Fluid transport and mechanical properties of articular cartilage: A review

SURVEY

ARTICLE -.

FLUID

TRANSPORT

AND MECHANICAL

ARTICULAR VAN C.

Mow,*

MARK

CARTILAGE:

PROPERTIES A REVIEW

H. HOLMES* and W.

Rensselaer Polytechnic

Institute.

Troy.

MICH.AEL

OF

LN*

N1’ 12181. C.S.A.

.Abstract--This review is aimed at unifying our understanding of cartilage liscoelastic propertics in compression. in particular the role of compression-dependent permeability in controllinr interstitial fluid Ho\\ and its contribution to the observed viscoelastic etfects. During the pre\ious decade. Tt was shown that compression causes the Frmeability ofcartilage to drop in a functional manner described by i, = !q, cxpic.~f~ u here X, and .\I were defined as intrinsic permeability parameters and E is the dilatation of the solid matrix (F = crVu).Since permeability is inversely related to the diffusive drag coefftcient of relattve Iluid motion utth respect to the porous solid matrix, the measured load-deformation response of the tissue must thereforealso depend on the non-linearly permeable nature of the tissue. We have summarized in this review our understanding of this non-linear phenomenon. This understanding of these flow-dependent viscoelastic etTectsare put rnto the historical perspectiveofacomprehensive litcraturc rcvica ofcarlierattcmpts to model the compressive viscoelastic properties of articular cartilage.

ISTRODKCTION Diarthrodial joints, such as the knee and shoulder, are one of the three types of joints in the human body; the others are fibrous joints or synarthroses, which have no relative motion (e.g. the coronal suture of the skull), and cartilaginous joints or amphiarthroses, which have little or no relative motion (e.g. the vertebral bodyintervertebral disk-vertebral body joints). A diarthrodial joint is characterized by its large degree of motion; consequently. one of its primary functions is to facilitate body movement and locomotion (see for example Warwick and Williams, 1973; Sokoloff, 1978, 1980). Under normal conditions, it is an amazingly efficient bearing system, capable of providing a nearly frictionless performance with little, if any, wear for the entire lifespan of the individual. Although their individual anatomical forms and material properties vary considerably. there are two components common to all diarthrodial joints: synovial fluid and connective tissue. In the latter category are articular cartilage, meniscus, ligaments and tendons. The biomechanical characterizations of these soft connective tissues (for example, Hayes and Bodine, 1978; Kempson ct al., 1971; Kempson, 1980; Wooer al., 1979,198l; MOWet al., 1980; Roth and Mow, 1980; Lanir, 1983) and attempts to develop dynamic interaction models (Higginson et al., 1976; Higginson, 1978; Dowson et al.. 1981; Mow and Lai, 1979, 1980; Collins, 1982) are currently active areas of interdisciplinary research, incorporating concepts from biochemistry, continuum Re~riwcl IS MOJ 1983: in rehwf jorm 7 fkwmhrr 1983. *Department of Mechanical Engineering. Aeronautical Engineering and Mechanics. +Depnrtment oi Mathematical Sciences.

mechanics, and advanced mathematics. We intend in this article to review some of the experimental and theoretical research projects we have pursued over the past decade and to discuss some of the unresolved questions in the study of the structure and function of just one of the major connective tissues of the joint: articular cartilage. In particular. we intend to focus on the role of the non-linear permeability function in controlling the rate of interstitial fluid flow. which in turn governs the biphasic compressive viscoelastic properties of the tissue. Reviews of work on synovial fluid can be found in Lai cr al. (1978) and some of the dynamic interaction problems that may arise within diarthrodial joints between cartilage and synovial fluid can be found in Mow and Lai (1979, 1980). Articular cartilage is a white, dense, connective tissue, from 1 to 5 mm thick, that covers the bony articulating ends inside the joint. In biomechanical terms, it is a multi-phasic, non-linearly permeable, viscoelastic material, consisting of two principal phases: a solid organic matrix, which is composed predominately of collagen fibrils and proteoglycan macromolecules, and a movable interstitial fluid phase, which is predominately water (Linn and Sokoloff, 1965; Venn and Maroudas, 1977; Muir. 1980; Armstrong and Mow, 1982a.b). The importance of the integrity of the tissue in maintaining the health of the joint and of the deformational characteristics of the tissue as a whole has been appreciated for over at least half a century (Hirsch, 1944; Sokoloff, 1969; Mankin. 1974; Muir, 1977; Schofield and Weightman, 1978; Moskowitz rt al., 1979). However, only recently has the importance of the role of the movement of interstitial fluid through the porous matrix and the subsequent interaction of the interstitial fluid with the porous solid 377

378

VAN C. Mow,

MARK

H.

HOLMES and W. MICHAEL LAI

matrix in controlling the deformational characteristics of the tissue as a whole been established (Maroudas, 1975a.b; IMOWrt ul., 1980; Holmes et n/., in press). Principally, the fluid movement plays a fundamental role in: (1) the biological processes by augmenting the transport of nutrients into, and of waste products out of, the tissue (Ekholm and Norback. 1951; Maroudas er al., 1968; McKibbin and Maroudas, 1979; Honner and Thompson, 1971; Salter et al., 1980); (2) the deformational processes by controlling the mechanism, through a non-linear interaction process, of the rate of fluid transport through the deforming tissue (Maroudas, 1975b; Mansour and Mow, 1976; Lai and Mow, 1980; Lai et al., 1981; Mow YI al., 1982); and (3) the functional processes by providing the lubricant for the conformational gap between the articulating surfaces of the joint, through exudation and imbibition caused by the deformation of the tissue during joint articulation (McCutchen, 1962, 1978; Malcom. 1976; Torzilli and Mow, 1976a,b; Armstrong and Mow, 1980; Dowson et al., 1981; Collins. 1982; Mow and Mak, in press). The portion of cartilage mechanics that we will review here will be those derived from compressive theory describing the behavior of articular cartilage. Recent advances from these investigations have codified the various roles of interstitial fluid movement on the various compressive responses of the tissue. These fundamental and simple concepts may now be summarized in a coherent manner to promote an understanding of the role of interstitial fluid flow through articular cartilage, or any other biphasic tissue (meniscus, nasal cartilage, ligaments, etc.), under a more complex deformational field. Thus the objective of this review is to provide a comprehensive elucidation of the role of interstitial fluid flow in controlling the compressive deformational characteristics of cartilaginous tissues. COMPOSITION

Inters*i:tY Water -

1’

‘Proteogiycan Asregates

Fig. 1. Schematic representation of articularcartilage depicting a porous, composite organic solid matrix swollen with water. The major organic components of the matrix are collagen fibrils and proteoglycrtn aggregates (Mow and Lai, 19801.

30

I-

.:~:0,,5+022*

-I

35

Fig. 2. Variation of porosity C;,’YTand solid content V,/ V, with depth from the surface. Note that the top 204; of the tissue has the most fluid (Lipshitz er at.. 1976).

AND ULTRASTRUCTURE OF ARTICULAR CARTILAGE

As a biomaterial, articular cartilage may be considered as a porous composite organic solid matrix swollen by water, Fig. 1. In the water, a variety of mobile electrolytes, which maintain the charge neutrality condition of the ionized proteoglycan aggregates of this matrix, are distributed (Sokoloff, 1963; Schubert and Hamerman, 1968; Maroudas, 1970). The interstitial concentrations of the counterions and ions are governed by the Donnan equilibrium concept (Maroudas, 1970, Urban et al., 1979). The specific composition of cartilage varies with joint age and the topographical and depth location in the joint, as well as with the specific type ofjoint (Venn, 1978; Muir, 1980). In normal adult cartilage, the fluid content makes up approximately 85 o/0of the total mass by wet weight in the most superficial 25% of the cartilage, and then decreases in a nearly linear manner to about 70 7: at the subchondral bone (Lipshitz et al., 1976a), Fig. 2.

