**1. Introduction**

The properties and behaviour of articular cartilage (**AC**) have been studied from numerous aspects. A number of biomechanical models of the properties and behaviour of AC are available today. The traditional model presents cartilage as homogeneous, isotropic and biphase material (Armstrong et al., 1984). There also exist models of transversally isotropic biphase cartilage material (Cohen et al., 1992; Cohen et al., 1993), non-linear poroelastic cartilage material (Li et al., 1999), models of poroviscoelastic (Wilson et al., 2005) and hyperelastic cartilage material (Garcia & Cortes, 2006), models of triphase cartilage material (Lai et al., 1991; Ateshian et al., 2004), and other models (Wilson et al., 2004; Jurvelin et al., 1990). The published models differ, more or less, by the angle of their authors' view of the properties and behaviour of articular cartilage during its loading.

The authors base their theories on various assumptions concerning the mutual links between the structural components of the cartilage matrix and their interactions on the molecular level.

The system behaviour of AC very depend on nonlinear properties of synovial fluid (**SF**). Certain volumes of SF are moveable components during the mechanical loading in the peripheral zone of AC. Biomechanical properties of peripheral zone of AC are significantly influenced by change of SF viscosity due to mechanical loading.

The hydrodynamic lubrication systems and influences of residual strains on the initial presupplementation of articular plateaus by synovial fluid were not sufficiently analyzed up to now.

Our research has been focused on analyses of residual strains arising in AC at cyclic loading and on the viscous properties of SF. Residual strains in articular cartilage contribute the preaccumulation of articular surfaces by synovial fluid.

SF reacts very sensitively to the magnitude of shear stress and to the velocity of the rotation of the femoral and tibial part of the knee joint round their relative centre of rotation when the limb shifts from flexion to extension and vice versa. Shear stresses decrease aggregations of macromolecules of hyaluronic acid in SF.

Articular cartilage (AC) is a viscohyperelastic composite biomaterial whose biomechanical functions consist

Biomechanical Properties of Synovial Fluid in/Between Peripheral Zones of Articular Cartilage 209

peripheral zone of AC and on its surface (in the gap between the opposite AC surfaces). The viscosity of synovial fluid is caused by the forces of attraction among its molecules being fully manifested during its flow. In other words, viscosity is a measure of its internal resistance during the SF flow. In the space between the opposite AC surfaces, its flow

As was pointed out above, biomechanical effects play a non-negligible and frequently a

The principal components of synovial fluid are water, hyaluronic acid **HA**, roughly 3-4 mg/ml, D-glucuronic acid and D-N-acetylglucosamine (Saari et al., 1993 and others). By its structure, hyaluronic acid is a long polymer, which very substantially predetermines the viscous properties of synovial fluid. Its molecular structure is evident from Fig. 2. *Synovial fluid* also contains an essential growth hormone *prolactin* (PRL) and *glycoprotein lubricin*.

Fig. 3. Topography of the surface of articular cartilage verified by means of FAM (Force Atomic Microscope). The height differences of surface points range up to ca 200 nm - 2,4 μm.

In unloaded condition, they are flooded by synovial fluid

behaves like a non-Newtonian fluid.

primary role in regulating rheological properties.

Fig. 2. Molecular complex of hyaluronic acid (HA)


3. in protecting the structural components of cartilage from higher physiological forces.

The macromolecular structure of AC in the peripheral zone (Fig. 1.) has two fundamental biomechanical safety functions, i.e. to regulate the lubrication of articular surfaces and to protect the chondrocytes and extracellular matrix from high loading.

The rheological properties of SF play the key role in the achievement of the optimum hyaluronan concentration.

Fig. 1. Complex structural system of articular cartilage (collagen fibres of 2nd type are not drawn)

The properties of SF in the gap between the opposite surfaces of articulate cartilage are not homogeneous during loading. The properties of SF change not only during biomechanical loading, but also during each individual's life time. The viscous properties of this fluid undergo changes (in time) due to mechanical loading. As a consequence of its very specific rheological characteristics, SF very efficiently adapts to external biomechanical effects. Exact knowledge of the rheological properties of synovial fluid is a key tool for the preservation and treatment of AC. The significance of the specific role of SF viscosity and viscosity deviations from predetermined physiological values were first pointed out as early as the 1950s to 1990s (Johnson et al., 1955; Bloch et al., 1963; Ferguson et al., 1968; Anadere et al., 1979; Schurz & Ribitsch, 1987; Safari et al., 1990 etc.). The defects of concentrations of the dispersion rate components were noticed by Mori (Mori et al., 2002). In this respect, it cannot be overlooked that mechanical properties of SF very strongly depend on the molecular weight of the dispersion rate (Sundblad et al., 1953; Scott & Heatley, 1999; Yanaki et al., 1990; Lapcik et al., 1998) and also on changes in the aggregations of macromolecular complexes in SF during mechanical effects (Myers et al., 1966; Ferguson et al., 1968; Nuki & Ferguson, 1971; Anadere et al., 1979 and Schurz & Ribitsch, 1987).

Synovial fluid is a viscous liquid characterized by the apparent viscosity η. This viscosity depends on stress and the time during which the stress acts. SF is found in the pores of the peripheral zone of AC and on its surface (in the gap between the opposite AC surfaces). The viscosity of synovial fluid is caused by the forces of attraction among its molecules being fully manifested during its flow. In other words, viscosity is a measure of its internal resistance during the SF flow. In the space between the opposite AC surfaces, its flow behaves like a non-Newtonian fluid.

As was pointed out above, biomechanical effects play a non-negligible and frequently a primary role in regulating rheological properties.

The principal components of synovial fluid are water, hyaluronic acid **HA**, roughly 3-4 mg/ml, D-glucuronic acid and D-N-acetylglucosamine (Saari et al., 1993 and others). By its structure, hyaluronic acid is a long polymer, which very substantially predetermines the viscous properties of synovial fluid. Its molecular structure is evident from Fig. 2. *Synovial fluid* also contains an essential growth hormone *prolactin* (PRL) and *glycoprotein lubricin*.

Fig. 2. Molecular complex of hyaluronic acid (HA)

208 Biomaterials – Physics and Chemistry

1. in transferring physiological loads into the subchondral bone and further to the

3. in protecting the structural components of cartilage from higher physiological forces. The macromolecular structure of AC in the peripheral zone (Fig. 1.) has two fundamental biomechanical safety functions, i.e. to regulate the lubrication of articular surfaces and to

The rheological properties of SF play the key role in the achievement of the optimum

Fig. 1. Complex structural system of articular cartilage (collagen fibres of 2nd type are not

Ferguson, 1971; Anadere et al., 1979 and Schurz & Ribitsch, 1987).

The properties of SF in the gap between the opposite surfaces of articulate cartilage are not homogeneous during loading. The properties of SF change not only during biomechanical loading, but also during each individual's life time. The viscous properties of this fluid undergo changes (in time) due to mechanical loading. As a consequence of its very specific rheological characteristics, SF very efficiently adapts to external biomechanical effects. Exact knowledge of the rheological properties of synovial fluid is a key tool for the preservation and treatment of AC. The significance of the specific role of SF viscosity and viscosity deviations from predetermined physiological values were first pointed out as early as the 1950s to 1990s (Johnson et al., 1955; Bloch et al., 1963; Ferguson et al., 1968; Anadere et al., 1979; Schurz & Ribitsch, 1987; Safari et al., 1990 etc.). The defects of concentrations of the dispersion rate components were noticed by Mori (Mori et al., 2002). In this respect, it cannot be overlooked that mechanical properties of SF very strongly depend on the molecular weight of the dispersion rate (Sundblad et al., 1953; Scott & Heatley, 1999; Yanaki et al., 1990; Lapcik et al., 1998) and also on changes in the aggregations of macromolecular complexes in SF during mechanical effects (Myers et al., 1966; Ferguson et al., 1968; Nuki &

Synovial fluid is a viscous liquid characterized by the apparent viscosity η. This viscosity depends on stress and the time during which the stress acts. SF is found in the pores of the

2. in ensuring the lubrication of articular plateaus of joints and

protect the chondrocytes and extracellular matrix from high loading.

spongious bone,

hyaluronan concentration.

drawn)

Fig. 3. Topography of the surface of articular cartilage verified by means of FAM (Force Atomic Microscope). The height differences of surface points range up to ca 200 nm - 2,4 μm. In unloaded condition, they are flooded by synovial fluid

Biomechanical Properties of Synovial Fluid in/Between Peripheral Zones of Articular Cartilage 211

cartilage, on the analysis of the effects of shear stresses on changes in SF viscosity and on the

With respect to the project objectives, the focus of interest was on the confirmation of the rheological properties of hyaluronic acid with sodium anions (sodium hyaluronan, NaHA) in an amount of 3.5 mg ml-1 in distilled water without any other additives. The use of only NaHA was based on the verification of the association of HA macromolecules and on the manifestation of highly specific rheological properties of SF, which regulate its lubrication function. The rheological properties were verified using the rotation viscometer Rheolab QC (Anton Paar, Austria). Viscosity values were measured continuously within 8 minutes.

analysis of the residual strains arising in AC at cyclic loading.

