, David B. Burr2, Neil A. Sharkey3 and David P. Fyhrie4
Orthopaedic Surgery, UC Davis Medical Center, Olga, WA, USA
Department of Anatomy and Cell Biology, Indiana University School of Medicine, Indianapolis, IN, USA
Kinesiology, Orthopedics and Rehabilitation, Pennsylvania State University, University Park, PA, USA
Orthopaedic Surgery, Biomedical Engineering, UC Davis Medical Center, Sacramento, CA, USA
Most large joints, for example the knee, have very low friction under dynamic loading. The viscoelastic properties, fluid flow through, and surface decorating molecules of articular cartilage are keys to the efficient transmission of load through this soft tissue.
KeywordsCartilageViscoelasticityFluid FlowPermeabilityElectrostaticsPolymer MechanicsTribologyWearFrictionLubrication
[The bone ends] are covered with a smooth elastic crust, to prevent mutual abrasion; connected with string ligaments, to prevent dislocation; and enclosed in a bag that contains a proper fluid deposited there for lubricating the two contiguous surfaces (Hunter 1743).
Now that we have explored the mechanical behaviors of tendons and ligaments, constructs that transmit the forces needed to produce and constrain relative motion between bones, we will turn our attention to the tissue that directly transmits forces from one bone to another while at the same time allowing for smooth almost frictionless motion between them. These motions occur within our synovial joints and the predominant tissue responsible is cartilage. In this chapter we explore the mechanics of synovial joints; however, we must not forget what we have learned about bone, tendon and ligament because their presence heavily influences the characteristics of the new tissues we are going to study.
Synovial or diarthroidal joints are those that move freely and contain synovial fluid, like the knee and elbow joints, as opposed to joints that do not, such as the “tether” joints between the vertebral bodies in the spine. Figure 5.1 is a schematic diagram of a typical synovial joint. The synovial fluid is manufactured by cells in the synovial membrane, which lines the interior of the fibrous joint capsule. As explored previously, the joint capsule is aided by ligaments (not shown) and the surrounding musculotendinous tissue in holding the two bones of the joint together (i.e., “stabilizing” the joint). The bone surfaces within the joint capsule are covered by articular cartilage, also called hyaline cartilage. The purpose of this tissue is not to serve as a shock absorber, as some textbooks state, but to provide a suitable surface for lubrication and wear prevention. Some joints also contain fibrocartilage disks that help maintain the other joint components in appropriate positions and reduce stress by extending the load-bearing surfaces. Such disks are generally found in joints where there is substantial rotation or torsion but incongruous joint surfaces (e.g., the acromioclavicular joint). In the knee, these disks are called menisci. An important compositional difference between articular cartilage and meniscus is that the latter contains type I instead of type II collagen (Mow et al. 1992).
Schematic diagram of a typical synovial joint.
5.1.1 Functions of a Synovial Joint
Synovial joints provide essentially frictionless motion between limb segments while transmitting relatively high loads between them. Joints in animals are the equivalent of “bearings” in machines. Of course, these bearings do not support continuously rotating shafts, but components that start, stop, and oscillate. A similar engineering example might be the ball joints in the steering mechanism of an automobile.
We saw in Chap. 1 that the forces acting in the joints between bones are quite large relative to the external loads being supported; the stresses in the bones, tendons, and ligaments are therefore relatively high as well. The same may be said of the stresses in cartilage. The normal contact pressures exerted on the cartilage in various normal joints across species typically range between 0.5 and 5.0 MPa (Brand 2005).
