Biomechanics of Fractures

Movement and/or Displacement

Engineering mechanics is fundamentally the study of the effect of forces on a body or an object. In general terms, when a force or load is applied to a body or object, that body either moves (a quarterback throws a football) or deforms (the bend in a pole vault), or both occur (a club hits a ball) ( Fig. 6.4 ). Biomechanics is the application of mechanics principles to the human body. Both modes of force response (motion and deformation) are relevant in biomechanics. Under low-load conditions, the deformation of bones is fairly minimal, and the propulsion of the body by the muscles is often considered only in terms of motion (i.e., assumes that the bones do not bend or twist). Substantial stretching or compression occurs in soft tissues, such as ligaments or cartilage, so those tissues are usually analyzed in terms of deformation ( Table 6.1 ).

Bone fractures require consideration of both motion and deformation. In this chapter, we will introduce the engineering fundamentals necessary to understand how bones fracture. This includes understanding material responses, structural characteristics, and bone loading. We will also give an overview of common fractures and address the basic mechanisms that promote fracture healing. Finally, we will address the mechanics behind some critical factors that affect both fracture risk and proper healing.

Stress/Strain

The local response of a material to an applied load is usually discussed in terms of stress and strain. Stress (σ) is defined as the force (F) applied to an object, divided by the area (A) over which the force is applied:

Therefore two bones of different sizes may be loaded with the same force, but the smaller bone experiences a higher stress because of its smaller area. This makes the smaller bone more prone to failure than the larger bone, assuming that the material properties and shapes are similar ( Fig. 6.5 ).

Similarly, strain (ε) is defined as the change in length (Δ l ) of an object divided by the initial length (L) of that object:

The definitions just given for both stress and strain are for normal compressive and tensile stress (normal implies that the direction is perpendicular to the surface). In addition, there are equivalent concepts for shear stress and strain. Shear stress follows the same form as the previous equations, with the relevant force and area as indicated in Fig. 6.6 .

Shear strain is defined as the parallel deformation divided by the perpendicular distance, which can be calculated as the tangent of angle α in Fig. 6.6 .

Both stress and strain are actually three-dimensional entities fully described in terms of vectors or tensors. The same principles apply to the simplified two-dimensional analysis in a plane (anterior-posterior [AP] or medial-lateral [ML], for example) as for the full three-dimensional analysis.

Stress-Strain and Other Diagrams

Materials are often quantified and represented through the use of stress-strain diagrams. Stress-strain diagrams or graphs are typically generated from data obtained by performing a uniaxial tension or compression test of a small, well-formed sample of the material, such as the dog-bone shape shown in Fig. 6.7 . In a uniaxial test, the sample of the material is pulled (or pushed) along the long axis of the object (uniaxial). The stress and strain are then plotted, and the resulting curve provides significant insight into the behavior of the material.

Properties of Materials (Young's or Elastic Modulus, Yield and Failure, Energy/Toughness)

An example of a typical stress-strain diagram for steel is shown in Fig. 6.8 . Several material properties can be determined from this curve. The elastic modulus (A), or Young's modulus, of the steel is the slope of the linear portion of the curve (see point A in Fig. 6.8 ). The Young's modulus represents the general stiffness of the material. Stiffer materials (brittle) have a higher Young's modulus. The material's yield stress occurs at the end of the linear region, which is typically the point where irreversible damage begins occurring (see point B in Fig. 6.8 ). The yield stress of a material indicates that the material is beginning to fail or break. The ultimate, or failure, stress represents the stress at which the material ruptures or fractures. The end of the curve (at the point where the stress abruptly drops to zero) represents this material's failure stress or strength, and it occurs when the specimen breaks (see point C in Fig. 6.8 ). Finally, the area under the curve (see the shaded area in Fig. 6.8 ) represents the amount of energy expended in breaking the specimen. This is also known as the material toughness.