Pathological conditions can also have a significant effect on the water content ofcartilage. For example, in osteoarthritic human cartilage, the water content is above normal (Ballet and Nance, 1966; Maroudas, 1976,1979; McDevitt and Muir, 1976; Armstrong and Mow, 1982a.b)and the water concentration is higher in the central region of the tissue rather than in the superficial zone (Venn and Maroudas, 1977; Venn. 1978). This increase of water content has been found to be the earliest observable change in tissue in animal models of osteoarthritis (McDevitt and Muir, 1976) and this increase has been correlated to the observed decrease of the compressive equilibrium modulus in animal models as well as in human autopsy materials from aging joints (Armstrong and Mow, 1982b). The largest component of the organic solid matrix is collagen (Muir, 1980). Cartilage collagen, which accounts for about half of the tissue mass by dry weight, occurs in a highly specific ultrastructural arrangement, forming four principal layers (McCall, 1968; Clarke,

Properties 0i articular cartilage: a rsbleu

of collagen tibril ultrastructure Fig. 3. Representation throughout the depth of the cartilage depicting the distinct idwlized zones: (a) a superticial tangential zone composed of Jen\i‘j packed collagen ribrils in sheets parallel to the surface. tb)n randomly arrayed middle zone with abundant intertibrlllar space to accommodats the proteoglycan and water. and ICI~ radial deep zone where collagen tibrils appear to snactomose into larger bundles as they insert into thecalcified zoner. The cartilage. i.e. the non-calcified and calcified zones, rests upon a stitl’ subchondral plats supported by the soft wnceilous-marrow composite structure (Mow er [I/., 1974).

197-l: Mcachim and Stockwell, 1979). The superficial tangential zone, which is near the articular surface. consists of sheets of tightly woven collagen fibrils (Mow ef ~1.. 1974; Clarke, 1971, 1974; Ghadially et al., 19761. This region accounts for the highest concentration of collagen. Based on recent X-ray diffraction studies (Aspden and Hukins, 1951) and polarized light and electron microscopy studies (Bullough and Goodfellow. 1968; Sprerand Dahners, 1979), the fibers on the surface seem to be aligned to the split-line patterns.* The tibers of the middle zone, on the other hand. are randomly oriented and homogeneously dispersed. In the deep zone the fibers come together to form larger. radially oriented fiber bundles. These bundles enter the calcified zone. crossing the tidemark. to form an interlocking network that anchors the tissue to the bony substrate, Fig. 3. The major noncollagenous components of the solid phase of articular cartilage are proteoglycan macromoleculss, which in normal adult cartilage make up approximately 2&30”, of the total dry weight of the tissue. However, unlike the collagen and water content, the percentage of proteoglycans is lowest near the articular surface and increases with depth. These macromolecules consist of a protein core on which 50-100 glycosaminoglycan chains are bonded (Muir, 1950: Buckwalter and Rosenberg, 1982). The glycosa-

*The oreanization of the collagen fiber network of artitular cartilnee has been of interest to scientists for a long time. Early rn-icroanatomists believed that the extracellular substance of the tissue was homogeneous. Hultkrantz (1898) demonstrated the regular split-line patterns on the articular buriaces by puncturing them with an awl. The appearance of this regular array of’split lines’on the surfaceofcartilage was taken as ebidrnce of the surface collagen fiber ultrastructure. Tncnty-seven years later. BenninghotT(l925)used these split lincsand pol,irired light techniques to investigate thearrangement of the collagen fibers and the whole tissue. From this he postulated the well-known ‘arcade’ theory of collagen fiber architecture.

3-9

minoglycans are relatively short chains of repeatmg disaccharide units of sulphated hexosamines. In cartilage. proteoglycans are bound to a linear chain of hyaluronic acid to form an enormous aggregate with a molecular weight of up to 2 x 10R and a length of approximately 2 pm (Hardingham and Muir, 1974: Rosenberg t’l (II.. 1975). It appears that most of the proteoglycans in normal cartilage are in these aggregated forms. However, in aging and diseased tissues. the molecular architecture of this aggregate is altered in a number of ways (Brandt and Palmoski. 1976: Muir, 1977). PHYSICOCHE~IICAL C.~RTIL.\GE

BASIS OF FLUID

FLOQ

ASD

DEFORSIATIOS

In normal adult cartilagc.collagen is woven together to form a fibrous network in which the huge proteoglycan aggregates are trapped. Together they form a cohesive porous composite organic solid matrix. The collagen network is characterized by its great tensile stiffness (Kempson et (II., 1973; Woo er rll.. 1976, 1979. 1980: Roth and Mow, l980), but due to the high slenderness ratio of each segment of the collagen fibril. the network is relatively weak in compression. It also has a negligible net charge density at neutral pH (Bowes and Kenten. 1948). On the other hand, the proteoglycans contain a high concentration of fixed. negatively charged groups, giving them a propensity to expand and occupy a large solution domain. Since these macromolecules are enmeshed in the collagen matrix, which prevents them from reaching their fully extended solution volume, their closely spaced charged groups, 5-15 A, together with the high concentration of freely mobile counterions in the interstitial fluid (Donnan osmotic pressure), endow the tissue with a high capacity to swell and gain or lose water when the external ionic or mechanical environment is altered (Schubert and Hamerman, 1968; Pasternack rf ul.. 1974; Maroudas, 1976.1979; Hascall, 1977; \low et al.. 1981; Grodzinsky YItrl., I98 I; Myers et (II.. in press a.b). In human femoral heads, it has been estimated that the internal osmotic pressure can range up to 0.2 \IN m _ ’ (Maroudas, 1979). At equilibrium, this swelling force is balanced by the tensile strength of the surrounding collagen matrix. Consequently, even in an unloaded state, the in sifu collagen network of articular cartilage is under a tensile prestress (Maroudas, 1976). The hydrophilic nature of the proteoglycans and the architecture of the collagen network give the tissue its micro-porous characteristic. Very little of the water in cartilage is intracellular, and a sizable amount of this water, about 30 00, appears to be associated with the collagen fibrils (Torzilli rl (II., 1992). Therefore. most of the water is in the intermolecular space and is free to move through the tissue. The ‘pores’ through which it Rows have been estimated to have a diameter of - 60A (McCutchen, 1962); the derivation of this estimate will be concisely described below. The hydraulic permeability of the tissue, as well as the

VANC. Mow.

380

MARK

H.

HOLMES and W. MICHAEL

LM

aggegate equilibrium compression modulus, are highly dependent on both the water content and the uranic acid content, hence providing the physicochemica1 basis for the observed decrease of the tissue permeability with increased compression (Maroudas, 1975a.b; Mansour and Mow, 1976; Armstrong and Mow. 1982a), Figs 4a-d. During compression, as the mmhanical strain increases, the concentration of the organic material and the charge density both increase, since the interstitial fluid is forced to flow from the matrix. A new equilibrium state is reached when the charge density, collagen tension and applied load are in balance. It is generally believed that this equilibrium compressive

. O70

75

80 WATER

95

90

CGNTENT

95

100

(Z’

Fig. 4(c).