Fig. 5. SF apparent viscosity as related to time (velocity gradient 100s-1)

*property is of key importance for controlling the quality of the AC surface protection.* 

human body (37°C).

to the same values (ca 0.8 Pa s).

velocity gradient 2000 s-1 (in time 60 – 180s).

Samples were subjected to the effect of constant velocity gradient (100s-1 – 500s-1 – 1000s-1 – 2000s-1) in time 0 – 120s and 240 – 360s. Samples were subjected in the tranquility state in time 120 – 240s and 360 – 480s. The measurements were performed at the temperature of

Fig. 5. clearly shows that at the constant SF flow velocity gradient 100s-1 there is a distinct time-related constant values in viscosity. The verified synthetic synovial fluid possesses *pseudoplastic properties*. It is evident that the *macromolecules of hyaluronic acid (NaHA/HA) in a water dispersion environment principally contribute to the pseudoplastic behaviour of the fluid. This* 

Fig. 6. also shows that at the constant SF flow velocity gradient 2000s-1 there is a distinct time-related constant values in viscosity. The viscosity of SFafter unloading always returns

Fig. 7. shows that viscosity values of SF with increasing rate of flow velocity gradient 0 – 2000 s-1 (in time 0 – 60s) decrease. Viscosity values of SF are constant with constant rate of

**2.1 Rheological properties of synovial fluid** 

Prolactin induces the synthesis of proteoglycans and, in combination with glucocorticoids, it contributes to the configuration of chondrocytes inside AC and to the syntheses of type II collagen. The average molecular weight of human SF is 3 – 4 MDa.

Important components of SF are lubricin and some proteins from blood plasma (γ-globulin and albumin), which enhance the lubricating properties of SF (Oates*,* 2006). The importance of HA and proteins for the lubricating properties of SF was also described (Swann et al., 1985; Rinaudo et al., 2009).

 In the gap between AC surfaces, synovial fluid forms a micro-layer with a thickness of ca 50 μm. It fills up all surface micro-depressions (Fig. 3. and 4.**,** Petrtýl et al., 2010) and in accessible places its molecules are in contact with the macromolecules of residual SF localized in the pores of the femoral and tibial peripheral zone of AC.

Fig. 4. Topography of the articular cartilage surface of a man (58 years of age). The AC surface oscillates to relative heights of 2.5 μm**.** During fast shifts of the AC surface (due to the effect of dynamic shifting forces/dynamic bending moments or shear stresses), the AC surface is filled up with generated synovial gel (with less associated NaHA macromolecules) with *low viscosity*

SF is a rheological material whose properties change in time (Scott, 1999 and others). As a consequence of loading, associations of polymer chains of HA (and some proteins) arise and rheopexic properties of SF are manifested (Oates et al., 2006). Due to its specific rheological properties, SF ensures the lubrication of AC surfaces. The key component contributing to lubrication is HA/NaHA. In healthy young individuals, the endogenous production of hyaluronic acid (HA) reaches the peak values during adolescence. It declines with age. It also decreases during arthritis and rheumatic arthritis (Bloch et al., 1963; Anadere et al., 1979; Davies & Palfrey, 1968; Schurz & Ribitsch, 1987 and numerous other authors). Some AC diseases originate from the disturbance of SF lubrication mechanisms and from the defects of genetically predetermined SF properties. Therefore, the lubrication mechanisms of AC surfaces must be characterized with respect to the rheological properties of SF.

### **2. Contents**

The objectives of our research has been aimed on the definition of the biomechanical properties of SF which contribute to the lubrication of the opposite surfaces of articular cartilage, on the analysis of the effects of shear stresses on changes in SF viscosity and on the analysis of the residual strains arising in AC at cyclic loading.

### **2.1 Rheological properties of synovial fluid**

210 Biomaterials – Physics and Chemistry

Prolactin induces the synthesis of proteoglycans and, in combination with glucocorticoids, it contributes to the configuration of chondrocytes inside AC and to the syntheses of type II

Important components of SF are lubricin and some proteins from blood plasma (γ-globulin and albumin), which enhance the lubricating properties of SF (Oates*,* 2006). The importance of HA and proteins for the lubricating properties of SF was also described (Swann et al.,

 In the gap between AC surfaces, synovial fluid forms a micro-layer with a thickness of ca 50 μm. It fills up all surface micro-depressions (Fig. 3. and 4.**,** Petrtýl et al., 2010) and in accessible places its molecules are in contact with the macromolecules of residual SF

Fig. 4. Topography of the articular cartilage surface of a man (58 years of age). The AC surface oscillates to relative heights of 2.5 μm**.** During fast shifts of the AC surface (due to the effect of dynamic shifting forces/dynamic bending moments or shear stresses), the AC surface is filled up with generated synovial gel (with less associated NaHA macromolecules)

AC surfaces must be characterized with respect to the rheological properties of SF.

The objectives of our research has been aimed on the definition of the biomechanical properties of SF which contribute to the lubrication of the opposite surfaces of articular

SF is a rheological material whose properties change in time (Scott, 1999 and others). As a consequence of loading, associations of polymer chains of HA (and some proteins) arise and rheopexic properties of SF are manifested (Oates et al., 2006). Due to its specific rheological properties, SF ensures the lubrication of AC surfaces. The key component contributing to lubrication is HA/NaHA. In healthy young individuals, the endogenous production of hyaluronic acid (HA) reaches the peak values during adolescence. It declines with age. It also decreases during arthritis and rheumatic arthritis (Bloch et al., 1963; Anadere et al., 1979; Davies & Palfrey, 1968; Schurz & Ribitsch, 1987 and numerous other authors). Some AC diseases originate from the disturbance of SF lubrication mechanisms and from the defects of genetically predetermined SF properties. Therefore, the lubrication mechanisms of

collagen. The average molecular weight of human SF is 3 – 4 MDa.

localized in the pores of the femoral and tibial peripheral zone of AC.

1985; Rinaudo et al., 2009).

with *low viscosity*

**2. Contents** 

With respect to the project objectives, the focus of interest was on the confirmation of the rheological properties of hyaluronic acid with sodium anions (sodium hyaluronan, NaHA) in an amount of 3.5 mg ml-1 in distilled water without any other additives. The use of only NaHA was based on the verification of the association of HA macromolecules and on the manifestation of highly specific rheological properties of SF, which regulate its lubrication function. The rheological properties were verified using the rotation viscometer Rheolab QC (Anton Paar, Austria). Viscosity values were measured continuously within 8 minutes.

Fig. 5. SF apparent viscosity as related to time (velocity gradient 100s-1)

Samples were subjected to the effect of constant velocity gradient (100s-1 – 500s-1 – 1000s-1 – 2000s-1) in time 0 – 120s and 240 – 360s. Samples were subjected in the tranquility state in time 120 – 240s and 360 – 480s. The measurements were performed at the temperature of human body (37°C).

