineralization scenarios are tested to establish how the fraction of extrafibrillar minerals evolves with the overall mineralization.

 

Another improvement in our modeling with respect to earlier models is the incorporation of the experimentally observed shape of the HA platelets in a genuine 3D formulation. In some earlier works (Akiva et al., 1998, e.g.), the 3D elastic properties of mineralized collagen fibrils were constructed using different one-dimensional models for the Young- and the shear moduli in different directions. Other authors proposed more advanced 3D formulations based on continuum micromechanics but used less realistic shape assumptions for the HA minerals (needle-like, Akkus, 2005, or spherical, Fritsch and Hellmich, 2007). In the present approach we compute the so-called Eshelby’s tensor (Eshelby, 1957), accounting for the shape of the inclusions in a particulate composite, through numerical integration[*] using the viscoplastic self consistent code (VPSC6) code developed by Lebensohn and Tomé, 2003. This allows us to solve a 3D equivalent inclusion problem for particles shaped as general ellipsoids embedded in a stiffness matrix with arbitrary anisotropy and thus provide better estimates for the orthotropic elastic properties of mineralized collagen fibrils and fibril arrays.

In this Section, we consider in detail a hierarchical modeling approach consisting of consecutive homogenization steps. In terms of Figure 1, we find the effective elastic properties of bone at each of the hierarchy levels, from Level 1 up to Level 4, in a bottom-up order. For different hierarchy levels, we employ different continuum micromechanics methods in order to model the specific microstructure in a realistic but still reasonably simple way.

 

2.1 Effective properties of the collagen-water composite

 

 

Within the mineralized collagen fibrils, the collagen molecules are arranged in a staggered fashion (Level 1 in Figure 1) with axial period of 67 nm consisting of overlaps (~27 nm) for neighboring molecules and axial gaps (~40 nm) between two successive molecules (Hodge and Petruska, 1963). The molecular packing is quasi hexagonal (Miller, 1984), the intermolecular spaces are filled with water and contain a small amount of noncollagenous proteins such as proteoglicans (Miller, 1984).

 

Because the transverse properties of isolated collagen molecules are not known, we model the collagen triple helices as a hexagonal array of perfectly aligned long isotropic cylindrical fibers in an isotropic water-NCP matrix, the elasticity of which is due to the hydrogen bonds linking the collagen molecules as well as to crosslinks provided by the noncollagenous proteins. Accurate estimates for the effective properties of long-fiber composites with arbitrary volume fraction of the fibers and arbitrary contrast between the mechanical properties of the phases have been developed by Torquato, 1998, based on earlier works by Silnutzer, 1972, and Milton, 1981.

 

Let the indices 1 and 2 denote the matrix (water) and the dispersed phase (collagen “fibers”), respectively. Then, the effective 2D shear- and bulk moduli,  and , in a plane perpendicular to the orientation of the collagen molecules are given by (Torquato, 1998):

                                         (1)

where

                                                                                  (2)

with  and  being the shear moduli and ,  the plane-strain bulk moduli of water and collagen in the cross-section plane, respectively;  and  represent the volume fractions of water and collagen molecules, respectively. The scalar parameters  and  are defined by three-fold integrals depending on the cross-section’s microstructure – in our case a hexagonal array of identical circles – and on the volume fractions of the phases. For hexagonal arrays of cylinders, their values have been numerically computed and tabulated for different volume fractions by McPhedran and Milton, 1981 for  and Eischen and Torquato, 1993, for .

 

To construct the transversely isotropic stiffness tensor of the collagen-water composite, we need to find three additional constants for the properties along the collagen molecules, namely the longitudinal Young modulus, , the longitudinal 3D Poisson’s ratio, , and the shear modulus, . To estimate  and , we use the Hill’s lower bounds for long-fiber composites (Hill, 1964), which for  read:

               (3)

where  and  are the conventional Young moduli and ,  stand for the 3D Poisson’s ratios of the water and the collagen molecules, respectively. The shear modulus along the collagen molecules, , is independent on the other elastic constants but can be evaluated via numerical simulations, which for a water-collagen composite give . With the obtained values for the elastic constants, one can construct the elastic stiffness tensor of the homogenized collagen-water composite .