5.1.2 Joint Diseases
Joint disease and osteoporosis are the two most common skeletal pathologies in modern society. One or the other seems to give almost everyone problems eventually. Their human and economic costs are enormous, and are increasing as other diseases are controlled and people live longer. Although these conditions do not kill directly, they do so indirectly in most painful and debilitating ways. (Immobilization itself, from the pain of arthritis, can be life-threatening to an elderly person.) The causes of both osteoporosis and arthritis are both poorly understood, but the difficulties with understanding them are different. Mechanically, osteoporosis is rather simple: there is not enough bone tissue, and therefore the bone structure is weak and fractures occur. The complexity is in the bone remodeling biology that leads to diminished bone structure and altered bone quality. In the case of an arthritic joint, on the other hand, both the mechanics of joint function and the cell-level biology are poorly understood. However, it seems clear that most kinds of arthritis involve feedback loops between mechanical damage processes and inflammatory processes (Fig. 5.2). For example, mechanical overuse may create fatigue damage in cartilage. This damage may incite an inflammatory process that activates bone and cartilage cells not only to repair the damage, but also to attempt to reshape the tissues to improve the mechanical situation. If too much begins to happen at once, however, the circumstances become chaotic. Enzymes intended to get rid of a damaged tissue also destroy healthy tissue. The size and shape of a joint can be altered rather easily in young people by bone and cartilage modeling, but after growth stops, these processes become sluggish and, apparently, are not well matched to mechanical needs. New cartilage and bone production formed by the joint result in misshapen “spurs” (called osteophytes) around the periphery of the joint, which often get in the way and cause pain, or cause damage to adjacent muscle (e.g., in the case of some rotator cuff tears in the shoulder).
Block diagram of potential feedback loops in synovial joint physiology. Joint loads and anatomy determine tissue stresses, which produce tissue property-dependent strains that influence bone and chondral modeling and remodeling. If tissues are damaged, inflammatory responses as well as modeling and remodeling of bone and cartilage, may be activated, influencing joint anatomy and tissue properties. In some instances, the system may be stable and self-correcting; in others, it may become unstable and self-destructive.
Alternatively, an infection or a defect in the patient’s immune system may cause an inflammatory response in a joint when mechanical loading is normal. This response can lead directly to tissue destruction, which so jeopardizes mechanical properties that normal loads produce damage-level strains. Thus, there are many possibilities for the development of “vicious circles” in which mechanical and cellular processes in the bone and cartilage tissues, which normally operate in a controlled way at modest levels, become rampant and unstable.
Understanding joint mechanics is essential to understanding various forms of arthritis. The way synovial joints work is not entirely clear, but much is known and it is a fascinating story.
5.2 Mechanical Properties of Cartilage
5.2.1 Initial Points
The general structure and composition of cartilage were introduced in Chap. 2. There are three important initial points to be made about the importance of this structure to the mechanical role that cartilage plays in joint mechanics.
126.96.36.199 Cartilage Is 70 % Water
The extracellular matrices (ECM) of all connective tissues (including bone and cartilage) are three-dimensional crosslinked networks of insoluble collagen fibers with other more soluble components enmeshed in, integrated with or chemically attached to the collagen fibers. The void dimensions of cartilage ECM are about 50 Å, much smaller than the 50–300 μm void size of bone. The very small distances between molecular components in cartilage very strongly affects mechanical properties. Cartilage ECM is about 95 % of the total tissue volume and the remaining 5 % is filled with cartilage cells called chondrocytes. Water completely fills the entire tissue volume and is in very intimate association with the matrix and, of course, the chondrocytes. The chondrocytes create and assemble the matrix initially and maintain the framework of collagen and proteoglycans. Cartilage is essentially an extracellular matrix inflated with water that is held in place by strong electrochemical interactions with the proteoglycans. If anything interferes with the proteoglycan content (e.g., the release of destructive enzymes by the chondrocytes), or with the ability of chondrocytes to make normal ECM (e.g., a genetic defect), the mechanical functioning of the cartilage will deteriorate (Kuettner 1992) (Fig. 5.3).
In the simplest sense, the three components of cartilage are water, extracellular matrix and chondrocytes.
