Calcif Tissue Int (1992) 51:333-339
Calcified Tissue International 9 1992 Springer-Verlag New York Inc.
Editorial
Bone Strength: The Bottom Line The human skeleton is a mechanically optimized biological system whose composition and organization reflect the functional demands made upon it. According to a set of physical relationships derived by German anatomists during the late 19th century [1, 2], and presently defined as Wolff's Law [3], the patterns of trabecular alignment and the geometry of the cortical compartments are the results of functional adaptation to normal physiological loads. Under these conditions, the human skeleton can fulfill all of its required activities including locomotion, protection, and the ability to participate in metabolic pathways associated with mineral homeostasis. Consequently, the molecular, cellular, and metabolic changes that occur in bone are directed to maintain this mechanical environment or to adapt to any new loading conditions that it may experience. Failure to do this leads to the development of pathological fractures and the skeletal pain associated with a variety of metabolic bone diseases. To understand the interactions between bone physiology and bone integrity, a basic knowledge of biomechanics is needed. This communication will review the fundamental principles of bone biomechanics with specific reference to fracture risk.
Basic Concepts Any basic knowledge of biomechanics must begin with an understanding of the terms stress and strain. These terms are used to describe the phenomenon whereby, when a force is applied to bone, the bone will not only be deformed from its original dimensions but an internal resistance will be generated to the counter applied force. This internal reaction is known as stress; it is equal in magnitude but opposite in direction to the applied force. As stress is distributed over the cross-sectional area of the bone, it is expressed in units of force per unit area. The standard international unit for stress is the Pascal which is equivalent to one Newton of force distributed over 1 square meter. Strain is a term used to describe the changes in dimension which bone experiences under the influence of an applied force. Strain is dimensionless and is therefore reported as a fraction or a percentage. It is equivalent to the change in length divided by the original length of the bone section. Although an applied force can be directed at bone from any angle producing any set of complex stress patterns, all stresses can be resolved into three types: tension, compression, and shear (Fig. 1). Tension is produced in bone when two forces are directed away from each other along the same straight line. The resistance to a loading situation of this type is produced by intermolecular attractive forces which prevent the bone from being torn apart. In nature, it is unusual for a bone to experience a pure tensile force (i.e., it
would have to literally be pulled apart). However, an example of a tensile force producing a failure in bone occurs when a tendon or ligament that is inserted into bone undergoes acute loading and, instead of failing within its own substance (tearing), it detaches itself from the bone by actually pulling a piece of bone off with it (Fig. 2). Compression results from two forces that are directed towards each other along the same straight line. The common vertebral compression fracture sustained in osteoporotic patients is an example of the failure of bone as a result of this type of loading configuration. Lastly, when two forces are directed parallel to each other but not along the same line, shear stresses are produced. In nature, the three basic stress types can combine as a result of a variety of complex loading configurations and lead to different fracture patterns (Fig. 3). Bending, for example, results from a combination of tensile and compressive forces (tension on the convex side; compression on the concave side) and is one of the more common loading conditions that will lead to clinical fractures. Torsion or twisting produces shear stresses along the entire length of a bone and can result in spiral fractures. Comminuted fractures appear as shattered bones and this occurs because the amount of force applied to the bone is so great that a variety of fracture lines have to be generated in order to dissipate the energy transmitted. Because stress and strain are properties related to the quality of the material experiencing the load, the quality of bone can influence the magnitude of the stresses and strains generated. In normal, well-mineralized bone tissue, an applied stress will generally result in small strains. In poorly mineralized tissue, such as osteomalacic bone, bone will experience larger strains in response to the same stress. Moreover, since in nature, forces are applied to bone not only from perpendicular and horizontal directions but also from oblique angles, situations will arise in which a variety of complex mechanical relationships will be generated. When bones show different mechanical properties if loads are transmitted from different directions, they express a property known as anisotropy [4]. In general, bones resist loads best when they are oriented in the direction of customary loading. For example, the femur is much better adapted to resisting compressive loads than bending loads [5]. Thus, the same stresses generated in the femur when you jump up and down (compressive stresses) may fracture the femur if they are oriented from a transverse direction (bending stresses). Similarly, in the vertebrae, the ability to resist loads is greatest when the stresses are compressive [6]. However, in the proximal femur (hip joint area), loads are best resisted when they are transmitted along lines that are parallel to the trabecular systems [7] (Fig. 4).
Biomechanical Properties The biomechanical properties of bone can be described at
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TENSION COMPRESSION SHEAR
Fig. 1. Schematic representation of the three basic types of stress: tension, compression, and shear.
two levels. First, the material properties of bone must be considered and these properties are defined by the tissuelevel qualities of bone which are independent of structure or geometry. Second, the structural properties of bone, which function as a whole anatomical unit, should be examined. Classically, the material properties of bone are defined by performing standardized mechanical tests on uniform, machined specimens of intact bone. Structural properties are determined on whole sections of intact bone whose normal geometry has been maintained. It is important to recognize that a fracture sustained in a patient may represent the failure of bone at either the material or structural level or both.