 

2.2 Collagen fibril reinforced with intrafibrillar HA platelets

 

Because the Young modulus of HA significantly exceeds that of the collagen-water composite discussed above, the homogenized elastic properties of a single collagen fibril reinforced with aligned HA mineral platelets should strongly depend on the shape and the volume fraction of the platelets. Jäger and Fratzl, 2000, argued that the upper bound for the mineral volume fraction within the fibrils, , must be , with a most likely value in fully mineralized bone . On the other hand, in small deformations, the very soft collagen matrix would drastically reduce the interactions between neighboring mineral platelets. Therefore, to find the effective properties of a mineralized collagen fibril, one can employ the Mori-Tanaka homogenization scheme for two-phase composites reinforced with non-interacting, aligned ellipsoidal inclusions (Mori and Tanaka, 1973). The Representative Volume Element (RVE) of a mineralized fibril in our model is shown in Figure 2.

The Mori-Tanaka model for the overall properties of the fibril can be written as follows (Benveniste, 1987):

                                                                (4)

where

                                                                                       (5)

 

is the so-called strain concentration factor for an isolated single ellipsoidal inclusion in an infinite elastic matrix; ,  and  denote the stiffness tensors of the homogenized fibril, the collagen-water composite and the HA mineral, respectively;  is the fourth-order identity tensor,  is the fourth-order Eshelby’s tensor depending only on the shape of the HA inclusions and on the elastic constants of the collagen matrix;  and  are the volume fractions of the collagen-water matrix and the mineral platelets, respectively. The tensor product contracted over two indices is denoted by (:) while the inverse of a matrix (fourth-order tensors are represented in matrix notation because of symmetries) is indicated by .

 

2.3 Mineralized collagen fibril reinforced with extrafibrillar minerals

 

As already mentioned in the Introduction, the most recent experimental studies on the extrafibrillar mineralization in bone (Sasaki et al., 2002, Hassenkam et al., 2004, Kindt et al., 2007) found that i) each collagen fibril is individually coated with HA mineral particles of different sizes and shapes ii) the HA minerals strongly adhere to the fibrils surface and iii) the periodicity in the arrangement of part of the extrafibrillar HA shells along the fibrils is about the same as the D-period (COMMENT: explain what is D-period) (~67 nm) of the underlying “naked” collagen fibrils. On the other hand, to explain the observed decrease in the lateral spacing of the collagen molecules with the increase in the mineral content, Lees, 2003, suggested that the HA mineral is initially deposited in the extrafibrillar space and prevents the collagen fibrils from merging. These findings are also supported by the work of Zhang et al., 2003, who could experimentally mimic the formation of bundles of parallel mineralized collagen fibrils through self-assemlby and obtained cylindrical fibrils of pure collagen and water individually coated with crystalline HA shells with their c-axes oriented along the fibril axes.

The observation that part of the extrafibrillar HA shells have the same periodicity as the D-period of the underlying collagen fibril can be explained by considering that the 40 nm gaps between consequent collagen molecules are also present at the fibrils’ surface and follow the staggered patterns shown in Figure 3 b. According to Piaz and Miller, 1974, a single collagen fibril is composed itself by subfibrils arranged in a subsequent (Figure 3b top) or alternate (Figure 3b bottom) staggered pattern, that must also be present at the fibril surface. It is reasonable to assume that during the mineralization process, these gaps, arranged in a periodic staggered fashion on the surface as shown in Figure 3b, are initially filled with HA and with increase in the overall mineralization, the extrafibrillar minerals located in the gaps would first thicken above the level of the fibril surface and then start growing along the fibrils axis, thus forming a kind of reinforcing “rings” of HA around the fibril. As a result of this process and the deposition of additional extrafibrillar shells on the surface, the fibril would become considerably stiffer in directions perpendicular to the fibril axis while the increase in its bending rigidity would be less pronounced, thus preserving the fibril’s flexibility.