Variation of apparent permeability in m’:N’s units as a function of native uater content of aging human patellar specimens (Armstrong and Liow, 1982a).

01

0

Fixed-charge Fig. 4(a). Variation

02 OS density (m equiv./cm5)

of permeability

in cm’sg-’

units as a

function of tixcd-charge density in m equiv. cm-” of the tissue (Maroudas. 197Sb).

o,

0 .o

75

80 ‘NAT’R

@5 CiVEFIT

90

35

100

(7.)

Fig. 4(d). Variation of intrinsicequilibrium modulus in MPa units as a function of native water content of aging human patellar specimens (Armstrong and Mow, 1982a).

3

90

m

80

Cortiloge OS %

hydration of

initiol

60

55

expressed weight

Fig. 4(b). Variation of permeability in cm3 sg-’ units as a function of hydration as percentage of initial weight of the tissue (Maroudas, 1975b).

stiffness of the tissue is derived from the Donnan osmotic effect associated with the closely spaced, negatively charged sulphate and carboxyl groups on the glycosaminoglycans. as well as from the resistance to bulk compression of the neutral composite proteoglycan-collagen solid matrix (Maroudas, 1975b; Parsons and Black, 1979; Armstrong and Mow, 1982b; Myers and Mow, 1953). The ‘flow-independent shear rigidity’ of cartilage is derived from the collagen fibriis that apparently carry the tensile stress rendered by the spatial nature of the collagen ultrastructure within the tissue when it is sheared, Fig. 5 (Hayes and Bodine, 1978; Mow et al., 1982, 1983). Because the proteoglycan aggregates alone do not provide signilicant shear resistance, they must interact with the collagen network to produce a solid matrix with

Properties of articular

Fig. 5. Schematic depiction of functional role of collagen tibrils in resisting the shear deformation of a unit block of articularcartilage. The existence ofa composite solid matrix is nccessar); for the collagen fibrils to play an active role during shear.

sufhcient shear strength (Armstrong et al., 1981; Mow er LI/.,in press 1983). Similarly, the tensile stitfness of cartilage is derived primarily from the intrinsic collagen fibril properties. the strength of collagen crosslinking. and, secondarily, from collagen-proteoglycan interactions. These equilibrium behaviors are ‘tlowindependent’. When interstitial fluid flow becomes significant, the frictional drag or diffusive resistance of relative fluid flow causes the observed flow-dependent viscoelastic effects, i.e. creep and stress relaxation. In compression, it has been shown that this biphasiceffect is the dominant mechanism responsible for the observed compressive viscoelastic behavior of cartilaginous tissues (Mow er al., 1980, Lai et ~1..198 I; Holmes rr LJ., in press). From the discussion of these rather simple notions, it is apparent that the general deformation of articular cartilage is very complex, involving the mechanical entanglement of collagen and proteoglycan, the tensile resistance of the collagen network, the compressive resistance of the proteoglycans, and the diffuse resistance generated as the fluid flows through the matrix. The details of the internal equilibration mechanisms between the collagen network, proteoglycan swelling pressure, and fluid flow are presently largely unresolved, although some progress has been made in the last few years (Maroudas, 1979; Mow et al., 1980; Grodzinsky er ol., 1981). In this review we intend to present some of the findings concerning the role of interstitial fluid flow in cartilage deformation and, in particular, the non-linear interaction between cartilage deformation and interstitial fluid flow as it is manifested in the non-linear, intrinsic permeability function. The physicochemical characteristics of the counterion within the interstitial fluid, e.g. valency and concentration, also have a significant effect on the response of articular cartilage to imposed stresses and strains. For example, an increase of interstitial Na’ concentration causes a charge-shielding effect on the closely spaced ( - 5-I 5 A) anionic groups on the glycosaminoglycans. This, in turn, can cause the proteoglycans to contract from their extended solution volume (Pasternack et al., 1974). Macroscopically, this results in a measurable

cartilage:

a review

381

shrinkage in tissue size and a decrease m tis,ue mass caused by the decrease in tissue hydration, %loreover. recent experimental and theoretical evidence shows that the modulation of intra- and inter-molecular electrostatic repulsion forces among the proteoglycan charge groups (Mow and Lai, 1980), and or disruption of collagen-proteoglycan electrostatic interactions (Muir, 1980) by changes in the counterion environment in the interstitial fluid. can atfcct compressive (Sokoloff, 1963; Maroudas. 1979; Parsons and Black. 1979) tensile (Kempson. 1980; Myers t’r LJ.. in prrsh a-b), and shear (Hayes and Bodine, 1978; Mow t’t tr!., 1983) properties of the tissue. It is clear from these studies that both the movement of interstitial fluid and the rates of ion transport in cartilage arc of predominant importance in the functioning of the tissue.

FLL’ID

FLOH:

SLTRITIO\

The importance ofinterstitial fluid flow. particularly as it affects the nutrition of cartilage, was appreciated as early as the nineteenth century. Later the role of fluid flow was more popularly expounded by Benninghoff (1924.1925)and by Ekholm and Norback (1951). However. even though there was a great deal of speculation that the deformation of cartilage was strongly influenced by the exudation and imbibition of interstitial fluid, its exact role was not understood in the early studies on the elasticity of cartilage (Bsr, 1926; Giicke, 1927: Hirsch, 1944). For example. Hirsch conjectured that the ‘circulation of tissue juices’ decreased as the cartilage lost its elasticity. thereby reducing the mechanism for its nutrition. At the same time, he could not explain why he obtained a creep-like response and an incomplete recovery in his indentation tests of cartilage, giving rise to a phenomenon which became known as ‘imperfect elasticity’. He apparently did not realize the importance of the exuded fluid that he observed on the articular surface around the indenting probe. Nevertheless, Hirsch hypothesized that there should be a strong connection between mechanical strain and tluid Row, being aware of the ideas of Hildebrand (1921) and Policard (1936) that fluid movement can be induced from a ‘suction and pressure effect’ of repeated loading of the cartilage surface. The role of a pumping mechanism for the transport of nutrients through cartilage has been questioned by Maroudas et a/. (1968) and by McKibbin and Maroudas (1979). During their intermittent compression experiments, they found that the penetration of methylene blue into cartilage is no more than they believed would be expected from simple diffusion with no intermittent compression. Consequently. they concluded that for small solutes, such as glucose. diffusion is the controlling mechanism, whereas a mechanical pumping action probably governs the transport of solutes of a larger molecular weight. such as serum albumin. One of the arguments used in support of a

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H. HOLMES and W. MICHAEL LAr

pumping mechanism is that when the joint is incapacitated, so there is no rhythmic stressing, the cartilage shows signs of degeneration (Honner and Thompson. 1971: Salter et al., 1980; Brandt. 1981). However. there is also the possibility that the synovial fluid ‘stagnates’ at the articular surfaces in such a situation, thereby reducing the nutrients available for transport by diffusion. The experimental evidence to date supports the fact that low molecular weight solutes are transported by simple diffusion, but there has not yet been a complete coherent theoretical analysis to assess the relative importance of the mechanical pumping effect vs the ditfusive transport mechanism for cartilage (Lai and Mow, 1978).