Material Properties When a uniform section of bone is tested under controlled laboratory conditions, and the applied forces and deformations are known, the four basic mechanical properties of bone can be derived from a plot of the stress-strain relationship. These properties are strength, stiffness, energyabsorptive capacity, and deformation. By convention, stress is plotted on the ordinate (y-axis) and strain on the abscissa (x-axis). Figure 5 shows a stress-strain plot of an idealized material. Considering the fact that the three types of stress can combine to produce different stress patterns as a result of different types of externally applied loads (tension, compression, bending, and torsion), the terms used to define the parameters of the y-axis can include any of these loading conditions. Thus, in effect, the stress-strain curve is actually a load versus deformation relationship whereby the Y-axis could be labeled as torque, compressive load, tensile force, or shear. At low levels of stress there is a linear relationship between the applied load and the resultant deformation. This proportionality is known as the modulus of elasticity or Young's modulus. It is a measure of the stiffness or rigidity of bone and is equivalent to the slope of the linear part of the curve. It is calculated by dividing the stress by the strain at any point along this straight line. This linear part of the curve is also known as the elastic region. The physiological significance of this property relates to the fact that forces applied to bone at any point along this line will only deform the bone
temporarily. After the load is removed, it will return to its original shape (when an "elastic" band is stretched, it returns to its original shape when the force required to stretch it is removed). At the point where the curve becomes nonlinear, the elastic region ends and the stress at this point is known as the elastic limit. Further loading beyond this point will result in a permanent deformation in the material and this property is known as plasticity. This part of the curve is known as the plastic region. The strength of bone tissue is determined by calculating the maximum stress at the point where the bone fails (i.e., the height of the curve on the Y-axis). Depending on the loading conditions, one can refer to this as being tensile, bending, compressive, or torsional strength. The strain at the point of failure is known as the ductility. The area of the curve is a measure of the energy absorptive capacity, strain energy, or toughness of the bone (synonymous terms). This energy is dissipated when the bone fractures and is lost at the point of failure. Note that energy stored by the bone up to the point where it reaches its elastic limit is known as the resilience. This energy is recovered if the bone returns to its original shape after the load is removed. These relationships can be illustrated by examining the mechanical behavior of bone in a patient with severe Paget's disease (Fig. 6). Initially, each time this patient would take a step on the lower limb, the proximal femur would undergo mild deformation but return to its original shape when the gait switched to the other leg. Thus, the bone had exhibited elastic deformation and resilience. However, after years of walking on this diseased bone, internal changes had taken place in the bone material. Figure 6B shows the typical coxa vara deformity in a femur that has undergone plastic deformation; some of the resilience has been lost. Continued loading of this bone caused further deformation to take place and finally, one day, the bone experienced its ultimate bending load and failed. Figure 6C shows a subtrochanteric fracture. Because the bone exhibited significant deformation prior to the time of fracture, it is considered to be a ductile material. Not all materials (and, for that matter, not all bones) have significant plastic properties. Instead, a material may exhibit elastic deformation but upon reaching its yield point, will fail. This type of material is considered to be brittle. A piece of chalk is a perfect example of such a material. When you try to bend a piece of chalk it breaks before it has a chance to deform and remain bent. Thus, the "chalk bones" of patients with osteopetrosis are brittle because they show no plastic properties when they are loaded (Fig. 7). As will be discussed later, the interaction between the mineral and matrix phases of bone will dictate its mechanical properties. The patient shown in Figure 7 was studied by total body dual photon absorptiometry and shown to have a bone mineral density that was 11 SDs above the age-matched norm; nevertheless, she had brittle bones.
Structural Properties When whole bones are subjected to experimental or physiological loading conditions, their mechanical behavior can be dependent not only on the mass of the tissue and its material properties but also on its geometry and architecture. Of particular interest in the study of metabolic bone diseases are fractures of the vertebral bodies. Here, the vertebrae fracture as a result of predominantly axial compressive loads. The reduced load-bearing capacity of each vertebra is related to the material properties of the bone as well as the way by which the vertebral trabeculation is altered
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Fig. 2. Anterior-posterior radiograph and associated schematic drawing of a fracture produced by a pure tensile force that occurred in a skiing accident. In this patient, an acute tensile force produced by the anterior cruciate ligament caused an avulsion fracture of bone to occur in the tibial plateau.
4,
/
t
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! ]al tension )eculae
I
9.1 ~ ....
Fig. 3. Fracture patterns in a cylindrical section of bone subjected to different complex loading configurations. Bending (a combination of compression and tension) produces an essentially transverse fracture with a small fragment on the concave side; torsion produces a spiral fracture; axial compression causes an oblique fracture; and tension produces a purely transverse fracture [After: Carter DR, Spengler DM (1982) Biomechanics of fractures. In: Bone in clinical orthopaedics. WB Saunders, Philadelphia, pp 305-334].
through the processes of postmenopausal and age-related bone loss. Studies have shown that it is the removal of the horizontal trabeculae or lateral support cross-ties that alter the architectural arrangement and lead to reduced loadbearing capacity [8]. As a result, the vertical trabeculae begin to behave as columns and, as such, are subjected to critical buckling loads [9, 10]. A 50% reduction in crosssectional area contributed by these horizontal trabeculae will be associated with a 75% reduction in load-bearing capacity of the vertebral body [11].
Fig. 4. Diagram of trabecular systems of the proximal femur.
To carry this discussion further, let us consider why it may be that sodium fluoride treatment can lead to increased spinal bone density without a significant decrease in vertebral fracture rate [12]. If sodium fluoride is given to an osteoporotic woman who has already lost a substantial proportion of her horizontal vertebral trabeculae, the new fluorotic bone will be deposited on the remaining horizontal trabeculae and vertical trabecular columns. Although this may slightly increase the vertebrae's ability to resist buckling loads (because the trabeculae are made thicker), there is still no change in the extent of horizontal cross-tie stabilization. As a result, despite an increase in bone mass and measurable density, the ability of the bone to withstand mechanical loads has been altered only minimally.
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Point of failure Ultimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . strength ~ ' ~ - ~ ~ / ~ Yield
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STRAIN Fig, 5. A standard stress/straincurve of bone loaded in bending. The linear portion of the curve represents the elastic region and the slope of this part of the curve is used to derive the stiffness of the bone. Loading in this region will result in nonpermanent deformation, and the energy returned to the bone when the load is removed is known as resilience. The nonlinear portion of the curve represents the plastic region in which the bone will be permanently deformed by the load. The junction of these two regions defines the yield point and the stress here is known as the elastic limit. The maximum stress at the point of failure is known as the ultimate strength of the bone. The maximum strain at this point represents the bone's ductility. The area under the curve is known as the strain energy, and the total energy stored at the point of fracture defines the toughness of the material.
Most clinical fractures occur as a result of a combination of axial compression, bending, and torsion. In bending or torsion, the cross-sectional area of a structure is more important in resisting loads than is its mass or density [4]. Ideally, in bending or torsion, bone should be distributed as far away from the neutral axis of the load as possible. The geometric parameter used to describe this phenomenon in bending is the "areal moment of inertia" [4]. Similarly, in torsion, deformation would be resisted more efficiently if bone were distributed further away from the neutral torsional axis. This property is known as the " p o l a r moment of inertia" [4] (Fig. 8). During aging, the outer cortical diameter of bone increases and the cortical wall diameter becomes thinner [13, 14]. This results from the combined effects of increased endosteal resorption and periosteal bone formation. Although the net effect may be cortical thinning, the increased diameter of the bone distributes the material further from the neutral axis and improves its resistance to bending and torsional loads through its areal and polar moment properties. It is now possible to understand why fractures occur at specific sites in the skeleton as a result of aging and metaboric bone disease. Table 1 shows the relative contents of trabecular bone in various parts of the skeleton. Conditions such as osteoporosis will result in the loss of horizontal trabeculae and lead to fractures of the vertebrae and intertrochanteric regions of the hip. These parts of the skeleton rely heavily on trabecular bone to support loads [7, 15]. On the other hand, it is unusual to find osteoporotic fractures occurring in the mid-radius or femoral shaft [16]. This is because these areas are largely composed of cortical bone,
and, as the diameter of the bone expands with aging, there is a resistance to torsional and bending loads. The femoral neck, however, is not protected in the same way as the diaphyses of the long bones. Although this part of the skeleton is largely composed of cortical bone, the unique anatomy of the proximal femur is such that the periosteum is absent from the femoral neck, that part of the femur that is within the hip joint [17, 18]. Consequently, during the aging process, endosteal bone resorption leads to cortical thinning without the associated periosteal response required for appositional bone growth to occur. Hence, the cortex thins without an increase in cross-sectional diameter and the bone becomes susceptible to fracture. This is the only part of the skeleton that demonstrates this phenomenon.