 

With the above considerations in mind, we can find the mechanical properties of a collagen fibril coated with extrafibrillar minerals as follows. We first find the overall properties of a fibril that is fully coated with a HA layer having uniform thickness. As the HA coating is much stiffer and harder than the core collagen fibril, the effective mechanical properties of the coated fibril will be dominated by the HA properties as long as the coating is sufficiently thick. For simplicity, we can apply the Mori-Tanaka method for a composite consisting of a single inclusion (the collagen fibril) embedded in a HA matrix:

                                                              (6)

with strain concentration factor

                                                                                         (7)                              

where  is the stiffness tensor of the coated fibril,  is the Eshelby tensor for the needle-like fibril;  and  are the volume fractions of the fibril and the coating, respectively. The remaining quantities were defined in Section 2.2.

 

The actual effective properties of fibrils with extrafibrillar minerals must be within the upper bound for a fully coated fibril, , and the lower bound for a “naked” fibril, . In terms of our modeling, it is clear that the effective properties of a partially coated fibril will be closer to the upper bound in certain directions, like tension/compression in directions normal to the fibril’s surface, and closer to the lower bound in others. In fact, for the extrafibrillar reinforcements shown in Figure 3b, only two deformation modes would be not stiffened by the extrafibrillar mineralization, namely, tension/compression along the fibril axis (axis 1) and shear in the 12 plane  in direction perpendicular to the sheet (the local frame is identical to that shown in Figure 4). The 13 shear must be a stiff deformation mode because of the additional resistance to shear in this plane due to the specific staggered geometry of the extrafibrillar and intrafibrillar minerals (Figure 3b).

 

Let  be the fraction of the fibril surface coated with extrafibrillar minerals. If all deformation modes of the fibril were equally strengthened by the extrafibrillar mineralization, a simple Voigt-like estimate for the effective elastic constants of the mineralized fibril, , gives:

                                                                                            (8)

To account for the non-reinforced modes, from the stiffness tensors  and  we first compute the Young moduli along the fibril axis,  and , as well as the shear moduli,  and , in the 12 plane corresponding to coated and “naked” fibril, respectively. Then we obtain the effective Young modulus  and shear modulus  in the soft deformation modes using asymptotic homogenization method (Andrianov et al, 2002), which in our case gives the lower Hashin-Strikman bound for the mechanical properties, known to be more realistic than the simple Reuss estimate. The effective moduli and  are estimated as follows:

                                             (9)

                                            (10)

Then one can immediately substitute  in the stiffness matrix  while to replace  it is necessary to first find the compliance matrix , replace the old value for  with the new estimate and invert the updated compliance matrix again to obtain the corrected stiffness . This approach is simple and gives a satisfactory estimate of the properties of the mineralized collagen fibrils decorated with extrafibrillar minerals, introducing only insignificant error for some of the Poissons ratios.

 

To be able to use the above described model, we need an additional formula for the extrabibrillar mineral content. By definition, the total mineral volume fraction is  with  and  being the total volume of the mineral and the tissue volume, respectively. On the other hand,  can be expressed as  with the volume, , and the volume fraction, , of the mineralized fibrils. Then, the total mineral volume can be expressed as . Introducing an equivalent thickness  for the HA coating rings (Figures 3b), we can write the volume of the extrafibrillar minerals as

 

                                                         (11)

 

where  and are the radius and the length of the fibril and  is the fraction of the coated fibril surface.

 

Taking the volume of the fibril as  and using , we can write the volume fraction of the extrafibrillar minerals to the total mineral volume as

 

                                                                                 (12)

The volume fraction of the minerals within the fibrils, , referred to the fibril volume, , can be found starting with the basic expression

 

                                                                                                         (13)

where  is the volume of the intrafibrillar minerals. Using  and Eq. (16), after some manipulation we obtain

                                                                                                               (14)

Equation (14) establishes a relationship between extrafibrillar and intrafibrillar mineralization.

2.4 Effective properties of a bundle of aligned mineralized collagen fibrils

 

We now consider the overall properties of a bundle of parallel, closely packed mineralized collagen fibrils (Figure 4, left). The HA platelets within all the fibrils are assumed to have the same orientation and shape and the adjacent fibrils are held together by glue-like noncollagenous proteins (NCP) (Fantner et al., 2005). This type of microstructure is observed in several important bone tissues like the parallel-fibered bone for example. More importantly, such a bundle can be viewed as a basic unit for the structure of any bone tissue at micron scale (Level 4 in Figure1) where different types of fibril arrays can be modeled as consisting of bundles of parallel fibrils with different orientation and volume fractions.