CONTISL’USl Single

phase:

~IODELISC

OF ARTICULAR CARTILAGE

elastic

In studying the deformational characteristics of articular cartilage under mechanical loading, one of the central concerns has been the determination of the ‘elastic modulus’ of this thin layer of tissue at the ends of the bone in a diarthrodial joint. Because of its anatomical form and its thinness, the indentation experiment was the choice of many investigators (Hirsch, 1944; Sokoloff, 1966; Kempson et al., 1971; Colctti et al., 1972; Hayes et al., 1972; Hori and Mockros, 1976). The first attempts to find the Young’s modulus usually involved the application of the Hertz solution for the contact between two elastic bodies of infinite depth. Although the justification for this is usually vague, Hirsch (1944), who was aware of the possible influence of the multiphasic nature of cartilage on its deformational characteristics, justified his analysis on Gildemeister’s (1914) work on the contact between gels. Another approach was taken by Sokoloff (1966), who considered cartilage to be comparable, in terms of its mechanical response, to medium-hard rubber. For this latter material, the ‘mean instantaneous deformation’ is apparently not very sensitive to the thickness if it is thick enough, in this case more than about 2 mm (Livingston rr a[., 1961). Since the cartilage specimens used by Sokoloff were approximately 3 mm thick, he assumed that the Young’s modulus E for cartilage could be determined from the relation E

P =2.67w,a1

P(1 -v)

4W, a

E _

P(l -G) 2w,ai(a,

h, v) ’

where the function K comes from the solution of the integral equation. From their analysis. Hayes et al. concluded that the depth of indentation is very sensitive to the aspect ratio, a/h, since the radius of the indenter is less than the thickness in most indentation tests, Fig. 6. To illustrate, with the assumption that cartilage is incompressible, it can be shown that for cartilage on a rigid foundation, the instantaneous Young’s modulus was 30 :,, less than that calculated by Sokoloff. This shows that in Sokoloff‘s calculations, the stitfness of the underlying bone was lumped into the cartilage stiffness, giving an erroneously high Young’s modulus value (Mow er al., 19S2). Singlr phase: rkcoelasric The assumption that cartilage is purely elastic applies, at best, only at equilibrium because there

(1)

where P is the constant applied load, we is the depth of penetration, and a is the radius of the plane-ended, cylindrical indenter. This comes from the solution of the elastic punch problem for a linearly elastic medium of infinite depth at equilibrium and with a shear modulus of /L=p.

incompressible, so v = 1,‘2and E = 311.Inserting these values into equation (2). one obtains equation (1) used by Sokoloff for E. From this. for human patella. Sokoloff found the ‘instantaneous Young’s modulus’ (which uses w,, at 0.S s after application of the load) to be 2.28 MPa. Also. the’equilibrium Young’s modulus’, which uses we at one hour, was found to be 0.69 MPa. Moreover, he observed that the deformation of articular cartilage was non-linear, even for strains of IO 9,. which is contrary to the basic assumptions employed in deriving the expressions for E and /I, i.e. equations (1) and (2). The analysis for a plane-ended, as well as for a spherical-ended, indenter on a layer of linearly elastic material attached to a rigid foundation was carried out by Hayes et al. (1972) and by Hori and Mockros (1976). Using the approach of Lebedev and Ufliand (1958) of reducing the problem to the inversion of a Fredholm integral equation of the second kind. Hayes it al. were able to determine the displacement field of the elastic layer at equilibrium. For the case of a plane-ended indenter, the Young’s modulus was found to be given as

(2)

In his analysis, Sokoloff further assumed cartilage is

Subchondral Bone Fig. 6. Representation ofa plane-ended indenter applied to a thin layer ofarticular cartilage. Cartilage has been modeled as a single-phase elastic medium (Sokoloff. 1966). a layered, single-phase elastic medium (Hayes et (II.. 1972: Kempson et NI., 1979). a single-phase viscoelastic medium (Coletti er ul.. 1972; Parsons and Black, 1977). and a layered, biphasic medium (Mow rr al.. 1991t.

Properties of articular would bc no dissipative et%ct due to the movement of the interstitial fluid. However, it was not until the early 1960’s that the systematic study of the role of fluid flow on the function of cartilage actually began. Elmore e’t al. ( 1963). in one of the Hurststudies, showed that the creep response observed in the indentation testing of cartilage is largely due to the eRlux of interstitial fluid from the tissue. Their obsenation resolved the problem of the ‘imperfect’ elasticity of cartilage that had been observed 3Oyr earlier by Hirsch. They also observed that after removal of the load, complete recovery occurs only if the fluid is present to enter the cartilage. This study was extended by Linn and Sokoloff (1965), who correlated the magnitude of the creep response and the amount of fluid exuded from the tissue. and by Sokoloff (1966). who studied the topographical variations of these properties. This latter problem has also been examined by Hirsch (1944). Kempson er ol. (1971). and Cameron t’r 01. (1975). While most investigators had by then realized that interstitial fluid flow is intrinsically linked to cartilage deformation, no theoretical attempts were made to consider these effects in a continuum model. Most investigators, except Torzilli and Mow (1976a.b) continued to use single-phase viscoelastic models to describe the compressive creep response of cartilage. In the indentation tests of cartilage, after a sudden application of static load, a rapid compression takes place which is followed by a slow creep process toward equilibrium over the next 60 min or so. One method used to account for this is to use a lumped-parameter, single-phase, spring-and-dashpot, viscoelastic model without regard for the interstitial fluid flow and internal redistribution of the organic matrix and the compaction within the cartilage specimen (Camosso and Marotti, 1962; Hayes and Mockros. 1971; Coletti ZI al., 1972; Parsons and Black, 1977, 1979). Kempson (‘Ial. (1971,19SO), in examining the results of their indentation experiments, used an expression for what they refer to as a ‘two-second creep modulus’ for cartilage, based on a study by Waters (1965) on the indentation of thin sheets ofvulcanized natural rubber. The latter assumes that the classical Hertzian solution for a semi-intinite elastic layer can simply be multiplied by an appropriate dimensionless function to account for a tit-rite depth. For a plane-ended indenter, he concluded that (4) where 4 was determined experimentally. This result is functionally equivalent to equation (3), but this empirical approach says nothing about the displacement tield of the elastic layer and little about physical constraints, such as boundary conditions. In any case, from the measurement of the indentation, two seconds after application of the load, Kempson used equation (4) to determine E. The resulting value, the’two-second creep modulus’, is supposed to include the initial elastic response as well as a small portion of the creep. This is,

cartilage:

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however. inconsistent with the equilibrium condition used to derive equation (4). and it clearly violates the assumption that the material is purely elastic. Further, the expression is quite sensitive to the Poisson’s ratio. see equation (3). and had to be guessed at in order to calculate E from equation (4). It is not surprising that Simon (1971). from his indentation tests on canine tibia1 cartilage, obtained inconsistent results using the ‘two-second creep modulus’ concept. A similar diffculty was encountered by Hori and Mockros (1976) in using equation (3); they found there was considerable dispersion in their values for E. which they suggested was due to the non-linear, anisotropic and inhomogeneous nature of articular cartilage. This type of analysis has been extended by Parsons and Black (1977) to include linear viscoelasticity, using a generalized Kelvin model to describe the transient portion of the creep curve under a constant indenting load. The concepts of ‘unrelaxed and relaxed modulus’ were introduced to determine the instantaneous and equilibrium values of the modulus using equation (3) where the Poisson’s ratio was assumed to be 0.4. With appropriate reinterpretation (Mak PI al., 1987; Mow YL al.. 1982). these unrelaxed and relaxed moduli can be used to estimate the intrinsic elastic moduli of the solid organic matrix of the biphasic model for cartilage of Mow et al. (1980) without an a priori assumption on the value of the Poisson’s ratio. The retardation spectrum also can be reinterpreted to yield the permeability of the solid matrix. Thus we see that the evolution of our understanding of cartilage deformation, experimental and theoretical, leads us naturally into a multiphasic model for the tissue. A comprehensive application of viscoelastic theory to cartilage in tension has been made by Woo er al. (1979, 1980). From their experiments on the stretching properties of articular cartilage, they developed a quasi-linear viscoelastic model, which assumes that the kernel of the stress-strain-history integral is a function of strain and time. This is similar to biomechanical models for other types of soft tissues (Fung, 1981). From it Woo er al. found that they could accurately predict the relaxation and cyclic behavior, but at the same time, they were unable to obtain a reasonable value for the elastic stress if the strain rates were low. At higher strain rates. the biphasic viscoelastic effects would begin to dominate, and the movement of interstitial fluid must therefore be taken into account. Recently, Li er al. (1983) produced experimental evidence, which correlated well with a biphasic analysis of cartilage strips in tension, that interstitial fluid flow could be important if the strain rates of the tensile experiment are sufficiently high, Fig. 7. Mulliphnsic