Stress Fractures In order for a fracture to occur, it is not always necessary that the ultimate stress be exceeded. Repeated loading and unloading of the bone can cause it to fail even if loads are below this level. This phenomenon is known as fatigue failure and the fractures that result from this type of loading are known as stress fractures. Fatigue failure occurs when each loading cycle produces a minute amount of microdamage which accumulates with repetitive loads [19]. Biological materials such as bone have repair mechanisms for coping with microdamage as it occurs. Under normal conditions of healing, microdamage will occur but not accumulate because it will be repaired in a timely fashion. However, in bones where the normal repair mechanisms are impaired or attenuated (e.g., in certain metabolic bone diseases), or are repetitively loaded over short periods without sufficient time for a reparative response (e.g., during intense basic military training), bones may exhibit fatigue failure after several cycles of loading [19]. A high incidence of stress fractures in patients who are chronically maintained on glucocortocoids is due to the combined effects of osteoporosis and the impaired healing of microfractures. Sodium fluoride may have a similar effect on bone and this may explain the increased incidence of appendicular fractures observed in certain fluoride-treated patients [20, 21]. Biomechanically, stress fractures most likely occur when bone is loaded repeatedly in the plastic region [22]. Here, deformation in the material has already occurred and repeated damage will eventually lead to the material's reaching its ultimate point of failure. However, fatigue failure in the elastic region is also possible, particularly in bones that are more brittle. This will require a large number of loading repetitions over a short duration [22]. Stress fractures may occur on either the tension or the compression side of the bone. It should be emphasized that a stress fracture on the tension side, resulting in a crack in the cortex, is serious because it may rapidly go on to a complete fracture. Stress fractures that occur on the compression side of bones appear to result from a slower process, in which repair mechanisms may be more easily mobilized, leading to healing before a complete fracture occurs [23].
Bone Biochemistry and Material Properties Bone is a two-phase composite material made up of a mineral phase deposited in a protein-rich matrix. By weight, it is composed of approximately 70% mineral, 22% protein, and
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Fig. 6. The mechanical behavior of bone in a patient with severe Paget's disease: (A) Anterior-posterior radiograph of the proximal femur showing significant osteolytic pagetoid changes. When the bone is subjected to normal physiological loads, any deformation that occurs will be nonpermanent and bone will return to its original shape after the load is removed. (B) Plastic deformation of a pagetoid femur showing a permanent change (varus angulation) in the shape of the bone. The bending stresses have exceeded the elastic limit of the bone. (C) Subtrochanteric fracture of a pagetoid femur. The bending stresses have exceeded the ultimate strength of the bone.
18% water [24]. Currey [25] proposed an analogy for bone in which he compared it to fiberglass, a glass, fiber-reinforced epoxy resin matrix. He suggested that the mineral phase of bone is similar to the glass fiber in fiberglass because it acts to reinforce the matrix material. Burstein et al. [26] tested this hypothesis by investigating the individual contributions of the collagen and mineral phases to the elastic and plastic properties of bone. By sequentially demineralizing machined strips of cortical bone and subjecting them to bending loads, the mineral phase was shown to contribute the major proportion of the elastic properties (stiffness) of bone whereas the plastic region of the stress-strain curve was shown to be solely a function of the matrix. The ultimate yield strength of
bone was shown to be related to both its mineral content, as well as the way by which the mineral is distributed within the collagenous matrix. Because the ductility of bone is largely a function of its plastic properties, ductility will be affected predominantly by conditions that alter the collagenous matrix. In osteoporosis, for example, the biochemical composition of bone is unchanged although its amount is reduced. Therefore, in osteoporosis, all mechanical properties of bone should be affected equally and bone will fail under smaller loads. On the other hand, in a condition such as osteogenesis imperfecta, the collagen component of bone is primarily affected. Stiffness and strength will be reduced proportionately but the ability of the bone to deform under load will be
Editorial
338 Table 1. Trabecular bone content at various skeletal sites Vertebrae Hip (intertrochanteric) Hip (femoral neck) Distal radius Mid-radius Femoral shaft
66-90% 50% 25% 25% 1% 5%
'I
Fig. 9. Cartoon demonstrating that two brick walls of the same mass and dimensions will have different mechanical properties if one is constantly being excavated and patched whereas the other undergoes controlled remodeling. This situation may be analogous to bone that is in a state of high turnover versus bone that experiences normal homeostatic remodeling. Fig. 7. Proximal humeral fracture in a patient with osteopetrosis. Because of the brittle nature of this bone, the fracture occurred without any plastic deformation. [Reprinted with permission from Labat ML, Bringuier AF, Chandra A, Einhorn TA, Chandra P (1990) Retroviral expression in mononuclear blood cells isolated from a patient with osteopetrosis (Albers-Schrnberg Disease). J Bone Miner Res 5:425-435].
greatly compromised. In osteomalacia, both stiffness and strength should be affected because it is the mineral phase that is deficient in an otherwise normal collagenous matrix. Plastic deformation should be essentially normal.
Bone Biochemistry and Structural Properties f
F'-----..~
o Bending s t i f f n e s s = E I
@
I
T
3e
T
J
Torsional stiffness = GJ E= Elastic modulus G = Shear modulus
I = Areal moment of inertia d = Polar moment of inertia
Fig. 8. Diagram of the structural stiffnesses of long bones loaded in bending and torsion. Note that the elastic modulus and the shear modulus, measurements of stiffness in bending and torsion, are related to the areal and polar moments of inertia, respectively. These moments of inertia are increased, and thus the structural stiffnesses are increased, when bone is distributed further away from the neutral axis of bending or torsion.
