# Quantification of Cellular Elasticity

During the past decades, a number of studies have proved the link between mechanical properties and cellular functions, emphasizing alterations in cellular mechanical properties correlated with development of various diseases, in particular, in cancer. Recent progress in cancer cell motility and invasion research has resulted in a greater understanding of the role of mechanical properties in malignant transformation. There is evidence that cancer progression is characterized by disruption and/or reorganization of cytoskeleton (i.e., a cellular scaffold [1]). This is accompanied by various molecular alterations influencing the mechanical properties of cells [2].

## Materials Properties and Theoretical Models

### Basic Terms Used in Material Mechanics

Quantitative (or even qualitative) characterization of mechanical properties of various materials is important in many fields of our life. The mechanical properties, such as Young’s modulus, the index of plasticity, the ultimate strength, the tensile strength, and the elongation, are the examples of parameters describing material mechanical properties. They are usually derived on a

*Cellular Analysis by Atomic Force Microscopy *Malgorzata Lekka

Copyright © 2017 Pan Stanford Publishing Pte. Ltd.

ISBN 978-981-4669-67-2 (Hardcover), 978-1-315-36480-3 (eBook) www.panstanford.com

basis of stress-strain relationship, where stress is the applied load force per unit area while strain is the ratio between deformation and initial length [3-5]. A scheme of stress-strain relation is presented in Fig. 4.1.

Figure 4.1 Relation between stress and strain for a hypothetical material.

In the low-strain region, many materials obey the Hooke’s law saying that a stress s is proportional to a strain e (s = *E?* e). The proportionality factor is Young’s modulus *E* (frequently called elastic modulus). In this region, materials are *elastic,* i.e., if some deformation is produced in a material by applied stress, stress release brings deformation back to zero [4]. A classic example of elastic material is a metal spring. Here, the applied stress changes linearly with a strain following the above-mentioned Hooke’s law:

where *k* is the elasticity constant of a spring (i.e., spring constant), *x* is the displacement of the end of the spring from its equilibrium position (i.e., when applied stress equals to zero), and *F* is the force exerted on the material.

As strain increases, many materials deviate from this linear dependence (material becomes *non-elastic**).* The point of transition is termed *a yield point.* Once it is reached, the applied stress produces a certain fraction of deformations that is permanent and non-reversible (material becomes *plastic).* Further strain increase leads to a material *failure.* There are several theories developed to describe plastic material properties (i.e., *material **plasticity)* based on three common assumptions:

- (a) During plastic deformation, the sample volume does not change.
- (b) The directions of principal strains and stresses overlap.
- (c) In each point, the maximum shear stress is equal to a specific constant.

The simplest theory describing plasticity considers tensions induced in the material under deformations following the Hooke’s law [5]. Lack of proportionality beyond the elastic limit (a yield point) in the stress-strain curve is explained assuming that Young’s modulus is not constant. It is expressed as a function of a strain e in the following manner:

where ?[ is the yield limit, j = j(e) describes the plasticity function that is derived from experimental data. This function becomes zero for elastic deformations and increases for the plastic ones. Hence, the common dependence between stress and strain (s = *E?* e) can be rewritten into

This equation is suitable only for tensions. Any phenomena such as torsion or bending (or both) require more complicated theoretical approaches.

Fluids have characteristic resistance to flow called *viscosity, *which results from the frictional energy loss [6]. The more viscous fluid induces larger frictional energy losses. Figure 4.2 illustrates schematically the flow of the fluid. The largest flow velocity is on the fluid surface while close to boundaries the velocity drops to zero.

To characterize the material viscoelastic properties, the mechanics of solid bodies (where the Hooke’s law is valid and the stress is proportional to the strain) cannot be used. Solid material undergoes only a finite amount of deformations under the applied stress. In fluid mechanics, such approach fails because fluid continuously deforms under an applied shear stress. Thus, here, another law applies, linking shear stress *t* with a shear rate, under assumption that velocity profile is linear:

where *dy/dt* defines the shear rate and the proportionality factor h is the viscosity coefficient. The fluids fulfilling Eq. (4.5) are called *Newtonian liquids.* For *a non-Newtonian Hquid,* the shear stress-strain rate relation is not linear and, furthermore, viscosity depends upon the applied shear stress [7].

Figure 4.2 Fluid velocity profile showing linear dependence as a function of distance from a boundary (e.g., fixed, bottom plate).

Some materials, such as polymers or cells, behave as both viscous fluids and elastic solids, i.e., they are *viscoelastic.* Their distinguishable feature is the dependence of mechanical properties on time [8, 9].