12 Biomechanics of Cartilage
Diarthrodial joints are covered with hyaline articular cartilage (AC) enabling the transmission of high loads at low friction. Hence, AC plays a very important mechanical role in our musculoskeletal system. Unfortunately, AC can degenerate idiopathically or due to trauma or joint misalignment leading to osteoarthritis with a potential need for joint replacement. For decades, methods have been investigated to heal or regenerate AC with different strategies. To evaluate such methods, it is important to understand the biomechanical way of functioning of AC. Its complex functionality is accomplished by the biphasic composition of this tissue. It contains chondrocytes, which secrete an extracellular matrix with the main constituents collagen and proteoglycans. Besides this solid matrix, AC possesses a high water content as its fluid component. While collagen is mainly responsible for the mechanical strength of the solid matrix, the proteoglycans are able to store water by osmotic swelling providing the resistance to compression and creating the time-dependent behavior, which results in creep and relaxation phenomena. Any change in composition of the extracellular matrix resulting from trauma or degenerative processes leads to changes in biomechanical properties. Therefore, the biomechanical characterization of AC is an important and useful tool to assess its functional quality and can be used in any kind of research on AC such as basic questions on the functionality of AC, degenerative processes, repair, and tissue engineering aiming to replace destroyed cartilage. A variety of different test methods for the biomechanical characterization of AC are available depending on which properties are of interest regarding a special research question. This chapter gives an overview of practical methods to assess the compressive and time-dependent properties including friction of AC.
12.1 Unconfined Compression and Tensile Testing
The simplest way to determine the material properties of AC is unconfined compression or tensile testing of geometrically standardized specimens. From such experiments, parameters can be derived such as stiffness (K in N/mm), deduced from the force-elongation diagram, and Young′s modulus (E in N/mm2 or MPa), determined from the stress-strain diagram. Although both tensile and unconfined compression testing lead to the same parameters, the results are different, depending on the sample and test conditions. For example, the cartilage tensile modulus has been reported to be one to two orders of magnitude greater than the corresponding compressive modulus. 1 Testing AC under compressive loads commonly involves cylindrical specimens, which can be easily harvested by punches or hollow drills, or cubic specimens, whereas for tensile tests mostly rectangular strips or dumbbell-shaped samples are used, which are more complicated to create. A beneficial attribute associated with the compressive testing method is its comparatively simple test setup (Fig. 12.1a) where a flat-ended rod is used to compress the specimen to a previously defined load, stress, deformation, or strain. Due to the varying Poisson′s ratio of different soft tissues, the cross section of the cartilage specimen changes differently during compressive (ν = 0.2) 2 and tensile (ν > 0.5)1 loads. Compared to unconfined compression, tensile testing is more elaborate, because it implies the problem of clamping soft tissue, which is a general pitfall in soft tissue biomechanics. Therefore, the test setup for tensile testing (Fig. 12.1b) commonly consists of two clamps to ensure adequate fixation of the samples during tensile loading.
During testing, it should be considered that both test methods do not intend to represent physiological or in vivo loading conditions. This involves the integrity of the cartilage sample with respect to the loading characteristics, as well as the ambient conditions (e.g., temperature, nutrition, and moistness). Due to its anisotropic and biphasic material behavior, AC shows a typical strain-rate–dependent response during unconfined compression and tensile loading where an increased strain rate is associated with strengthening the transient stiffness and Young′s modulus 3, 4 (Fig. 12.2a). Another phenomenon can be observed when loading the cartilage specimen to a certain point with instant unloading, leading to a so-called hysteresis loop, which is common for biphasic soft tissues like cartilage 5 (Fig. 12.2b).
Jargon Simplified:
Load (F in N): Usually results in a deformation of the tested material.
Stress (σ in N/mm2 or MPa): Measure of the internal force acting on a material. Defined as the measure of the force that is applied via a certain area.
Deformation (∆l in mm): Change in the dimensions of a material to external forces or stress.
Strain (ε in %): Deformation of a material related to its initial length or height.
Poisson′s Ratio (ν): Tapering or bulging ratio of a material in the direction perpendicular to the applied tensile or compressive stress, relative to its initial state.
12.2 Creep/Relaxation and Dynamic Compression
Jargon Simplified
Aggregate Modulus (HA in N/mm2 or MPa): Modulus of a biphasic material in its equilibrium state reflecting the properties of the solid matrix. Can be calculated by dividing the equilibrium stress (σt→∝) by the corresponding strain (ε):
Hydraulic Permeability (k in m4/Ns): Parameter to describe the fluid flow through a porous material.
When repeating the previously introduced hysteresis loop several times, their characteristics will change in a way that the differences of the loading and unloading curve start to converge (Fig. 12.3a). Additionally, when using a time-resolving illustration of the stress and strain curves of these loading and unloading ramps, a characteristic phase shift of the material response could be observed (Fig. 12.3b). This phase shift also happens during sinusoidal or other alternating loading schemes.