Recent work has further elucidated the contributions of the collagen phase to cortical bone's mechanical properties [27]. Using genetically engineered constructs of D N A sequences, transgenic mice expressing an abnormal type I collagen gene product have been developed. As the matrix phases of these bones have been rendered biochemically abnormal, they represent conditions that may be analogous to human collagen diseases. As predicted by the above model, these experiments have shown that reduced synthesis of type I collagen leads to a reduction in strength-related properties and plastic deformation when static loading tests are performed early in the life cycle of the animal. However, within weeks, an adaptive change in bone geometry occurs leading to cortical expansion and a normalization of structural properties due to increases in the areal and polar moments of inertia [27]. This mechanically compensated relationship among the structural geometry, bone composition, and biomechanical properties exemplifies the important interactions between form and function. In other words, the skeleton "knows something is wrong" with its material composition and adapts structurally to compensate for it.
Editorial
Afterthoughts We can now begin to understand why only certain patients with low bone mass express the osteoporotic syndrome whereas others do not. Although studies have shown that approximately 759"0--80% of the variance in the ultimate strength of bone is accounted for by bone density [16], a variety of changes in the material composition or structural geometry of the skeleton can offset the effects of altered bone mineral content. In addition, cellular activity may also have a role to play in the mechanical properties of bone, although this aspect of bone biomechanics has not been studied carefully. It is easy to understand how bone, which is in a state of high turnover and is constantly undergoing breakdown and repair, may have reduced mechanical properties when compared with bone that exhibits normal homeostasis. The resorptive cavities produced by osteoclastic action may act as stress risers for the initiation of crack propogation, even before osteoblasts have had a chance to fill them in with new bone [28, 29]. By analogy, two brick walls of the same mass and dimensions will have different mechanical properties if one is constantly excavated and patched with cement and the other undergoes controlled remodeling (Fig. 9). In the next few years, new pharmaceutical agents and biophysical modalities may be introduced in an effort to treat metabolic bone diseases. These treatments may independently affect bone biochemistry, cellular activity, or bone's adaptive response to mechanical strain. The principles and concepts outlined above should enable the physician and scientist to plan thoughtful experiments and rational therapeutic programs for the implementation of these advances. In the final analysis, the desired outcome of all treatment regimens is to improve bone integrity and reduce the incidence of fractures. That's the "bottom line." Thomas A. Einhorn, M.D. Department of Orthopaedics Mount Sinai School of Medicine New York, New York USA
References 1. von Meyer GH (1867) Die Architektur der Spongiosa. Arch Anat Physiol Wissenhaftliche Med (Reichert und DuboisReymonds Archiv) 34:615-628 2. Culmann C (1866) Die Graphische Statik, 1. Auflage. Mayer and Zeller, Zurich 3. WolffJ (1892) Das Gaesetz der Transformation der Knochen. A. Hirchwald, Berlin 4. Melton LJ, Chat EYS, Lane JM (1988) Biomechanical aspects of fractures. In: Riggs BL, Melton LJ (eds) Osteoporosis: etiology, diagnosis and management. Raven Press, New York, pp 111-131 5. Burstein AH, Reilly DT, Martens MJ (1976) Aging of bone tissue: mechanical properties. J Bone Jt Surg 58A:82-86 6. Galante J, Rostoker W, Ray RD (1970) Physical properties of trabecular bone. Calcif Tissue Res 5:236-246
339 7. Brown TD, Ferguson AB Jr (1978) The development of a computational stress analysis of the femoral head, J Bone Jt Surg 60A:619--629 8. Mosekilde Li, Viidik A, Mosekilde LE (1985) Correlation between the compressive strength of iliac and vertebral trabecular bone in normal individuals. Bone 6:291-295 9. Higdon A, Ohlsen EH, Stiles WB, Weese JA, Riley WF (1985) 10. Townsend PR (1975) Buckling studies of single human trabeculae. J Biomech 8:199-201 11. Yamada H (1970) Strength of biological materials. Evans FG (ed) Williams and Williams Co, Baltimore 12. Riggs BL, Hodgson SF, O'Fallon M, Chat EYS, Wahner HW, Muhs BSN, Cedel SL, Melton LJ III (1990) Effect of fluoride treatment on the fracture rate of postmenopausal women with osteoporosis. N Engl J Med 322:802-809 13. Smith RW, Walker RW (1980) Femoral expansion in aging women: implications for osteoporosis and fractures. Henry Ford Hosp Med J 28:168-170 14. Ruff CB, Hayes WC (1982) Superiosteal expansion and cortical remodeling of the human femur and tibia with aging. Science 217:945-948 15. McBroom RJ, Hayes WC, Edwards WT,Goldberg RP, White AA (1985) Prediction of vertebral body compressive fracture using quantitative computed tomography. J Bone Jt Surg 67A: 1206-1214 16. Smith CB, Smith DA (1976) Relations between age, mineral density and mechanical properties of human femoral compacta. Acta Orthop Scand 47:496-502 17. Phemister DB (1939) The pathology of ununited fractures of the neck of the femur with special reference to the head. J Bone Jt Surg 21:681-693 18. Pankovich AM (1975) Primary internal fixation of femoral neck fractures. Arch Surg 110:20--26 19. Stanitski CL, McMaster JH, Scranton PE (1978) On the nature of stress fractures. Am J Sports Med 6:391-396 20. Einhorn TA, VigoritaVJ (1987) Unique histology of the fracture callus in a sodium fluoride (NaF)-treated osteoporotic patient with hip fracture. In: Christiansen C, Johansen JS, Riis BJ (eds) Osteoporosis. Norhaven, Viborg, pp 262-265 21. Boivin G, Grousson B, Meunier PJ (1991) X-ray microanalysis of fluoride distribution in microfracture calluses in cancellous iliac bone from osteoporotic patients treated with fluoride and untreated. J Bone Miner Res 6:1183-1190 22. Chamay A (1970) Mechanical and morphological aspects of experimental overload and fatigue in bone. J Biomech 3:263-270 23. Baker J, Frankel V, Burstein A (1972) Fatigue fractures: bitmechanical considerations. J Bone Jt Surg 54A: 1345-1346 24. Lane JM (1979) Biochemistry of fracture healing. AAOS Montery Seminar, Chicago. Am Acad Orthop Surg, pp 141-165 25. Currey JD (1989) Strain dependence of the mechanical properties of reindeer antler and the cumulative damage model of bone fracture. J Biomech 22:469--476 26. Burstein AH, Zika JC, Heiple KG, Klein L (1977) Contribution of collagen and mineral to the elastic-plastic properties of bone. J Bone Jt Surg 57A:956-961 27. Jepsen KJ, Mansoura MK, Kuhn JL, Wu H, Jaemisch R, Bonadio JF, Goldstein SA (1992) An in vivo assessment of the contribution of type I collagen to the mechanical properties of cortical bone. Trans Orthop Res Soc 32:93 28. Goodier JN (1933) Concentration of stress around spherical and cylindrical inclusions and flaws. J Appl Mecb 55:39 29. Currey JD (1962) Stress concentration in bone. Quart J Micro Sci 103:111-133
Calcified Tissue International 9 1992 Springer-Verlag New York Inc.
