# Finite Element Modeling of Cells Adhering to a Substrate: An Overview
## 2 Physical Modeling of Cells in Adhesion Phenomena
The substrate is highlighted in white, the cell material in beige, and the bead in light gray. \(F\) (i.e., red arrow), which is the external force acting on the bead. stiffness \(E\) exhibited by the cell's material. However, this approach yields high dispersion and discrepancy in cell rigidity values when testing operations are conducted on the same cell type [79; 80]. The explanation for the inhomogeneity in cell rigidity resulting from the analytical approach was identified by [PERSON] et al. [73] in the spurious nature of [PERSON]'s modulus \(E\) included in Equation (2a,b), which does not represent the actual cell [PERSON]'s modulus \(E_{cell}\), rather an apparent modulus \(E_{app}\) affected by geometric factors related to the experimental technique and the cell type. The bead radius \(R\), the bead embedding half angle \(\gamma\), and the cell height \(h\) were recognized in [73] as the driving geometric parameters of the deviation of the apparent [PERSON]'s modulus \(E_{app}\) obtained by experimental measurements from the intrinsic cell [PERSON]'s modulus \(E_{cell}\) (see Figure 5). In order to provide a valid mathematical formulation for overcoming this issue, [PERSON] et al. [73] introduced the correction functions \(\alpha\) and \(\beta\) dependent on \(R\), \(\gamma\), and \(h\). These functions establish a relationship between \(E_{cell}\) and \(E_{app}\) and make it possible to obtain the actual cell stiffness once the bead force and the induced translation or, equivalently, the associated rotations are known as follows:  
\[E_{app}^{\delta}=\alpha(R,\gamma,h)E_{cell}, \tag{3a}\] \[E_{app}^{\theta}=\beta(R,\gamma,h)E_{cell}, \tag{3b}\]  
where \(E_{app}^{\delta}\) and \(E_{app}^{\theta}\) are the apparent cell rigidity returned by Equation (2a,b), respectively. Since the cell's height \(h\) can be expressed as \(h=h_{u}+R(1-\cos\gamma)\) using \(R\), \(h\), and the under-bead cell thickness \(h_{u}\), some authors pointed out that the mathematical formulations of functions \(\alpha\) and \(\beta\) actually depend on the adimensional factor \(h_{u}/2R\) and \(\gamma\)[73]. In order to quantitatively evaluate the correcting functions, an FEM framework studied the effect of geometrical factors \(h_{u}\), \(R\), and \(\gamma\) on the cellular response induced by an external force during tweezer micromanipulation. The finite element framework considered 300 testing scenarios corresponding to different combinations of the factor's triplet (\(h\), \(R\), \(\gamma\)), thus simulating the corresponding bead motion that can extract the force-bead displacement values useful to determine the scalar value assumed by correcting functions in every testing condition. In the parametric FE framework proposed by [PERSON] et al. [73], tweezer micromanipulation experiments were modeled by considering a deformable cell representative volume element (RVE), which interacts with a rigid spherical bead partially embedded in the cellular  
Figure 5: (**a**) Schematization of the probing bead partially embedded in the outer layer of the cell; (**b**) schematic section view of cell representative volume element (RVE) obtained by cutting the cell’s material with a perpendicular plane to the cell’s top surface passing through the center of the bead partially embedded in the cellular material. The bead is highlighted in gray, and the cell representative volume element is in beige. The bead embedding half angle \(\gamma\) is the angle defined by the vertical dashed line perpendicular to the cell surface and the solid line connecting the bead center with the point where the cell profile meets the cell top surface at the right end of the interaction region. The under-bead cell thickness \(h_{u}\) is related to the cell height h by the formula \(h=h_{u}+R(1-\cos\gamma)\). material (see Figure 5b). The cell RVE has a rectangular parallelepiped shape characterized by a cell length \(l\) equal to \(10R\), a height \(h\), and a width \(w\) equal to \(1/2\).