In this case, we cannot apply directly the modeling used in Sections 2.1 and 2.2. The method of Torquato (Section 2.1) is valid only for isotropic phases while the individual collagen fibers possess markedly orthotropic properties (Weiner and Wagner, 1998). On the other hand, the volume fraction of the fibrils in the RVE of the bundle is much higher than 50%, given that their diameter is ~100 nm while the separation distance between two adjacent fibrils is only 1-2 nm (Gupta et al., 2006). In addition, the interactions between neighboring fibrils cannot be neglected, which prevents us from using the Mori-Tanaka method if we model the fibrils as inclusions embedded in a NCP matrix. Instead, we use an “inverse Mori-Tanaka” homogenization by simply inverting the matrix and the inclusions. Thus, we consider a bundle of aligned fibrils as a two-phase composite where the matrix properties are those of a single mineralized fibril and the needle-like inclusions represent the interfibrillar spaces filled with NCPs (Figure 4 rigth). In fact, this approximation is more realistic than it may seem at a first glance because it is known that adjacent collagen fibrils have the tendency to merge together (Weiner and Wagner, 1998), and at certain places are bonded with “bridges” containing HA minerals (Hasenkam et al., 2004).

 

With the above considerations in mind, we can express the effective stiffness tensor of a bundle of aligned fibrils, , as:

                                                              (15)

with concentration factor

                                                                                        (16)

where  and  denote the stiffness tensors of the mineralized collagen fibrils and the NCP, respectively;  is the Eshelby tensor for the needle-like NCP inclusions;  and  are the volume fractions of the fibrils and the interfibrillar spaces, respectively.

 

2.5 Effective properties of fibril arrays with narrow orientation distribution of the fibrils

 

 

Once the stiffness tensor  for the effective properties of aligned fibrils is known, one can model other types of fibril arrays (Level 4 in Figure 1) provided that the orientation distribution of the fibril packets and the volume fractions corresponding to different orientations are known.

 

Consider a fibril array where the fibrils are oriented along (i) different directions with  being the stiffness tensor for a fibril bundle along the i-th orientation and  the volume fraction of the bundle. Here we consider only the case where the fibrils have narrow orientation distribution, like in the rotated plywood structure of lamellar bone proposed by Weiner et al., 1991, for example. Then, we can safely assume that the deformation within the fibril array due to external loading will be more or less uniform and therefore, the effective properties of a fibril array without pores, , defined in a fixed coordinate frame, can be written as the following volume average:

                                                                                                      (17)

where  is a 6x6 rotation matrix constructed from the conventional 3x3 rotation matrix as shown in Doghri, 2000, for example, and the symbol  denotes the transpose of a matrix. The introduction of  comes from the fact that the effective elastic properties in a given direction are found with respect to a local frame (as shown in Figure 4, left) coinciding with the axes of symmetry of the bundle, and that frame in general does not coincide with the fixed coordinate system where  is defined.

 

The Mori-Tanaka method can be further applied for modeling of porosity at micron scale by considering the pores as ellipsoidal inclusions but we do not consider this topic here. As for fibril arrays with broad orientation distribution like woven-like patterns, one can also use the Mori-Tanaka method for finding the overall properties as it was done for textile tissues by Gommers et al., 1997.