Fig. 5. clearly shows that at the constant SF flow velocity gradient 100s-1 there is a distinct time-related constant values in viscosity. The verified synthetic synovial fluid possesses *pseudoplastic properties*. It is evident that the *macromolecules of hyaluronic acid (NaHA/HA) in a water dispersion environment principally contribute to the pseudoplastic behaviour of the fluid. This property is of key importance for controlling the quality of the AC surface protection.* 

Fig. 6. also shows that at the constant SF flow velocity gradient 2000s-1 there is a distinct time-related constant values in viscosity. The viscosity of SFafter unloading always returns to the same values (ca 0.8 Pa s).

Fig. 7. shows that viscosity values of SF with increasing rate of flow velocity gradient 0 – 2000 s-1 (in time 0 – 60s) decrease. Viscosity values of SF are constant with constant rate of velocity gradient 2000 s-1 (in time 60 – 180s).

Biomechanical Properties of Synovial Fluid in/Between Peripheral Zones of Articular Cartilage 213

Fig. 7. Viscosity values of SF with increasing rate of flow velocity gradient 0 – 2000 s-1 (in time 0 – 60s) decrease. Viscosity values of SF are constant with constant rate of velocity

SF represents a mobile dispersion system in which *synovial gel is generated due to non-Newtonian properties of SF.* Within this system, the macromolecules of hyaluronic acid can be intertwined into a three-dimensional grid, which continuously penetrates through the dispersion environment formed by water. The pseudoplastic properties of SF are manifested through mechanical effects (for example while walking or running), Fig. 8., Fig. 9. *Physical netting occurs, which is characterized by the interconnection of sections of polymer chains into knots or knot areas.* Generally speaking, the association of individual molecules of hyaluronic acid (HA/NaHA) occurs in cases of reduced affinity of its macromolecular chains to the solvent. In other words, the *macromolecules of hyaluronic acid (HA) form a spatial grid structure in* 

Mutually inverse shifts and inverse rotations of the opposite AC surfaces cause inverse

The greatest magnitudes of SF velocity vectors due to the effect of shear stresses τxy, (or the effects of shifting forces respectively) are found near the upper and lower AC surface. They are, however, mutually inversely oriented. Fig. 10. displays the right-oriented velocity vector direction near the upper surface, and the left-oriented one near the lower AC surface. The magnitudes of velocity vectors decrease in the direction towards the central SF zone. In this thin neutral zone, the velocity vector is theoretically zero in value. A very thin layer (zone) of SF in the vicinity of the central zone, with very small to zero velocities, can be

At very small velocities of SF flows, the *viscosity of the neutral central zone is higher than the viscosity in the vicinity of AC surfaces*. Under the conditions of very low viscosity, the SF material in the vicinity of AC surfaces is characterized by a low friction coefficient. Friction

gradient 2000 s-1 (in time 60 – 180s)

*a water solution* (Fig. 9.).

appointed **neutral SF zone.** 

flows of SF on its interface with the AC surface (Fig. 10.).

reaches values of ca 0.024 – 0.047 (Radin et al., 1971).

Fig. 6. SF apparent viscosity as related to time (velocity gradient 2000s-1)

Due to the fact that the lubrication abilities of SF strongly depend on the magnitude of viscosity, and SF viscosity depends on the SF flow velocities, the effects of the magnitudes and directions of shifting forces or shear stresses respectively on the distributions of the magnitudes and directions of SF flow velocity vectors in the space between the opposite AC surfaces had to be analyzed.

The kinematics of the limb motion (within one cycle) shows that during a step the leg continuously passes through the phases of flexion – extension – flexion (Fig. 8.). The effect of shifting forces (or shear stresses respectively) is predominantly manifested in the phases of flexion, while normal forces representing the effects of the gravity (weight) of each individual mostly apply in the phases of extension, Fig. 8. The distributions of the magnitudes of SF flow velocity vectors depend on the shifts of the tibial and femoral part of the knee joint, Fig. 9.**,** reaching their peaks in places on the interface of SF with the upper and lower AC surface, Fig. 10. The velocities of SF flows very substantially affect the SF behavior contributing to the lubrication of AC surfaces and their protection.

At rest the bonds are created among the macromolecules of hyaluronic acid (HA) leading to the creation of associates. *By associating molecular chains of HA (at rest) into a continuous structure, a spatial macromolecular grid is created in SF which contributes to the growth in viscosity and also to the growth in elastic properties.* 

The associations of HA molecules are the manifestation of cohesive forces among HA macromolecular chains. SF represents a dispersion system (White, 1963) in which the dispersion rate is dominantly formed by snakelike HA macromolecules. The dispersion environment is formed by water. Cohesive forces among NaHA polymer chains in SF are of physical nature. The density (number) of bonds among HA macromolecules is dominantly controlled by mechanical effects. Fig. 9. In relation to the magnitudes of velocity gradients, NaHA macromolecules are able to form "thick" synovial gel which possesses elastic properties characteristic of solid elastic materials, even though the dispersion environment of synovial gel is liquid.

Fig. 6. SF apparent viscosity as related to time (velocity gradient 2000s-1)

behavior contributing to the lubrication of AC surfaces and their protection.

surfaces had to be analyzed.

*and also to the growth in elastic properties.* 

of synovial gel is liquid.

Due to the fact that the lubrication abilities of SF strongly depend on the magnitude of viscosity, and SF viscosity depends on the SF flow velocities, the effects of the magnitudes and directions of shifting forces or shear stresses respectively on the distributions of the magnitudes and directions of SF flow velocity vectors in the space between the opposite AC

The kinematics of the limb motion (within one cycle) shows that during a step the leg continuously passes through the phases of flexion – extension – flexion (Fig. 8.). The effect of shifting forces (or shear stresses respectively) is predominantly manifested in the phases of flexion, while normal forces representing the effects of the gravity (weight) of each individual mostly apply in the phases of extension, Fig. 8. The distributions of the magnitudes of SF flow velocity vectors depend on the shifts of the tibial and femoral part of the knee joint, Fig. 9.**,** reaching their peaks in places on the interface of SF with the upper and lower AC surface, Fig. 10. The velocities of SF flows very substantially affect the SF

At rest the bonds are created among the macromolecules of hyaluronic acid (HA) leading to the creation of associates. *By associating molecular chains of HA (at rest) into a continuous structure, a spatial macromolecular grid is created in SF which contributes to the growth in viscosity* 

The associations of HA molecules are the manifestation of cohesive forces among HA macromolecular chains. SF represents a dispersion system (White, 1963) in which the dispersion rate is dominantly formed by snakelike HA macromolecules. The dispersion environment is formed by water. Cohesive forces among NaHA polymer chains in SF are of physical nature. The density (number) of bonds among HA macromolecules is dominantly controlled by mechanical effects. Fig. 9. In relation to the magnitudes of velocity gradients, NaHA macromolecules are able to form "thick" synovial gel which possesses elastic properties characteristic of solid elastic materials, even though the dispersion environment

Fig. 7. Viscosity values of SF with increasing rate of flow velocity gradient 0 – 2000 s-1 (in time 0 – 60s) decrease. Viscosity values of SF are constant with constant rate of velocity gradient 2000 s-1 (in time 60 – 180s)

SF represents a mobile dispersion system in which *synovial gel is generated due to non-Newtonian properties of SF.* Within this system, the macromolecules of hyaluronic acid can be intertwined into a three-dimensional grid, which continuously penetrates through the dispersion environment formed by water. The pseudoplastic properties of SF are manifested through mechanical effects (for example while walking or running), Fig. 8., Fig. 9. *Physical netting occurs, which is characterized by the interconnection of sections of polymer chains into knots or knot areas.* Generally speaking, the association of individual molecules of hyaluronic acid (HA/NaHA) occurs in cases of reduced affinity of its macromolecular chains to the solvent. In other words, the *macromolecules of hyaluronic acid (HA) form a spatial grid structure in a water solution* (Fig. 9.).

Mutually inverse shifts and inverse rotations of the opposite AC surfaces cause inverse flows of SF on its interface with the AC surface (Fig. 10.).