There are often trade-offs between the properties just described. A material with a higher Young's modulus (stiffness) often has lower toughness (energy). Similarly, high modulus and high strength are not necessarily correlated. A material may be brittle (with high modulus) but breaks at a low failure strength and, hence, has a low toughness. In placing a stiff metal at an interface with bone or ligament, it is possible for there to be too much stiffness, thus inhibiting the motion that stimulates mechanobiologic feedback necessary for tissue growth. Such a phenomenon is known as stress shielding and will be discussed in later sections. Also, an abrupt change between a low-stiffness ligament and a higher-stiffness bone requires specialized tissue material properties and architecture within the transition zone. It is this transition-zone interface that often causes difficulties in healing of ligaments, with the low-stiffness region experiencing large tissue deformations leading to fibrous healing.

Other Material Properties (Viscoelasticity, Anisotropy, Creep and Relaxation, Fatigue, S - N Curve)

Several other important material properties cannot be determined from simple stress-strain curves. Most biologic tissues display some form of viscoelasticity, which simply indicates that they respond differently to forces depending on how fast the load is applied. In other words, viscoelastic materials have different stress-strain curves when they are applied at a fast rate (e.g., cutting maneuvers in sports) rather than a slow rate (e.g., yoga stretching). The amount of viscoelasticity in a material is represented by how much its stress-strain curves change for different rates of loading. Viscoelastic properties are often reported in terms of a dynamic modulus or a creep time–constant. Experimentally, viscoelastic properties are determined either by cyclically loading the material at different rates or by conducting relaxation tests, where a sample is held at a constant load and the long-term change in deformation, or creep, is recorded.

The unloading curve will be different from the loading curve because more energy is absorbed during loading than is dissipated during unloading. This variation between loading and unloading is called hysteresis and ultimately means that energy is lost during the loading of viscoelastic materials ( Fig. 6.9 ).

Soft tissues, such as cartilage and ligaments, demonstrate high viscoelasticity, as shown in Fig. 6.9 . Bone also has a viscoelastic response, but it is not as pronounced. Stretching before exercising helps protect muscle, tendons, and ligaments and demonstrates their viscoelastic properties. When measured in vitro, muscle-tendon units achieved sustained elongations (e.g., flexibility) after repetitive stretching, whereas greater peak tension and greater energy absorptions—and thus an increased risk of tendinous injury—occurred at faster stretch rates, as shown in Fig. 6.9 .

Rotator cuff healing demonstrates how soft tissues are vulnerable to a combination of mechanical and biologic factors. In multiple studies, the amount of retraction of the tear at the time of surgery is an independent predictor of healing. Massive tears, especially in the anterior rotator cuff, likely have a detrimental effect on healing tissues by increasing regional strains. In one animal model of supraspinatus tears, the tension required to repair the tendon increased early after detachment and progressively increased with the chronicity of the tear. Repair tension also correlated with an increase in stiffness and cross-sectional area but a decrease in the failure properties (e.g., failure load and stress) and viscoelastic peak stress. In another study, diabetes was accompanied by less fibrocartilage and less organized collagen, which translated into a significantly reduced load to failure and stiffness. Nicotine causes a delay in tendon–bone healing, with poor expression of type I collagen and inferior biomechanical properties. Given that the tendon–bone interface plays a key role in surgical failure of tendon repair, recent developments in tissue engineering have aimed to mimic the structure and function of soft-to-hard-tissue junctions. Scaffolds must be stratified with interconnected and preintegrated layers to capture the multitissue matrix organization seen in natural interfaces. Most recently, spatially patterned structural and chemical cues have been incorporated into these scaffolds and triggered mechanobiologic pathways for cell differentiation and growth, helping create a gradient of mechanical properties and mineral distribution on the scaffold.