Editorial
Bone Strength: The Bottom Line The human skeleton is a mechanically optimized biological system whose composition and organization reflect the functional demands made upon it. According to a set of physical relationships derived by German anatomists during the late 19th century [1, 2], and presently defined as Wolff's Law [3], the patterns of trabecular alignment and the geometry of the cortical compartments are the results of functional adaptation to normal physiological loads. Under these conditions, the human skeleton can fulfill all of its required activities including locomotion, protection, and the ability to participate in metabolic pathways associated with mineral homeostasis. Consequently, the molecular, cellular, and metabolic changes that occur in bone are directed to maintain this mechanical environment or to adapt to any new loading conditions that it may experience. Failure to do this leads to the development of pathological fractures and the skeletal pain associated with a variety of metabolic bone diseases. To understand the interactions between bone physiology and bone integrity, a basic knowledge of biomechanics is needed. This communication will review the fundamental principles of bone biomechanics with specific reference to fracture risk.
Basic Concepts Any basic knowledge of biomechanics must begin with an understanding of the terms stress and strain. These terms are used to describe the phenomenon whereby, when a force is applied to bone, the bone will not only be deformed from its original dimensions but an internal resistance will be generated to the counter applied force. This internal reaction is known as stress; it is equal in magnitude but opposite in direction to the applied force. As stress is distributed over the cross-sectional area of the bone, it is expressed in units of force per unit area. The standard international unit for stress is the Pascal which is equivalent to one Newton of force distributed over 1 square meter. Strain is a term used to describe the changes in dimension which bone experiences under the influence of an applied force. Strain is dimensionless and is therefore reported as a fraction or a percentage. It is equivalent to the change in length divided by the original length of the bone section. Although an applied force can be directed at bone from any angle producing any set of complex stress patterns, all stresses can be resolved into three types: tension, compression, and shear (Fig. 1). Tension is produced in bone when two forces are directed away from each other along the same straight line. The resistance to a loading situation of this type is produced by intermolecular attractive forces which prevent the bone from being torn apart. In nature, it is unusual for a bone to experience a pure tensile force (i.e., it
would have to literally be pulled apart). However, an example of a tensile force producing a failure in bone occurs when a tendon or ligament that is inserted into bone undergoes acute loading and, instead of failing within its own substance (tearing), it detaches itself from the bone by actually pulling a piece of bone off with it (Fig. 2). Compression results from two forces that are directed towards each other along the same straight line. The common vertebral compression fracture sustained in osteoporotic patients is an example of the failure of bone as a result of this type of loading configuration. Lastly, when two forces are directed parallel to each other but not along the same line, shear stresses are produced. In nature, the three basic stress types can combine as a result of a variety of complex loading configurations and lead to different fracture patterns (Fig. 3). Bending, for example, results from a combination of tensile and compressive forces (tension on the convex side; compression on the concave side) and is one of the more common loading conditions that will lead to clinical fractures. Torsion or twisting produces shear stresses along the entire length of a bone and can result in spiral fractures. Comminuted fractures appear as shattered bones and this occurs because the amount of force applied to the bone is so great that a variety of fracture lines have to be generated in order to dissipate the energy transmitted. Because stress and strain are properties related to the quality of the material experiencing the load, the quality of bone can influence the magnitude of the stresses and strains generated. In normal, well-mineralized bone tissue, an applied stress will generally result in small strains. In poorly mineralized tissue, such as osteomalacic bone, bone will experience larger strains in response to the same stress. Moreover, since in nature, forces are applied to bone not only from perpendicular and horizontal directions but also from oblique angles, situations will arise in which a variety of complex mechanical relationships will be generated. When bones show different mechanical properties if loads are transmitted from different directions, they express a property known as anisotropy [4]. In general, bones resist loads best when they are oriented in the direction of customary loading. For example, the femur is much better adapted to resisting compressive loads than bending loads [5]. Thus, the same stresses generated in the femur when you jump up and down (compressive stresses) may fracture the femur if they are oriented from a transverse direction (bending stresses). Similarly, in the vertebrae, the ability to resist loads is greatest when the stresses are compressive [6]. However, in the proximal femur (hip joint area), loads are best resisted when they are transmitted along lines that are parallel to the trabecular systems [7] (Fig. 4).
Biomechanical Properties The biomechanical properties of bone can be described at
Editorial
334
TENSION COMPRESSION SHEAR
Fig. 1. Schematic representation of the three basic types of stress: tension, compression, and shear.
two levels. First, the material properties of bone must be considered and these properties are defined by the tissuelevel qualities of bone which are independent of structure or geometry. Second, the structural properties of bone, which function as a whole anatomical unit, should be examined. Classically, the material properties of bone are defined by performing standardized mechanical tests on uniform, machined specimens of intact bone. Structural properties are determined on whole sections of intact bone whose normal geometry has been maintained. It is important to recognize that a fracture sustained in a patient may represent the failure of bone at either the material or structural level or both.