The greatest magnitudes of SF velocity vectors due to the effect of shear stresses τxy, (or the effects of shifting forces respectively) are found near the upper and lower AC surface. They are, however, mutually inversely oriented. Fig. 10. displays the right-oriented velocity vector direction near the upper surface, and the left-oriented one near the lower AC surface. The magnitudes of velocity vectors decrease in the direction towards the central SF zone. In this thin neutral zone, the velocity vector is theoretically zero in value. A very thin layer (zone) of SF in the vicinity of the central zone, with very small to zero velocities, can be appointed **neutral SF zone.** 

At very small velocities of SF flows, the *viscosity of the neutral central zone is higher than the viscosity in the vicinity of AC surfaces*. Under the conditions of very low viscosity, the SF material in the vicinity of AC surfaces is characterized by a low friction coefficient. Friction reaches values of ca 0.024 – 0.047 (Radin et al., 1971).

Biomechanical Properties of Synovial Fluid in/Between Peripheral Zones of Articular Cartilage 215

Fig. 9. Diagram of the distribution of magnitudes and directions of SF flow velocity vectors in the gap between AC surfaces. Associations of NaHA/HA macromolecules decline in places with the greatest SF flow velocity gradient, i.e. in zones adjoining each AC surface.

Fig. 10. Rotation of the tibial and femoral part of the knee joint during the transition from flexion to extension. During the rotation of the femoral part of the knee joint (due to the effect of the left-hand rotation moment M) round the current (relative) centre of rotation (which is the intersection of longitudinal axes of the femur and tibia), point **A** moves to position **A´**. During a simultaneous rotation of the tibial part of the knee joint (due to the effect of the right-hand rotation moment M) round the same current (relative) centre of

rotation, point **B** moves to position **B´** 

The SF flow velocity gradient decreases in the direction towards the neutral zone

Fig. 8. Orientation diagram of the magnitudes of angles between the axes of the femoral and tibial diaphysis during the "flexion – extension – flexion" cycle of the lower limb in relation to the time percentage of the cycle

The total *thickness of the gap* between the opposite AC surfaces is only ca 50 μm, including height roughness of the surfaces near both peripheral layers 2 x 2.5 μm**,** Fig. 4., Fig. 9. (Petrtýl et al., 2010). In quiescent state, the AC surfaces are flooded with SF (synovial gel) while during the leg motion (from flexion to extension and vice versa) synovial sol with the relatively low viscosity is generated in SF in peripheral zones of AC. In other words, *due to the effect of shear stresses* τxy the viscosity η of SF decreases and synovial sol is generated. Aggregations of macromolecules of hyaluronic acid decrease. *The most intense aggregations* are in places of the smallest SF velocities, i.e. in neutral (central) zone of SF between the AC surfaces.

Fig. 8. Orientation diagram of the magnitudes of angles between the axes of the femoral and tibial diaphysis during the "flexion – extension – flexion" cycle of the lower limb in relation

The total *thickness of the gap* between the opposite AC surfaces is only ca 50 μm, including height roughness of the surfaces near both peripheral layers 2 x 2.5 μm**,** Fig. 4., Fig. 9. (Petrtýl et al., 2010). In quiescent state, the AC surfaces are flooded with SF (synovial gel) while during the leg motion (from flexion to extension and vice versa) synovial sol with the relatively low viscosity is generated in SF in peripheral zones of AC. In other words, *due to the effect of shear stresses* τxy the viscosity η of SF decreases and synovial sol is generated. Aggregations of macromolecules of hyaluronic acid decrease. *The most intense aggregations* are in places of the smallest SF velocities, i.e. in neutral (central) zone of SF between the AC

to the time percentage of the cycle

surfaces.

Fig. 9. Diagram of the distribution of magnitudes and directions of SF flow velocity vectors in the gap between AC surfaces. Associations of NaHA/HA macromolecules decline in places with the greatest SF flow velocity gradient, i.e. in zones adjoining each AC surface. The SF flow velocity gradient decreases in the direction towards the neutral zone

Fig. 10. Rotation of the tibial and femoral part of the knee joint during the transition from flexion to extension. During the rotation of the femoral part of the knee joint (due to the effect of the left-hand rotation moment M) round the current (relative) centre of rotation (which is the intersection of longitudinal axes of the femur and tibia), point **A** moves to position **A´**. During a simultaneous rotation of the tibial part of the knee joint (due to the effect of the right-hand rotation moment M) round the same current (relative) centre of rotation, point **B** moves to position **B´** 

Biomechanical Properties of Synovial Fluid in/Between Peripheral Zones of Articular Cartilage 217

AC is composed of cells (chondrocytes), of extracellular composite material representing a reinforcing component – collagen 2nd type (Benninghoff, 1925) and of a non reinforcing, molecularly complex matrix (Bjelle, 1975)*.* A matrix is dominantly composed of glycoprotein molecules and firmly bonded water. In the peripheral zone, there is synovial fluid unbound

The principal construction components of the matrix are glycoproteins. They possess a saccharide component (80-90 %) and a protein component (ca 20 - 10 %). Polysaccharides are composed of molecules of chondroitin-4-sulphate, chondtroitin-6-sulphate and keratansulphate. They are bonded onto the bearing protein, which is further bonded onto the hyaluronic acid macromolecule by means of two binding proteins. Keratansulphates and chondroitinsulphates are proteoglycans which, through bearing and binding proteins and together with the supporting macromolecule of hyaluronic acid, constitute the proteoglycan (or glycosaminoglycan) aggregate. As the saccharide part contains spatial polyanion fields, the presence of a large number of sulphate, carboxyl and hydroxyl groups results in the

The proteoglycan aggregate, together with *bonded water*, creates an amorphous extracellular material (matrix) of cartilage, which is bonded onto the reinforcing component – collagen 2nd type. Glycosaminoglycans are connected onto the supporting fibres of collagen type II by means of electrostatic bonds. In articular cartilage, nature took special efforts in safeguarding the biomechanical protection of chondrocytes in the peripheral zone. In the biomechanical perspective, chondrocytes are protected by glycocalix (i.e. a spherical saccharide envelope with firmly bonded water). Glycocalix is composed of a saccharide envelope bonded onto chondrocytes via transmembrane proteoglycans, transmembrane glycoproteins and adsorbed glycoproteins. *The glycocalix envelopes create gradually the* 

Our research has been focussed on analyses of viscoelastic strains of the upper peripheral cartilage zone, on the residual strains arising at cyclic loading, on the analyses of strain rate

The peripheral cartilage zone consists of chondrocytes packaged in proteoglycans (GAGs) with firmly bonded molecules of water. In the intercellular space, there is unbound synovial fluid which contains water, hyaluronic acid, lubricin, proteinases and collagenases. Synovial fluid exhibits non-Newtonian flow characteristics. As was pointed out above, under a load

Fig. 12. Peripheral zone of articular cartilage without (a)/with (b) loads. The peripheral incompressible zone is integrated with the incompressible zone in the middle (transitional)

*incompressible continuous layer during the loading in peripheral zone of AC* (Fig. 12.).

and on the creation of a peripheral incompressible cartilage cushion.

the synovial fluid is relocated on the surfaces of AC.

zone and low (radial) zone

creation of *extensive fields of ionic bonds with water molecules.* 

by ions.

Due to pseudoplastic properties of SF in the space between the opposite AC surfaces (Fig. 9.), non-physiological *abrasive wear* of the surfaces of AC peripheral zones is efficiently prevented.

*The SF solution process in the gap between the AC surfaces is not an isolated phenomenon.* It is interconnected (during walking, running etc.) with residual SF in the pores of the intercellular matrix in peripheral zones of the tibial and femoral part of AC. Under high loads, an *integrated unit* is generated which, after the *formation of an incompressible "cushion"*, is able to transfer extreme loads thus protecting the peripheral and internal AC structures from their destructions.

### **2.2 Residual strains during the cyclic loading in the articular cartilage**

In agreement with our analyses, the properties and behaviors of articular cartilage in the biomechanical perspective may be described by means of a complex viscohyperelastic model (Fig. 11.). The biomechanical compartment is composed of the Kelvin Voigt viscoelastic model (in the peripheral and partially in the transitional zone of AC) and of the hyperelastic model (in the middle transitional zone and the low zone of AC). The peripheral zone is histologically limited by oval (disk shaped) chondrocytes. The viscohyperelastic properties of AC are predetermined by the specific molecular structures.

The mechanical/biomechanical properties of articular cartilage are topographically non homogeneous. The material variability and non homogeneity depends on the type and the size of physiological loading effects (Akizuki et al., 1986; Petrtyl et al., 2008).