theories

It is apparent from experimental results. such as those obtained from indentation tests, that the movement of the interstitial fluid plays a fundamental role in the dynamic deformational behavior of the tissue. In other words, to obtain a realistic rheological model for

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C. i%tow.MARK H.

HOLMES and W. MICHAEL LAI

c

fluid phase and the solid phase, and ps is the apparent density of the solid. The coeRicient z is defined as the solid content, which is given by the ratio of solidity r’,/ C; to porosity C;/ C; of the tissue. Here I’,. C’,.and V, are the solid, fluid and total volume of the tissue, respectively. The momentum balance equations for the two components of the mixture. omitting inertial forces, are divd-k’(v’-vyl)

= 0,

@a)

) = 0.

(6b)

and div 0’ + K (9 - vJ

LONGITUDINAL

STRAIN

(%I

Fig. 7. The stress-strain data for superficial-zone articular cartilage at three difkrent strain rates. The strain rate and the number of specimens tested at each strain are shown on the figure. Those data are for the small-strain region. For small strains, the biphasic model predicts that the load per unit area on the specimen is

2

=

irE

A

+

’

h2F:

-4 k,H,

A0 -

c’ A, exp( -2,2”2 ~~k,,r/h*)

1 k=---(1 +a)*K’

“=I

where E, and II, are the Young’s the solid matrix and k,

modulus of permeability A ,, .

A,.

modulus and aggregate and h are the specimen and thickness. respectively. and A,, and sI,are constantsdependent on the moduli of the solid matrix (Li et al.. 1983).

biomechanical studies on articuiar cartilage. it is necessary to account for the fluid component as a distinct phase of the system within the tissue. This

means that, at the very least, cartilage should be modeled as a biphasic (two-phase) material, with the solid matrix and the interstitial fluid as the two phases. This point of view has been taken by a number of investigators such as Fessler (1960). McCutchen (1962), Zarek and Edwards (1964), Torzilli and Mow (1976a.b). and Higginson cl al. (1976). It should be pointed out that even though a two-phase model is adequate for the description of the motion found in most mechanical testing done so far, a more involved model of articular cartilage would be multiphasic, taking into account the influence of the mobile electrolytes and the collagen-proteoglycan solid phase as a fiber-reinforced composite porous matrix (Myers et (II., in press). A biphasic model for articular cartilage was begun by Torzilli and Mow (1976a.b) and extended by MOW and Lai (1979) and Mow et al. (1980). based on the mixture theory of Craine et al. (1970) and Bowen (1976). In essence, this model depicts articular cartilage as a soft, porous and permeable, elastic solid filled with water. Assuming that both phases are intrinsically incompressible, the continuity equation for this binary mixture is divv’+cxdivv”+rl(v’-v/)-grad

In p’= 0,

where d and u/are the apparent stress tensors for the solid and fluid phases, respectively. The second term in equations (6) represents the diffusive drag arising from the relative velocities between the fluid and solid components. Even in the unloaded state, the diffusive coefIicient K has values on the order of 10L5N .s rnmS, which helps explain why the inertial terms are omitted in equations (6). Under slow flow conditions, the diffusive coefficient is related to the permeability k of the tissue by the inverse relation (Lai and MOW, 1980)

(5)

where v’ and v” are, respectively, the velocities of the

In the simple version of the biphasic theory used to model articular cartilage, known as the KLM model for cartilage, the solid phase is assumed to be isotropic and linearly elastic, and the interstitial fluid is inviscid (Mow and Lai, 1980). Accordingly, assuming small strains, the isotropic stress-strain relationship for the first order theory for the solid phase is d = - apl

+

i.,eI + Q,e,

@a)

and for the fluid phase it is flf = -PI,

(W

where p is the apparent fluid pressure, A,, pr are the intrinsic elastic moduli of the solid matrix in the mixture, and e is the infinitesimal strain tensor describing the deformation of the solid matrix. It is also assumed that the tissue is spatially homogeneous, SO i., and y, are constants. Because of the generality of the formulation of this biphasic model, other terms can be included in the linearized constitutive laws, equations (8). which can incorporate, for example, a viscoelastic solid matrix, diffusive couples and capillary forces. Also, the isotropic case is presented here, but formulae for the more general case of biphasic anisotropic stress-strain laws can be found in Mow and Lai (1979). These extensions are important to remember because the assumptions of the linear KLM model are only approximately correct. For example, cartilage appears to have a significant anisotropic structure, as manifested by the split-line patterns and anisotropic tensile properties (WOO ef a!.. 1976; Kempson, 1980; Roth and Mow, 1980). Also, it has been found that in shear the organic solid matrix is slightly viscoelastic (Hayes and Bodine, 1978; MOW et al., 1982). So a more accurate model of articular

Properties of articuhr cartilage: a review cartilage would have to include terms to describe the mtrinsic viscoelastic behavior of the solid matrix. However, in compression the predominant mechanism giving rise to the observed viscoelastic behavior of cartilage appears to arise primarily from the diffusive drag caused by the interstitial fluid flow through the solid matrix, and depends only secondarily on the v iscorlastic properties of the solid matrix itself. In fact. we have found that the linear KLM theory can adequately describe the observed creep and stressrslaxation behavior of articular cartilage and nasal cartilage, as well 3s meniscus. under isothermal and constant electrolytic conditions (Mow and Lai. 1980; Mow er al.. 1980; Mow and Schoonbeck. 1982; Favenesi t’r (11.. 1983).

