Bone Biomechanics

David B. Burr, PhD

Joseph M. Wallace, PhD

Dr. Burr or an immediate family member serves as a paid consultant to or is an employee of Agnovos. Neither Dr. Wallace nor any immediate family member has received anything of value from or has stock or stock options held in a commercial company or institution related directly or indirectly to the subject of this chapter.

INTRODUCTION: BASIC CONCEPTS OF BONE MECHANICS

The mechanical properties of bone are defined both by its structure (mass and architecture) and by the properties of the material of which it is made (the bone matrix). These properties can be termed structural (or extrinsic) or material (intrinsic). Extrinsic properties are usually measured by causing the bone to deform and measuring how much load it can bear before fracture. The resulting load-deformation curve is used to define the bone’s strength, stiffness, and energy to fracture. When the load, or force, is adjusted for area (force/area = stress) and deformation is normalized by the original length of the structure (change in length/original length = strain), a similar curve results that defines the intrinsic properties of the tissue itself, independent of its size or geometry (Figure 1). This is called the stress-strain curve, and tissue properties analogous to structural properties can be estimated: the ultimate stress (tissue strength), modulus of elasticity (tissue stiffness), and modulus of toughness (fracture energy of the tissue).

Load and deformation are linearly proportional for some time, until the yield point is reached. At this point, the slope of the curve begins to decrease, meaning that lower loads are required to create substantially more deformation. The area under the linear portion of the load-deformation (or stress-strain) curve is called the elastic region because the structure will return to its original shape when the load is removed, elastically returning most of the energy required for deformation. However, the shift of fluids in the bone when it is loaded will cause some loss of elastic energy each time the bone is loaded within this range; this loss of energy is called hysteresis. Following yield, bone is permanently, or plastically, deformed and will not return to its original shape upon unloading. Within this postyield region, the bone has been irreversibly damaged (microcracking, slippage at cement lines, damage at the nanoscale to mineral crystals or collagen and its cross-linked structure, or combinations of these). If the load continues to increase, an ultimate (maximum) load is reached, and continued loading will result in fracture (the failure load). The brittleness of the bone is defined by the amount of strain that occurs after the yield point. Fracture after the yield point without much additional strain/deformation is typical of a brittle structure (the opposite, in which strain increases without an increase in stress, is defined as ductility).

The strength of bone may be defined as the load (or stress) at the yield point, or as the ultimate load/stress. Because there may be continued deformation after the maximum load is reached, causing the failure load to be lower than the ultimate load, it is best to use ultimate load or ultimate stress as the strength criterion. Furthermore, there is less variability in the ultimate load than in the failure load. Stiffness or rigidity is the load required to deform the bone a given amount. This is the same thing as the slope of the linear portion of the load-deformation curve. Elastic modulus (or Young modulus, named for Thomas Young [1773-1829], the physician-physicist who coined the term “modulus of elasticity”) is the equivalent measurement taken from the slope of the linear portion of the stress-strain curve. The work or energy required to yield or fracture the structure is determined by measuring the area under the load-deformation curve out to the yield or failure points. The modulus of resilience and
modulus of toughness are the analogous tissue-level measurements made from the stress-strain curve.

FIGURE 1 Schematic representations of force-displacement and stress-strain curves. In Panel A, applied load is plotted as a function of displacement. Important properties related to strength, stiffness, ductility or ability to deform, and ability to dissipate energy can be derived from this curve. An important note is that this characteristic graph represents the structural behavior of a sample and is influenced by the size and shape of the sample, as well as its material properties. By normalizing for the geometry of the sample, a complimentary stress-strain curve is obtained (panel B). Similar information can be calculated from this curve, but these parameters are estimated material properties which are no longer dependent on size and shape. Of note here is the yield point, where behavior turns from being linear-elastic to nonlinear. A typical method to locate this point is to use the 0.2% offset method in the stress-strain domain as shown.

Bone can be loaded in several different modes. When a structure is loaded in tension, it becomes longer in the load direction, and when loaded in compression, it becomes shorter. When loaded, the relationship between the change in transverse width, and the change in length is known as Poisson ratio. For geometric reasons, this ratio must be between 0 and 0.5 in ordinary materials. For incompressible materials (eg, cartilage), Poisson ratio is 0.5. Typical values for solids, including bone, are about 0.3, but in cortical bone the ratio can vary between 0.28 and 0.45.