Fig. 11. Mechanical diagram of the complex viscohyperelastic model of articular cartilage. The mechanical compartment is composed of the Kelvin Voigt viscoelastic model (in the peripheral and transitional zone of AC) and of the hyperelastic model (in the middle transitional zone and the low zone of AC)

Due to pseudoplastic properties of SF in the space between the opposite AC surfaces (Fig. 9.), non-physiological *abrasive wear* of the surfaces of AC peripheral zones is efficiently

*The SF solution process in the gap between the AC surfaces is not an isolated phenomenon.* It is interconnected (during walking, running etc.) with residual SF in the pores of the intercellular matrix in peripheral zones of the tibial and femoral part of AC. Under high loads, an *integrated unit* is generated which, after the *formation of an incompressible "cushion"*, is able to transfer extreme loads thus protecting the peripheral and internal AC

In agreement with our analyses, the properties and behaviors of articular cartilage in the biomechanical perspective may be described by means of a complex viscohyperelastic model (Fig. 11.). The biomechanical compartment is composed of the Kelvin Voigt viscoelastic model (in the peripheral and partially in the transitional zone of AC) and of the hyperelastic model (in the middle transitional zone and the low zone of AC). The peripheral zone is histologically limited by oval (disk shaped) chondrocytes. The viscohyperelastic

The mechanical/biomechanical properties of articular cartilage are topographically non homogeneous. The material variability and non homogeneity depends on the type and the

Fig. 11. Mechanical diagram of the complex viscohyperelastic model of articular cartilage. The mechanical compartment is composed of the Kelvin Voigt viscoelastic model (in the peripheral and transitional zone of AC) and of the hyperelastic model (in the middle

**2.2 Residual strains during the cyclic loading in the articular cartilage** 

properties of AC are predetermined by the specific molecular structures.

size of physiological loading effects (Akizuki et al., 1986; Petrtyl et al., 2008).

prevented.

structures from their destructions.

transitional zone and the low zone of AC)

AC is composed of cells (chondrocytes), of extracellular composite material representing a reinforcing component – collagen 2nd type (Benninghoff, 1925) and of a non reinforcing, molecularly complex matrix (Bjelle, 1975)*.* A matrix is dominantly composed of glycoprotein molecules and firmly bonded water. In the peripheral zone, there is synovial fluid unbound by ions.

The principal construction components of the matrix are glycoproteins. They possess a saccharide component (80-90 %) and a protein component (ca 20 - 10 %). Polysaccharides are composed of molecules of chondroitin-4-sulphate, chondtroitin-6-sulphate and keratansulphate. They are bonded onto the bearing protein, which is further bonded onto the hyaluronic acid macromolecule by means of two binding proteins. Keratansulphates and chondroitinsulphates are proteoglycans which, through bearing and binding proteins and together with the supporting macromolecule of hyaluronic acid, constitute the proteoglycan (or glycosaminoglycan) aggregate. As the saccharide part contains spatial polyanion fields, the presence of a large number of sulphate, carboxyl and hydroxyl groups results in the creation of *extensive fields of ionic bonds with water molecules.* 

The proteoglycan aggregate, together with *bonded water*, creates an amorphous extracellular material (matrix) of cartilage, which is bonded onto the reinforcing component – collagen 2nd type. Glycosaminoglycans are connected onto the supporting fibres of collagen type II by means of electrostatic bonds. In articular cartilage, nature took special efforts in safeguarding the biomechanical protection of chondrocytes in the peripheral zone. In the biomechanical perspective, chondrocytes are protected by glycocalix (i.e. a spherical saccharide envelope with firmly bonded water). Glycocalix is composed of a saccharide envelope bonded onto chondrocytes via transmembrane proteoglycans, transmembrane glycoproteins and adsorbed glycoproteins. *The glycocalix envelopes create gradually the incompressible continuous layer during the loading in peripheral zone of AC* (Fig. 12.).

Our research has been focussed on analyses of viscoelastic strains of the upper peripheral cartilage zone, on the residual strains arising at cyclic loading, on the analyses of strain rate and on the creation of a peripheral incompressible cartilage cushion.

The peripheral cartilage zone consists of chondrocytes packaged in proteoglycans (GAGs) with firmly bonded molecules of water. In the intercellular space, there is unbound synovial fluid which contains water, hyaluronic acid, lubricin, proteinases and collagenases. Synovial fluid exhibits non-Newtonian flow characteristics. As was pointed out above, under a load the synovial fluid is relocated on the surfaces of AC.

Fig. 12. Peripheral zone of articular cartilage without (a)/with (b) loads. The peripheral incompressible zone is integrated with the incompressible zone in the middle (transitional) zone and low (radial) zone

Biomechanical Properties of Synovial Fluid in/Between Peripheral Zones of Articular Cartilage 219

value (as was the case during the initial, first loading cycle), but at point B, or at the value of the residual strain ε**t2**. *The first residual strain provides the initial presupplementation of articular plateaus with synovial fluid.* Fig. 13. manifests that the envelope curve OBDF slightly grows during cyclic loading to stabilize after a certain time *at a steady value characterizing long-term strain (during the time of cyclic loading) and long-term presupplementation of articular space with synovial fluid.* After cyclic loading stops (i.e. after AC unloading) during the last loading cycle, as seen in Fig. 1., the strain relaxation follows the convex curve, and strains asymptotically approach to the time axis *t* (or zero). After the termination of the last loading cycle, SF (in the form of synovial sol) is sucked back into the peripheral layer of AC. The mechanism of viscous strain time growth and viscous strain relaxation creates a highly efficient *protective pump* functioning not only to discharge and suck back synovial fluid, but

Stresses in the peripheral zone may be expressed for the Kelvin Voigt model by the

( ) ( ) ( ) *d t t Et dt* ε

where η is the coefficient of viscosity, E is the modulus of elasticity, ε(t) is the strain of AC

Equation (1) is a first order linear differential equation for an unknown function *ε(t).* The solution to the non-homogeneous equation (1) under the given initial conditions determines

0

στ

*t t e ed*

Let us further consider the case where articular cartilage is loaded by a constant load

() 1 *Et t <sup>c</sup> t e E*

Equation (3) implies that the strain of AC is a function of time depending on the magnitude of the constant stress *σc* also (for example by shifting an individual's weight onto one foot). The presence of residual strain (marked by a thick line in Fig. 13.) ensures the accumulation of synovial fluid between articular plateaus. It means that during each step (during cyclic loading) articular plateaus are presupplemented with the lubrication medium – synovial fluid. The magnitudes of residual strains of AC play a key role in the presupplementation of AC surface plateaus with synovial fluid. The magnitudes of residual strains may be determined from the functions expressing strain during the strain time growth and from the functions expressing strain during the strain relaxation of AC, this may be performed separately for

For the 1st phase of the first loading cycle, for 0 1 *t tt* ∈< > ; , (Fig. 13.) the concave curve is

σ

<sup>−</sup> <sup>⎡</sup> <sup>−</sup> <sup>⎤</sup> = − <sup>⎢</sup> <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup>

η

1 1 <sup>1</sup> () ( ) *<sup>t</sup> Et <sup>E</sup>*

η

η

 ε

> τ

0 <sup>1</sup> ( )  τ

<sup>−</sup> <sup>⎡</sup> <sup>⎤</sup> <sup>=</sup> <sup>⎢</sup> <sup>⎥</sup> ⎢⎣ ⎥⎦ <sup>∫</sup> (2)

 η (1)

(3)

 = + η

σ

the time related strain of articular cartilage. In our case, it is in the form:

ε

ε

is the strain rate of cartilage tissue in the peripheral zone.

also to pump (accumulate) it into the articular space.

constitutive equation:

and *d t*( ) *dt* ε

( ) *<sup>c</sup>* σ τ σ

= = *const* (Fig. 13.) :

each loading cycle of cartilage (Fig. 13.).

defined by function (3) for the articular cartilage strain.

