8.1.1 Ultramicroscopic Findings

Bone is composed of cells embedded in a fibrous, organic ground substance (extracellular matrix). The extracellular matrix of mature bone consists of about 20% water. The dry material consists of 30% to 40% organic and 60% to 70% inorganic components. The organic component consists of 90% collagen, with the remaining 10% being composed essentially of glycosaminoglycans and glycoproteins.

The collagen of the bones is exactly the same as the material employed in other tissues of the body subjected to tensile stress, such as ligaments or connective tissue. In primary woven bone, the collagen fibrils are not aligned in any preferred direction. In mature lamellar bone, by contrast, the collagen fibrils can form a meshed or spiral arrangement, aligned preferentially in the longitudinal direction of the long bones.

The rigidity and strength characteristic of bone material are effected by the deposition of mineral substances in the organic ground substance (matrix). These inorganic salts comprise about 85% calcium phosphate, 6% to 10% calcium carbonate, and small admixtures of other alkaline salts. After generation of the organic matrix, these salts are first deposited in an amorphous phase; subsequently crystals bearing a close resemblance to hydroxyapatite Ca₁₀(PO₄)₆(OH)₂ are formed. These crystals are between 20 and 100 nm long and between 1.5 and 3.0 nm in diameter (1 nm = 10⁻⁹ m). They are aligned in their long axis to the collagen fibrils and are attached to the fibrils in several layers.

8.1.2 Microscopic Findings

The bone cells comprise the bone-building cells (osteoblasts), the bone-resorbing cells (osteoclasts), and the mature cells enclosed in the extracellular matrix (osteocytes). Osteoblasts produce an early stage of bone tissue, the osteoid. In the course of this phase the osteoblasts remain enclosed in the matrix and become osteocytes. Within a few days the osteoid is mineralized to about 70%; mineralization is com pleted within a few months. Osteoclasts resorb the collagen component of the bone tissue by an enzymatic process. The mineral component is dissolved by creation of an acidic environment.

During the growth phase, as well as during the entire lifetime, bone tissue is continuously remodeled as osteoclasts resorb bone and osteoblasts rebuild new bone, often in a different architecture. A distinction is made between primary bone (created for the first time) and secondary bone (created by reconstruction). Almost all bone present at birth, and the callus produced for the purpose of stabilization around fractures, consists of so-called woven bone. The collagen fibers of the woven bone have diameters in the region of 0.1 μm (1 μm = 10⁻⁶ m) and exhibit no preferred direction. Unordered trabeculae, the so-called primary spongy bone, develop from woven bone. Under the influence of mechanical loading, the primary spongy bone is transformed into spatially ordered, secondary trabecular (spongy) bone. The lamellar bone material of cortical bone also originates from woven bone. In lamellar bone, mineral and collagen are deposited in layers of about 5 μm thickness. The collagen fibers of lamellar bone have diameters in the region of 2 to 3 μm and definitely preferred orientations. The architecture of lamellar bone bears some similarity to that of technical fiber composite materials.3,4 Lamellar, cortical bone is interspersed with osteons. Osteons are created when bone is resorbed by osteoclasts, in the form of a thin channel, which is subsequently filled with new bone deposited in layers by osteoblasts.

8.1.3 Macroscopic Findings

The small difference in the density of woven bone and lamellar bone material is due to minor differences in their organic and inorganic composition. Major differences in the density of bone tissue samples are due to the porosity of the tissue. In the cross section of long bones, an outer layer of compact bone, known as cortical bone, is visible. The internal volume of long bones is usually filled with a structure composed of thin rods (trabeculae) and plates. Bone displaying this architecture is termed trabecular (spongy) bone. The architecture of trabecular bone is not unlike that of technical foams with open pores. There is no abrupt transition from cortical to trabecular bone. The outermost bony layer of the long bones beneath the articular cartilage, for example, is called cortical bone, despite the fact that its thickness at this location is virtually the same as that typical of trabeculae.