Material Properties When a uniform section of bone is tested under controlled laboratory conditions, and the applied forces and deformations are known, the four basic mechanical properties of bone can be derived from a plot of the stress-strain relationship. These properties are strength, stiffness, energyabsorptive capacity, and deformation. By convention, stress is plotted on the ordinate (y-axis) and strain on the abscissa (x-axis). Figure 5 shows a stress-strain plot of an idealized material. Considering the fact that the three types of stress can combine to produce different stress patterns as a result of different types of externally applied loads (tension, compression, bending, and torsion), the terms used to define the parameters of the y-axis can include any of these loading conditions. Thus, in effect, the stress-strain curve is actually a load versus deformation relationship whereby the Y-axis could be labeled as torque, compressive load, tensile force, or shear. At low levels of stress there is a linear relationship between the applied load and the resultant deformation. This proportionality is known as the modulus of elasticity or Young's modulus. It is a measure of the stiffness or rigidity of bone and is equivalent to the slope of the linear part of the curve. It is calculated by dividing the stress by the strain at any point along this straight line. This linear part of the curve is also known as the elastic region. The physiological significance of this property relates to the fact that forces applied to bone at any point along this line will only deform the bone
temporarily. After the load is removed, it will return to its original shape (when an "elastic" band is stretched, it returns to its original shape when the force required to stretch it is removed). At the point where the curve becomes nonlinear, the elastic region ends and the stress at this point is known as the elastic limit. Further loading beyond this point will result in a permanent deformation in the material and this property is known as plasticity. This part of the curve is known as the plastic region. The strength of bone tissue is determined by calculating the maximum stress at the point where the bone fails (i.e., the height of the curve on the Y-axis). Depending on the loading conditions, one can refer to this as being tensile, bending, compressive, or torsional strength. The strain at the point of failure is known as the ductility. The area of the curve is a measure of the energy absorptive capacity, strain energy, or toughness of the bone (synonymous terms). This energy is dissipated when the bone fractures and is lost at the point of failure. Note that energy stored by the bone up to the point where it reaches its elastic limit is known as the resilience. This energy is recovered if the bone returns to its original shape after the load is removed. These relationships can be illustrated by examining the mechanical behavior of bone in a patient with severe Paget's disease (Fig. 6). Initially, each time this patient would take a step on the lower limb, the proximal femur would undergo mild deformation but return to its original shape when the gait switched to the other leg. Thus, the bone had exhibited elastic deformation and resilience. However, after years of walking on this diseased bone, internal changes had taken place in the bone material. Figure 6B shows the typical coxa vara deformity in a femur that has undergone plastic deformation; some of the resilience has been lost. Continued loading of this bone caused further deformation to take place and finally, one day, the bone experienced its ultimate bending load and failed. Figure 6C shows a subtrochanteric fracture. Because the bone exhibited significant deformation prior to the time of fracture, it is considered to be a ductile material. Not all materials (and, for that matter, not all bones) have significant plastic properties. Instead, a material may exhibit elastic deformation but upon reaching its yield point, will fail. This type of material is considered to be brittle. A piece of chalk is a perfect example of such a material. When you try to bend a piece of chalk it breaks before it has a chance to deform and remain bent. Thus, the "chalk bones" of patients with osteopetrosis are brittle because they show no plastic properties when they are loaded (Fig. 7). As will be discussed later, the interaction between the mineral and matrix phases of bone will dictate its mechanical properties. The patient shown in Figure 7 was studied by total body dual photon absorptiometry and shown to have a bone mineral density that was 11 SDs above the age-matched norm; nevertheless, she had brittle bones.
Structural Properties When whole bones are subjected to experimental or physiological loading conditions, their mechanical behavior can be dependent not only on the mass of the tissue and its material properties but also on its geometry and architecture. Of particular interest in the study of metabolic bone diseases are fractures of the vertebral bodies. Here, the vertebrae fracture as a result of predominantly axial compressive loads. The reduced load-bearing capacity of each vertebra is related to the material properties of the bone as well as the way by which the vertebral trabeculation is altered
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Fig. 2. Anterior-posterior radiograph and associated schematic drawing of a fracture produced by a pure tensile force that occurred in a skiing accident. In this patient, an acute tensile force produced by the anterior cruciate ligament caused an avulsion fracture of bone to occur in the tibial plateau.
4,
/
t
prin( coral
! ]al tension )eculae
I
9.1 ~ ....
Fig. 3. Fracture patterns in a cylindrical section of bone subjected to different complex loading configurations. Bending (a combination of compression and tension) produces an essentially transverse fracture with a small fragment on the concave side; torsion produces a spiral fracture; axial compression causes an oblique fracture; and tension produces a purely transverse fracture [After: Carter DR, Spengler DM (1982) Biomechanics of fractures. In: Bone in clinical orthopaedics. WB Saunders, Philadelphia, pp 305-334].
through the processes of postmenopausal and age-related bone loss. Studies have shown that it is the removal of the horizontal trabeculae or lateral support cross-ties that alter the architectural arrangement and lead to reduced loadbearing capacity [8]. As a result, the vertical trabeculae begin to behave as columns and, as such, are subjected to critical buckling loads [9, 10]. A 50% reduction in crosssectional area contributed by these horizontal trabeculae will be associated with a 75% reduction in load-bearing capacity of the vertebral body [11].
Fig. 4. Diagram of trabecular systems of the proximal femur.
To carry this discussion further, let us consider why it may be that sodium fluoride treatment can lead to increased spinal bone density without a significant decrease in vertebral fracture rate [12]. If sodium fluoride is given to an osteoporotic woman who has already lost a substantial proportion of her horizontal vertebral trabeculae, the new fluorotic bone will be deposited on the remaining horizontal trabeculae and vertical trabecular columns. Although this may slightly increase the vertebrae's ability to resist buckling loads (because the trabeculae are made thicker), there is still no change in the extent of horizontal cross-tie stabilization. As a result, despite an increase in bone mass and measurable density, the ability of the bone to withstand mechanical loads has been altered only minimally.
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Point of failure Ultimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . strength ~ ' ~ - ~ ~ / ~ Yield
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STRAIN Fig, 5. A standard stress/straincurve of bone loaded in bending. The linear portion of the curve represents the elastic region and the slope of this part of the curve is used to derive the stiffness of the bone. Loading in this region will result in nonpermanent deformation, and the energy returned to the bone when the load is removed is known as resilience. The nonlinear portion of the curve represents the plastic region in which the bone will be permanently deformed by the load. The junction of these two regions defines the yield point and the stress here is known as the elastic limit. The maximum stress at the point of failure is known as the ultimate strength of the bone. The maximum strain at this point represents the bone's ductility. The area under the curve is known as the strain energy, and the total energy stored at the point of fracture defines the toughness of the material.
