‘Universal’ microstructural patterns in cortical and trabecular, extracellular and extravascular bone materials: Micromechanics-based prediction of anisotropic elasticity
Introduction
Bone materials are characterized by an astonishing variability and diversity. Their hierarchical organizations are often well suited and seemingly optimized to fulfill specific mechanical functions. This has motivated research in the fields of bionics and biomimetics. The aforementioned optimization is primarily driven by selection during the biological evolution process. However, apart from the fact that selection is quite unlikely to push bone skeletal and material design to a well-defined optimum (Nowlan and Prendergast, 2005), it is of great importance to notice that selection is realized at the level of the individual plant or animal (and not at the material level). Therefore, material optimization in the strictest sense of the word does not take place. Rather, ‘architectural constraints’ (Seilacher, 1970, Gould and Lewontin, 1979) merely due to once chosen material constituents and their physical interactions imply the fundamental hierarchical organization patterns or basic building plans, which remain largely unchanged during biological evolution. Firstly, these building plans are expressed by typical morphological features which can be discerned across all bone materials. Katz et al. (1984) distinguish five levels of hierarchical organization, which have been quite generally accepted in the scientific community:
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The macrostructure at an observation scale of several mm to cm, where cortical (or compact) bone and trabecular (or spongy) bone can be distinguished [Fig. 1(a) and (b)];
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The microstructure at an observation scale of several to several mm, where cylindrical units called osteons build up cortical bone, and where the single trabecular struts or plates can be distinguished [Fig.1(c) and (d)];
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The ultrastructure (or extracellular solid bone matrix) at an observation scale of several , comprising the material building up both trabecular struts and osteons [Fig. 1(e)].
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Within the ultrastructure, collagen-rich domains [light areas in Fig. 1(e)] and collagen-free domains [dark areas in Fig. 1(e)] can be distinguished at an observation scale of several hundred nanometers. Commonly, these domains are referred to as fibrils and extrafibrillar space.
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Finally, at an observation scale of several ten nanometers, the so-called elementary components of mineralized tissues can be distinguished. These are:
Plate-shaped mineral crystals consisting of impure hydroxyapatite (HA; ) with typical 1–5 nm thickness, and 25–50 nm length (Weiner and Wagner, 1998) [Fig. 1(f)].
Long cylindrically shaped collagen molecules with a diameter of about 1.2 nm and a length of about 300 nm (Lees, 1987), which are self-assembled in staggered organizational schemes (fibrils) with characteristic diameters of 50–500 nm (Cusack and Miller, 1979, Miller, 1984, Lees et al., 1990, Lees et al., 1994b, Weiner et al., 1997, Weiner and Wagner, 1998, Rho et al., 1998, Prostak and Lees, 1996), [Fig.1(g)]; several covalently bonded fibrils are sometimes referred to as fibers.
Different non-collagenous organic molecules, predominantly lipids and proteins (Urist et al., 1983, Hunter et al., 1996) and
Water.
The collagen matrix introduced in the first concept does not refer to molecular collagen with a stiffness of several GPa (Harley et al., 1977, Cusack and Miller, 1979; Sasaki and Odajima, 1996a, Sasaki and Odajima, 1996b; Lorenzo and Caffarena, 2005, Vesentini et al., 2005), but to a ‘collagen–water composite’ or ‘wet collagen’, with significantly smaller stiffness. However, there is no general agreement either on the magnitude of this stiffness: experiments reveal a few MPa stiffness for collagen fibrils self-assembling under laboratory conditions (Christiansen et al., 2000), several tens of MPa stiffness for unmineralized turkey leg tendon (Landis et al., 1995; used in the model of Jäger and Fratzl, 2000), several hundreds of MPa for demineralized bone (Bowman et al., 1996, Catanese et al., 1999), and 1.5 GPa for leather-type skin (Hall, 1951; used in the models of Currey, 1969, Katz, 1980, Katz, 1981, Sasaki, 1991, Mammone and Hudson, 1993).
As regards the second concept, recent research work strongly suggests the ‘mineral matrix’ to be a mineral foam or porous polycrystal with typically several nm-sized water-filled ‘nano-pores’. This recent research work comprises material science contributions (Benezra Rosen et al., 2002), cellular solid-type mechanical energy considerations (Hellmich and Ulm, 2002a) relying on a comprehensive experimental data base encompassing the pioneering work of Lees et al., 1983, Lees, 1987, Lees et al., 1994a, Lees et al., 1995, and continuum micromechanics models (Hellmich and Ulm, 2002b, Hellmich et al., 2004a, Fritsch et al., 2006).
