Elsevier

Journal of Theoretical Biology

Volume 244, Issue 4, 21 February 2007, Pages 597-620
Journal of Theoretical Biology

‘Universal’ microstructural patterns in cortical and trabecular, extracellular and extravascular bone materials: Micromechanics-based prediction of anisotropic elasticity

https://doi.org/10.1016/j.jtbi.2006.09.013Get rights and content

Abstract

Bone materials are characterized by an astonishing variability and diversity. Still, because of ‘architectural constraints’ due to once chosen material constituents and their physical interaction, the fundamental hierarchical organization or basic building plans of bone materials remain largely unchanged during biological evolution. Such universal patterns of microstructural organization govern the mechanical interaction of the elementary components of bone (hydroxyapatite, collagen, water; with directly measurable tissue-independent elastic properties), which are here quantified through a multiscale homogenization scheme delivering effective elastic properties of bone materials: at a scale of 10 nm, long cylindrical collagen molecules, attached to each other at their ends by 1.5nm long crosslinks and hosting intermolecular water inbetween, form a contiguous matrix called wet collagen. At a scale of several hundred nanometers, wet collagen and mineral crystal agglomerations interpenetrate each other, forming the mineralized fibril. At a scale of 510μm, the extracellular solid bone matrix is represented as collagen fibril inclusions embedded in a foam of largely disordered (extrafibrillar) mineral crystals. At a scale above the ultrastructure, where lacunae are embedded in extracellular bone matrix, the extravascular bone material is observed. Model estimates predicted from tissue-specific composition data gained from a multitude of chemical and physical tests agree remarkably well with corresponding acoustic stiffness experiments across a variety of cortical and trabecular, extracellular and extravascular materials. Besides from reconciling the well-documented, seemingly opposed concepts of ‘mineral-reinforced collagen matrix’ and ‘collagen-reinforced mineral matrix’ for bone ultrastructure, this approach opens new possibilities in the exploitation of computer tomographic data for nano-to-macro mechanics of bone organs.

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:

  • 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)];

  • The microstructure at an observation scale of several 100μm 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)];

  • The ultrastructure (or extracellular solid bone matrix) at an observation scale of several μm, comprising the material building up both trabecular struts and osteons [Fig. 1(e)].

  • 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.

  • 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; Ca10[PO4]6[OH]2) 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.

Secondly, the aforementioned universal microstructural patterns across the hierarchical organization of bone concern the mechanical interaction between the elementary components, governing the elasticity of the material at all different observation scales, as was probably first pointed out by Katz (1980). In the open literature, there is no general agreement on the interaction of collagen and mineral: The concept of a ‘mineral-reinforced collagen matrix’ (Currey, 1969, Katz, 1980, Katz, 1981, Sasaki, 1991, Mammone and Hudson, 1993, Jäger and Fratzl, 2000, Kotha and Guzelsu, 2003) is seemingly opposed to that of a mineral matrix with collagen inclusions (Crolet et al., 1993, Aoubiza et al., 1996, Benezra Rosen et al., 2002, Hellmich and Ulm, 2002a, Wang and Qian, 2006).

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 510μm, 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, d standing for the characteristic length of inhomogeneities within the RVE (see Fig. 2), and L, L 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 CultraMTII and CexvasMTIII 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.

References (142)

  • J. Crolet et al.

    Compact bone: numerical simulation of mechanical characteristics

    J. Biomech.

    (1993)
  • J. Currey

    The relationship between the stiffness and the mineral content of bone

    J. Biomech.

    (1969)
  • S. Cusack et al.

    Determination of the elastic constants of collagen by Brillouin light scattering

    J. Mol. Biol.

    (1979)
  • M. Ding et al.

    Quantification of age-related changes in the structure model type and trabecular thickness of human tibial cancellous bone

    Bone

    (2000)
  • L. Dormieux et al.

    Micromechanical approach to the behavior of poroelastic materials

    J. Mech. Phys. Solids

    (2002)
  • S. Eppell et al.

    Shape and size of isolated bone mineralites measured using atomic force microscopy

    J. Orthopaed. Res.

    (2001)
  • A. Fritsch et al.

    Porous polycrystals built up by uniformly and axisymmetrically oriented needles: homogenization of elastic properties

    C. R. Méc.

    (2006)
  • T. Hassenkam et al.

    High-resolution AFM imaging of intact and fractured trabecular bone

    Bone

    (2004)
  • C. Hellmich et al.

    Are mineralized tissues open crystal foams reinforced by crosslinked collagen?—some energy arguments

    J. Biomech.

    (2002)
  • C. Hellmich et al.

    Mineral-collagen interactions in elasticity of bone ultrastructure—a continuum micromechanics approach

    European J. Mech. A—Solids

    (2004)
  • K. Hofstetter et al.