3.1 Collagen-water composite

 

 

 

Because of the extreme smallness of the collagen molecules, only their longitudinal Young modulus has been evaluated by various experimental and theoretical approaches. The Young modulus of collagen molecules obtained via Electron microscopy (Hofmann et al., 1984) ranges from 3 to 5.1 GPa, X-ray diffraction yields  (COMMENT: was does that mean: 2.8 to 3?) GPa (Sasaki and Odajima, 1996) while MD simulations (Vesentini et al., 2005) predict  ((COMMENT: was does that mean, see above) GPa. Here we choose , which seems to be a good compromise between simulations and experiment. The Poisson’s ratio of the collagen molecule is not known exactly. We set  to have an overall Poisson’s ratio for the collagen-water composite about 0.35 as estimated by Katz, 1971. For the water-protein matrix we choose Poisson’s ratio of  corresponding to a nearly incompressible material. The associated Young modulus is taken as  GPa in order to match the measured bulk modulus for water,  GPa. With these values we determine the shear moduli, , the 3D bulk moduli  and the 2D bulk moduli , where i =1, 2 stand for water and collagen, respectively. From geometry, the volume fraction, , of the collagen molecules modeled as cylindrical fibers with hexagonal packing is

                                                                                                               (18)

Where  and  denote the fiber radius and the lateral separation between the fibers, respectively.

 

For , the separation between two collagen molecules is 2.5 Å (about the length of one H-bond) for a diameter of the collagen molecule of 1.5 nm. The corresponding 3-point statistics parameters are  (McPhedran and Milton, 1981) and  (Eischen and Torquato, 1993). Substituting the above values in Eqs (1-3) and assuming that the collagen molecules are aligned along axis 1 of a Cartesian coordinate system, the elastic constants of the collagen-water system expressed in Voigt notation are obtained as:

  GPa                                                          (19)

Because the properties of the water-protein matrix are subject to uncertainty, we also test a higher Young modulus for the matrix,  GPa with corresponding Poisson’s ratio of , keeping the collagen properties and volume fraction unchanged. This results in a somewhat stiffer collagen-water composite:

 Gpa                                                            (20)                            

 

3.2 Properties of HA crystallites and extrafibrillar proteins

 

As in the case of collagen, there are no reliable experimental measurements for the properties of HA nanocrystals in bone. The measured properties of synthetic HA obtained through various processes vary wildly with Young modulus ranging from 6 (Martin and Brown, 1995) to 147 GPa (Snyders et al.. 2007). For the Young modulus of biological HA, we first take a typical value of  GPa (Gupta et al., 2006), and Poisson’s ratio , as measured by Gilmore and Katz, 1982, and computed with ab initio calculations by Snyders et al., 2007. We also use the Young modulus for single crystals of HA,  GPa, given by Gilmore and Katz, 1982. For the sake of simplicity, we assume that the properties of the HA crystals are isotropic.

 

The mechanical properties of the extrafibrillar proteins are not known at present. Given that they consist of flexible, coiling macromolecules, their Young modulus must be lower than that of the collagen with its relatively stiff triple-helix molecules. We assume that the extrafibrillar NCPs have isotropic properties with Young modulus  GPa (Gupta et al., 2006) and Poisson’s ratio , a typical value for soft polymers with flexible molecules. The volume fractions of the mineralized fibrils, , and the interfibrillar proteins, , are determined with Eq. (18) assuming that the fibrils have a diameter of 100 nm and are packed in a hexagonal array with separation between the fibrils of 1.5 nm (Gupta et al., 2006). This gives  and .

 

3.3 Mineralization parameters

 

 

The volume fraction of the overall mineral content is assumed to vary from 32 to 52 vol%, which comprises a large amount of literature data for wet bone summarized by Fritsch and Hellmich, 2007, as well as the data given by Currey, 1999. The most important issues concerning the mineralization in fibril arrays are the ratio between the extrafibrillar and the intrafibrillar mineral content, the evolution laws for mineralization outside and inside the fibrils as functions of the total mineral content and the relation between the shape of the intrafibrillar HA crystals and the mineral content in the fibrils. To our knowledge, these dependencies are not known beyond some contradictory data about the extrafibrillar mineral content. Using X-ray diffraction methods, Katz and Li, 1973, and Sasaki and Sudoh, 1997, determined that between 70 and 80% of the total mineral content must be within the fibrils. On the other hand, electronic micrographs methods (Lees et al., 1994) and AFM measurements (Sasaki et al, 2002), predict that that as much as 70-77% of the mineral is extrafibrillar.

