Biomechanics of Bone
The primary responsibility of the skeleton is to provide structural support for the body. In this role, the skeleton is the basis of posture, opposes muscular contraction resulting in motion, withstands functional load bearing, and protects internal organs. The skeleton's structural success can be jeopardized by genetic disorders such as osteogenesis imperfecta, metabolic diseases such as Paget's, or the bone loss that parallels the aging process (i.e., osteopenia). To appreciate the structural risks that accompany bone diseases, it is essential that several interdependent concepts of the biomechanical properties of bone be considered.
STRAIN
When a force (newton = N = force that will cause a 1 kg mass to accelerate at 1 m/s2; essentially, the force of a 100 g apple in Earth's gravitational pull) is applied to any material, such as bone, it deforms. The amount of deformation in the material, relative to its original length, is called “strain”. Strain is a dimensionless unit formally defined as the change in length divided by its original length (ε = ΔL/L). When a material is pulled, it gets longer (tensile strain); when it is pushed together, the material shortens (compressive strain). Shear strain is the angle, measured in radians, through which a material has been deformed by forces acting parallel, rather than opposed, to the material. Shear strain arises when layers within a material slide against another, as might occur with torsion or bending (1). Absolute magnitudes of peak compressive strains in bone during vigorous activity can be on the order of 3500 microstrain in compression (0.35% strain), 1200 microstrain in tension, and 1500 microstrain in shear. To determine the material properties of bone, a machined sample is subjected to a known tensile or compressive load. To aid in comparing this specific test with other materials or studies from other laboratories, it is preferable to report the magnitude of the force in terms of the cross-sectional area of the material on which it is acting (3). The force per unit area is the stress (σ = Force/Area), and is reported in newtons per square meter (N/m2), or pascals (Pa). A pascal is essentially the stress caused by the weight of one apple acting on a square meter tabletop. The compressive stress caused in the third metacarpal of a thoroughbred racehorse during a gallop is on the order of 63,000,000 pascals, or 63 MPa (4). Now imagine 63 X 106 apples on that same table.
|
|
FIG. 1. The stress-strain curve of two different materials is shown. In material A (solid line), the material is stiffer, as depicted by the steeper slope of the line (E = σ/ε; where E is elastic modulus, σ is stress, and ε is strain. Thus for a given stress, there is less strain. However, the yield strain of material A is lower than that of material B (dashed line), as depicted by the point where the relation between stress and strain is no longer linear (Yield Failure). As well, the ultimate strain of material A is lower than that of material B, as depicted by the point where there is catastrophic failure of the material [i.e., there is no longer any stress necessary to cause strain (Ultimate Failure)]. Material A is more brittle than material B. The area beneath each curve reflects the toughness of each of the materials. |
MODULUS
The degree to which a material deforms depends not only on the magnitude of the forces and moments (turning, twisting, or rotational effect of a force; M = Nm) applied to the structure, but also on the stiffness of the constituent materials (5). In the case of bone, stiffness is determined by the relative proportions of the hydroxyapatite crystals and the collagen fibers that make up the composite (6). During the initial stages of a test to define a bone's material properties, there is a linear increase in strain as the stress increases (Fig. 2). This is known as the elastic region. Should the load be removed during this phase of the test, the specimen will return to its original size and shape almost immediately, without incurring permanent damage. The linear relation between strain and stress is called Hooke's Law (E = σ/ε; where E is the elastic modulus): Ut tensio, sic vis: as the extension, so the force (7).
The slope of the elastic region of the stress-strain curve reflects the stiffness of the material, otherwise known as the modulus of elasticity (E): The stiffer the material, the steeper the slope of the line. One hundred newtons pulling on a tendon will result in far greater strain than the same force pulling on a sample of bone of the same dimensions (8). Whereas the stress is identical, the modulus of the bone is much higher than the tendon (the slope of the stress-strain curve is much steeper), and thus the bone deforms much less (i.e., is less stiff).
An isotropic material, like steel, has equivalent material property values in all directions. A material like bone, which has distinct mechanical properties in different directions, is anisotropic. The modulus of mature cortical bone is on the order of 18 gigaPascals (GPa) in the longitudinal direction, 12 GPa in the transverse direction, and 3.3 GPa in shear (9). The material properties of cancellous bone are even more complex, as trabecular orientation, connectivity, and bulk density (volume fraction of the bone) greatly influence the stiffness (10). Depending on location, the elastic modulus of trabecular bone can range from 0.1 to 3.5 GPa (11). The degree of mineralization (e.g., immature or woven bone) or porosity (e.g., old bone) will compromise the stiffness of the bone and thereby lower the elastic modulus.