During loading, the chondrocytes with GAGs encapsulation (in the peripheral zone) create a continuous incompressible mezzo layer with protected chondrocytes. Simultaneously, an incompressible peripheral zone arises in the middle of the transitional zone and in the low (radial) zone of AC. There are dominantly hyperelastic properties in the transitional and the low radial zone (Fig. 11.). Stress states can be simulated by the modified Cauchy stress tensor for incompressible hyperelastic material.

Viscous properties in the peripheral zone of articular cartilage result from the interaction between the molecules of the extracellular matrix and the molecules of free (unbound) synovial fluid. The transport of SF molecules through the extracellular space and the lack of bonding of these molecules onto glycosaminoglycans create the basic condition for the viscous behaviors of cartilage. High dynamic forces are dominantly undertaken by the AC matrix with firmly bonded water in its low and middle zone with a simultaneous creation of an incompressible tissue, a cushion (Fig. 1.).

The articular cartilage matrix with viscoelastic properties functions dominantly as a *protective pump* and a regulator of the amount of SF permanently maintained (during cyclic loading) between articular plateaus. The importance of the *protective pump* is evident from the function of retention of AC strains during cyclic loading. Due to slow down viscoelastic strain, part of accumulated (i.e. previously discharged) SF from the preceding loading cycle is *retained in articular cartilage* (Fig. 13.).

Fig. 13. Application of Kelvin Voigt viscoelastic model for the expression of step by step increments of strains εti in the peripheral zone of AC during cyclic loading (e.g. while walking or running)

Fig. 13. in its upper part (a) shows the loading cycles e.g. during walking, while in the lower part (b) strains during the strain time growth and during strain relaxation are visible. The strain time growth occurs during the first loading (see the first concave curve OA of the strain growth). At the time t1after unloading strain relaxation occurs (see the convex shape of the second curve AB). At the time t2 the successive (second) loading cycle starts. The strain time growth during the successive loading cycle, however, does not start at a zero

During loading, the chondrocytes with GAGs encapsulation (in the peripheral zone) create a continuous incompressible mezzo layer with protected chondrocytes. Simultaneously, an incompressible peripheral zone arises in the middle of the transitional zone and in the low (radial) zone of AC. There are dominantly hyperelastic properties in the transitional and the low radial zone (Fig. 11.). Stress states can be simulated by the modified Cauchy stress

Viscous properties in the peripheral zone of articular cartilage result from the interaction between the molecules of the extracellular matrix and the molecules of free (unbound) synovial fluid. The transport of SF molecules through the extracellular space and the lack of bonding of these molecules onto glycosaminoglycans create the basic condition for the viscous behaviors of cartilage. High dynamic forces are dominantly undertaken by the AC matrix with firmly bonded water in its low and middle zone with a simultaneous creation of

The articular cartilage matrix with viscoelastic properties functions dominantly as a *protective pump* and a regulator of the amount of SF permanently maintained (during cyclic loading) between articular plateaus. The importance of the *protective pump* is evident from the function of retention of AC strains during cyclic loading. Due to slow down viscoelastic strain, part of accumulated (i.e. previously discharged) SF from the preceding loading cycle

Fig. 13. Application of Kelvin Voigt viscoelastic model for the expression of step by step increments of strains εti in the peripheral zone of AC during cyclic loading (e.g. while

Fig. 13. in its upper part (a) shows the loading cycles e.g. during walking, while in the lower part (b) strains during the strain time growth and during strain relaxation are visible. The strain time growth occurs during the first loading (see the first concave curve OA of the strain growth). At the time t1after unloading strain relaxation occurs (see the convex shape of the second curve AB). At the time t2 the successive (second) loading cycle starts. The strain time growth during the successive loading cycle, however, does not start at a zero

tensor for incompressible hyperelastic material.

an incompressible tissue, a cushion (Fig. 1.).

is *retained in articular cartilage* (Fig. 13.).

walking or running)

value (as was the case during the initial, first loading cycle), but at point B, or at the value of the residual strain ε**t2**. *The first residual strain provides the initial presupplementation of articular plateaus with synovial fluid.* Fig. 13. manifests that the envelope curve OBDF slightly grows during cyclic loading to stabilize after a certain time *at a steady value characterizing long-term strain (during the time of cyclic loading) and long-term presupplementation of articular space with synovial fluid.* After cyclic loading stops (i.e. after AC unloading) during the last loading cycle, as seen in Fig. 1., the strain relaxation follows the convex curve, and strains asymptotically approach to the time axis *t* (or zero). After the termination of the last loading cycle, SF (in the form of synovial sol) is sucked back into the peripheral layer of AC. The mechanism of viscous strain time growth and viscous strain relaxation creates a highly efficient *protective pump* functioning not only to discharge and suck back synovial fluid, but also to pump (accumulate) it into the articular space.

Stresses in the peripheral zone may be expressed for the Kelvin Voigt model by the constitutive equation:

$$
\sigma(t) = \eta \frac{d\varepsilon(t)}{dt} + E\varepsilon(t) \tag{1}
$$

where η is the coefficient of viscosity, E is the modulus of elasticity, ε(t) is the strain of AC and *d t*( ) *dt* εis the strain rate of cartilage tissue in the peripheral zone.

Equation (1) is a first order linear differential equation for an unknown function *ε(t).* The solution to the non-homogeneous equation (1) under the given initial conditions determines the time related strain of articular cartilage. In our case, it is in the form:

$$\boldsymbol{\omega}(t) = e^{\frac{-\boldsymbol{1}\_{\mathcal{E}t}}{\eta}} \left[ \frac{1}{\eta} \Big| \int\_{\boldsymbol{\eta}} \sigma(\tau) e^{\frac{\boldsymbol{1}\_{\mathcal{E}t}}{\eta}} d\tau \right] \tag{2}$$

Let us further consider the case where articular cartilage is loaded by a constant load ( ) *<sup>c</sup>* σ τ σ= = *const* (Fig. 13.) :

$$
\omega(t) = \frac{\sigma\_c}{E} \left[ 1 - e^{\frac{-1}{\sigma} \mathcal{E}(t - t\_0)} \right] \tag{3}
$$

Equation (3) implies that the strain of AC is a function of time depending on the magnitude of the constant stress *σc* also (for example by shifting an individual's weight onto one foot). The presence of residual strain (marked by a thick line in Fig. 13.) ensures the accumulation of synovial fluid between articular plateaus. It means that during each step (during cyclic

loading) articular plateaus are presupplemented with the lubrication medium – synovial fluid. The magnitudes of residual strains of AC play a key role in the presupplementation of AC surface plateaus with synovial fluid. The magnitudes of residual strains may be determined from the functions expressing strain during the strain time growth and from the functions expressing strain during the strain relaxation of AC, this may be performed separately for each loading cycle of cartilage (Fig. 13.).

For the 1st phase of the first loading cycle, for 0 1 *t tt* ∈< > ; , (Fig. 13.) the concave curve is defined by function (3) for the articular cartilage strain.

Biomechanical Properties of Synovial Fluid in/Between Peripheral Zones of Articular Cartilage 221

<sup>1</sup> ( ) ( ) *Et t <sup>c</sup> d t e*

η

1

( ) *Et t <sup>t</sup> t e*

=

〈*t*1*; t*2〉

*dt*

is decreasing.

ε

1 <sup>1</sup> ( ) ( ) *Et t t d t E e*

The above described analyses lead to the formulation of the following key conclusions: Synovial fluid is a viscous pseudoplastic non-Newtonian fluid. Apparent viscosity of SF decreases with increasing rate of flow velocity gradient. SF does not display a decrease in viscosity *over time at a constant flow velocity gradient* (as it is typical for thixotropic material). The rheological properties of synovial fluid essentially affect the biomechanical behaviour of SF between the opposite AC surfaces and in the peripheral AC zone also. During the shifts of the femoral and tibial part of AC in opposite directions the velocities of SF flows decrease in the direction towards the neutral central zone of the gap between the AC surf*aces. Nonlinear abatement in viscosity* in the direction from the *neutral ("quiescent") layer of* SF towards the opposite AC surfaces contributes to the lubrication quality and very efficiently

The viscoelastic properties of the peripheral zone of AC and its molecular structure ensure the regulation of the transport and accumulation of SF between articular plateaus. The hydrodynamic lubrication biomechanism adapts with high sensitivity to biomechanical stresses. The viscoelastic properties of AC in the peripheral zone ensure that during cyclic loading some amount of SF is always retained accumulated between articular plateaus, which were presupplemented with it in the previous loading cycle. During long-term

The limit strain value of AC during loading is always greater than its limit strain value after unloading. Shortly after loading, the strain rate is always greater than before unloading. In

harmonic cyclic loading and unloading, the strains stabilize at limit values.