 

Using Eq. (14) one can deduce that the results of Katz and Lee, 1973 and Sasaki and Sudoh, 1997, namely that in a mature bone between 20 and 30% of the minerals are extrafibrillar, are consistent with the model of Jäger and Fratzl, 2000, for the mineral content within the fibrils. For overall mineral volume fraction , fibril volume fraction  and extrafibrillar mineral fraction , Eq. (14) gives the volume fraction of HA within the fibrils as , the most probable value in a fully mineralized cortical bone according to Jäger and Fratzl, 2000.

4.1 Elastic moduli of a bundle of aligned fibrils

 

4.1.1 Elastic moduli of aligned fibrils without extrafibrillar mineralization

 

Initially, we consider the elastic moduli for a hypothetical bundle of “naked” aligned fibrils without extrafibrillar minerals. We first assume that the shape of the HA platelets remains unchanged for the studied volume fractions and the minerals are ellipsoids with shape determined by the typical dimensions of the HA platelets of 50x25x3 nm. Then we consider that the longest dimension of the platelets (oriented along the fibril axis) grows proportionally with increasing the mineral volume fraction. Thus, for mineral content 32 vol% we assume that the platelets have average dimensions of 40x25x3 nm, i.e., they fill completely the gaps within the collagen fibrils but do not penetrate in the overlap zones. For mineral content of 52 vol%, their longest dimension must evolve linearly to 65x25x3 nm while their width and thickness remain unchanged. The resulting elastic moduli of bundles of aligned fibrils without extrafibrillar minerals are shown in figures 5 and 6. Hereafter, the subscripts 1,2 and 3 are with respect to the coordinate system shown in Figure 4. The collagen-water properties are given by Eq. (19) and the Young modulus of HA is taken as  GPa.

It is seen that the preferential growth of the HA platelets along the fibril axis changes only the longitudinal Young modulus of the fibrils and the obtained values in this case are more realistic than the values obtained assuming constant platelet shape. The preferential growth of the HA crystals along the fibril axis seems to explain the experimental fact that the longitudinal Young modulus in parallel-fibered and fibro-lamellar bone varies considerably for relatively small changes in the total mineral content (Currey, 1999).

 

The bundles of aligned “naked” fibrils possess excessive orthotropy and only two deformation modes are stiff enough to match the experimentally observed stiffness in bone – these are tension/compression along the fibril axis and shear in the 12 plane. The other deformation modes are too soft to insure sufficient load bearing capacity of the bone tissue, which indirectly proves the importance of the extrafibrillar mineralization for the mechanical properties.

 

4.1.2 Elastic moduli of aligned fibrils with extrafibrillar mineralization

Now we consider the elastic moduli of a bundle of aligned fibrils with extrafibrillar mineralization. We test two scenarios – first we assume that the extrafibrillar mineralization develops simultaneously with the intrafibrillar mineralization in a way that keeps the extrafibrillar mineral fraction fixed at  for . Next, we assume that the volume fraction of the extrafibrillar minerals grows linearly with the overall mineral content starting from  (for ) to  for . The equivalent thickness of the HA rings is assumed to be  nm so that at mineral volume fraction , 76% of the fibrils’ surface is coated with HA, which is a reasonable value. The assumed minimum fibril surface fraction corresponds to HA reinforcing rings 5 nm thick and 59 nm long. The values for the extrafibrillar fraction, , the surface fraction occupied by the coating rings,  and the mineral volume fraction within the fibrils, , for the two scenarios ( constant  and evolving ) for different total volume fractions  are calculated with Eqs. (12) and (14) and are listed in Table 1.

	[-]	0.32	0.36	0.40	0.44	0.48	0.52
constant	[-]	0.27	0.27	0.27	0.27	0.27	0.27
	[-]	0.468	0.526	0.584	0.643	0.701	0.76
	[-]	0.265	0.299	0.332	0.365	0.398	0.431
	lp nm	40	45	50	55	60	65
evolving	[-]	0.10	0.134	0.168	0.202	0.236	0.27
	[-]	0.173	0.261	0.363	0.481	0.613	0.76
	[-]	0.327	0.354	0.378	0.399	0.417	0.431
	lp nm	43	46.6	49.8	52.6	55	56.8

Table 1: Parameters for extrafibrillar mineralization with equivalent thickness of the coating rings  nm; lp denotes the longest average dimension of the HA platelets.