The material properties of articular cartilage depend on the detailed relationships between water and the macro-molecular components of the ECM. Fibrillar type II collagen and proteoglycans are the primary macro-molecules of adult articular cartilage ECM, however, water is 70 % (by weight) of the tissue. The water is very tightly bound to the ECM by the physical chemical properties of the macromolecules. Collagen is more than 50 % of the dry weight (or 10–20 % of the wet weight) of articular cartilage. In adults, approximately 90 % of cartilage collagen is of type II which is found in only a few other tissues such as the vitreous body of the eye. Type II is a fibril-forming collagen and the molecules are made of three identical α1(II) polypeptide chains. Each of the α1(ΙΙ) molecules is itself a 300 nm-long triple helix initially formed with short telopeptides at each end that are enzymatically removed before assembly. Once the collagen molecules leave the cell they assemble into fibrils extracellularly. The resulting network of type II collagen fibrils gives the ECM tensile strength and is essential to maintaining the tissue’s volume and shape. The tensile strength of the collagenous matrix is increased by enzymatically produced covalent, intermolecular crosslinks.
Proteoglycans (PGs) are highly abundant in the ECM of all connective tissues but especially in cartilages. In articular cartilage these highly hydrophilic macromolecules are embedded within the fibrillar network in an under-hydrated (compressed) form. They give articular cartilage the ability to undergo large, non-damaging, reversible compression. The predominant PG of articular cartilage is aggrecan. It contains a protein core (215 kDa) to which glycosaminoglycans and oligosaccharides are covalently attached. Ninety percent of the total mass of aggrecan is from the glycosaminoglycans keratan sulfate and chondroitin sulfate.
Aggrecan attaches noncovalently to the very high molecular weight biopolymer hyaluronan. A link protein acts to stabilize this interaction between aggrecan and hyaluronan making it nearly as stable as a covalent bond. The combination of aggrecan and hyaluronan forms an extraordinarily large space-filling molecule called aggregate. In free solution, aggrecan swells to about 50 times its dry volume. However, in healthy cartilage the collagen network constrains swelling to about ten times the dry volume (i.e., 20 % of free swelling). This permits the accumulation of high concentrations of compressed aggrecan in the tissue. Aggrecan is a highly negatively charged molecule. The highly concentrated negative charges repel each other and also strongly attract water molecules creating a swelling pressure. Under compressive load, water is pushed out of aggrecan and the whole tissue. Compression of the proteoglycans increases the negative charge density in the tissue which, in turn, increases electrical repulsion between the negative charges and the attraction of water into the tissue. The process of water expression continues until there is equilibrium between loading forces and the electrical and osmotic swelling stresses. Removing the load permits water to return to the tissue and it expands until the expansion force of the proteoglycans is balanced by tension in the fibrillar network. The numbers of negatively charged groups in the proteoglycans will rise and fall in proportion to the pH of the synovial fluid. Cations in the synovial fluid (e.g., Na+ and Ca++) also are attracted to and shield the negative proteoglycan charges reducing the effective charge density; divalent cations having effects much more than twice as strong as monovalent cations (June et al. 2009). For these and other reasons, the chemistry of cartilage has substantial effects on its mechanical properties.
In addition to the swelling properties of the PG’s, their high negative charge concentration has strong effects on the diffusion of solutes. The negative charges of aggrecan are fixed to the molecule and don’t diffuse independently. As a result, they create a high density of fixed negative charge that can retard the diffusion of negatively charged (anionic) macromolecules into the tissue while attracting and concentrating positively charged (cationic) molecules (Nimer et al. 2003). The effect of the fixed negative charges on the diffusion of neutral macromolecules with a mixture of positive and negative regions (zwitterions) is more complex.