η ε

The strain rate of articular cartilage shortly after the unloading (during the strain relaxation)

 ε

 σ η

= ε

*dt*

decelerates. The strain rate shortly after the load is the highest.

decreasing. The strain rate in the same interval of

is distinctly higher than to the end of interval of

during the strain relaxation in interval of

∈〈*t1; t2*〉

protect the uneven micro-surfaces of AC.

**relaxation** 

The strain rate ( ) () 0 *d t <sup>t</sup> dt* ε = ε

time in interval of *t*

**3. Conclusions** 

From equation (10) is evident, that the strain rate of articular cartilage in interval of *t*

**2.2.2 The strain rate of articular cartilage in peripheral zone during the strain** 

The strain of peripheral zone in time *t1* during unloading is given by equation (11):

ε

0

1

< . It means that the strain function *ε(t)* in interval of *t*

〈*t*1*; t*2〉

η

is given by equation (12):

. Strain rate *d t*( )

<sup>1</sup> ( )

<sup>−</sup> <sup>−</sup>

1

〈*t1; t2*〉

<sup>−</sup> <sup>−</sup> <sup>⎡</sup> <sup>⎤</sup> = −⎢ <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup>

η (10)

(11)

is decreasing also. Strain rate

(12)

*dt* ε

∈〈*t0; t1*〉

∈〈*t*1*; t*2〉is

with increasing

<sup>−</sup> <sup>−</sup>

Discrete strain at the time t0 is 0 0 *<sup>t</sup>* ε= , at the time t1 discrete strain is:

$$\mathcal{L}\_{\gamma\_1} = \frac{\sigma\_c}{E} \left[ 1 - e^{\frac{-1}{\eta} \mathcal{E} (t\_1 - t\_0)} \right] \tag{4}$$

For the 2nd phase of the first loading cycle (for 1 2 *t tt* ∈< > ; ) (Fig. 13.), the convex curve AB is defined by the function for articular cartilage strain:

$$
\varepsilon(t) = \varepsilon\_{\text{rel}} e^{\frac{-1}{\eta} \varepsilon(t-t\_1)} \tag{5}
$$

Discrete strain at the time *t0* is 0 0 *<sup>t</sup>* ε= , at the time *t1* discrete strain is:

$$
\omega\_{t\_1} = \varepsilon\_{t\_1} e^{\frac{-1}{\eta} \mathcal{E}(t\_1 - t\_1)} \tag{6}
$$

The magnitudes of strains during cyclic loading at the starting points of loading and unloading of articular cartilage may be expressed by recurrent relations. For the time ti with an odd index, the strain at the respective nodal points is:

$$\varepsilon\_{\iota\_{(2\iota+1)}} = \frac{\sigma\_{\iota}}{E} \left[1 - e^{\frac{-E}{\eta}(\iota+1)}\right]\_{\iota=0,1,2\ldots} \tag{7}$$

where *l* is the length of the time interval <ti ; ti+1> . For the time t**i** with an even index, the strain is :

$$\mathcal{L}\_{\varepsilon\_{z\_{1i}}} = \frac{\sigma\_c}{E} \left[ e^{\frac{-E\_l}{\eta}} - e^{\frac{-E\_c}{\eta}(k+1)l} \right]\_{k=0,1,2...} \tag{8}$$

where *l* is the length of the time interval **<**ti ; ti+1**>** , i = 0, 1, 2, … During long-term cyclic loading and unloading, for k → ∞ the strain εt(2k+1) asymptotically approaches the steady state σc/E ; for k → ∞ the strain εt2k asymptotically approaches the steady state *El <sup>c</sup> <sup>e</sup> E* σ η − . It is evident that for k → ∞ it holds true that:

$$
\sigma\_{t\_{\text{(in)}}} = \frac{\sigma\_c}{E} > \sigma\_{t\_{\text{1i}}} = \frac{\sigma\_c}{E} e^{\frac{-E\_i}{\eta}} \tag{9}
$$

#### **2.2.1 The strain rate of articular cartilage in peripheral zone during the strain time growth**

Strain *ε(t)* of AC during the strain time growth in the interval of t∈〈t0; t1〉 is given by equation (3). Because ( ) () 0 *d t <sup>t</sup> dt* ε = ε > (in the indicated interval) the function *ε(t)* is increasing. The strain rate of AC during the strain-time growth in interval of *t*∈〈*t0; t1*〉 is given by equation (10):

*dt*

From equation (10) is evident, that the strain rate of articular cartilage in interval of *t*∈〈*t0; t1*〉 decelerates. The strain rate shortly after the load is the highest.

η

= ε

#### **2.2.2 The strain rate of articular cartilage in peripheral zone during the strain relaxation**

The strain of peripheral zone in time *t1* during unloading is given by equation (11):

$$\varepsilon(t) = \varepsilon\_{t\_i} e^{\frac{-1}{\eta}\varepsilon(t-t\_i)} \tag{11}$$

The strain rate ( ) () 0 *d t <sup>t</sup> dt* ε = ε < . It means that the strain function *ε(t)* in interval of *t*∈〈*t*1*; t*2〉is

decreasing. The strain rate in the same interval of 〈*t*1*; t*2〉 is decreasing also. Strain rate during the strain relaxation in interval of 〈*t*1*; t*2〉is given by equation (12):

$$\frac{d\varepsilon(t)}{dt} = \varepsilon\_{t\_i} e^{\frac{-1}{\eta}\varepsilon(t-t\_i)} \left[ -\frac{E}{\eta} \right] \tag{12}$$

The strain rate of articular cartilage shortly after the unloading (during the strain relaxation) is distinctly higher than to the end of interval of 〈*t1; t2*〉 . Strain rate *d t*( ) *dt* ε with increasing time in interval of *t*∈〈*t1; t2*〉is decreasing.

#### **3. Conclusions**

220 Biomaterials – Physics and Chemistry

<sup>1</sup> <sup>1</sup> *Et t <sup>c</sup> <sup>t</sup> e E*

For the 2nd phase of the first loading cycle (for 1 2 *t tt* ∈< > ; ) (Fig. 13.), the convex curve AB

<sup>1</sup> ( ) *Et t <sup>t</sup> t e*

=

2 1

=

ε ε

1 *<sup>k</sup>*

σ

*E* <sup>+</sup>

*E*

state σc/E ; for k → ∞ the strain εt2k asymptotically approaches the steady state

σ *t t e*

 ε

ε

0 *<sup>t</sup>* ε

(2 1)

*t*

2

*t*

ε

*k*

ε

ε

σ

<sup>−</sup> <sup>⎡</sup> <sup>−</sup> <sup>⎤</sup> = − <sup>⎢</sup> <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup>

= , at the time t1 discrete strain is:

η

1 0 <sup>1</sup> ( )

1 <sup>1</sup> ( )

<sup>−</sup> <sup>−</sup>

= , at the time *t1* discrete strain is:

2 1

( 1)

( 1)

*El c c*

 σ η

<sup>−</sup> <sup>+</sup>

η

0 ,1,2...

0 ,1,2...

