6.1. General Theoretical and Technical Approximation
The biomechanical behavior of Articular cartilage is better understood when the tissue is viewed as a biphasic medium [
24]. In general, there exist models to describe viscoelastic behavior built up as a mixture of fluid (damper) and solid (spring) components (e.g., ).
Figure 10. Simple model consisting of a mixture of spring and damper for describing viscoelastic behavior (= Kelvin–Voigt model). (σ → stress, ε → strain, E → is a modulus of elasticity; η → is the viscosity).
In the simple Kelvin–Voigt model is shown. The overall strain
ε corresponds to the elastic strain
εe and viscous strain
εv.
The force equilibrium (kinetic relationship) delivers the stress in the Kelvin–Voigt model:
The constitutive relationship will be generated from the material equation for a spring:
(This corresponds to Hook’s law), and the material context for a damper:
(This corresponds to Newtonian fluid behavior).
From the force equilibrium, it is possible to create the general equation for the model:
For a complex tissue like AC, the model is too simple and needs more information to approximate the biomechanical behavior.
6.2. Specific Models of Articular Cartilage
To date, the most successful theory for cartilage compressive viscoelastic behaviors is the
biphasic poroelastic theory developed by Mow et al. [
38]. This biphasic theory models AC as composite materials consisting of a solid phase (solid like components of the cartilage, proteoglycans, collagen, cells and lipids are lumped together to constitute the solid phase) and a fluid phase (interstitial fluid that is free to move through the matrix) [
5]. The porous solid matrix is elastic and permeable to fluid. Three major (internal) forces act within the loaded tissue: (1) the stress developed within the deformed collagen—proteoglycan macromolecules (solid) matrix; (2) the pressure that is developed within the fluid phase, furthermore, (3) the frictional drag acting between the fluid phase and the solid phase as they flow past each other [
5]. In a biphasic medium, all of the three internal forces act in concert to balance applied external forces, thus giving rise to a viscoelastic effect [
5].
By fitting the mathematical biphasic model to the measured displacement, two material properties of the cartilage are determined: the aggregate modulus and permeability [
15]. In cartilage biomechanics, instead of Young’s modulus, the aggregate modulus is often used to describe the tissue, because it can be directly calculated from the equilibrium data in a confined compression test (e.g., ) [
5]. The aggregate modulus is a measurement of the stiffness of the tissue at the equilibrium when all fluid flow has slowed down. The higher the aggregate modulus, the less the tissue deforms under a given load [
15]. The aggregate modulus of cartilage is typically in the range of 0.5 to 0.9 MPa [
15,
39].
In addition to the aggregate modulus, the hydraulic permeability k of the cartilage is also determined from a stress-relaxation or creep test (, and ) of confined or unconfined compression. It can be determined by curve-fitting the creep or relaxation curve generated in the test [
5,
17,
40,
41]. The permeability k indicates the resistance to fluid flow through the cartilage (solid) matrix.
In the simplest version of the biphasic theory, the stress—strain law for the solid matrix, is assumed to be isotropic and linearly elastic [
5]. The frictional drag acting on the solid phase can be given by Darcy’s law [
38]:
This means that the average fluid velocity vave through a soil sample is proportional to the pressure gradient ∇pand permeability k.
The pressure gradient is approximated by ():
Permeability is not constant through and also varies with deformation of the tissue. The permeability of articular cartilage is highest inching the joint surface and lowest getting closer to the deep zone [
42,
43,
44,
45,
46]. The permeability influences also the deformation rate. If
k is high, the fluid can flow out of the matrix easily, and the equilibrium is quickly reached. A lower value for k causes a slower transition from the rapid early displacement to the equilibrium. As cartilage is compressed, its permeability decreases. Under increasing load, the fluid flow will decrease because of the decrease in permeability which accompanies compression [
15]. These qualitative results are helpful for interpreting data from tests of physiological and OA related cartilage [
15].
Lai et al. [
36]. published the
triphasic cartilage modeling framework, expanding the generalized biphasic models by incorporating an ionic phase [
36]. They described how swelling pressure, Donnan osmotic pressure, and solid matrix stress are related to one another.
A distinction is made between three phases in cartilage: a solid phase with a tightly packed collagen network and a high concentration of proteoglycan aggregates, a liquid phase with a high water content and a ion phase in which the environment to ensure electrical neutrality is dominated by dissolved electrolytes (anions, cations).
The physico-chemical view of cartilage and other hydrated charged tissue are based on the Donnan theory for aqueous polyelectrolyte solutions [
47]. The Donnan equilibrium and Gibbs–Donnan equilibrium, respectively, describes the equilibrium between two solutions that are separated by a membrane. The membrane allows a passage of different charged ions of the solutions, and is thus a selectively permeable membrane. The effect is related to the presence of impermeant ions, which means that they are unable to pass through a semipermeable membrane upon one side of a boundary on the distribution of permeant ions across the boundary. The membrane permeability and impermeability, respectively, is related to the size of the particular ion, which can be too large to pass through the membrane pores from one to the other side. In the case that the concentration of those ions passing freely, the membrane is equal the total number of charged molecules on either side of the membrane is equal [
36]. In the case that there exists a selective permeability of the membrane, an electrical potential between the two sides of the membrane will be developed [
36]. The result is that the two solutions vary in osmotic pressure. That means one solution have in the end more of a certain type or types of ion that does the other solution [
47].
According to this model, cartilage is partly a porous but solid material and partly a non-compressible liquid. The pore diameter is between 2.5 nm and 6.5 nm [
38], its permeability (total weight minus dry weight/total weight) is between 60% and 85%. The micropores make sure that molecules only can pass up to a size of 20,000 daltons. There exist a selection process according to spatial molecular properties. When compressed, the diameter increases the pores, which reduces the permeability of the cartilage.
According to biphasic and triphasic theories, equilibrium modulus of cartilage includes the contributions from two sources: the Donnan osmotic pressure and the ‘‘intrinsic stiffness’’ of the solid matrix without charge effect [
5]. The apparent and intrinsic tissue properties are defined as the tissue properties with and without the osmotic pressure effects—that is, the properties in triphasic framework as intrinsic properties and those in biphasic model as apparent properties [
5,
23]. The triphasic mixture theory has been successfully used to describe the flow-dependent and flow-independent viscoelastic behaviors, swelling behaviors, and electro-kinetic behaviors of charged, hydrated AC [
5].
An overview about further and more complex models is given in [
23].