188.8.131.52 Cartilage Is Viscoelastic
Cartilage is a water-filled biopolymer that has strong interactions between its constituent components. As a result, its mechanical properties are viscoelastic due to the flow of water in the matrix and time dependent reconfiguration of the biopolymers under mechanical loading. The elastic modulus of the tissue varies greatly with the rate of loading, demonstrating a broad spectrum of interactions between water and the biopolymers comprising the tissue. At very slow rates of loading, the resulting modulus is only a few MPa, a characteristic that can be reasonably modeled as the slow percolation of water through an elastic matrix. At physiologic rates of dynamic loading the modulus of cartilage is much larger, approaching 500 MPa (Higginson and Snaith 1979; Radin et al. 1970; Unsworth 1981; Yamada 1973). The mechanical behavior of cartilage under these more typical dynamic conditions cannot be explained using simple fluid flow models; additional interactions between the constituent biopolymers are needed to explain this more extreme viscoelastic behavior. This range of behaviors should be borne in mind when reading the rest of this chapter, because many of the descriptions of cartilage stiffness measurements refer to behavior at very slow rates of loading. For example, indentation tests are usually quasistatic (extremely slow loading rate) and the “equilibrium” or “aggregate” modulus used in many models refers to the stiffness associated with an infinitely slow rate of loading.
184.108.40.206 Cartilage Is Not a Shock Absorber
Finally, while it is much less stiff than cortical bone, cartilage does not serve to “cushion” or reduce impulsive forces in joints. It cannot do this because it is so thin that its capacity to absorb energy is insignificant compared to (a) eccentric contractions of muscles and (b) energy absorption in the bones on either side of the joint. (Exercise 5.1 asks you to demonstrate this to yourself). Instead, the role of cartilage is to provide a self-renewing, well-lubricated, compliant, load-bearing surface.
5.2.2 Structure of Articular Cartilage
Figure 5.4 is a schematic diagram of the structure of articular cartilage in relation to its foundation of subchondral bone. The collagen fibers in the cartilage have been described as forming arches (the “arcades of Benninghoff”) (Benninghoff 1925) such that their orientation is parallel to the surface in the superficial tangential zone and perpendicular to it in the deeper zones. This description of the collagen orientation in articular cartilage is not really correct, but rather the collagen appears as arches because the orientation of the collagen fibrils gradually change throughout the cartilage matrix, being more parallel to the surface superficially, and orthogonal to the surface in the deeper cartilage layers. The chondrocytes tend to line up in columns parallel to the collagen fibers in the middle and deep zones. As in the growth plate, the deepest layer of cartilage calcifies.1 The junction between calcified and uncalcified cartilage is called the tidemark and shows up as a line in stained histologic sections. The calcified cartilage layer may reduce the stress concentration that would exist at the bone–cartilage interface by virtue of their different stiffnesses. Beyond these general observations, the structure of cartilage remains poorly understood, but even this degree of understanding reveals much about the origins of the anisotropy and inhomogeneity in cartilage’s mechanical properties.
Schematic diagram of collagen fiber orientations in articular cartilage.
Because cartilage consists of an organic biopolymeric matrix and water, it can be useful to think of it as a multiphasic system. The simplest multiphasic system is biphasic, with water being one phase and the collagen–proteoglycan biopolymer matrix the other. The volume fraction of the water phase can be called the porosity of the cartilage. Although the water molecules are restrained to a significant degree by the matrix molecules, they may nevertheless slowly migrate through the tissue under the influence of a pressure gradient. This may be modeled using Darcy’s law:
where Q is the flow rate per unit cross-sectional area, dP/dx is the pressure gradient, and k is the hydraulic permeability coefficient (sometimes called the apparent permeability). Typical values of k are shown in Table 5.1. Permeability diminishes in the deepest layers of the articular cartilage, and also varies with the composition of the liquid phase. Clearly, the resistance of the tissue to deformation depends to some extent on its permeability.