*k*

=

−

> (in the indicated interval) the function *ε(t)* is

*k*

=

η

1 *Et t* ( )

η

The magnitudes of strains during cyclic loading at the starting points of loading and unloading of articular cartilage may be expressed by recurrent relations. For the time ti with

*<sup>E</sup> k l <sup>c</sup>*

⎡ ⎤ = − ⎢ ⎥ ⎣ ⎦

where *l* is the length of the time interval <ti ; ti+1> . For the time t**i** with an even index, the

*E E l kl <sup>c</sup>*

where *l* is the length of the time interval **<**ti ; ti+1**>** , i = 0, 1, 2, … During long-term cyclic loading and unloading, for k → ∞ the strain εt(2k+1) asymptotically approaches the steady

> *t t e E E* <sup>+</sup>

**2.2.1 The strain rate of articular cartilage in peripheral zone during the strain time** 

increasing. The strain rate of AC during the strain-time growth in interval of *t*

 ε

Strain *ε(t)* of AC during the strain time growth in the interval of t∈〈t0; t1〉 is given by

=>= σ

− − <sup>+</sup>

 η

*e e*

η

⎡ ⎤ = − ⎢ ⎥ ⎣ ⎦

*e*

<sup>−</sup> <sup>−</sup>

(4)

(5)

(6)

(7)

(8)

(9)

*El <sup>c</sup> <sup>e</sup> E* σ η

∈〈*t0; t1*〉is

−

. It

0 *<sup>t</sup>* ε

is defined by the function for articular cartilage strain:

an odd index, the strain at the respective nodal points is:

is evident that for k → ∞ it holds true that:

equation (3). Because ( ) () 0 *d t <sup>t</sup>*

given by equation (10):

(2 1) *k k* <sup>2</sup>

*dt* ε = ε ε

Discrete strain at the time t0 is 0

Discrete strain at the time *t0* is 0

strain is :

**growth** 

The above described analyses lead to the formulation of the following key conclusions: Synovial fluid is a viscous pseudoplastic non-Newtonian fluid. Apparent viscosity of SF decreases with increasing rate of flow velocity gradient. SF does not display a decrease in viscosity *over time at a constant flow velocity gradient* (as it is typical for thixotropic material). The rheological properties of synovial fluid essentially affect the biomechanical behaviour of SF between the opposite AC surfaces and in the peripheral AC zone also. During the shifts of the femoral and tibial part of AC in opposite directions the velocities of SF flows decrease in the direction towards the neutral central zone of the gap between the AC surf*aces. Nonlinear abatement in viscosity* in the direction from the *neutral ("quiescent") layer of* SF towards the opposite AC surfaces contributes to the lubrication quality and very efficiently protect the uneven micro-surfaces of AC.

The viscoelastic properties of the peripheral zone of AC and its molecular structure ensure the regulation of the transport and accumulation of SF between articular plateaus. The hydrodynamic lubrication biomechanism adapts with high sensitivity to biomechanical stresses. The viscoelastic properties of AC in the peripheral zone ensure that during cyclic loading some amount of SF is always retained accumulated between articular plateaus, which were presupplemented with it in the previous loading cycle. During long-term harmonic cyclic loading and unloading, the strains stabilize at limit values.

The limit strain value of AC during loading is always greater than its limit strain value after unloading. Shortly after loading, the strain rate is always greater than before unloading. In

Biomechanical Properties of Synovial Fluid in/Between Peripheral Zones of Articular Cartilage 223

Anadere, I.; Chmiel, H. & Laschner, W. (1979). Viscoelasticity of "normal" and pathological

Armstrong, C.G.; Lai, W.M. & Mow, V.C. (1984). An analysis of the unconfined compression

Ateshian, G.A.; Chahine, N.O.; Basalo, I.M. & Hung, C.T. (2004). The correspondence

Benninghoff, A. (1925). Form und Bau der Gelenkknorpel in ihren Beziehungen zur Funktion, *Zeitschrift für Zellforschung und Mikroskopische Anatomie*, Vol. 2, No. 5, pp. 783-862 Bjelle, A. (1975). Content and Composition of Glycosaminoglycans in Human Knee Joint

Bloch, B. & Dintenfass, L. (1963). Rheological study of human synovial fluid. *Australian and New Zealand Journal of Surgery*, Vol. 33, No. 2, (November 1963), pp. 108-113 Cohen, B.; Gardner, T.R. & Ateshian, G.A. (1993). The influence of transverse isotropy on

*Orthopaedic Research Society*, Orthopaedic Research Society, pp. 185, Chicago, IL Cohen, B.; Lai, W.M.; Chorney, G.S.; Dick, H.M.; Mow, V.C. (1992). Unconfined compression

Davies, D.V. & Palfey, A.J. (1968). Some of the physical properties of normal and

Ferguson, J.; Boyle, J.A.; McSween, R.N.; Jasani, M.K. (1968). Observations on the flow

Garcia, J.J.; Cortes, D.H. (2006). A nonlinear biphasic viscohyperelastic model for articular cartilage, *J. of Biomechanics*, Vol. 39, No. 16, pp. 2991-2998, ISSN 0021-9290 Johnson, J.P. (1955). The viscosity of normal and pathological human synovial fluids. *J.* 

Jurveli, J.; Kiviranta, I.; Saamanen, A.M.; Tammi, M. & Helminen, H.J. (1990). Indentation

Lapcik, L. Jr.; Lapcik, L.; De Smedt, S.; Demeester, J. & Chabrecek, P. (1998). Hyaluronan:

Li, L.P.; Soulhat, J.; Buschmann, M.D.; Shirazi-Adl, A. (1999). Nonlinear analysis of cartilage

Myers, R.R.; Negami, S. & White, R.K. (1966). Dynamic mechanical properties of synovial

loading. *J. of Biomechanics*, Vol. 23, No. 12, pp. 1239-1246, ISSN 0021-9290 Lai, W.M.; Hou, J.S. & Mow, V.C. (1991). A Triphasic Theory for the Swelling and

Society of Mechanical Engineers, pp. 187-190, ISBN 0791811166

of articular cartilage. *J. Biomech. Eng.*, Vol. 106, No. 2, (May 1984), pp. 165–173, ISSN

between equilibrium biphasic and triphasic material properties in mixture models of articular cartilage. *J. of Biomechanics*, Vol. 37, No. 3, (March 2004), pp. 391-400,

Cartilage: Variation with Site and Age in Adults. *Connective tissue research*, Vol. 3,

cartilage indentation behavior - A study of the human humeral head. In: *Transactions* 

of transversely-isotropic biphasic tissue, In: *Advances in Bioengineering*, American

pathological synovial fluids. *J. of Biomechanics*, Vol. 1, No. 2, (July 1968), pp. 79-88,

properties of the synovial fluid from patients with rheumatoid arthritis. *Biorheology*,

stiffness of young canine knee articular cartilage—Influence of strenuous joint

Deformation Behaviors of Articular Cartilage. *J. Biomech. Eng.*, Vol. 113, No. 3,

Preparation, Structure, Properties, and Applications. *Chemical reviews*, Vol. 98, No.

in unconfined ramp compression using a fibril reinforced poroelastic model. *Clinical Biomechanics*, Vol. 14, No. 9, (November 1999), pp. 673-682, ISSN 0268-0033 Mori, S.; Naito, M. & Moriyama, S. (2002). Highly viscous sodium hyaluronate and joint

lubrication. *International Orthopaedics*, Vol. 26, No. 2, (April 2002), pp. 116-121, ISSN

synovial fluid. *Biorheology*, Vol. 16, No. 3, pp. 179-184

No. 2-3, (January 1975), pp. 141-147, ISSN 0300-8207

0148-0731

ISSN 0021-9290

ISSN 0021-9290

0341-2695

Vol. 5, No. 2, (July 1968), pp. 119-131

*Biochem*, Vol. 59, No. 3, (April 1955), pp. 633-637

(August 1991), pp. 245-351, ISSN 0148-0731

fluid. *Biorheology*, Vol. 3, pp. 197-209

8, (December 1998), pp. 2663-2684, ISSN 0009-2665

this way, the hydrodynamic biomechanism quickly presupplements the surface localities with lubrication material. Shortly after unloading, the strain rate is high. During strain relaxation, it slows down. This is the way how the articular cartilage tissue attempts to retain the lubrication material between the articular plateaus of synovial joints as long as possible during cyclic loading.

Analogically to the low and the middle zone of AC where an incompressible zone arises under high loads whose dominant function is to bear high loads and protect chondrocytes with the intercellular matrix from destruction, in the peripheral zone as well a partial incompressible zone arises whose function is to bear high loads and protect the peripheral tissue from mechanical failure. The appearance of the incompressible tissue in all zones is synchronized aiming at the creation of a single (integrated) *incompressible cushion*. The existence of an incompressible zone secures the protection of chondrocytes and extracellular material from potential destruction.