As a rough categorization, the apparent density of cortical bone is above 1.5 g/cm³, and the apparent density of trabecular bone between 0.1 and 1.0 g/cm³. The qualifier apparent points to the fact that this density measurement allows for porosity; it does not describe the true density of the compact material (that is, as measured without voids). The apparent density is obtained by dividing the mass of a bone sample, cleaned of marrow and body fluids, by its volume. The apparent density should be distinguished from the density measured in the material in its compressed state (that is, without pores). With regard to this latter density, cortical and trabecular bone would exhibit only minor differences because both bone types are constructed essentially from lamellar bone. Deformation and strength of bones depend strongly on the apparent density.

8.4.1 Stress and Strain of Inhomogeneous, Anisotropic Materials

A material that is of uniform composition throughout its whole volume is termed homogeneous. Bone material is inhomogeneous because it is composed of a crystallized mineral and a fibrous organic component. A material whose mechanical properties do not depend on the direction of load or deformation is termed isotropic. Due to the spatial alignment of the organic fibers in lamellar bone, the architecture of the layered lamellae in cortical bone, and the spatial configuration of the trabeculae in trabecular bone, the mechanical properties of bone (moduli of elasticity and shear, etc.) depend on the orientation in which they have been measured. Thus, bone material is anisotropic.

Unlike in isotropic materials, the strain occurring in anisotropic materials is not necessarily in the same direction as the applied stress. Under tensile stress a beam made of an anisotropic material may, for example, not only lengthen but also exhibit an angular deformation ( Fig. 8.4 ). The deformation of inhomogeneous, anisotropic materials under the influence of external forces cannot be characterized by the three material constants (modulus of elasticity E, shear modulus G, and Poisson’s ratio μ), as in the case of homogeneous, isotropic materials. To describe the relation between stress and strain, Hooke’s law and the law describing shear deformation

have to be generalized. In this generalized law the modulus of elasticity E and the shear modulus G are replaced by a term c, known as the stiffness matrix. In the general case, this term depends on 21 material constants. (We are dealing here with a symmetric 6 · 6 tensor with 21 independent elements.) The stress σ is replaced by a mathematical expression (a vector with six components) which depends on the three normal and the three shear stresses. The strain ε is replaced by a mathematical expression (vector with six components) which depends on the linear strains in the three spatial directions and on three angular deformations. In this formulation, the generalized Hooke’s law is written as

The index i assumes values from 1 to 6. The mathematical sum sign £ indicates that, to calculate the i-th component σ_i of the stress vector, all products c_ij · ε_j have to be added together. The index j runs from j = 1 to j = 6. Thus, in contrast to the case of isotropic materials, the stress in a certain spatial direction will depend on the strain and the angular deformation in all three spatial directions. Conversely, a normal stress effects not only a deformation in the direction of the stress but also an angular deformation. This was the very case in the example shown in Fig. 8.4 .

In the technical domain the generalized Hooke’s law forms the basis for a description of the mechanical properties of fiber-reinforced materials. But even in such applications it proves difficult to determine the complete set of 21 material constants experimentally. When dealing with bone material, additional difficulties arise. Due to changes in bone density and architecture, the mechanical properties of bone change from one volume element to the next. In addition, bone is nonlinearly elastic: that is, the material properties depend on the magnitude of the applied strain and on the deformation velocity. In technical materials there is no equivalent to the in vivo repair and adaptation processes encountered in bone.

These comments are aimed at making it clear that, other than for those situations where a crude, approximate description of the mechanical behavior of bone will suffice, a detailed description has to be based on intricate theory and on experimental determination of a large set of material constants. For this reason, lines of reasoning are often complex and hard to follow. Nevertheless, a detailed description is indispensable if, for example, the relation between mechanical stimuli and bone adaptation is to be investigated. To this end, the state of stress and deformation under the application of external forces and moments must be known as accurately as possible. Estimates based on the properties of a homogeneous, isotropic material are inadequate.

