Modelling Soft Biological Tissues

The diversity of mechanical properties encountered in soft biological tissues is huge. Soft organic tissues are in general characterized by very complex mechanical behaviour. They show non-linear, anisotropic, viscoelastic and in some cases also viscoplastic behaviour. They often have a layered or an even more complicated structure. The mechanical properties are inhomogeneous, i.e. they depend on the position in the material. The perfusion of the organs and their constituting tissues also plays an important role regarding the elastic properties.

The stress strain relationship

There are mainly two sources of elasticity in soft biological tissues. The first source of elasticity is due to changes of internal energy whereas the second one is due to changes of entropy. Change of entropy occurs in tissues whenever changes of orientation or waviness of fibers during loading or unloading occur.A typical load-elongation and load-time diagram for soft tissue is shown in the figure below.

With repeated loading cycles the load-deformation curves shift to the right in a load-elongation diagramm and the hysteretic effects diminish . In a load-time diagramm the load-time curves shift upwards with increasing repetition number. By repeated cycling, eventually a steady state is reached at which no further change will occur unless the cycling routine is changed. In this state the tissue is said to be preconditioned. Any change of the lower or upper limits of the cycling process requires new preconditioning of the tissue. Preconditioning occurs due tointernal changes in the structure of the tissue. Hysteresis, non-linearity, relaxation and preconditioning are common properties of all soft tissues, although their observed degrees vary.

The hysteresis in the stress strain relationship clearly shows the viscoelastic behaviour of soft biological tissue. In a viscoelastic material the history of strain affects the actually observed stress. As well, loading and unloading occur on different stress-strain paths. The hysteresis of most biological tisssues is is assumed to show only little dependence on the strain rate within several decades of strain rate variation. This insensitivity to strain rate over several decades is not compatible with simple viscoelastic models consisting e.g. of a single spring and dashpot element. With such a simple viscoelasticity approach the material model will show a maximum hysteresis loop at a certain strain rate whereas all other strain rates will show a smaller hysteresis loop. A model consisting of a discrete number of spring-dashpot elements therefore produces a discrete hysteresis spectrum with maximum dissipation at discrete strain rates. If the relaxation times of the different elements are chosen adequately a series of spring-dashpot elements might be used as an approximation to a continuous relaxation spectrum. Living tissues often show a viscoelastic behaviour as shown qualitatively in the figure below.

In the above figure the viscoelastic material properties are characterized by storage and loss modulus, which are concepts only valid for linear elasticity.

With a series of spring-dashpot elements abritrary viscoelastic material properties can be modelled.