Properties of lateral femoral condyle cartilage in several different species
Aggregate modulus (MPa)
Permeability, 10 − 15 m4/N s
5.2.4 Indentation Testing
It is difficult to form mechanical test specimens of cartilage because it is only found in thin (2–3 mm) layers. Therefore, many of the early mechanical tests involved indentation of the surface as a means of gaining compressive stiffness data. Such experiments were based on the equation derived by Boussinesq (1885) for a hemispherical indenter applied to an elastic half-space (Fig. 5.5). If an indenter of radius r penetrates to a depth d under load P, the elastic modulus is given by
Diagram of indentation experiment. An indenter having a hemispherical end of radius r is pressed into the surface of the tissue for a distance d by a load P.
where ν is Poisson’s ratio of the indented material. Assuming that the cartilage was indeed of infinite depth and that it was incompressible (so that ν = 0.5), Sokoloff (1966) obtained E = 2.3 MPa for the elastic modulus of articular cartilage. Later, when a mathematical solution was obtained for indentation of an elastic layer on a rigid foundation (i.e., cartilage on relatively rigid bone) by a plane-ended cylinder, Hayes et al. (1972) found that E was only 1.6 MPa. As more work was done, it became clear that this value, too, was probably inaccurate because cartilage is not truly incompressible and fails to meet the assumptions of the tests in other ways as well (e.g., it is anisotropic, inhomogeneous, and viscoelastic). Table 5.1 shows data on the compressive modulus and Poisson’s ratio of articular cartilage for various species. These data were obtained from compression tests on machined specimens using techniques similar to those described later in this section.
Box 5.1 The Different Moduli for Simple and Poroelastic Materials (Fig. 5.6)
An isotropic linear elastic material such as discussed in the previous section has two independent material properties, the Young’s modulus (E) and the Poisson’s ratio (ν). These are measured in the “usual way” in unconfined compression of a specimen as Ε = σ/εy and ν = εx/εy. A material filled with fluid (like cartilage) has similar properties, but they depend on whether the fluid is in or out of the solid matrix (Korhonen et al. 2002)! If you compress a fluid filled material quickly it will shorten and widen in the same manner as does a simple solid. However, if you hold the compressive displacement for a long time the fluid can percolate out of the matrix. As a result of fluid percolation, the stress to hold the compression decreases, the width of the specimen decreases and the volume of the matrix decreases with time—eventually reaching a steady-state value that depends only on the properties of the drained solid matrix. In many of the simpler theories for fluid filled materials it is the drained modulus and drained Poisson’s ratio of the solid matrix that are needed to predict the mechanical response of the matrix to loading. Fluid effects are governed by the permeability of the solid matrix which determines how the solid and fluid interact. A little curve ball in the use of the drained moduli for the solid in a fluid filled material is that many theories (in particular the KLM or poroelastic theory discussed below) often use a confined mechanical test that prevents the widening of the specimen to measure mechanical properties. The modulus calculated from a confined test is called the aggregate modulus (HA). The aggregate modulus has a defined relationship with the drained Young’s modulus and drained Poisson’s ratio of the matrix as:
In a simple compression test the Young’s modulus is E = σ/ε y and the Poisson’s ratio is ν = ε x /ε y .
Whether one measures the aggregate modulus directly using a confined compression test or calculates it from the drained moduli of the matrix is all the same mathematically.
5.2.5 Tensile Tests
Woo and co-workers (1976) pioneered the tensile testing of cartilage. They sliced articular cartilage into miniature “dumbbells” having a thickness of 0.25 mm and clamped them into a tensile test apparatus. Strain was measured by videotaping the specimen as it was pulled apart and measuring, in each video frame, the distance between two lines drawn across the undeformed specimen (see Fig. 4.11 for further details on this approach).
When cartilage is loaded, the way the force is shared between the fluid and solid phases is highly dependent on the nature of the loading. In tension, it is primarily the solid phase of the cartilage that carries the load; in particular, it is the collagen fibers. Figure 4.1 shows a typical tensile load-deformation relationship for a collagenous soft tissue loaded slowly. As explored in Chap. 4, these tissues exhibit a “toe” region with an initially low modulus, followed by a region of increasing stiffness. Fibers in the toe region straighten and become stretched in the linear region; in the case of cartilage, it is probably the collagen fibrils more than the proteoglycan molecules that are behaving in this way.