8.4.2 Material Properties of Cortical Bone

When, in the following, the mechanical properties of bone are characterized by only a few material constants, it must be kept in mind that this constitutes a crude approximation to reality. Table 8.1 gives an overview of the properties of human cortical bone. The numbers quoted are only approximate. The ranges of variation are due, in part, to the biological variation of the individual bones investigated and, in some cases, to varying examination methods. The stress σ_u, which effects the destruction of the sample, is called the ultimate (or maximum) stress.

The anisotropy of bone material becomes apparent when samples cut in different directions from the cortical layer of a long bone are compared. As an illustration, Table 8.2 quotes measured results for the modulus of elasticity and the ultimate stress for samples cut longitudinally and transversely with respect to the long axis of a bone.

Bone material is viscoelastic; the values of the material constants depend on the deformation velocity (that is, whether deformation occurs over a short or long time interval). As an example, Fig. 8.5 shows the dependence of the modulus of elasticity E and of the ultimate stress σ_u on the deformation velocity. The observed increase of E and σ_u indicates that with increasing deformation velocity the material is becoming harder and stronger.

Like many other materials, cortical bone exhibits mechanical fatigue under repeated loading. The stress at which fracture of a sample is observed decreases with increasing numbers of load cycles. Carter et al 10– 12 reported that material fatigue depended predominantly on the magnitude of the strain, with bone density or ash content having little influence on fatigue strength. According to these authors, the ultimate stress after 10⁷ load cycles is in the region of 7 · 10⁶ N/m². This is considerably lower than the ultimate stress under a single load cycle as quoted in Table 8.1 .

Fig. 8.6 shows the number of cycles to fracture in dependence on the magnitude of the strain as measured by Carter et al.10 To check how high the risk of a fatigue fracture is when walking, running, or performing strenuous athletic exercise, estimated regions of the strain in the lower extremity are also shown in this diagram. It is concluded that fatigue fractures are not to be expected when walking. When running over long distances, occurrence of fatigue fractures is possible, especially if an individual running style with high repetitive loading coincides with low individual bone strength. Indeed, fractures of the lower extremity occasionally observed in young soldiers after running long distances or marching with heavy packs are interpreted as fatigue fractures (sometimes termed stress fractures). 13

The mechanical properties of cortical bone change with age. 14 These changes are assumed to be caused by structural changes in the organic and inorganic bone components. According to Burstein et al 15 and Duchemin et al, 16 the modulus of elasticity, ultimate stress, and ultimate strain decrease with age. In other words, bone becomes softer and more brittle with increasing age. In the material investigated by Burstein et al,15 no gender-related difference in the mechanical properties was noted. The observation that microfissures in cortical bone are not rare findings agrees with the results of Burstein and coworkers. The prevalence of such fissures increases with age ( Fig. 8.7 ). 17

8.4.3 Architecture and Material Properties of Trabecular Bone

Trabecular bone is composed of a three-dimensional mesh of interconnected bone rods and plates, arranged similarly to that of open-pore technical foam materials. Except in woven bone, the rods and plates are oriented in preferred directions that are readily discernible to the naked eye on bone sections. Below the cartilaginous layer of joints, for example, the trabeculae are preferentially oriented perpendicular to the joint surface. There appears to be a second preferred direction oriented at about 90° to the fi rst, where the rods and plates are interconnected, thus reinforcing the mesh. It is understandable from these observations that the elastic properties and the strength of trabecular bone are determined by the density, the distance, and the relative orientation and interconnection of the rods and plates. It follows furthermore that measured material properties of trabecular bone depend on the direction of testing. Trabecular bone is an anisotropic material. 18

With increasing age the compressive strength of trabecular bone decreases. 19 At the same time the ratio of rods to plates changes ( Fig. 8.8 ) 20: the number of plates decreases while the number of rods remains virtually constant. In specimens of trabecular bone originating from the acetabulum of the hip joint or from the central volume of vertebral bodies, fractured trabeculae in different stages of the healing process are a common finding.20– 22 The prevalence of such fractures increases with age ( Fig. 8.9 ). 21 It is hypothesized that these fractures are caused by mechanical overloading; alternatively, they may result from a normal remodeling process.