YIELD FAILURE
When the increase in strain is no longer proportional to the stress, the elastic region ends, and with it the ability of the material to resume its original shape (Fig. 1). The bone specimen has moved into the plastic region where permanent damage has begun to accrue. In terms of bone, yield failure arises through ultrastructural microcracks within the hydroxyapatite and the disruption of the collagen fibrils. The yield strain of cortical bone is on the order of 6800 microstrain (12), suggesting that a safety factor of 2 exists between peak strains caused by normal functional activity and the point where damage is inevitable. The yield stress is approximately 130 MPa. In other words, by the time there are 130 X 106 apples on the table, the table surface begins to crack.
ULTIMATE FAILURE
As loading continues in the plastic region, the material will eventually reach ultimate failure, at which point the specimen fails catastrophically (13). The point at which the bone breaks can be viewed as either the ultimate strain (10,000-15,000 microstrain in tension) or the ultimate stress (140 MPa in tension, 200 MPa in compression, and 65 MPa in shear). Because of this disparity, it should become clear that the cause of fracture in "normal" bone material is most likely due to tensile or shear failure.
The amount of post-yield strain that occurs before ultimate failure is a measure of the material's ductility, reflecting its ability to resist the propagation of cracks (3). A ductile material is one that can change form without breaking; tendon is more ductile than bone. A material that undergoes little post-yield behavior before ultimate failure is considered brittle (e.g., glass or ceramic). Whereas osteoporosis is often referred to as the brittle-bone disease, in reality little experimental evidence supports an actual reduction in the ductility of bone material examined from patients with this disease (14). There is just less bone to resist a given force (stress during functional activity is just that much closer to yield stress). Perhaps a better example of brittle bone is provided by osteogenesis imperfecta, in which a qualitative or quantitative deficiency in type I collagen in the material (15) reduces its ductility, condemning it to ultimate failure soon after the material has begun to yield.
TOUGHNESS
The stress-strain curve yields another important property of the material. The area under the curve reflects the amount of work, or energy per unit volume, possessed by the material at any given point on the curve. At ultimate failure, the area under the curve defines the energy required to break the object, or toughness. A major contributor to the toughness of bone is its composite nature of Haversian (secondary osteon), circumferential, and interstitial lamellae. The analogy of a bundle of straws versus a plastic stick illustrates how the architecture of a composite, anisotropic structure outperforms a single, uniform isotropic material in resisting loads and avoiding yield and ultimate strain. The plastic stick breaks with relatively little bending because high strains are generated within the periphery of the material. A bundle of straws composed of the same mass and subjected to the same bending conditions will continue to strain rather than break, as each independent element slips relative to adjacent bundles.
Bone, as an organ, has a requirement to be both stiff and tough. There is, however, an inevitable trade-off between these two attributes, as they must be attained by a balance between the resistance to crack propagation provided by collagen and the resistance to deformation provided by the mineral. Comparatively small changes in the mineral content of bone tissue can have significant effects on its properties as a material, as demonstrated by Currey (16) in his determination of the mechanical properties of bones with diverse functional responsibilities. By comparing the bovine femur, the deer antler, and the whale tympanic bulla, he illustrated that the mineral content changed to accommodate a specific functional responsibility. In the extreme, the mineral content ranged from 86% in the bulla, which requires high acoustic impedance, to 59% in the antler, which must resist high impact and torsional loads. The consequence of this high mineral content is revealed by comparing the relative toughness of these bones; the bulla is only 3% as tough as the antler.
BONE AS A COMPOSITE MATERIAL
The composite structure of bone allows it to withstand compressive and tensile stresses, as well as bending and torsional moments. The inorganic phase of bone, with hydroxyapatite crystals arrayed in a protein matrix; provides the ability to resist compression. Individual calcium phosphate crystals of multiple sizes are imbedded in and around the fibrils of the collagen type I lattice (17). Hydroxyapatite, although effectively resisting compressive loads, has a poor ability to withstand tensile loads. As in concrete, a material that excels at resisting compression but is poor in resisting tension, tensile elements (e.g., steel reinforcing rods) are added to create a composite material that can cope with complex loading environments. In the case of bone, this tensile strength arises from collagen fibrils organized into lamellae.