PER%lE.ABILITY

The permeability of the tissue is a macroscopic measure of the ease with which fluid can flow through the matrix. One of the first studies of permeability was made by McCutchen (1962). who examined the uniaxial compressive properties of cartilage by creeptesting cylindrical plugs of bovine shoulder cartilage. For the surface layer, he found the average value of the permeability to be 5.g x IO-” mJ/N.s and he observed a decrease in the permeability with depth from the articular surface. This dependence on depth was also examined by Maroudas t’t al. (1968). who found the permeability for human femoral condyle cartilage to increase from the supcrticial region to the middle region by about 35 “,,. then to decrease from the middle region to the deep region of the tissue by about X0”,. This inhomogeneity has been attributed to the dense network ofcollagen fibers in the superficial zone (Muir cr a/., 1970) and to the increase in charge density in the deep zone (Maroudas rr al., 1969). In fact, Maroudas (1975b) found that the decrease of permeability below the surface layer correlates significantly well with the observed increase of the fixed-charge density, Fig. 4a. This dependence of the permeability on the ionic content of the bathing solution was first demonstrated by Edwards (1967) on cartilage from the hips of dogs. He found that by increasing the ionicconcentration. by changing from normal saline to normal Ringer’s, the permeability increased by a factor of about three. The dependence of the permeability of articular cartilage c-n the compressive strain in the tissue was first shown by Mansour and Mow (1976). More recently, additional data were obtained, and this compressionstrain-dependent permeability was shown empirically to depend on the applied compressive strain and the applied pressure gradient in an exponential manner, Fig. 8. It was found that there is a non-linear decrease in the permeability with increased compressive strain, which leads to a significant reduction in the ability of the fluid to flow through the matrix. This latter measurement was a simple mechanical method to verify the correlation of permeability to fixed-charge density obtained earlier by Maroudas er al., since with

$85

the fixed-charge density of the tissue compression. increased while the permeability decreased. The approach in all these studies of cartilage permeability is basically the same. Fluid is forced to flow through a cylindrical disc of cartilage that has been removed from the bone by applying a direct fluid pressure P,, across the tissue. For a one-dimensional flow through a sample of thickness h, the apparent permeability k0 measured in these permeation experiments is determined from the empirical Darcy’s law, which states that (9) where Q is the volume flux of permeated fluid through a permeating area A of the specimen. The difficulty in determining the permeability of soft biological tissues with this procedure is that the permeation process gives rise to a drag force (i.e. the force exerted by the fluid on the solid as it flows through the specimen) of significant magnitude to compact the soft, permeable solid matrix in a non-uniform manner. This compaction decreases the permeability within the tissue, and the decrease varies with the distance from the surface. making the measured value of k,, an average. lumpedparameter value. This effect is particularly important for soft tissues such as articular cartilage, where substantial compaction of the matrix can easily occur. Experimentally, this is manifested by the dependence of Lo on the driving pressure P, that was first observed for articular cartilage by Mansour and Mow (1976). Note that for porous, permeable elastic materials where the solid matrix is very stiff, e.g. soils or sintered metallic materials, this effect is negligible or nonexistent. Thus this flow-limiting effect appears to be particularly important only in soft biological multipbasic materials. To separate the effects of the clamping strain E, used to hold the specimen discs in the permeation experiment and the applied pressure PA, Mow and Lai (1980) obtained a family of permeability curves L,(E,: PA) with PA as the parameter. In doing this. it was empirically demonstrated that there is an exponential decrease of the permeability function with sc. Figure 8 shows how the permeability changes as a function of E,

for various parametric values of the constant pressure drop PA used to maintain steady permeation. The empirical exponential law used to curve-fit the data was k, = A(P,)exp[-W,)s,)].

( 10)

where A(P,) and a(P,) are also shown in Fig. 8. To incorporate this result into the biphasic theory, Lai and Mow (1980) introduced the concept of intrinsic permeability, k, defined as X-= lim k,(s,; P,4) = li,exp(Ms),

(II)

P,-0

where k, = A(0). .V = a(0). and E = -E, is the dilatation field. It is important to note that this function is

386

VAN

C. Mow, MARK H. HOLMES and W.

PERMEABlLlTY

vs. APPLIED

p,:

I6

0

IMICHAEL

Applied

STRAIN

.

0 fJF,q

MN/,,?

l

OS?2

MN/m2

24 Compressive

LAI

32 SlroYn

40

l

C

(%I

Fig. 8(a). Experimental curves of permeability versusapplied compressive strain at various levels of applied pressure for a sample of bovine articular cartilage. The solid curves are least square best fits of the indicated exponential decay law, equation (10) (Mow er 01.. 1980).

1

05

I I5

I IO

I 20

) P 1

Pressvre (MN/m21 Fig. 8(c). Curve of exponent z(PA) vs P, for the same bovine specimen (Mow et aI., 1980).

through a circular cylinder is used IO

05

PressuretMN/m*l

I5

20

Q=-gy

A

Fig. 8(b). Curve of the amplitude function A(P,) and experimental data points vs P, for the same bovine specimen (Mow er al., 1980).

defined for the theoretical limit of PA -+ 0, which can be obtained parametrically from Fig. 8 by extrapolation. For normal bovine cartilage immersed in normal Ringer’s solution, we find from the permeation experiment that k, = 1.7 x lO_” m4/N*s and M = 4.3. Consequently, even for small compressive strains, there is a significant nonlinear interaction between the fluid phase and the solid matrix as the tissue deforms. It is of interest to note how the estimates of the ‘uniform pore size’ of these cartilage specimens, which appear in the literature, have been obtained (McCutchen, 1962; Maroudas, 1973). Usually the Poiseuille formula for steady-volume rate of flow, Q,

-’ .

naf

P

PA

Here /Ais the viscosity of the permeating fluid, ai and Ii are the radius and length of each pore, Fig. 9a, and P,, is the applied pressure at the upstream end of the tube. For a specimen whose permeated area is A, and thickness h. and with n number of pores, the area1 porosity and tortuosity factor would be given by flA A,/A, and di = ii/h, respectively. For a un=c;_, iform Poiseuille model whose pore structure is homogeneous and isotropic,

and ,=!!=’ h

ii’

where a and I are the assumed constant radius and length of the ‘circular cylindrical pore’. Inserting these model assumptions into equation (12) above and

Properties of articular

cartilage: a review

lations show that the pore size decreases practically linearly with the bulk compression. This leads one to conclude that bulk compression of cartilage is really achieved by expelling the fluid from the intrrstitium and that the organic matrix is essentially incompressible. One must warn. however. that this type of heuristiccalculation may be grossly inaccurate by itself and should not be used strictly by itself but rather should be used to supplement other. more rigorously derived results such as those presented in this review. Fig. 9(a). A uniform tube Poireuille flow model for assessing the ‘average pore size’ ofsrticular cartilage. In this model, all circular cylindrical tubes are uniform in size (I, = I and ,-ii = .41. It is assumed that the porosity is isotropic so that the areal porosity is equal to the volume porosity.

X x 0.669

MN/m*

PERMEABILITY

03 APPLIED

COMPRESSIVE

STRAIN

(%)

Fig. 9(b). Variation of average pore size as assessed by the uniform tube model. The method allows assessment of change of interstitial pore size with compression and applied pressure. Average solid content z and average porosity /I were assumed IO be 0.22 and 0.81 respectively. and d = tortuosity Sactor.

this volume rate of flow with that defined by Darcy’s law, equation (9). the&average pore radius’may be shown to be

equating

SOS-LINE.aK

FLLID

SOLID ISTERACTIO\

DL-RISC

CO\lPREsSIOS

The above discussions pertain to steady-state behavior ofcartilage during compression or permeation. They explain the need to consider a nonlinearly permeable biphasic model for articular cartilage under compression. This concept of nonlinear, straindependent permeability must be tested in other deformational conditions to verify its general validity (Lai ef al., 1981; Holmes c~ ul., in press). A number of different testing procedures have been used to accomplish this. They generally involve the nonsteady or dynamic deformational behavior of cartilage; one method is the in sifu stress-relaxation indentation test described earlier (Mak er nl., 1982). and another is the uniaxial confined-compression experiment. A particular example of thcsc latter experiments arises when cylindrical ostrochondral plugs arc compressed against a free-draining, rigid. porous filter at the articular surface. The pores of the filter are small enough so that the filter may be used to compress the solid matrix of the tissue, yet large enough to allow ‘free’ exudation of the Huid across the surface. At the same time, the plug of material is inserted into a snugly fitting confining chamber to prevent significant amounts of lateral expansion as the tissue is being compressed normally across the articulating surface of the specimen, Fig. 10. This experimental protocol was imposed in an attempt to achieve the theoretical condition of one-dimensional confined compression -

(13) Here k, is the experimentally determined apparent permeability, equation (10) or (I 1). With known values of tissue porosity p-O.8 and water viscosity, Fig. 9b shows the range of average pore radii for normal bovine articular cartilage in terms of the tortuosity factor 6. There is no estimate for the tortuosity factor-the path length actually traveled by a fluid particle as it traverses the specimen of thickness h. It would not be unreasonable to assume that this factor ranges between one and two. For example, for d = 2, we see from Fig. 9b that for uncompressed tissues (from the intrinsic permeability curve), a - 56 A and with 40”” (clamping or bulk) compression a - 30A (assuming S = 2 for both cases). These simple calcu-

Fig. 10. Schematic depiction of the confined-compression biphasic experiments. The rigid porous loading (with either prescribed tractions or displacements) allows exudate to flow freely into the pores of the tilter. The confining chamber is used to approximate the one-dimensional theoretical requirement assumed in the analysis.

