# Rheological Models

Rheology usually describes the behavior of materials whose mechanical properties are characterized by both elastic ad viscous components, like most of biological objects. However, rheology does not always describe their mechanical properties (e.g., elastic modulus) due to their high structural complexity.

Focusing on mechanical properties of a single cell introduces more complications in the theoretical description of cellular mechanics. Various attempts to model single cell have been reported so far. Some of these models treat a cell as a uniform object, composed of fluid surrounded by elastic membrane, while the others consider also the structural elements present within the cell interior [9-11].

## Mechanical behavior of soft materials

Mechanical models applied to soft materials (also to a biological cell) must combine both elastic and viscous components [12]. For example, the simplest two-compartment model takes into account an elastic cortical membrane and a viscoelastic cytoplasm [11]. At small deformations, such soft materials behave as the Hook elastic body *(modeled as a spring* for which a dependence between a stress s and a strain e is linear) and as the Newtonian fluid *(represented as a dashpot* where stress s is linearly proportional to time derivative of a strain e). The equations describing the behavior of such body are as follows [13]:

where *E* is the modulus of elasticity, h is the viscosity, and *t* is the time. The combination of these elements gives a simple description of viscoelasticity.

The simplest rheological models (the Maxwell, Voigt, and Kelvin ones) used to describe viscoelastic properties of living cells are presented in Fig. 4.3.

Figure 4.3 Schematic representations of the simplest rheological models, composed of a spring (an elastic component) and a dashpot (a viscous element).

*The Maxwell model* consists of a viscous element (the Newtonian liquid) linked in series with an elastic element (the Hookean spring). The basic equation for the model is

where *E* is the modulus of elasticity, h is the viscosity, s is the stress, e is the strain, and *t* is the time.

When such system is quickly loaded, the viscous element is too slow to react and only the spring responses to deformation. On the contrary, if the load is applied slowly, the deformation of the viscous element starts to dominate and interfere with the deformation of a spring. In cellular biology, the Maxwell model is often used to describe the deformation of neutrophils using the micropipette aspiration technique [14].

*The Voigt model* describes the material consisting of the elastic spring and the viscous dashpot elements connected in parallel (assuming that there is no bending in such arrangement). The strain experienced by the spring is the same as that experienced by the dashpot. In this model, the stress s depends on a strain e according to the followingrelation:

where *E* is the modulus of elasticity, h is the viscosity, and *t* is the time.

The good example of the Voigt model application is description of viscoelastic response of fibroblasts exposed to tension transmitted through adherent junctions [15].

*The Kelvin model* (or more precisely *the Kelvin-Voigt model) *describes both stress and strain relaxations. It consists of a module described by the Voigt model (i.e., spring and dashpot linked in parallel), linked in series to an elastic spring. During the deformation, part of the energy is dissipated. The equation that describes mechanical properties of such material is as follows:

where *E _{2}* is Young’s modulus of a spring present in the Voigt model,

*E*is Young’s modulus of the spring linked in series. The Kelvin-Voigt model has been used, for example, to model mechanical response of collagen molecules delivering information on both Young’s modulus and viscosity [16].

_{1}*More complex models.* The above-mentioned models are the simplest ones. They are not always describing mechanical behavior of biological samples in a fully satisfactory way. Thus, more complex models are employed (a so-called *generalized Maxwell* [17] or *generalized Kelvin* models [13, 18]).

The generalized Kelvin model consists of an isolated spring E_{0} and *N* Kelvin units (E_{;}-, *hi)* linked in series, described by the following relation:

The stress at each unit is the same external stress *a(t)* while the total strain e(t) is the sum of internal strains in each element. For *N* elements, the stress-strain relation is following:

The specific creep function for the generalized Kelvin model is

where *6 _{i}* =

*h/E*and

_{i}*t*is the characteristic creep time of a single Kelvin unit.

The generalized Maxwell model consists of *N* different Maxwell units (E_{;}> *hi)* linked in parallel (it is important to note that the absence of the isolated spring ensures fluid-type behavior).

For the generalized Maxwell model the strain is the same for all constituting elements and the total stress is given by the following equation:

The relaxation function is

where *T _{i}* =

*h/E*and

_{i}*t*is the characteristic relaxation time for single Maxwell unit.

The generalized models introduce more elements to be considered to fully describe the response of real materials, and, simultaneously, they result in larger number of material parameters to be evaluated. In some cases, their determination might be a difficult, if not an impossible, task.