The collagen orientation between adjacent lamellae can rotate by as much as 90 degrees, permitting the tissue to resist forces and moments acting from several different directions, much like the added strength in plywood realized by the distinct orientation of the fibers in each specific ply (18). Whereas the ultrastructural organization is, to a certain extent, defined by the genome, the functional environment also contributes to the distribution of lamellae as well to the osteons that house them (19). This directed deposition of collagen adds to the anisotropy of the bone. Given that >80% of functional strains are due to bending (and thus a high percentage of strain is tensile), the structural quality of the bone may ultimately be determined by the quality of the collagen and the organization of the microarchitecture. Recent studies have shown that collagen itself deteriorates with age and undoubtedly contributes to the declining material properties of the skeleton (20).
Alterations in either the organic (e.g., collagen) or inorganic (e.g., hydroxyapatite) matrix components can bring about changes in bone strength. Mutations in the collagen gene give rise to several genetic skeletal problems, some of which increase fracture risk. In some forms of osteogenesis imperfecta, mutations in the primary structure of type I procollagen lead to brittle bone (21). Another disorder of collagen resulting in excessively fragile bone is fibrogenesis imperfecta ossium (22), a rare disease in which remodeling results in a disorganized, collagen-deficient tissue. Although the number of hydroxyapatite crystals contributes to the ability to resist compression, density is not everything. Fluoroapatite, which incorporates into the mineral phase of bone during fluoride poisoning, is denser than hydroxyapatite, but results in brittle bone that can fracture relatively easily (23).
AREAL PROPERTIES OF BONE
Areal properties, which define the overall mass and pattern of the structure, are as important as material properties to the ultimate success of the skeleton. Size, density, and architecture effectively describe areal properties at the gross level. Other, more subtle properties also are key contributors to the structural efficacy of bone, including the long-bone curvature, the girth and geometry of the cross-sectional area, and the trabecular organization (e.g., connectivity).
Axial loading results in very little strain for a given load: imagine how strong a pencil is when you press straight down on the long axis of the shaft. At the same time, it is important to consider how easily the pencil is snapped when it is subject to bending. Consider now the neck of the femur while climbing a flight of stairs: the functional demands on bones, as opposed to pencils, are very complex, and subject not only to axial conditions, but also to bending and torsional moments (1). With the diversity of the functional environment, it is clear that a strategy of minimal mass will not serve as a successful structure. Instead, the structure of the bone must be designed to resist a wide variety of loading conditions, perhaps to control and regulate the loading environment rather than to minimize the strain (24).
Even the simplest of loading cases create complex strain and stress environments in a material, including bone. Axial loads applied to a slightly curved beam will cause tensile strain on the convex side, and compressive strain on the concave side. The strains are greatest at these extremes and decrease to zero in the middle of the beam. This area is called the neutral axis, where strain approaches zero. The flexural rigidity of the material, EI, represents the amount of force per unit cross-sectional area required to deform the material a given amount, where E is elastic modulus (see earlier), and I is the second moment of area (25). The second moment, or moment of inertia, reflects the contribution of each bit of material to the stiffness in each position in the cross-section of the beam (I = ly2dA, where y is the distance of each element of area A from the neutral axis). Therefore the further the material is relative to the neutral axis, the better placed it will be to resist bending (Fig. 2). This areal property is a powerful means of rapidly increasing flexural rigidity for a small investment of material. For example, in the elderly, it was shown that the subtle increases in the second moment of area, which are achieved through periosteal expansion, may to a certain degree structurally compensate for the bone loss and cortical thinning that parallels the aging process (26).
|
|
|
FIG. 2. The cross-sectional areas of these three cylinders are identical (Aa = Ab = Ac). The elastic modulus of the material that makes up these cylinders also is identical (Ea = Eb = Ec). Therefore for an axial force (i.e., pushing or pulling the ends of the bar), the stress is identical (o,A = o-e = oc). However, because the geometry of the cross sections is different, the ability of each of these cylinders to resist bending and/or torsion is strongly dependent on the distance of the material relative to the center of the cylinder. The relative resistance to bending of these cylinders is IA = 100%, IB = 400%, and Ic = 700%a. Subtle changes in a bone's cross-sectional geometry will contribute heavily to the bone's structural properties. |
An appreciation of the biomechanical attributes of bone is critical to an improved understanding of both the pathogenesis of metabolic bone disease and the emerging possibility of controlling bone mass and structure through mechanical stimuli. As important as inherent mechanical properties may be, it also is essential to appreciate that bone is extremely sensitive to its mechanical environment, and to a large extent, it is this functional milieu that defines skeletal structure and ultrastructural organization. This "form follows function" aspect of skeletal tissue is known as Wolf s Law and helps us to understand how mechanically based interventions such as exercise serve as an anabolic agent to bone in some cases, and how disuse, cast immobilization, and bed rest put the skeleton at risk (24). Clinicians must be aware that the most devastating complication of bone disease is the structural collapse of the skeleton. There is every reason to believe that mechanical strategies can retard, prevent, or even reverse the structural demise of the skeleton, and that therapeutic options such as these should be actively pursued. As important, scientists beginning to generate bone with molecular triggers must assure that the new bone is adequate to the functional demands placed on it. Skeletal science must strive to incorporate an understanding of the biomechanical functions of the skeleton and consider not only the engineering basis of the bone material, but also the biologic response of the bone tissue to the potent mechanical stimuli that arise from function.