The mineralization parameters for linear evolution of  with the total mineral content  are visualized in Figure 7 and one can notice that in this case, the mineral content within the fibrils grows only slightly with the total mineralization while the increase in the fibril surface fraction covered with HA is more significant.

The mineralization scenario shown in Figure 7 also suggests that for mineral grades below 30% volume fraction, the intrafibrillar mineralization would develop much faster than the extrafibrillar one, while for mineral levels above 30 vol%, the rate of intrafibrillar mineralization saturates and is compensated by a higher growth rate of the extrafibrillar HA.

 

In these simulations, the collagen-water properties are given by Eq. (19) and the Young modulus of HA is  GPa. The results are shown in Figures 8 and 9.

It is seen that the assumption of constant proportion  of the extrafibrillar minerals for all mineral volume fractions results in a degree of orthotropy that changes with the overall mineral content, while linear evolution of  yields a self-similar pattern for the evolution of the elastic moduli that keeps the same degree of orthotropy with increasing the mineral content. The literature data for the 3D elastic constants of bone (Espinoza, 2005) suggest that the degree of orthotropy does not change much for considerable changes in the values for the elastic constants. Therefore, we conclude that the fraction of extrafibrillar minerals and the longest dimension of the HA platelets within the fibrils develop with increasing the total mineral content in a way similar to the evolution laws visualized in Figure 7. From Figures 8 and 9 one can see that the elastic moduli for high degree of mineralization are slightly underestimated, most probably because the properties of the collagen-water composite and HA are somewhat stiffer than the assumed so far values.

 

We therefore perform simulations where the collagen-water properties are given by Eq. (20), the Young modulus for HA is taken as  GPa and the extrafibrillar mineralization parameters are the same as in the bottom half of Table 1. The results for the elastic moduli are shown in Figures 10 and 11. For comparison, we superpose the moduli of a fibril bundle without extrafibrillar minerals where the longest dimension of the HA platelets evolves linearly with the mineralization from 40 to 65 nm.

From Figures 10 and 11 is seen that mineralized collagen fibrils with extrafibrillar mineralization have superior mechanical properties compared to fibrils without extrafibrillar minerals. The only moduli that are not considerably enhanced by the extrafibrillar minerals are the longitudinal Young modulus E₁₁ and the shear modulus G₁₂. However, in a bundle with EF minerals, the same Young modulus is achieved with only slight increase in the intrafibrillar mineral content and in the dimensions of HA platelets along the fibril axis. The latter range from 43 to 56.8 nm compared to an increase from 40 to 65 nm for the “naked” fibrils. Also, one can notice that the softest deformation modes, the 23 and 31 shear, have practically the same moduli and are the most strengthened by the extrafibrillar minerals.

 

The values obtained for the longitudinal Young modulus are in a good agreement with those obtained for hydrated fibrolamellar bone via micro-tensile testing (Gupta et al., 2006), nanoindentation on individual trabecula (Hengsberger et al., 2001) and nanoindentation on osteons (Swadener et al., 2001). In our simulations, E₁₁ increases from 10 to 25 GPa compared to the experimentally measured range from 5 to 23 GPa (Gupta et al., 2006). The values lower than 10 GPa can be explained not only with total mineral content lower than 32% but also with imperfect packing (square array of aligned fibrils with total mineralization of 32% has longitudinal Young modulus of 8.9 GPa), misalignment of the fibrils in the sample with respect to the tensile axis and lower than 10% extrafibrillar mineral content. As for the cases where the experimentally measured longitudinal Young modulus of hydrated bone tissues is higher than 25 GPa, one can explain such values by longer than 59 nm dimensions of the HA platelets.