    Development and experimental validation of a continuum micromechanics model for the elasticity of wood

    Eur. J. Mech. A—Solids

    (2005)
  • S. Hollister et al.

    Application of homogenization theory to the study of trabecular bone mechanics

    J. Biomech.

    (1991)
  • S. Hollister et al.

    A homogenization sampling procedure for calculating trabecular bone effective stiffness and tissue level stress

    J. Biomech.

    (1994)
  • B. Huo

    An inhomogeneous and anisotropic constitutive model of human dentin

    J. Biomech.

    (2005)
  • I. Jäger et al.

    Mineralized collagen fibrils: a mechanical model with a staggered arrangement of mineral particles

    Biophys. J.

    (2000)
  • E. Katz et al.

    Structure and function of bone collagen fibrils

    J. Mol. Biol.

    (1973)
  • J. Katz et al.

    On the anisotropic elastic properties of hydroxyapatite

    J. Biomech.

    (1971)
  • B. Khuri-Yakub

    Scanning acoustic microscopy

    Ultrasonics

    (1993)
  • M. Knothe Tate

    Whither flows the fluid in bone?—an osteocyte's perspective

    J. Biomech.

    (2003)
  • W. Kreher

    Residual stresses and stored elastic energy of composites and polycrystals

    J. Mech. Phys. Solids

    (1990)
  • W. Kreher et al.

    Residual stresses in polycrystals as influenced by grain shape and texture

    J. Mech. Phys. Solids

    (1993)
  • W. Landis et al.

    Mineral and organic matrix interaction in normally calcifying tendon visualized in three dimensions by high-voltage electron microscopic tomography and graphic image reconstruction

    J. Struct. Biol.

    (1993)
  • W. Landis et al.

    Mineralization of collagen may occur on fibril surfaces: evidence from conventional and high-voltage electron microscopy and three-dimensional imaging

    J. Struct. Biol.

    (1996)
  • T. Lang et al.

    Volumetric quantitative computed tomography of the proximal femur: Precision and relation to bone strength

    Bone

    (1997)
  • T. Lang et al.

    Measurement of bone mineral density at the spine and proximal femur by volumetric quantitative computed tomography and dual-energy X-ray absorptiometry in elderly women with and without vertebral fractures

    Bone

    (2002)
  • S. Lees et al.

    A study of dense mineralized tissue by neutron diffraction

    Int. J. Biol. Macromol.

    (1984)
  • S. Lees et al.

    A generalized packing model for type I collagen

    Int. J. Biol. Macromol.

    (1984)
  • A. Lorenzo et al.

    Elastic properties, Young's modulus determination and structural stability of the tropocollagen molecule: a computational study by steered molecular dynamics

    Med. Eng. Phys.

    (2005)
  • J. Mammone et al.

    Micromechanics of bone strength and failure

    J. Biomech.

    (1993)
  • R. McCarthy et al.

    Ultrasound speed in equine cortical bone: effects of orientation, density, porosity and temperature

    J. Biomech.

    (1990)
  • T. Mori et al.

    Average stress in matrix and average elastic energy of materials with misfitting inclusions

    Acta Metallurgica

    (1973)
  • M. Morris et al.

    Fluid spaces in canine bone and marrow

    Microvascular Res.

    (1982)
  • R. Müller et al.

    Morphometric analysis of noninvasively assessed bone biopsies: comparison of high-resolution computed tomography and histologic sections

    Bone

    (1996)
  • A. Ascenzi et al.

    The compressive properties of single osteons

    Anat. Rec.

    (1968)
  • Baylink, D., Wergedal, J., 1971. Bone Formation and Resorption by Osteocytes. Academic Press, New York, USA, pp....
  • J. Black et al.

    Haversian osteons: Size, distribution, internal structure, and orientation

    J. Biomed. Mater. Res.

    (1974)
  • J. Catanese et al.

    Heterogeneity of the mechanical properties of demineralized bone

    J. Biomech.

    (1999)
  • J. Currey

    Role of collagen and other organics in the mechanical properties of bone

    Osteoporosis Int.

    (2003)
  • I. Eckardt et al.

    Quantitative measurements of the mechanical properties of human bone tissues by scanning acoustic microscopy

    Ann. Biomed. Eng.

    (2001)
  • M. Epple

    Solid-state chemical methods to investigate the nature of calcified deposits

    Z. Kardiol.

    (2001)
  • Cited by (301)

    View all citing articles on Scopus
    1

    Present address: Laboratoire des Matériaux et des Structures du Génie Civil, Ecole Nationale des Ponts et Chaussées, 77455 Marne-la-Vallée, France.

    View full text