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A Four-Component Mixture Theory Applied to Cartilaginous Tissues
Cartilaginous tissues like, for example, intervertebral disc tissue, exhibit swelling or shrinking behaviour. The swelling of these tissues influences the behaviour of the cells. This swelling is caused by mechanical, chemical or electrical forces. An example of a mechanical force is the weight of the body applied on the human spine. A chemical force is for example a concentration difference over the tissue. An electrical force can be applied by an electrical potential field. In all cases, the macroscopic swelling and shrinking is caused by inflow or outflow of fluid. The forces cause a fluid flow, a solute flow and/or an electrical current. By the interaction between these fixed charges and the freely moving and electrically charged particles inside the tissue, physical phenomena occur, like osmosis, streaming potentials, diffusion potentials, electro-phoresis and electro-osmosis (table 1). In order to develop more insight into the mechanism, this swelling and compression behaviour has been modelled.
An intervertebral disc is a cartilaginous structure that is located between two vertebral bodies (figure 1). The tissue consists of a gelatinous centre, the nucleus pulposus, surrounded by several concentrically arranged lamellae, the annulus fibrosus. In these lamellae, fibre networks can be seen. The orientation of these fibres differs per lamella. Obviously, these fibres make the material anisotropic.
figure 1: Schematic view of the intervertebral disc and its surroundings.
The material mainly consists of a collagen network embedded in a hydrated proteoglycan gel. Collagen is a rod-like protein molecule built of long polypeptide chains. Proteoglycans are large molecules consisting of many glycosaminoglycans linked to core proteins. These glycosaminoglycans are made up of long chains of polysaccharides. Due to the physiological pH and the ionic strength of the interstitial fluid, the carboxyl and sulfate groups of the polysaccharides are ionised. The density of these charges is the fixed charge density. Due to this ionisation the proteoglycans are capable of retaining water up to a 50-fold of their own weight.
Cartilaginous tissues mainly consist of four components: a fibre network (collagen fibres and proteoglycans), a fluid, and positively and negatively charged particles. Due to this consistency, physical effects as described in table 1 can occur. In order to describe these effects in a physical model, these four components have to be included.
The interactions of the components are described by a macromodel based on a continuum approach. In this approach, the material is divided into representative elementary volumes that have a size such that they are large enough to be treated as homogeneous. At the same time, they are small enough to model the differences in material properties. In each representative elementary volume, the structure properties are averaged and the volume fractions of each component are determined (figure 2).
figure 2: Continuum approach of the tissue.
Next, the representative elementary volume is considered to be a homogeneous material. This means that at every point in the material, a fraction of every component is present. In this way, the tissue is modelled as a continuum.
We describe this continuum by a four-component mixture theory: the tissue is modelled as a charged porous solid, which is saturated with a mono-valent salt solution. In the model, four components are distinguished: a charged porous solid (s), a fluid (f), cations (+) and anions (-). Because of the distinction between the cations and the anions, the electrical phenomena can be modelled.
As we assume that all components are intrinsically incompressible and that there are no (chemical) reactions between the components, the material behaviour is described by four mass balances for the solid, the fluid, the cations and the anions, the momentum equation, in which the inertial terms and the body forces are neglected and the electro-neutrality of the mixture. We also need some constitutive equations: Hooke's law for the solid skeleton, an extended Darcy's law for the fluid flow, and an extended Fick's law for the ion fluxes. These extended laws depend on the pressure gradient, the concentration gradients and the electrical potential fields. In this way a set of coupled, time-dependent non-linear differential equations is established. Using an implicit time-discretization and a mixed finite element method, the system is solved.
The behaviour of soft biological tissues was demonstrated by confined swelling and compression experiments. In a uniaxial confined swelling and compression experiment, a cylindrical sample was enclosed in an impermeable confining ring. A mechanical load was applied on top of the sample via a loading piston made out of glass. Inside the piston was a chamber filled with a 0.15 M NaCl solution (figure 3 ). This chamber was closed at the bottom with a dense glass filter. Inside the chamber, a Ag/AgCl electrode was placed. At the bottom of the sample was a glass filter through which a NaCl solution flowed. The permeability of the glass filter was much larger than the permeability of the sample. Thus, the boundary conditions were well defined for the fluid flow and for the ion concentrations along the filter-sample interface. A chemical load was applied by altering the salt concentration in the glass filter. A similar electrode as in the piston was mounted in the fluid channel. The deformation of the sample and the electrical potential difference over the sample were measured.
We use a one-dimensional finite element model of the tissue to validate the four-component mixture model. We assume the stiffness, the permeability and the diffusion coefficients to be independent of the strain. The results are shown in figure 4.
figure 4: Simulated and measured data.
Although we assumed the material parameters to be constant, we were able to fit the experiments reasonably well. The deformation in the first part of the experiment (until 20 h) is fitted reasonably by the four-component mixture theory. Then, the sample deformation is overestimated. This is because the permeability is too large. This is caused by neglecting the deformation dependency of the permeability. When the tissue is compressed, the permeability should decrease because the pores are squeezed.
The values for the diffusion coefficients were fitted by looking at the last part of the curve for the electrical potential difference (t>20 h). The measured electrical potential difference was smaller than the computed one, because the concentration of the bathing solution changed slower in the experiment than in the simulation. In the simulation we assumed that at a certain moment the external salt concentration changes from one concentration immediately to another. In the experiment, this concentration went gradually from one value to another. This effect decreased the peaks in the electrical potential signal, because the electrical potential difference was lowered rapidly in the first couple of seconds. Qualitatively, the signals were predicted reasonably well.
We were able to measure the electrical potential differences over the samples that were caused by a change in the mechanical loads or by a change in the chemical load. The evolution in time of the tissue deformation as well as the evolution in time of the electrical potential differences were simulated by the four-components mixture theory. Furthermore, the estimated values for the stiffness, permeability, and the diffusion coefficients were in the same range as reported by other studies.
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