Of course, human bone is seldom or never loaded in pure compression or tension, but is usually subject to bending, which produces a combination of compression and tension on different bone cortices. Under bending loads in vivo, shear stresses are also created in which one region as bone slides past an adjacent region. Bone is weaker in shear (˜67 MPa for the femoral midshaft) than in either compression (˜205 MPa) or tension (˜135 MPa), so when bending loads are applied to bone to the point of failure, bone fracture may be accelerated by these shear stresses. This is also true when using three-point bending tests of bone ex vivo, where bone fails due to shear rather than tension or compression. One reason to use a four-point bending test is that the region between the two loading points exhibits no shear forces, and the bone fails due to tensile loading at its weakest point. In tension or compression, shear stress is maximized at a 45° angle from the direction of loading. Shear stresses are also created when bone is twisted (torsion).

CROSS-SECTIONAL PROPERTIES AS ESTIMATES OF STRENGTH AND RIGIDITY

If one can assume that the bone tissue is normal (as in osteoporosis, but perhaps not in other conditions such as osteogenesis imperfecta or types I and II diabetes), then estimates of bone’s strength can be made from its geometry.

Bones are thick-walled tubes, which allows them to be strong, but not heavy. Long bones are mostly loaded in bending, and the amount that they bend under a load (their deflection) is defined by the bending moment, length of the bone, elastic modulus, and the second moment of the area (or moment of inertia) about the bending axis. For any given bending moment, the bending of the bone can be reduced by decreasing its length, making it stiffer (larger modulus), or increasing its moment of inertia. The cross-sectional moment of inertia is a measure of the distribution of material about a cross-sectional axis and is calculated as the sum of the square distance of each unit of area from the axis around which bending occurs (or the radial distance from the centroid of the section for a polar moment of inertia, which is proportional to torsional rigidity). For a circular tube, bending rigidity around
a particular axis is proportional to the fourth power of its distance from the axis around which bending occurs.

where R_o is the distance from the bending axis to the periosteal surface, and R_i is the distance from the axis to the endosteal surface. Adding a small amount of bone to the periosteal surface can increase bone’s bending rigidity substantially as the new material is located far away from the bending axis and, therefore, contributes more greatly to the moment of inertia (Figure 2). Consequently, if only one-third of the amount of bone that is removed from the endosteal surface is added to the periosteal surface, the rigidity of the bone is maintained. Thus, periosteal apposition can increase bone strength even when bone mass and bone mineral density (BMD) have declined. This also means that a small amount of bone loss near the marrow cavity may have a negligible effect on bone’s overall strength and rigidity.

FIGURE 2 Illustrations demonstrating that the rigidity of bone is partly dependent on its geometry. Cross-sections A, B, and C all have the identical cortical area, but distributed in different ways. As the outer diameter of the bone is expanded, the moment of inertia and relative bending strength increase by more than fourfold though the amount of bone has not changed. The axial strength of a bone is dependent only on the amount of bone, not its distribution, so relative axial strength does not change. The two figures in the second row demonstrate what happens when 10 mm² of bone is removed from the endocortical surface of bone (hatched area). Adding only 3 mm² of bone back onto the periosteal surface (black area) maintains the original bone bending strength, although axial strength would be slightly lower.^a

BONE’S MECHANICAL PROPERTIES

The mechanical properties of cortical bone vary from location to location in the skeleton, but are also determined by the direction in which the properties are measured. This is partly dependent on the orientation of osteons in cortical bone and on the orientation of lamellae within osteons and interstitial bone. Bone is stiffer in the longitudinal direction (˜17 gigapascals, or GPa) than in the transverse direction (˜12 GPa).¹ Materials that have different mechanical properties in different directions are termed anisotropic (an isotropic material would have the same properties in all directions; an orthotropic material has different properties in every direction). Bone is transversely isotropic (but also anisotropic) because its mechanical properties in all transverse directions are the same, but differ from properties measured in the longitudinal direction. Strength in the transverse direction is about 40% of that in the longitudinal direction.¹

For cancellous bone, these intrinsic properties become more difficult to measure because cancellous bone is composed of individual trabeculae, each of which may have its own stiffness. This combination of individual trabecular stiffnesses, together with the orientation of the different trabeculae (the trabecular architecture), will determine the structural stiffness of the cancellous mesh, which can vary between 0.1 and 4.5 GPa. The strength of cancellous bone, which is dependent on apparent density and orientation of the trabeculae that collectively compose it, is much less than that in cortical bone, varying between about 1 and 20 MPa.