Another important characteristic of a material is its fatigue performance. Most engineering components fail not from a single high-magnitude loading event exceeding the material failure strength but from the accumulation of repeated loading cycles occurring at a lower-magnitude load. These lower-load cycles form microcracks that expand over time and eventually lead to full-component failure ( Fig. 6.10 ). Because the number of cycles a material can withstand depends on the intensity or stress level, the fatigue performance of a material is often quantified through an S - N curve, where S is stress and N is the number of cycles, as shown in Fig. 6.11 . These graphs show the number of cycles a material can be expected to withstand at a given stress level. The fatigue or endurance limit is defined as the stress below which cyclic or fatigue failure will not occur. Engineers design implants and devices to withstand a fatigue limit well over their expected loading so they can be confident that the device will not prematurely fail over time in use. The safety factor is determined by the expected maximum in situ stress divided by the maximum allowable stress.

The caveat is that exact knowledge of the expected loading is complicated for a device in the human body that is capable of a multitude of activities and loading. For example, an Olympic long jumper will place higher stress on his or her lower limb on landing compared with a high school long jumper or competitor in high hurdles. To our benefit, bones and musculoskeletal tissues are well suited to their normal activities and also are known to adapt to the level of loading they are experiencing. For example, Olympic long jumpers may experience adaptation in their tissues to be able to resist the higher loading they are experiencing. However, there may be periods when activity has increased markedly, yet the tissue has not yet had time to adapt. This vulnerable period can lead to stress fractures where the bone material has not had time (or physiologic ability) to heal microcracks or adapt to the higher loading being experienced. A well-known situation for stress fractures is new military recruits participating in intense activity during their basic training period.

Within a biologically active tissue such as bone, fatigue is more complicated because microcracks and even full fractures heal over time. In fact, the ability to repair microcracks is considered a unique property of bone, one that is mediated through osteoclasts. Medications or conditions that impair osteoclasts (such as catabolic substances) can diminish the ability of the bone to repair microcracks and lead to brittle failure over time. As noted, certain classes of injuries, including stress fractures, are still fundamentally fatigue fractures ( Fig. 6.12 ). Of course, most mechanical failures of implanted hardware, such as plates, nails, or joint replacements, are also failures in the fatigue mode ( Fig. 6.13 ).

The material properties discussed in this section have been presented as if the materials were isotropic, that is, as if the properties were the same regardless of their orientation to the load being applied. This is true for most metals and traditional engineering materials. However, many materials, including most native tissues, have anisotropic properties, meaning they have higher stiffness and strength in certain directions and orientations relative to the load ( Fig. 6.14 ). Soft tissues, such as ligaments and cartilage, have higher stiffness along their longitudinal axis, enforced by collagen fibrils along their length (or in multiple discrete bundles). Cancellous bone has anisotropic effects because of the microstructural arrangement of the trabecular network, which is different at separate anatomic sites. Most anatomic tissues are organized to have stronger and stiffer properties in the major direction of load. For example, within the proximal femur, four major groups of bony trabeculae distribute the compressive and tensile forces from the femoral head into the diaphysis through the femoral neck; on the medial side of the femoral neck, the compressive trabeculae are organized in transverse layers, whereas the trabeculae on the lateral side are oriented longitudinally to resist tension ( Fig. 6.15 ). It is a matter of research whether occasional peak- or maximum-loading events are more efficient in evoking an adaptation to add bone compared with sustained lower-level loading.

Finite Element Analysis Primer

Fundamental theorems in mechanics allow us to analyze deformation and stress in a variety of relatively simply shaped structures. However, finding analytical solutions for stress analysis in more complex geometries becomes much more difficult. For unstructured geometries, such as human bones, analytical stress solutions are impossible without substantial simplifications. For this reason, finite element analysis (FEA) is frequently used for biomechanical stress analysis.

FEA is a technique in which a complex geometry is subdivided into smaller pieces (the “finite elements”) for which the mechanical response can be described. As a simple example, consider Fig. 6.16 . The simple cylinder under tensile loading in Fig. 6.16A has a known deformation response, but for the more complex geometry in Fig. 6.16B , a simple equation is no longer applicable. However, the complex geometry can be subdivided into three elements that resemble the simple cylinder, shown in Fig. 6.16C . This creates a linear system (set) of equations, which can be solved to reveal the deformation of the complex structure.