4.2 3D elastic constants of bone

 

 

In Voigt notation, the elastic constants of a bundle of aligned parallel fibrils (in a coordinate frame oriented as in Figure 4 with the fibril axis along axis 1) with total mineral content of 52%, extrafibrillar mineral fraction of 27% and dimensions of the HA platelets of 56.8x25x3 nm read:

 GPa                                                       (21)

The local properties  are relevant for a length scale of several microns while the ultrasonic measurements of cortical bone found in the literature are for samples with characteristic length of several mm, where the mineralized collagen fibrils are organized in a much more complex manner to form rotated plywood structures, osteons, Haversian canals etc. (Figure 1 Levels 5 and 6). The explicit modeling of the microstructure at higher levels of hierarchy within the frame of our approach is possible but is beyond the scope of this work. Here we use a simple phenomenological model to reproduce the properties observed at macroscopic scales. Let us assume that in a first approximation, cortical bone can be represented as a composite of aligned fibrils where 67% of the fibrils are oriented like the bundle in Figure 4 and 33% of the fibrils are rotated about their axes (parallel to coordinate axis 1) at an angle . The volume fraction of 33% for the rotated fibrils was chosen so that the simulated elastic constant C₆₆ matches the value obtained via ultrasonic measurements by Espinoza Orías, 2005. The elastic constants of this composite can be found using Eq. (17) and are listed in Table 2 along with experimental measurements for hydrated bone femur from different sources.

Coefficient	This work	Espinoza Orías, 2005	Van Buskirk et al., 1981	Ashman et al., 1984
C11	27.68	27.331.64	25.004.30	27.601.74
C22	18.23	19.662.09	18.402.70	20.201.79
C33	16.06	16.752.27	14.102.40	18.001.60
C44	4.45	4.640.43	5.380.46	4.520.37
C55	5.36	5.650.53	6.300.67	5.610.40
C66	6.22	6.220.31	7.000.67	6.230.48
Source	-	Wet femur	Wet femur	Wet femur
Species	-	Human	Bovine	Human

Table 2: Elastic constants of cortical bone.

From Table 2 is seen that our simple phenomenological estimate yields surprisingly good predictions for the macroscopic 3D elastic constants despite the fact that the bone microstructure and porosity at higher hierarchy levels has been discarded. While we do not pretend to have explained in detail the macroscopic 3D elastic constants of bone, the obtained values indicate that our modeling is able to quantitatively describe the essential features of bone elasticity at micron and submicron length scales and is rich enough to serve as a basis for a physically relevant modeling of the bone properties at all length scales.

We have shown that extrafibrillar mineralization considerably enhances the overall mechanical properties of the mineralized collagen fibrils in bone when the extrafibrillar crystals strongly adhere to the fibrils surfaces and partially coat each individual collagen fibril. Part of the extrafibrillar minerals most probably form effective coating “rings” with the same period and staggered geometry as the underlying “naked” fibrils and start growing in the 40 nm gaps between the successive collagen molecules situated at the fibrils surface. 

 

We have established that typically, if the mineral content within the fibrils does not exceed 43 vol% in fully mineralized bone (Jäger and Fratzl, 2000), no more than 30% of the total mineral content is extrafibrillar, in accord with the experimental findings of Katz and Li, 1973, and Sasaki and Sudoh, 1997. More importantly, it is shown that the percentage of the extrafibrillar minerals must evolve proportionally to the overall mineral content in order to have a relatively small change in the degree of orthotropy of the elastic constants for different degrees of mineralization. In this case, the average gain in stiffness from extrafibrillar mineralization for a typical mineral content of 44 vol%, compared to fibrils where the whole mineral content is within the fibrils, is about 2 times for the weakest Young modulus (in direction perpendicular to the HA platelets) and about 3 times for the weakest shear moduli. The obtained 3D elastic constants for bundles of parallel fibril arrays are close to the elastic constants for macroscopic samples obtained via ultrasonic measurements.

 

The model predictions for the longitudinal Young modulus of arrays of parallel mineralized collagen fibers are in a good agreement with the experimental data from micro-tensile testing with the assumption that for mineral content of more than 30 vol%, the HA platelets within the fibrils grow preferentially in the fibrils direction and the growth of the longest dimension of the HA platelets is proportional to the increase in the overall mineral content.

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[*] Analytical solutions for the Eshelby’s tensor exist only for inclusions embedded in an isotropic and a transversely isotropic matrix (Mura, 1987).

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