Although strength is often equated with resistance to fracture, bone can be quite strong, but not very fracture resistant. An example is osteopetrotic bone. Although this bone is highly dense and can be loaded to a high stress, it cannot deform much before failure. Thus, it has a short postyield region (it is brittle), indicating that it cannot develop damage and absorb energy efficiently before it fractures. Therefore, a strong and stiff bone is not necessarily a tough and fracture-resistant one if it is unable to absorb much energy before it breaks. In fact, great strength (or stiffness) and great toughness are often mutually exclusive² (Figure 3). These relationships also show that although there is often a close relationship between BMD and mechanical properties, the relationship between BMD and fracture risk is more complex.

FIGURE 3 Schematic representation of the trade-off between strength and toughness. In many materials, ductility or toughness is sacrificed for high stiffness and strength (and vice versa). Bone has a somewhat unique ability to balance these two competing properties. This balance can be compromised in cases of disease where the tissue becomes overly stiff at the expense of the ability to dissipate energy and is brittle. Alternatively, some diseases have a negative impact on stiffness and strength and deform a great deal under relatively low load levels, leading to bone deformity.

THE BASIS FOR BONE’S MATERIAL PROPERTIES

Properties of the bone tissue itself (material properties) are also critical to bone’s mechanical integrity. Although conventional wisdom concentrates mostly on the collagen and mineral components of the tissue, because these are the most prevalent components, noncollagenous proteins (NCPs), and water also contribute.

Bone mineral is generally considered to give bone its strength and stiffness, but the nature of the mineral crystal itself also is important. Crystallinity (the size/shape/perfection of crystals, an increase of which is associated with older tissue and/or lower bone remodeling rates) is associated with lower ultimate strain and toughness.³^,⁴ Carbonate replaces some of the phosphate in the hydroxyapatite crystal as the tissue ages, but the effect of this substitution is uncertain. It changes the amount of elastic and plastic deformation that occurs before fracture,⁴^,⁵ but reports about whether this increases⁶ or decreases⁴ elastic modulus differ. This may depend on the relationship between crystal size and crystal lattice perfection. Furthermore, too much mineral (osteopetrosis) can make the tissue brittle even though it may be strong; too little mineral (osteomalacia) will reduce elastic modulus. Both conditions impair the overall mechanical properties of bone, and increase fracture risk. Therefore, bone mineral must be kept within a fairly narrow window to achieve the best mechanical function.

The type I collagen in bone is often considered to provide bone’s ductility, but equally important is the extent and manner in which the collagen fibrils are cross-linked. Two of the common trivalent (mature) cross-links, pyridinoline (PYD) and deoxypyridinoline (DPD) do not correlate with bending strength,⁷^,⁸ but the ratio of the two does correlate with increased strength and stiffness by stabilizing the collagen network.⁹^,¹⁰ These cross-links may also make the bone tougher,¹¹ because as they rupture with loading, they provide a means to dissipate energy in the tissue. Intermolecular bonds formed by nonenzymatic processes may have different effects. These cross-link-like bonds (they are not truly cross-links) result in the accumulation of advanced glycation end-products (AGEs), which are common in both types I and II diabetes and in chronic kidney disease. They make the bone tissue brittle and more likely to break.⁹^,¹² AGEs reduce the amount of energy that can be absorbed prior to fracture by restricting movement in the collagen matrix, so may especially increase the risk of hip fractures that result from an impact such as a fall.

There are several water compartments in bone. The loosely bound water found at the mineral-collagen interface serves an important function by increasing the ductility of this interface, dissipating energy and reducing shear stresses during loading.¹³ Some of this water is bound to the collagen, and some bound to the mineral, and their effects may be different. Collagen-bound water correlates positively with toughness,
whereas mineral-bound water correlates negatively with elastic modulus.¹⁴ The age-related reduction in bound water accounts for 71% of the reduction in bone toughness.¹⁵ Some of this may be regulated by glycosaminoglycans (GAGs), which are known to be hydrophilic. One of these, biglycan, which is found in the perilacunar matrix, is thought to have shear-sensing properties. The reduction in GAGs with age may contribute to an overall 40% reduction in toughness in the later decades of life.¹⁵^,¹⁶

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