Cartilage produces the same sort of curve seen in Fig. 4.1 except that its upper portion remains nonlinear because the collagen fibers are not brought into full alignment. Its stress-strain relationship can therefore be described by an equation of the form
where σ is stress, ε is strain, and A and B are coefficients. To find the modulus at any point on the curve, one merely differentiates to obtain
Figure 5.7 shows the shapes of such stress-strain curves for bovine articular cartilage as a function of depth below the surface and orientation relative to the split line direction (recall Fig. 2.20). Table 5.2 shows the elastic modulus at a stress of 2.5 MPa for each position and orientation. Note that the tensile modulus is greater parallel to the split lines. Near the articular surface fluid pressure caused by loading and the tensile stresses in the matrix at the edge of a loaded region have components that are perpendicular to the load (parallel to the surface) and tend to tear the surface of the tissue. As a result, the tangential orientation of the collagen at the tissue surface is a useful direction for resisting the tensile stress. Ultimate tensile stresses for articular cartilage range from 9 to 18 MPa and ultimate tensile strains from 60 % to 120 % (Williamson et al. 2003).
Stress-strain curves for bovine cartilage specimens as functions of orientation and depth. Loading rate approximately 1.2 % per second (Woo et al. 1976).
Data for bovine cartilage
Slice # (relative depth)
Tensile modulus (MPa)
Parallel to split lines
Normal to split lines
5.2.6 Poroelastic Theory of Articular Cartilage
When a cartilage specimen is loaded in tension or compression, drops of fluid can be seen to exude and collect on its surfaces (Fig. 5.8). This is a clear sign that cartilage liquid flows under the influence of pressure gradients produced by mechanical deformation of the solid phase. Movement of the fluid phase produces some of the viscoelastic behavior of the tissue, especially for slow loading. Additionally, the permeability of the tissue changes when the solid phase is deformed. Mansour and Mow (1976) studied this behavior in 200-μm-thick specimens of cartilage clamped between two porous metal platens. Through these porous platens a pressure gradient was imposed across the specimen: the applied pressure, PA. The resulting flow was measured and the permeability coefficient k calculated. By compressing the specimen to different strain levels (ε), the relationship between permeability and tissue strain could be studied. Typical results show that an interaction occurs between the deformations of the fluid and solid phases when cartilage is loaded. The curves can be fit to the equation
Permeability vs. compressive strain for cartilage. The family of curves shows how permeability decreases as increasing pressure is applied. From top to bottom, the pressures were 0.07, 0.17, 0.34, 0.69, 1.03, and 1.72 MPa. Previously published as Fig. 4 in Mow, V.C., Kuei, S.C., Lai, W.M., and Armstrong, C.G. Biphasic creep and stress relaxation of articular cartilage in compression. Theory and experiment. Journal of Biomechanical Engineering 1980; 102:73–84 (Mow et al. 1980).
where A and α both decrease as the applied pressure increases. To understand this interplay, Mow and co-workers developed what has become known as the KLM biphasic theory of cartilage mechanics (Mow et al. 1980). This theory assumes that cartilage consists of a soft, porous, elastic solid, the pores of which communicate and are filled with water. The materials that compose both phases are assumed to be incompressible and the fluid to be inviscid. The details of this theory are beyond the scope of this book, but three examples are presented to show where it can and cannot describe the compressive viscoelastic behavior of cartilage.