Most clinical fractures occur as a result of a combination of axial compression, bending, and torsion. In bending or torsion, the cross-sectional area of a structure is more important in resisting loads than is its mass or density [4]. Ideally, in bending or torsion, bone should be distributed as far away from the neutral axis of the load as possible. The geometric parameter used to describe this phenomenon in bending is the "areal moment of inertia" [4]. Similarly, in torsion, deformation would be resisted more efficiently if bone were distributed further away from the neutral torsional axis. This property is known as the " p o l a r moment of inertia" [4] (Fig. 8). During aging, the outer cortical diameter of bone increases and the cortical wall diameter becomes thinner [13, 14]. This results from the combined effects of increased endosteal resorption and periosteal bone formation. Although the net effect may be cortical thinning, the increased diameter of the bone distributes the material further from the neutral axis and improves its resistance to bending and torsional loads through its areal and polar moment properties. It is now possible to understand why fractures occur at specific sites in the skeleton as a result of aging and metaboric bone disease. Table 1 shows the relative contents of trabecular bone in various parts of the skeleton. Conditions such as osteoporosis will result in the loss of horizontal trabeculae and lead to fractures of the vertebrae and intertrochanteric regions of the hip. These parts of the skeleton rely heavily on trabecular bone to support loads [7, 15]. On the other hand, it is unusual to find osteoporotic fractures occurring in the mid-radius or femoral shaft [16]. This is because these areas are largely composed of cortical bone,
and, as the diameter of the bone expands with aging, there is a resistance to torsional and bending loads. The femoral neck, however, is not protected in the same way as the diaphyses of the long bones. Although this part of the skeleton is largely composed of cortical bone, the unique anatomy of the proximal femur is such that the periosteum is absent from the femoral neck, that part of the femur that is within the hip joint [17, 18]. Consequently, during the aging process, endosteal bone resorption leads to cortical thinning without the associated periosteal response required for appositional bone growth to occur. Hence, the cortex thins without an increase in cross-sectional diameter and the bone becomes susceptible to fracture. This is the only part of the skeleton that demonstrates this phenomenon.
Stress Fractures In order for a fracture to occur, it is not always necessary that the ultimate stress be exceeded. Repeated loading and unloading of the bone can cause it to fail even if loads are below this level. This phenomenon is known as fatigue failure and the fractures that result from this type of loading are known as stress fractures. Fatigue failure occurs when each loading cycle produces a minute amount of microdamage which accumulates with repetitive loads [19]. Biological materials such as bone have repair mechanisms for coping with microdamage as it occurs. Under normal conditions of healing, microdamage will occur but not accumulate because it will be repaired in a timely fashion. However, in bones where the normal repair mechanisms are impaired or attenuated (e.g., in certain metabolic bone diseases), or are repetitively loaded over short periods without sufficient time for a reparative response (e.g., during intense basic military training), bones may exhibit fatigue failure after several cycles of loading [19]. A high incidence of stress fractures in patients who are chronically maintained on glucocortocoids is due to the combined effects of osteoporosis and the impaired healing of microfractures. Sodium fluoride may have a similar effect on bone and this may explain the increased incidence of appendicular fractures observed in certain fluoride-treated patients [20, 21]. Biomechanically, stress fractures most likely occur when bone is loaded repeatedly in the plastic region [22]. Here, deformation in the material has already occurred and repeated damage will eventually lead to the material's reaching its ultimate point of failure. However, fatigue failure in the elastic region is also possible, particularly in bones that are more brittle. This will require a large number of loading repetitions over a short duration [22]. Stress fractures may occur on either the tension or the compression side of the bone. It should be emphasized that a stress fracture on the tension side, resulting in a crack in the cortex, is serious because it may rapidly go on to a complete fracture. Stress fractures that occur on the compression side of bones appear to result from a slower process, in which repair mechanisms may be more easily mobilized, leading to healing before a complete fracture occurs [23].
Bone Biochemistry and Material Properties Bone is a two-phase composite material made up of a mineral phase deposited in a protein-rich matrix. By weight, it is composed of approximately 70% mineral, 22% protein, and
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Fig. 6. The mechanical behavior of bone in a patient with severe Paget's disease: (A) Anterior-posterior radiograph of the proximal femur showing significant osteolytic pagetoid changes. When the bone is subjected to normal physiological loads, any deformation that occurs will be nonpermanent and bone will return to its original shape after the load is removed. (B) Plastic deformation of a pagetoid femur showing a permanent change (varus angulation) in the shape of the bone. The bending stresses have exceeded the elastic limit of the bone. (C) Subtrochanteric fracture of a pagetoid femur. The bending stresses have exceeded the ultimate strength of the bone.
18% water [24]. Currey [25] proposed an analogy for bone in which he compared it to fiberglass, a glass, fiber-reinforced epoxy resin matrix. He suggested that the mineral phase of bone is similar to the glass fiber in fiberglass because it acts to reinforce the matrix material. Burstein et al. [26] tested this hypothesis by investigating the individual contributions of the collagen and mineral phases to the elastic and plastic properties of bone. By sequentially demineralizing machined strips of cortical bone and subjecting them to bending loads, the mineral phase was shown to contribute the major proportion of the elastic properties (stiffness) of bone whereas the plastic region of the stress-strain curve was shown to be solely a function of the matrix. The ultimate yield strength of
bone was shown to be related to both its mineral content, as well as the way by which the mineral is distributed within the collagenous matrix. Because the ductility of bone is largely a function of its plastic properties, ductility will be affected predominantly by conditions that alter the collagenous matrix. In osteoporosis, for example, the biochemical composition of bone is unchanged although its amount is reduced. Therefore, in osteoporosis, all mechanical properties of bone should be affected equally and bone will fail under smaller loads. On the other hand, in a condition such as osteogenesis imperfecta, the collagen component of bone is primarily affected. Stiffness and strength will be reduced proportionately but the ability of the bone to deform under load will be
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338 Table 1. Trabecular bone content at various skeletal sites Vertebrae Hip (intertrochanteric) Hip (femoral neck) Distal radius Mid-radius Femoral shaft
66-90% 50% 25% 25% 1% 5%
'I
Fig. 9. Cartoon demonstrating that two brick walls of the same mass and dimensions will have different mechanical properties if one is constantly being excavated and patched whereas the other undergoes controlled remodeling. This situation may be analogous to bone that is in a state of high turnover versus bone that experiences normal homeostatic remodeling. Fig. 7. Proximal humeral fracture in a patient with osteopetrosis. Because of the brittle nature of this bone, the fracture occurred without any plastic deformation. [Reprinted with permission from Labat ML, Bringuier AF, Chandra A, Einhorn TA, Chandra P (1990) Retroviral expression in mononuclear blood cells isolated from a patient with osteopetrosis (Albers-Schrnberg Disease). J Bone Miner Res 5:425-435].
greatly compromised. In osteomalacia, both stiffness and strength should be affected because it is the mineral phase that is deficient in an otherwise normal collagenous matrix. Plastic deformation should be essentially normal.
