# Finite Element Modeling of Cells Adhering to a Substrate: An Overview
## 2 Physical Modeling of Cells in Adhesion Phenomena
Assuming that during spreading cell morphologies resemble truncated spheres and spherical caps means that the contact region between cell and substrate can be considered a circular area. Since the figure represents the cell side view, the contact area in the figure corresponds to the line defined by the interaction interface between cell and substrate schematizations. At the macroscopic level, cell morphology is described by two parameters: the contact radius \(R\) and the contact angle \(\theta\). The contact radius \(R\) is obtained by dividing the diameter of the circular contact area between the cell and the substrate by two (i.e., by dividing by two the length of the line defined by the contact interface between the cell and substrate in the figure). The contact angle \(\theta\) is the angle identified by the plane tangent to the upper surface of the substrate and the plane tangent to the cell-free contour that passes through the point where the cell-free contour meets the substrate. Contact radius \(R\) is colored orange. Contact angle \(\theta\) and planes tangent to the substrate’s top surface and the cell’s free contour are colored dark brown. cell's resistance to deformation, thus reducing the accuracy of simulations, the authors of Ref. [69] investigated how reduced values of membrane rigidity effect the stress-strain field in the cell. [PERSON] et al. [73] successfully used FEM to analyze the dispersion in cell stiffness experimental values gathered from tweezing micromanipulation tests (i.e., optical and magnetic tweezers experiments). The dispersion of experimental data also occurred in tests relative to a given type of adherent cell (i.e., epithelial or endothelial cells) performed with the same probing technique. Both tweezer micromanipulation techniques induce the cell's material response by applying an external force to the center of a solid spherical bead, which is partially embedded in the outer layers of the cellular material (Figure 4). The force acting in the direction parallel to the cell surface causes the bead translational movement until, at the equilibrium, the cell's material counterbalances the effect of the external stimulus by resisting deformation. In general, optical tweezer micromanipulation employs silica beads with a radius ranging between 1 and 2.7 \(\upmu\)m, whereas magnetic tweezer tests utilize ferromagnetic beads with a radius between 0.6 and 2.2 \(\upmu\)m [74; 75; 76; 77]. In both testing techniques, beads are coated with an RGD (Arg-Gly-Asp) peptide, enabling a connection between bead and integrin proteins, which spans the cellular membrane and ensures the interaction between beads and cytoskeletal load-carrying architecture. At the equilibrium point of the cell-bead system, it can be assumed that in the tweezing micromanipulation, the bead is totally contained in the cell. Furthermore, the cell behaves as an infinite elastic domain of incompressible material characterized by a linear elastic constitutive relationship. In view of this, analytical formulas were developed in [78; 79; 80] to relate the external force \(F\) to the cell's material stiffness \(E\) and the resulting bead displacement \(U\) or, equivalently, the corresponding associated bead rotation \(\theta\) as follows:  
\[F=\frac{1}{2}ES\delta, \tag{2a}\] \[F=\frac{1}{2}ES\theta, \tag{2b}\]  
where \(S\) is the surface area of the bead (i.e., \(S=4\pi R^{2}\) with \(R\) bead radius) and \(\delta\) is the normalized bead translation (i.e., \(\delta=U/R\)). By substituting the experimental values of force and displacement in these analytical expressions, it is possible to roughly estimate the  
Figure 4: Schematic of the tweezer micromanipulation technique. An adherent cell is mechanically tested by applying to the bead center an external force acting in the direction parallel to the substrate plane. The bead is partially embedded in the outer layer of the cell material, and the cell-bead interaction is ensured by the link between the RGD peptide-functionalized bead surface and membrane integrins.