REFERENCES
1. Mow VC, Hayes WC. Basic orthopaedic biomechanics. New York: Raven Press, 1991:93-142.
2. Rubin CT, Lanyon LE. Dynamic strain similarity in vertebrates; an alternative to allometric limb bone scaling. J Theor Biol 1984;107: 321-327.
3. Carter DR, Hayes WC. Bone compressive strength: the influence of density and strain rate. Science 1976194:1174-1176.
4. Gross TS, McLeod KJ, Rubin CT. Characterizing bone strain distributions in vivo using three triple rosette strain gages. J Biomech 1992;25:1081-1087.
5. Einhorn TA. Biomechanics of bone. In: Bilezikian L, Raisz L, Rodan G, eds. Principles of bone biology. San Diego, CA: Academic Press, 1996:25-37.
6. Martin RB, Burr DB. Structure, function, and adaptation of compact bone. New York: Raven Press, 1989:18-56.
7. Vincent JFV. Structural biomaterials. London: Macmillan Press, 1982:1-33.
8. Wainwright SA, Biggs WD, Currey JD, Gosline JM. Mechanical design in organisms. New York: Wiley, 1976:64-109.
9. Reilly DT, Burstein AH. The elastic and ultimate properties of compact bone tissue. J Biomech 1975;8:393-405.
10. Goldstein SA. The mechanical properties of trabecular bone: dependence on anatomic location and function. J Biomech 1987;20:10551061.
11. Cowin S. Bone mechanics. Boca Raton, FL: CRC Press, 1989:129-157.
12. Carter DR. Mechanical loading histories and cortical bone remodeling. Calcif Tissue Inl 1984;36:519-S24.
13. Carter DR, Harris WH, Vasu R, Caler E. The mechanical and biological response of cortical bone to in vivo strain histories. American Society of Mechanical Engineers publication AMD 1981;45:81.
14. Turner CH, Burr DB. Basic biomechanical measurements of bone: a tutorial. Bone 1993;14:595-608.
15. Prockop KJ. Mutations that alter the primary structure of type I collagen. J Biol Chem 1990;265:15349-15352.
16. Currey JD. Mechanical properties of bone tissues with, greatly differing functions. J Biomech 1979;12:313-319.
17. Weiner S, Traub W. Bone structure: from angstroms to microns. FASEB J 1992;6:879-885.
18. Alexander M. Bones: the unity of form and function. New York, NY: Macmillan, 1994:25-57.
19. Skedros JG, Mason MW, Nelson MC, Bloebaum RD. Evidence of structural and material adaptation to specific strain features in cortical bone. Anat Rec 1996;246:47-63.
20. Oxlund H, Mosekilde L, Ortoft G. Reduced concentration of collagen reducible cross links in human trabecular bone with respect to age and osteoporosis. Bone 199619:479-484.
21. Misof K, Landis WJ, Klaushofer K, Fratzl P. Collagen from the osteogenesis imperfecta mouse model (oim) shows reduced resistance against tensile stress. J Clin Invest 1997;100:40-45.
22. Carr AJ, Smith R, Athanasou N, Woods CG. Fibrogenesis imperfecta ossium. J Bone Joint Surg Br 1995;77:820-829.
23. Kroger H, Alhava E, Honkanen R, Tuppurainen M, Saarikoski S. The effect of fluoridated drinking water on axial bone mineral density: a population-based study. Bone Miner 199427:33-41.
24. Rubin CT, Gross TS, Donahue HJ, Guilak F, McLeod KJ. Physical and environmental influences on bone formation. In: Brighton CT, Friedlaender G, Lane J. Bone formation and repair. 1st ed. Rosemont, IL American Academy of Orthopaedic Surgeons, 1994: 61-78.
25. Wainwright SA. Axis and circumference: the cylindrical shape of plants and animals. Cambridge, MA: Harvard University Press, 1988.
26. Ruff CB, Hayes WC. Subperiosteal expansion and cortical remodeling of the human femur and tibia with aging. Science 1982;217:945948.