In a more generalized sense, the elements used in biomechanics applications are typically shaped as tetrahedral or hexahedral elements (pyramid or brick, as shown in Fig. 6.17 ). Fig. 6.18 shows examples of a human femur meshed with both types of elements. In both of these cases, the ensuing analysis allows the calculation of stress and strain distributions throughout the bones. Knowledge of locations of particularly high stress indicates areas to evaluate as being more vulnerable to failure. Many examples of FEA related to fracture mechanics can be found ( Fig. 6.19 ). FEA has been used to predict fracture patterns, evaluate fracture fixation devices, and study fracture progression at a microstructural level. As these examples show, FEA provides an alternative or complementary approach to physical testing. FEA can provide information that cannot be measured experimentally and can facilitate large numbers of testing perturbations that would not be feasible for physical testing because of time and financial constraints.

Despite its increasing prevalence and obvious advantages, there are several pitfalls that should be considered when evaluating FEA applications in biomechanics. First, the output of such analysis is only as good as the input data. Inaccurate imaging data (from which the geometry is created), material (tissue) property data, or loading (amount, orientation, timing) data will all lead to incorrect results. FEA models should therefore include validation in the form of comparisons to a well-defined physical experiment. Otherwise, interpretation should be limited. Sensitivity tests can also be performed to evaluate the importance of model parameters that must be estimated or have high variability, such as material (tissue) properties. Additionally, FEA results can be sensitive to the element size used, so ideally, a convergence test is performed to establish that the model results do not change with further decreases in element size. Fig. 6.20 shows the effect of mesh size on the stress in a plate and the discontinuous stress distribution that can result from using elements that are too large.

Despite these potential pitfalls, FEA has contributed significantly to our understanding of fracture mechanics and biomechanics in general. It has become a standard tool in the design of implants and fixation devices. In research applications, FEA has been used to study the effect of trabecular microstructure on fracture risk and even to simulate the fracture-healing process. As orthopaedic research continues to delve deeper into multiscale and multidisciplinary questions, FEA will become an increasingly important tool.

Bone Mechanical Properties

Bone provides structural support and allows for mechanical functions of the body. Its unique material properties and shape provide stability and allow for motion through its muscle attachments. Bone is not merely a rigid, homogeneous material throughout the body; rather, its composition varies depending on its location in the body and its primary function. Table 6.2 demonstrates the variation in bone properties at different points in the body.

Bone is a living tissue composed of cells, water, and extracellular matrix. Mineral components make up 60% to 70% of the composition by weight, organic components make up 20% to 25%, and the remainder is water. The mineral components of bone consist of calcium hydroxyapatite and osteocalcium phosphate. Organic components of bone matrix consist of collagen, proteoglycans, proteins, growth factors, and cytokines. These organic components generally provide resistance to tension, whereas the mineral components provide resistance to compression and stiffness ( Fig. 6.21 ).

The strength of bone tissue is gained predominately by the bone mineral and trabecular characteristics of connectivity and thickness. In the orthopaedic literature, there is a large emphasis on the bone mineral density and its role in the risk of fracture. However, type I collagen makes up 90% of the organic matrix and also contributes significantly to the mechanical properties of bone. Its fibril-forming structure provides tensile strength. A clinical example of collagen's importance to bone strength and the risk for fracture can be seen in osteogenesis imperfecta, a condition that causes a decrease in the amount of normal type I collagen ( Fig. 6.22 ). The orthopaedic complications with this disease include bone fragility and increased fracture risk. Previous studies that induced collagen denaturation in cadaver femurs, without changing the bone mineral, resulted in decreased toughness and strength.