Porosity is defined as that volume fraction of a bone sample not filled by bone material. The porosity of a sample of cortical bone with no voids would be zero; the porosity of trabecular bone can approach (though not reach) the value 1. If the material properties of cortical bone and those of the rods and plates of trabecular bone are assumed to be approximately equal, the material properties of samples of cortical and trabecular bone (save for anisotropic effects) can be expected to depend continuously on the degree of porosity. The modulus of elasticity of trabecular bone with porosity p can be approximated by 23

This formula tells us that, with increasing porosity p (with increasing void space in the sample), the modulus of elasticity E exhibits a marked decrease. Quantitatively, the modulus of elasticity of trabecular bone ranges between 1.4 MPa and 9800 MPa. 24

Fig. 8.10 shows the ultimate stress of samples of trabecular bone in dependence on porosity. The ultimate stress is seen to decrease with increasing porosity. With increasing porosity (or decreasing apparent density) trabecular bone becomes softer and weaker. The ultimate stress observed in samples of trabecular bone covers the entire stress range depicted in Fig. 8.10 . Depending on the type of bone and the source of the samples, published data range between 0.2 MPa and 378 MPa.24 Samples of trabecular bone are usually not irreparably damaged by one single overload episode, but subsequently still exhibit some strength, though at a reduced level (see, for example, Keaveny et al 25). The residual strength of mechanically damaged bones (fractured osteoporotic vertebrae, for example) constitutes a vital safety factor for the functioning of the locomotor system. The ultimate strain of trabecular bone is about 4%, 26, 27 and is independent of porosity and apparent density.

When a sample of trabecular bone is viewed with the naked eye, the alignment of the trabeculae in preferred directions is obvious beyond any doubt. In the past, describing the intuitively observed anisotropy in quantitative, mathematical terms posed considerable difficulties. The remarkable progress made in recent years, however, allows the spatial anisotropy of the mesh to be described in numerical terms and these data to be linked with the anisotropy of the mechanical properties. Parameters describing the local anisotropy (“fabric”) of the spatial mesh can be defined. In parallel, the deformation of such meshes under the influence of external forces can be computed using the finite element method or, alternatively, it can be measured directly. The comparison of calculated and measured results reveals how well a specific anisotropy parameter actually describes the mechanical properties of a sample of trabecular bone. 28– 30

Fig. 8.11 illustrates three proposals for a quantitative description of the local anisotropy of trabecular bone. 29 The figure uses a two-dimensional model, but the concepts are applied in three dimensions. The sample of trabecular bone shown appears intuitively to be preferentially oriented in the vertical direction. An attempt can be made to describe the preferred direction by superimposing a line grid in different orientations ( Fig. 8.11a ) and observing the lengths of the grid lines (intercept length) intersecting the trabeculae. In the example shown, we expect long intercept lengths when the grid is oriented vertically and short lengths when the grid is oriented horizontally.

Alternatively ( Fig. 8.11b ), points can be selected at random within the bone volume and the direction in which straight lines through these points exhibit the longest intercept with the bone volume (volume orientation) is observed. In the example shown, these directions would preferentially point to the vertical; a second preferred direction would point approximately to the horizontal. As a further alternative ( Fig. 8.11c ), the orientation of local bone volume elements can be quantified if points are again selected at random within the bone volume and the distance of the bone surface as seen from these points in different directions is recorded (described as star length and star volume). In addition to orientation, these parameters also quantify the interconnection of the rods and plates in the trabecular bone.

Due to recent improvements in the spatial resolution of computed tomography, it is now possible to reconstruct the lattice of trabecular bone not only from in vitro bone samples, but also in vivo from tibia and radius (see, for example, Boutroy et al 31). If further progress allows in vivo imaging of trabecular architecture in other regions of the skeleton as well, we can expect important new insights into bone growth and remodeling.