A central issue of this paper is that probably both concepts are relevant for the mineral–collagen interaction in the bone ultrastructure, but clearly at different observation scales: in the line of the ‘mineral-reinforced collagen matrix’-concept we consider, at an observation scale of some tens of nanometers, a ‘collagen matrix material’ called ‘wet collagen’, consisting of 1.2 nm thick collagen molecules and intermolecular water. At a scale of several hundred nanometers, we envision the mineralized collagen fibril to be formed by wet collagen and by mineral crystal agglomerations, interpenetrating each other. At a scale of , however, the mineralized collagen fibrils themselves are embedded in an extrafibrillar mineral foam, in the line of the concept of a ‘mineral matrix with collagen inclusions’.
For relating the aforementioned vision of ultrastructural organization to effective elastic properties, we rely on homogenization theory (continuum micromechanics, Hill, 1963, Suquet, 1997, Zaoui, 1997, see Section 2), which is a well-established tool for structure–property investigations of bone or teeth, both at the microstructural level (Katz, 1980, Katz, 1981, Sevostianov and Kachanov, 2000, Hellmich et al., 2004b, Qin and Swain, 2004, Hellmich, 2005, Huo, 2005), and at the ultrastructural level (Hellmich and Ulm, 2002b, Hellmich et al., 2004a). Thereby, we invest into careful validation of our micromechanical development (described in Section 3) through independent experiments related to (i) the elasticity of the elementary components, and to (ii) composition and elasticity of different bone tissues from different animals and different anatomical locations both at the extracellular and extravascular level (described in Section 4). Since we avoid introduction of micromorphological features which cannot be experimentally quantified (such as e.g. the ‘arrangement of lamellae’ around osteons in ‘lamellar’ cortical bone), no material parameters are left for tuning or back-analysis. Hence, the capability and the limitations of our mathematically expressed construction plan for extracellular and extravascular bone materials can be directly assessed in terms of model prediction errors. They are given in Section 4, and they are the basis for the Discussion (Section 5).
Section snippets
Fundamentals of continuum micromechanics—linear elasticity
In continuum micromechanics (Hill, 1963, Suquet, 1997, Zaoui, 2002), a material is understood as a macro-homogeneous, but micro-heterogeneous body filling a representative volume element (RVE) with characteristic length , , d standing for the characteristic length of inhomogeneities within the RVE (see Fig. 2), and , standing for the characteristic lengths of geometry or loading of a structure built up by the material defined on the RVE. In general, the microstructure within one RVE is
Micromechanical representation of hierarchical organisation of bone materials
Across the hierarchical organization of bone materials, the following ‘universal’ microstructural patterns are considered in the framework of a multistep homogenization scheme (Fig. 3): the first homogenization step refers to an observation scale of several nanometers, where crosslinked collagen molecules form a contiguous matrix, which is ‘perforated’ by intermolecular, water-filled spaces. We call the homogenized material ‘wet collagen’ [Section 3.1 and Fig. 3(a)]. At the fibrillar
Strategy
In the line of Popper, who stated that a theory—as long as it has not been falsified—will be ‘the more satisfactory the greater the severity of independent tests it survives’, cited from Mayr (1997, p. 49), the verification of the micromechanical representation of cortical and trabecular bone materials at the ultrastructural (extracellular) and the extravascular level will rest on two independent experimental sets: The stiffness values and predicted by the micromechanical
Discussion
This contribution aimed at quantification of the mechanical effects of universal patterns across different extravascular and extracellular cortical and trabecular bone materials. Such patterns have been identified across the hierarchical organization of bone materials; from typical pore shapes in the micrometer to millimeter regime down to the level of the elementary constituents of bone, namely mineral crystals, collagen molecules, and water.
Acknowledgments
This work was supported in part by the EU Network of Excellence project Knowledge-based Multicomponent Materials for Durable and Safe Performance (KMM-NoE) under the contract no. NMP3-CT-2004-502243.
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Present address: Laboratoire des Matériaux et des Structures du Génie Civil, Ecole Nationale des Ponts et Chaussées, 77455 Marne-la-Vallée, France.