The structure of bone constituents also influences its material characteristics. At the microscopic level, bone is either lamellar or woven. Lamellar bone includes both cortical and cancellous bone, and woven bone includes immature or pathologic bone. Woven bone has a random orientation of collagen and mineral. It is often made during periods of rapid bone formation, such as in fracture healing. Because of the arrangement of the tissue constituents, woven bone is weaker and more flexible. Lamellar bone, such as cortical or cancellous bone, consists of a structure that is oriented along the lines of major stress, providing strength, whereas disorganized woven bone is not oriented according to applied stress. Fig. 6.23 shows the microscopic differences of woven and lamellar bone. As discussed later, weaker woven bone compensates by producing more material and placing it farther from the center, as demonstrated with typical callus formation. Remodeling into more streamlined lamellar bone provides economy of material, with lower weight, while increasing resistance to the loads applied during activities.

Although this chapter focuses primarily on bone properties and relation to fracture propensity and healing, the soft tissues surrounding bone have unique material properties as well. For example, tendons, which are composed primarily of collagen fibers surrounded by a sheath, are stronger per area than muscle and have the same tensile strength as bone. Tendons transmit the large forces generated by the muscle to the bone to create movement. A tendon stress-strain curve ( Fig. 6.24 ) exhibits some unique characteristics. There is an initial “toe region” that corresponds with the straightening of a slight crimp in the fibers, sometimes referred to as fiber recruitment. After the toe region, there is the standard linear region, but beyond this region, collagen fibers fail unpredictably.

Cortical Bone Properties and Microstructure

Cortical bone is the strong, dense bone that lines the outer surface of long bones. Cortical bone consists of layers called lamellae, which are mineralized collagen fibers. These mineralized collagen fibers provide structure to the bone, and the lamellar tissue layers are 3 to 7 µm thick. The collagen fibers are parallel within each layer, but the orientation varies in the adjacent layers. In certain cortical bone, the lamellae wrap in layers around a canal to form a Haversian system, as outlined in previous chapters.

The structure of cortical bone, especially in the diaphysis of long bones, provides load-bearing support. The mechanical properties of cortical bone change depending on the orientation of the loading. As described earlier in this chapter, an anisotropic material, like bone, defines a material where the properties vary and are direction dependent. In contrast, an isotropic material is one where the properties remain constant in all orientations. A common example of an anisotropic material is wood, which is easier to splinter along its grain than against it.

The anisotropic properties of bone are demonstrated in bench testing in the laboratory. In the longitudinal direction, the elastic modulus is 17 to 19 GPa, and the ultimate strength is 100 to 150 MPa, whereas these values are only 8 to 10 GPa and 9 to 11 MPa, respectively, in the transverse direction. Tables 6.3 and 6.4 highlight these direction-dependent differences seen in bone.

The material properties of cortical bone are also dependent on the rate at which the bone is loaded. Materials such as bone whose stress-strain characteristics are dependent on the applied strain rate are termed viscoelastic or time-dependent materials. However, as noted previously, the strain rate dependency of bone is relatively modest, with the elastic modulus and ultimate strength of bone approximately proportional to the strain rate raised to the 0.06 power.

Trabecular Bone Properties and Microstructure

Trabecular bone is the porous bone that is found at the ends of long bones, such as the femur and radius, and in vertebrae. The bone tissue that makes up the individual trabeculae is only modestly less stiff and strong than the bone tissue within cortical bone. Trabecular bone has a lamellar structure similar to cortical bone but has increased porosity, often between 70% and 80%. This increased porosity is quantified by measurements of the apparent density (i.e., the mass of bone tissue divided by the bulk volume of the test specimen, including mineralized bone and marrow spaces). The porous nature of the bone is of mechanical importance because the pores allow for energy absorption into the bone.

Stress-strain curves ( Fig. 6.25 ) for trabecular bone in compression exhibit an initial elastic region followed by yield. The slope of the initial elastic region ranges from one to two orders of magnitude less than cortical bone. Yield is followed by a long plateau region created as more trabeculae cumulatively fracture.