In addition, the characteristic time-dependent properties of AC could be assessed using two more loading tests:
Creep: Loading of a cartilage sample with a constant load. The samples thickness diminishes by fluid efllux until an asymptotic thickness is reached.
Stress-relaxation: Compression of a cartilage sample until a desired strain is reached. The applied strain is kept constant over time, whereas the applied load decreases until an asymptotic equilibrium force is noted.
Both load methods are able to reveal the viscoelastic behavior of cartilage over time. However, to gain the aggregate modulus (HA in MPa or kPa) and hydraulic permeability (k in m4/Ns), each test routine could be conducted in two different ways:
Indentation test (Fig. 12.4a). Here, a flat indenter is pushed into the cartilaginous surface. The respective decrease of thickness (creep) or load (relaxation) is then measured over time. There is no need for preparing complex, geometrically standardized cartilage samples, because the whole specimen could be mounted into the testing rig, while ensuring adequate positioning. However, the cartilage thickness must be determined in another test (e.g., needle test), because without knowing the sample thickness the application of a certain strain level and relative loading rate is impossible.
Confined compression testing (Fig. 12.4b). A coplanar, geometrically defined cartilage disk is compressed to a certain load or strain level and held constant until an equilibrium state is reached. In contrast, the test setup is more complex as it must be ensured to constrain the samples in their radial direction while allowing an adequate fluid flow in the axial direction.
When determining the viscoelastic material parameters, various constitutive mathematical models could be used to solve their according equations. 6, 7, 8 However, it should be kept in mind that each test method and each mathematical model can lead to different results for the time-dependent, viscoelastic parameters, because mathematical models mostly are simplified models and do not necessarily match reality 100%. Therefore, care must be taken when comparing results with already existing results in literature.
12.2.1 Example for the Experimental Characterization of the Viscoelastic Properties of Articular Cartilage: Stress-Relaxation under Confined Compression
After harvesting a coplanar cartilaginous disk (thickness: 1 mm; Ø: 5 mm), the sample could be tested using a special measuring chamber providing confined conditions (Fig. 12.4b). This means that the cartilage sample cannot extend in the radial direction and no fluid can flow through the sides of the cartilage disk. The measurement chamber is filled with saline solution to prevent dehydration. The load is applied via a porous Al2O3 cylinder allowing adequate fluid flow through the surface of the sample.
During testing, it is important to measure the deformation, especially the strain, directly at the surface of the cartilage sample as, for example, measurement errors related to deflection of the crosshead of a materials testing machine or of the load cell should be avoided. A dedicated external displacement sensor with high accuracy can be used for this critical issue.
The viscoelastic properties strongly depend on the extent of the applied compression. Therefore, it is recommended to perform a test routine with incremental steps, for example increasing the strain stepwise from 5 to 20% of its initial thickness (Fig. 12.5a). The decrease of load is recorded for about 10 minutes or at least until the equilibrium state of each strain step is reached. As previously mentioned, the strain rate during the compression phase of the test also considerably influences the peak load occurring directly after reaching the desired strain step. Therefore, the strain rate must be kept constant over the whole test series of the experiment. Thus it is recommended to use a strain-dependent loading rate (e.g., 10% *h 0/min [h 0 = initial height of the specimen]) to load each specimen equally.
For data analysis, the modulus (H) indicating a combined modulus of both the fluid and the solid phase of the viscoelastic material and the hydraulic permeability (k) can be extracted in a first approximation from the measured relaxation curves by fitting the data to a diffusion equation by using a least square fitting algorithm using the following equation6,7:
where σ(t) is the present stress in MPa, σ(t=∝) the equilibrium stress in MPa, H the above-mentioned combined modulus in MPa, ∆l/d the applied strain level, d the sample thickness in mm, and k the hydraulic permeability in m4/Ns. The determination of the values for H and k can be performed by any statistical software (Fig. 12.5).
The exemplary results of the nonlinear curve fitting, calculated with JMP (V.3.2; SAS Institute GmbH, Germany) indicate an H of 0.79 MPa and k = 3.05 * 10 to 15 m4/Ns. Furthermore, the graph (Fig. 12.5) shows that the adaptation of the model equation to the measured data is good, but not optimal. This means that the real process does not exactly follow the chosen diffusion model, which does not reflect the dynamic behavior of the cartilage sample directly after application of the load. Therefore, an adequate number of measured data points must be excluded from the fitting process until the correlation coefficient is sufficiently high (e.g., 0.95).
Key Concepts: Mathematical Modeling of Articular Cartilage
Due to the anisotropic behavior of cartilage, there are plenty of mathematical descriptions of the response of cartilage to a various number of external loads and environmental conditions. After measuring the in vitro response of articular cartilage, an adequate mathematical model must be chosen to determine the required material properties. Usually, the model parameters are then varied until the model converges with the experimental data. For this purpose, statistical software can be used providing a variety of fitting algorithms (e.g., least-square, Levenberg-Marquardt, Gauss-Newton). After running the fitting algorithm, the mathematical model results in a best possible representation of the experimental data. This could be checked using the Pearson correlation coefficient and setting its accuracy limit, for example, to r = 0.95.
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