VANC. Mow.

388

MARK

H. HOLMESand W. MICHAELLAI

and it has been found by Mow et al. (1980) and Armstrong and Mow (1982a.b) to be a significantly accurate method to assess the intrinsic material properties of the solid matrix, its aggregate modulus H, -_i,+ 2~,, and its linear permeability coefficient k,. In an extension of the linear KLM biphasic theory of cartilage to incorporate the nonlinear permeation process, equation (11). the equation for the onedimensional deformation of the solid matrix along the axis of the plug was shown to follow the non-linear diffusion equation (Mow and Lai, 1980; Holmes et a!., in press).

H,

2 =k, exp

(14)

where the intrinsic permeability function, equation (11). has been used to derive equation (14). Here u(z, t) is the axial component of the displacement vector describing the deformation of the solid matrix, Fig. 10. The axial component of velocity c(z, r) of the interstitial fluid phase is found from the continuity relationship, equation (5), with the boundary condition u = u =Oatz=h I+. t) = -a

a z

ufz,1).

(15)

A common experiment for the uniaxial compression configuration involves the determination of the creep behavior of the material. For the creep test, a constant load is suddenly applied across the porous filter, and the subsequent displacement is recorded. In the bipbasic theory, the appropriate boundary condition at the articular surface (z = 0) interfaced with the freedraining rigid porous filter is

H,gL i

0

for

-F. for

tG0 t> 0,

tidemark, i.e. the noncalcilied~lcified juncture. In other words, prior to this time, the compression is confined near the surface, so the creep response of the tissue in confined compression is unaffected by the boundary at the tidemark. For the long-time behavior, the displacement exponentially approaches its equilibrium value of IIF,,/H,~, which is determined by the aggregate modulus of the linearly elastic solid matrix. One of the most important aspects of this asymptotic result is that the early time response of the tissue, as observed by a creep experiment, is proportional to J;, irrespective of the load Fe. This behavior can be easily verified from the creep data. Figure 11 shows some of our recent experimental data on normal bovine cartilage verifying the asymptotic solution given by equation (17). The other important aspect is that, for small strains, the equilibrium stress and the equilibrium strain in compression are linearly related; this has been repeatedly verified experimentally (Mow et al., 1980; Armstrong and Mow, 1982a). The second confined compression test measures the uniaxial stress-relaxation behavior of cartilage. This experiment was first used by Lipshitz et al. (1976b) to assess the influence of fluid Row on the stressrelaxation behavior of cartilage, Fig. 12. In these experiments, a ramp-displacement function is imposed on the articular surface given by %3(t) =

Vet 1 V&l

for for

0 < t < to (compression phase) t, < c (relaxation phase). (19)

As with the creep problem, the resultant inhomogeneous displacement field must be determined by solving the nonlinear diffusion equation, equation (14). An accurate approximation of the solution to this problem can be obtained for the case of a slow rate of (16)

where Fe is the magnitude of the applied compressive traction. The resulting mathematical problem, which includes equation (14), is non-linear, and at present no analytic solution is known. Therefore either numerical or asymptotic methods are necessary to obtain a solution to the non-linear creep problem. We have done this by examining the short and long-time behavior of the solution (I-Iolmes, in press). It was found that, in the initial moments after applying the load, the creep displacement of the articular surface, denoted by ue(t), is given as

t IS-

60-

45-

30-

h2

uo(t)=j30J;,0s t e<-k,H,’

(17)

where for small compressive strains /I,, =?exp(-$$($)“‘.

(18)

This describes the surface displacement for cartilage with a thickness of 1.5 x 10m3m during the first 120s or so, which corresponds approximately to the time it takes the deformation, u(z, t), to diffuse down to the

Fig. 11. The measured displacement of the articular surface during the first 120 s of a confined uniaxial creep experiment on bovine articular cartilage with an applied load of I.2 x IO’ N m -I. The linearity of these data compares well with the predicted behavior from the biphasic theory as given by the asymptotic solution in equation (17).

Properties of articular cartilage: a review Control led Romoed Dtsolocement

-w t

Stress History UP

During Romp Dispiocement

Timt

Schematic

of Fluid EfllUlt

Flow During Romp Displacement

Efflux dir

n Equilibrium

Articulor Cortilagt

HI

Arliculor Surioct

Tidtmorh E

Fig. I?. The top graph shows a controlled ramp displacement for a confined-compression experiment. The middlecurve showsa typical stress rise during thecompression phaseand a typical stress-relaxation response in the relaxation phase. The bottom schematics show that the movement of interstitial fluid predominates the stress-history response. During the compression phase, fluid exudation gives rise to a peak stress (I,,, and during the relaxation phase, fluid redistribution gives rise to the relief of the compacted region at the surface, hence stress relaxation. (For more quantitative details, see Fig. 13.)

compression. For a slow rate, by which we mean

(20) a nearly uniform state of compression exists in the tissue (Holmes, 1983.1984). In thiscase, theasymptotic expansion of the solution for the stress history at the articular surface (-_= 0) is given by

U;_(I) -&o H,

~+c,exp(crMt)

1

h2

, ----
(214

where h* co =3koH,1,

representing the strain imposed in the experiment. Based on the known typical permeation values for normal bovine cartilage, this expression is valid for experiments in which the compressive phase lasts longer than loo0 s and, of course, for strains of no more than about 20 y<. Higher rates of compression or higher frequency excitations in compression would induce compressive strains beyond the validity of linear infinitesimal strain theory. For this slow rate it is also relatively easy to determine the solution during the relaxation phase (Holmes et al., in press). We find that immediately subsequent to the start of the relaxation process, at the articular surface,

(Zlb)

L(t)

c’=T.

M, =

Here the equilibrium compressive strain .eois given by h

< 1 (23a)

2ha, t,jnH,k,exp(--EoM)’

(2W

and

I

pot0

~

0
where

v,

Eo =

- a,-.V,J;_

(22)

up = -coH,[l

+coexp(eoM)].

(23~)

390

VAN C. Mow.