The density of trabecular bone varies depending on location, function, and age and is affected by diseases such as osteoporosis. The density of trabecular bone ranges from 0.1 to 1.0 g/cm ³ throughout the body, whereas the density of cortical bone is approximately 1.8 g/cm . A trabecular bone specimen with an apparent density of 0.2 g/cm ³ has a porosity of approximately 90%. As would be expected, the variation seen in bone density also appears in the mechanical properties of trabecular bone. The mechanical properties have greater variation than the density, with differences up to a factor of five from one location to another. However, it has also been shown by many investigators that density alone cannot predict all of the variations in the mechanical properties. Importantly, the way the bone is organized and the variations in the internal structure of tissue within the bone are also important determinants of the strength and stiffness of bone.

Age-Related Bone Property Changes

The tissue properties and microstructure of bone change with time, and the differences emerge in both the overall bone shape and its apparent mechanical properties. Overall, bone mass and bone strength in humans peak around age 30 to 35 years. Up until this age, more bone formation is occurring than bone resorption, resulting in a net increase in bone. After this age, bone resorption begins to predominate, governed by factors such as hormones, sex, activity, and disease. It is important for teenagers and young adults (particularly females) to build up a large bone mass during their peak years for bone accrual.

Age-related changes in bone include reduced toughness and increased brittleness because of changes to the collagen matrix, less mineralization, and changes in structure, with fewer trabeculae and less connectivity. The material properties of bone are partially dependent on bone turnover, which decreases with age, causing collagen to be more highly cross-linked, mineral to accumulate, and tissue to become less hydrated, which decreases the ductility of bone and increases the skeletal fragility. Therefore, because of the net bone loss and reduced mechanical properties, fracture can occur with lower loads as one ages, and as bone becomes more brittle, it is less able to absorb energy and also more prone to fracture.

Traditionally, bone loss and subsequent fracture risk is a concern with menopause in women, as a result of sex-steroid deficiency, and later in men as a result of age-related factors. This traditional understanding of bone loss fits well with cortical bone loss through time. Studies of cortical bone loss have shown that significant loss does not occur until midlife for women and is correlated with menopause and estrogen deficiency. In men, cortical bone loss occurs with age, beginning with a small amount of loss in young adulthood and increasing at a faster rate later in life. However, in trabecular bone, bone loss varies from the traditional theory. Quantitative studies of trabecular bone morphology document that the thickness of trabeculae decreases, whereas the spacing between trabeculae increases. There are surprisingly few fundamental differences in trabecular bone between the sexes other than that bone is lost faster in females between the ages of 50 and 85 years. Both males and females were found to have a loss of trabecular bone that began in young adulthood. This began as early as the third decade for women and the fourth decade for men. There is an acceleration of bone loss that occurs in both men and women. In women, this corresponds to menopause, but men also have an acceleration of bone loss around age 65 years, which is thought to be caused by the decrease in sex steroid levels as well. These age-related morphologic changes significantly reduce the strength of the vertebrae, the proximal femur, and other bones, contributing to the observed increased fracture incidence in the elderly.

At the tissue level, several age-related changes in the material properties of femoral cortical bone have been demonstrated. With bone tissue specifically, the tissue elastic modulus has been shown to decline 0% to 2% per decade, ultimate strength decreases 2% to 5% per decade, and ultimate strain decreases 6% per decade ( Fig. 6.26 ). The energy required to fracture through bone, which is the area under the stress-strain curve, decreases with aging bone, making it more prone to fracture. Because the elastic modulus does not decrease as much, the energy to failure is predominantly reduced as a result of age-related decreases in ultimate strain. Thus, with aging, the bone behaves more like a brittle material, and it has less capacity to absorb the energy and resist fracture propagation from a traumatic event.

Studies of skeletal aging typically focus on bone density and often do not consider the overall geometry and distribution of bone tissue. In a study of the femoral density and geometry of thousands of men and women, a loss of bone density was clearly seen in both sexes, in both the femoral neck and in the diaphysis. In addition, the femurs expanded in periosteal diameter, but the cortical thickness decreased.

Age-related changes are not restricted to changes in the material and structural properties of bone mineral. Age-related changes in bone occur in collagen, as well as the mineral component of bone. Collagen changes with age reduce a bone's energy-absorption capability. These changes are also implicated as a factor in the increased fracture risk of older patients.