220.127.116.11 Example 1, Confined Compression Creep Test
Mow et al. (1980) conducted slow creep tests of cylindrical specimens of cartilage using the scheme shown in Fig. 5.9. A cylindrical specimen of cross-sectional area A is cored from a joint surface and placed in a cylindrical metal well with solid walls and bottom. A porous metal loading platen is placed on top of the specimen. A constant load, F, is applied at time t = 0 and the specimen deformation is recorded until an equilibrium deformation is reached. In this situation, creep is controlled by the exudation of fluid through the porous loading platen against the specimen’s top surface. The movement of fluid to this interface is modeled to understand exudation. In the biphasic theory, movement of the fluid in the tissue is determined by the permeability, k, and the fluid pressure gradients of the specimen. The equilibrium deformation is determined by the equilibrium, or aggregate, modulus of the specimen, HA, and the applied loading. If the initial height of the specimen is h, and its displacement under stress σ = F/A is u(t), the biphasic theory solution for the ratio of the deformation to the initial specimen height as a function of time is
Schematic diagram of the experimental setup used for creep compression tests of cartilage. Fluid is allowed to flow through the porous loading platen.
where t is time and Λn = (n + 1/2)2π2 (Woo et al. 1987). Notice that as t → ∞, this equation simplifies to σ = HA (u/h). Figure 5.10 and Table 5.3 show the results of fitting this equation to slow creep data for articular cartilage and two other kinds of cartilaginous tissue: nasal cartilage and knee meniscus. The distinct differences in the properties of these tissues are striking. Notice the inverse relationship between the aggregate modulus and the equilibrium deformation. Also, the cartilage with the greatest water content has the greatest permeability.
Strain (u/h) vs. time for articular cartilage (A), meniscus (M), and nasal cartilage (N) for slow loading. Previously published as Fig. 4.17b in Woo, S.L.-Y., Mow, V.C., and Lai, W.M. Biomechanical properties of articular cartilage. In Handbook of Bioengineering (edited by R. Skalak and S. Chien), 4.1–4.44. McGraw-Hill, New York; 1987.
Compressive properties of three types of bovine cartilage
Aggregate modulus (HA) (MPa)
Permeability (k), 10–15 m4/N s
H2O by weight (%)
18.104.22.168 Example 2, Confined Compression Relaxation Test
In this experiment, the apparatus is the same as that depicted in Fig. 5.9 except that the instantaneous load application is replaced by a ramp displacement like that shown in Fig. 5.11. After a specified maximum displacement is reached, it is maintained while the reactive stress produced in the specimen is monitored. As Fig. 5.11 shows, there is a nonlinear increase in stress until maximum displacement is reached, then the stress “relaxes” to an equilibrium value. Fluid flows from the specimen early in the test (e.g., at A and B), but later the strain redistribution is accompanied entirely by internal fluid movements (e.g., at C and D).
(Left) Displacement applied to cartilage specimen in compression relaxation test. (Right) Stress-time curve for the test. Letters A–E show equivalent positions on the two curves. Previously published as Fig. 6 in Mow, V.C., Kuei, S.C., Lai, W.M., and Armstrong, C.G. Biphasic creep and stress relaxation of articular cartilage in compression. Theory and experiment. Journal of Biomechanical Engineering 1980; 102:73–84 (Mow et al. 1980).
The biphasic theory predicts the stress during both the compression and the relaxation phases of the experiment. The equation for the latter is
where V0 is the displacement rate, t0 is the time at which the displacement is stopped (B in Fig. 5.11), and K = n2π2kHA/h2 (Woo et al. 1987). The other symbols retain their former definitions. Note again that as t → ∞,
because V0 t0/h is the strain at the end of the slow loading period, ε. These equations fit experimental relaxation data quite well when t0 is large enough (on the order of a hundred seconds), and the values of HA and k that achieve these fits agree well with those obtained from creep experiments. There are advantages to both methods of obtaining the aggregate modulus. The relaxation test takes less time to reach equilibrium, but it requires an apparatus to achieve a displacement curve like that shown in Fig. 5.11.