MARK

H. HOLMES and W.

MICHAEL

L,u

Here CT, denotes the equilibrium compressive stress attained in the limit I - x given by

typically the maximum peak load is s 4-5 N and the equilibrium load is about 2-3 N for a 6.35 mm diameter plug ofcartilage. Because of the simplicity of the (231 analytical asymptotic solutions for the predicted stress 0, = c,,H,,. This shows that at the start of the relaxation phase, the history in these experiments, it is relatively easy to compressive stress decreases as the square root of time determine the material constants of the model. For the from the peak compressive stress a, achieved at the end case at hand, we used a non-linear regression analysis of the compressive phase. This approximation applies, with equation (2la)as the object function to determine roughly, to the first 100 s of the relaxation phase. As k, and ?lf. The aggregate modulus H,4is obtained from time increases, the compressive stress continues to the equilibrium stress measured after complete stress decrease, and in fact it approaches exponentially its relaxation has occurred. Figure I4 shows an actual set of data from our experiment where i = 0.0021 y,;s-’ equilibrium value of EdH 4determined by the linearly and co = 5OCQs. In this case. we find that k, = 2.51 elastic response of the solid matrix. These results show x 10-1s rn’1N.s. ‘41= 7.83, and H, = 0.41 MPa. that stress relaxation occurs because of an internal fluid redistribution process, which in turn is governed For the typical slow strain-rate stress-history curve shown in Fig. 14. we see that, as in Fig. 13, immediately by the lower permeability within the compressed following the onset of the ramp compression, a tissue, k, exp( - E,,M), giving rise to a more rapid rate parabolic load-vs-time curve is observed. After this, an of relaxation relative to the peak stress op. A summary of the results obtained from the asymptotic solutions is almost linear time dependence is seen during the shown in Fig. 13. These results illustrate the fundamencompressive portion of the experiment. If we extal role of the flow-limiting effect on the deformational trapolate this curve back to r = 0, and denote the intercept with the a-axis. by ue, we have from equation behavior of the non-linearly permeable tissue. It tends (21a) u0 = cc,. The contribution of the non-linear to make the tissue appear stiffer-as with other viscoelastic materials-with increasing rates of compermeability can be seen in Fig. I3 if one compares era pression by virtue of the increased frictional drag the normalized difference between = (cp-c,)/c,7 the peak stress up and equilibrium stress uz, with ue. If required to extrude the fluid from the tissue during the the permeability were constant, so M = 0 in equation compressive phase of the experiment. Thus the non(11). uR and u,, would be equal. However, in Fig. 14, it linear, strain-dependent permeability effect appears to can be seen that they are not the same; uR is about be ideal in the diarthrodial joint loading situation-it 100 y,, larger than ue. These stress-rise characteristics exudes fluid when required, but only with everare governed by the rate of compression I’,, thickness increasing resistance to flow. h, aggregate modulus H,, and the intrinsic permeWe have performed numerous stress-relaxation ability parameters k, and 121.The non-linear, flowstudies following a very strict experimental protocol to limiting parameter M, as defined by Lai et al. (1981). is insure the repeatability of the sensitive measurements responsible for the rise of the stress above and beyond (Holmes er a[., in press; McCormack, 1983). The the values that would be obtained with a constant maximum peak load in all these experiments occurs at permeability. For the strain rates required by the slowthe end of the compressive phase, and it increases with rate experiment, the deviations are relatively small increasing strain rate t. For the ‘slow’ rate experiment,

10 t

1. Time

(secl

Fig. 14. Non-linear

Fig. 13. Depictions of the asymptotic normalized stresshistory prrdiction of a slow-rate compression experiment if the tissue is non-linearly biphasic. The stress rise in the compressive phase depends on the sum of a linear time function and an exponential time function. The stress intercept u0 is different from the total stress decay uR = c,, exp(e,M) (Holmes et al., in press). For a linearly biphasic material. CT”= CJ~ since .LI = 0.

regression analysis, using equation (Zla) data from 3000s to 5000s. is used to determine k, and M. These values, together with H, determined at equilibrium, are used in an exact numerical solution (solid line) to predict :he complete stress history. The agreement is excellent over the entire 8C00 s period. This method provides a self-consistency check on the validity of the model as well as the asymptotic solutions (Holmes rt al.. in press).

and the experimental

Properties

of articular

durmg the compressive phase. However, during the relaxation phase. the non-linear permeability causes a significant change in the rate ofstress relaxation. From equation (73), it is seen that the rate of stress relaxation is proportional to M,, which depends inversely on the lower permeability value k. exp( -cc, M) of the compressed tissue. The mechanism of stress relaxation is indeed fluid redistribution within the tissue, and interstitiai fluid flow in the compressed tissue must pass through the matrix with this layer of lower permeability. Thus a higher stress-relaxation rate is seen relative to the peak stress op. It should be pointed out that the material parameters were determined using the data from the compressive phase of the experiment; hence, the relaxation phase provides an internal check on the self-consistency of the biphasic theory, and, as can be seen in Fig. 14, the agreement is excellent.

This review has focused on three specific aspects of our understanding of cartilage biomechanics. First, relevant biochemical, physicochemical, and ultrastructural knowledge is presented so that the reader can interpret the biomechanical properties of this tissue from a micromechanical point of view. Second, a detailed historical account is presented of prior attempts to model articular cartilage indentation behavior. An appreciation of the evolution of thought on the development of cartilage deformational theories, leading up to the development of the biphasic model for articular cartilage, is both instructive and enlightening. Third, a summary of our most recent investigations on the influence of the non-linear straindependent permeability effect on the compressive creep and stress-relaxation behavior of the tissue is presented. We found, by two separate, independent tests-the steady, direct permeability experiment and the transient compressive stress-relaxation experiment-that cartilage permeability must be described by the function k = k, exp(EM) where k, and Mare the intrinsic permeability parametersand Eis the dilatation given by E = trVu. This non-linear permeability function has profound effects on the compressive creep and stress-relaxation behavior of the tissue, since these phenomenological viscoelastic effects are predominately governed by the rate of interstitial fluid flow. The physicochemical basis for this nonlinear flow-limiting effect derives from the significant direct correlation between cartilage permeability and tixedcharge density from steady-permeation experiments. Compaction of the tissue will correspondingly increase its fixed-charge density and hence decrease its permeability. Thus the nonlinearly permeable biphasic model for cartilage should be used whenever detailed deformational responses of the tissue in compression or detailed material-parameter characterizations of the tissue are desired. Preliminary experimental results show that, for example, in the stress-relaxation exper-

19

cartilage: a review

I

iment, M is very sensitive to the observed stress peaks. Thus we expect that this non-linear, flow-limiting parameter M could provide a sensitive quantitative indicator of cartilage degeneration in diseases such as osteoarthritis. Future work in this area must now focus on the correlation ofchanges of these intrinsic material properties ofcartilage with data such as: (1) the natural variation of tissue structure and biochemistry; (2) the pathological. e.g. osteoarthritic, changes of collagen fibrillar ultrastructure and proteoglycan conformation found in diseased tissues; and (3) the variation of monovalent (Na’), divalent (Ca’ ‘), and other ions in the interstitium. These problems pose extraordinary challenges not only to biomechanicians but to a whole host of allied biomedical scientists. Arknowlrdgzmml-This material is based on research supported by National Science Foundation grant MEA 82-11968. National Institute of Arthritis, Diabetes, and Digestive and Kidney Diseases grants AM 19093 and AM 26440. and the Clark and Crossan endowment 81 Rensselaer Polytechnic Institute. Any opinions, findings and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily represent the wews of the National Science Foundation. We wish to thank Mr. Brendan McCormack and Ms. W. B. Zhu for their technical assistance in obtaining the stress-relaxation and creep data. Mrs. Lynne Nagengast and Ms. Joyce A. Brock for their assistance in typing this manuscript. and Ms. Rose .A. Boshoff for her editorial assistance.

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