# Finite Element Modeling of Cells Adhering to a Substrate: An Overview
## 2 Physical Modeling of Cells in Adhesion Phenomena
By exploiting the symmetry plane perpendicular to the cell top surface passing through the bead's center, it was possible to reduce the width of the modeled cell portion from \(1\) to \(1/2\), thus saving computational cost. Force values used in numerical simulations by [PERSON] et al. [73] are consistent with those gathered from experimental works [74, 81] and correspond to an average value of \(50\) pN. Responses of epithelial and endothelial cells to micromanipulation testing were simulated by adopting a continuum mechanics modeling approach, which schematizes the cellular region as a continuum quasi-incompressible homogeneous domain whose material behavior is described by the neo-Hookean hyperelastic strain energy function \(W\) as follows:  
\[W=a(I_{1}-3), \tag{4}\]  
The above mathematical formulation depends on the hyperelastic constant \(a\) and the trace \(I_{1}\) (i.e., the linear invariant) of the right Cauchy-Green strain tensor. According to the neo-Hookean constitutive behavior, the material intrinsic stiffness \(E_{cell}\) can be related to the hyperelastic constant \(a\) via the stretch ratio \(\lambda\) relative to the uniaxial test (i.e., \(\lambda=L/L_{o}\), with \(L\) extended length and \(L_{o}\) initial length) as follows:  
\[E_{cell}=2a\left(2+\lambda^{-3}\right), \tag{5}\]  
Assuming \(\lambda\) values close to \(1\), which corresponds to having a small extension of the cell's material during the uniaxial test, Equation (5) yields \(E_{cell}\cong 6a\). For small extensions, once the external force and the actual cellular stiffness defined by the neo-Hookean law (i.e., \(500\) Pa [82]) were given in input to the model, the FEM framework of [73] was capable of computing bead displacement for all the geometric parameter combinations. Knowing the simulated bead motion derived from each hypothesized testing condition and the applied force, Equation (2) was used to determine the apparent cell stiffness, which represents the corresponding material rigidity analytically calculated during experimental tests. Then, Equation (3) can be rearranged in order to return the scalar value of correction functions \(a\) and \(\beta\) corresponding to each combination (\(h_{u}/2R\),\(\gamma\)). Having at their disposal a set of values consisting of the quantitative estimate of \(\alpha(h_{u}/2R\),\(\gamma)\) and \(\beta(h_{u}/2R\),\(\gamma)\), [PERSON] et al. identified the most appropriate mathematical formulation of the correction functions \(\alpha\) and \(\beta\) able to match the calculated values [73, 79]. The FE framework proposed by [PERSON] et al. [80] successfully described the adhesion response of epithelial cells to the external stimulus induced by magnetic twisting cytometry (MTC). Cells probed by MTC are subjected to large deformation with respect to tweezer micromanipulation, as the MTC probing technique induces a higher rotation of the ferromagnetic bead. The cellular structure was schematized by [PERSON] et al. by adopting a continuum mechanics-derived homogeneous representation characterized by the neo-Hookean hyperelastic constitutive behavior. The proposed FE model allowed to quantify the intrinsic Young's modulus of the cell by providing values of correction functions for the apparent cell stiffness, which depend on the dimensionless geometric factor \(h_{u}/2\)R.