In the course of testing various kinds of specimens, it has been learned that the aggregate modulus of cartilage is inversely proportional to water content but increases in proportion to hyaluronic acid content. For example, diminishing the water content from 85 % to 75 % increases the modulus from about 0.3 to 1.0 MPa, and a 30 % increase in hyaluronic acid content can double the modulus. There is also a reduction in modulus with age, but this is not a predictive relationship because the individual variability at every age is very great.
22.214.171.124 Example 3, Unconfined Compression Relaxation Test
For rapid unconfined compression of a cylinder of cartilage modeled as a linear poroelastic cylinder of radius r filled with an inviscid fluid, the poroelastic solution for the stress is
where σ s (=Ε s ε s ) is the stress after complete relaxation of the tissue, and the coefficients of the series are Α ν (ν) = ((1 − ν)(1 − 2ν)/(1 + ν))(1/[(1 − ν)2 α ν 2 − (1 − 2ν)]), ν = Poisson’s Ratio (between 0 and 0.5), α ν = characteristic equation roots, Η a = Ε s ((1 − ν)/(1 + ν)(1 − 2ν)), Ε s (ε s ) = equilibrium Young’s modulus (strain) of the material after complete relaxation, and k = permeability. The characteristic equation is ϑ 1 (ξ) − ((1 − ν) ξ ϑ 0 (ξ)/(1 − 2ν)) = 0, where ϑ 0 (ξ) and ϑ 1 (ξ) are Bessel functions and α ν are the roots of the equation. This relaxation kernel is well known as the result of solving the poroelastic constitutive model for unconfined compression. As it turns out, 1.5 is the maximum size of the formula in the square brackets of Eq. 5.9 at time t = 0. That the maximum stress (σ(0)) can only be 1.5 times the equilibrium stress (σ s ) means that one should never observe peak stresses larger than this if the KLM biphasic theory holds true; however, it was observed many years ago (Armstrong et al. 1984) that the peak stresses during unconfined compression of cartilage far exceed the limit of 1.5 times the equilibrium stress, showing that the KLM biphasic theory is unable to accurately model the behavior of cartilage when loaded at rapid physiological rates.
5.2.7 Other Models for Cartilage Mechanics
126.96.36.199 Nonlinearly Elastic Matrix
Using a biphasic model with a matrix that has a phenomenological (i.e., empirical) mechanical stiffness that is high in tension and low in compression (Soltz and Ateshian 2000) can solve some of the limitations of the KLM model (i.e., its inability to predict the initial stress of an unconfined compression experiment). However, the use of this special nonlinearly elastic matrix still puts all of the tissue viscoelasticity into the fluid flow. Even though it will now be possible to reach a higher apparent stress during load, the relaxation spectrum is still too slow. The nonlinear material behavior can be modeled mechanistically (e.g. as being due to the fact that you can pull but not push on a fiber) or be simply introduced as a rule. In research a mechanistic model helps more with understanding how a tissue functions and, therefore, the former approach is preferred.
188.8.131.52 Viscoelastic Matrix
The key to a suitable model of early rapid stress relaxation is to use a rapidly relaxing matrix that is viscoelastic (Huang et al. 2001). The viscoelastic matrix is often introduced phenomenologically (i.e., the source of viscoelasticity is not explained) and can be combined with bi-, tri-, and other phasic fluid flow models. When a phenomenological viscoelastic material is used with a “phasic” theory the mechanisms of fluid flow and ionic interactions are approximately mechanistic, but the viscoelasticity remains unexplained. There have been small steps in the direction of mechanistic modeling of cartilage matrix viscoelasticity using polymer dynamics models (Fyhrie and Barone 2003; June et al. 2011). Polymer dynamics is a very complex field in itself (Boyd and Smith 2007) but in theory it can account for the rapid stress relaxation caused by biopolymer:biopolymer interactions in the tissue. The approach is promising and has been shown to be consistent with observations of cartilage stress relaxation in unconfined compression.