Bone Biochemistry and Structural Properties f
F'-----..~
o Bending s t i f f n e s s = E I
@
I
T
3e
T
J
Torsional stiffness = GJ E= Elastic modulus G = Shear modulus
I = Areal moment of inertia d = Polar moment of inertia
Fig. 8. Diagram of the structural stiffnesses of long bones loaded in bending and torsion. Note that the elastic modulus and the shear modulus, measurements of stiffness in bending and torsion, are related to the areal and polar moments of inertia, respectively. These moments of inertia are increased, and thus the structural stiffnesses are increased, when bone is distributed further away from the neutral axis of bending or torsion.
Recent work has further elucidated the contributions of the collagen phase to cortical bone's mechanical properties [27]. Using genetically engineered constructs of D N A sequences, transgenic mice expressing an abnormal type I collagen gene product have been developed. As the matrix phases of these bones have been rendered biochemically abnormal, they represent conditions that may be analogous to human collagen diseases. As predicted by the above model, these experiments have shown that reduced synthesis of type I collagen leads to a reduction in strength-related properties and plastic deformation when static loading tests are performed early in the life cycle of the animal. However, within weeks, an adaptive change in bone geometry occurs leading to cortical expansion and a normalization of structural properties due to increases in the areal and polar moments of inertia [27]. This mechanically compensated relationship among the structural geometry, bone composition, and biomechanical properties exemplifies the important interactions between form and function. In other words, the skeleton "knows something is wrong" with its material composition and adapts structurally to compensate for it.
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Afterthoughts We can now begin to understand why only certain patients with low bone mass express the osteoporotic syndrome whereas others do not. Although studies have shown that approximately 759"0--80% of the variance in the ultimate strength of bone is accounted for by bone density [16], a variety of changes in the material composition or structural geometry of the skeleton can offset the effects of altered bone mineral content. In addition, cellular activity may also have a role to play in the mechanical properties of bone, although this aspect of bone biomechanics has not been studied carefully. It is easy to understand how bone, which is in a state of high turnover and is constantly undergoing breakdown and repair, may have reduced mechanical properties when compared with bone that exhibits normal homeostasis. The resorptive cavities produced by osteoclastic action may act as stress risers for the initiation of crack propogation, even before osteoblasts have had a chance to fill them in with new bone [28, 29]. By analogy, two brick walls of the same mass and dimensions will have different mechanical properties if one is constantly excavated and patched with cement and the other undergoes controlled remodeling (Fig. 9). In the next few years, new pharmaceutical agents and biophysical modalities may be introduced in an effort to treat metabolic bone diseases. These treatments may independently affect bone biochemistry, cellular activity, or bone's adaptive response to mechanical strain. The principles and concepts outlined above should enable the physician and scientist to plan thoughtful experiments and rational therapeutic programs for the implementation of these advances. In the final analysis, the desired outcome of all treatment regimens is to improve bone integrity and reduce the incidence of fractures. That's the "bottom line." Thomas A. Einhorn, M.D. Department of Orthopaedics Mount Sinai School of Medicine New York, New York USA
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339 7. Brown TD, Ferguson AB Jr (1978) The development of a computational stress analysis of the femoral head, J Bone Jt Surg 60A:619--629 8. Mosekilde Li, Viidik A, Mosekilde LE (1985) Correlation between the compressive strength of iliac and vertebral trabecular bone in normal individuals. Bone 6:291-295 9. Higdon A, Ohlsen EH, Stiles WB, Weese JA, Riley WF (1985) 10. Townsend PR (1975) Buckling studies of single human trabeculae. J Biomech 8:199-201 11. Yamada H (1970) Strength of biological materials. Evans FG (ed) Williams and Williams Co, Baltimore 12. Riggs BL, Hodgson SF, O'Fallon M, Chat EYS, Wahner HW, Muhs BSN, Cedel SL, Melton LJ III (1990) Effect of fluoride treatment on the fracture rate of postmenopausal women with osteoporosis. N Engl J Med 322:802-809 13. Smith RW, Walker RW (1980) Femoral expansion in aging women: implications for osteoporosis and fractures. Henry Ford Hosp Med J 28:168-170 14. Ruff CB, Hayes WC (1982) Superiosteal expansion and cortical remodeling of the human femur and tibia with aging. Science 217:945-948 15. McBroom RJ, Hayes WC, Edwards WT,Goldberg RP, White AA (1985) Prediction of vertebral body compressive fracture using quantitative computed tomography. J Bone Jt Surg 67A: 1206-1214 16. Smith CB, Smith DA (1976) Relations between age, mineral density and mechanical properties of human femoral compacta. Acta Orthop Scand 47:496-502 17. Phemister DB (1939) The pathology of ununited fractures of the neck of the femur with special reference to the head. J Bone Jt Surg 21:681-693 18. Pankovich AM (1975) Primary internal fixation of femoral neck fractures. Arch Surg 110:20--26 19. Stanitski CL, McMaster JH, Scranton PE (1978) On the nature of stress fractures. Am J Sports Med 6:391-396 20. Einhorn TA, VigoritaVJ (1987) Unique histology of the fracture callus in a sodium fluoride (NaF)-treated osteoporotic patient with hip fracture. In: Christiansen C, Johansen JS, Riis BJ (eds) Osteoporosis. Norhaven, Viborg, pp 262-265 21. Boivin G, Grousson B, Meunier PJ (1991) X-ray microanalysis of fluoride distribution in microfracture calluses in cancellous iliac bone from osteoporotic patients treated with fluoride and untreated. J Bone Miner Res 6:1183-1190 22. Chamay A (1970) Mechanical and morphological aspects of experimental overload and fatigue in bone. J Biomech 3:263-270 23. Baker J, Frankel V, Burstein A (1972) Fatigue fractures: bitmechanical considerations. J Bone Jt Surg 54A: 1345-1346 24. Lane JM (1979) Biochemistry of fracture healing. AAOS Montery Seminar, Chicago. Am Acad Orthop Surg, pp 141-165 25. Currey JD (1989) Strain dependence of the mechanical properties of reindeer antler and the cumulative damage model of bone fracture. J Biomech 22:469--476 26. Burstein AH, Zika JC, Heiple KG, Klein L (1977) Contribution of collagen and mineral to the elastic-plastic properties of bone. J Bone Jt Surg 57A:956-961 27. Jepsen KJ, Mansoura MK, Kuhn JL, Wu H, Jaemisch R, Bonadio JF, Goldstein SA (1992) An in vivo assessment of the contribution of type I collagen to the mechanical properties of cortical bone. Trans Orthop Res Soc 32:93 28. Goodier JN (1933) Concentration of stress around spherical and cylindrical inclusions and flaws. J Appl Mecb 55:39 29. Currey JD (1962) Stress concentration in bone. Quart J Micro Sci 103:111-133
