# Finite Element Modeling of Cells Adhering to a Substrate: An Overview
## 3 Discussion
[73] pointed out that Equation (3a,b) can also be extended to the case of cells probed at different load frequency values \(f\) by replacing the scalar moduli, \(E_{cell}\) and \(E_{app}\), with the complex moduli, \(\tilde{E}_{cell}(f)\) and \(\tilde{E}_{app}(f)\), and maintaining the unchanged \(\alpha\) and \(\beta\) formulations that depend exclusively on geometrical parameters \(h_{u}\), \(R\), and \(\gamma\). By employing FE analysis, [PERSON] et al. [73] simulated several testing conditions occurring in optical and magnetic tweezer experiments. They defined correction functions that allow the establishment of a relationship between the intrinsic cell Young's modulus and the measured apparent cellular stiffness. Other authors have pointed out that adopting a simplified description of the cell's response to micromanipulation may introduce artifacts with respect to the actual cell-bead interaction mechanisms, thus leading to an inaccurate simulation of cell behavior. In this particular FE framework, beads directly interact with the membrane, which mainly sustains the applied load by redistributing tension across it [69]. This results in a limited transfer of stress from the bead-membrane contact interface towards the neighboring cytoskeletal region. Therefore, in the FE framework, a cell's capability to resist deformation is driven by the membrane's mechanical properties and is less affected by the cytoskeleton, which has to carry a lower load fraction with respect to the membrane. The leading role of the cellular membrane in the overall cell's mechanical response was highlighted by showing that reducing membrane stiffness by 50% leads to an increase by one order of magnitude in both the cell's maximum effective stress and strain in the Y direction. Conversely, the actual physical interaction mechanism involves bead binding to specific receptor (integrin) sites, which span the cell membrane and link to the cytoskeletal load-bearing structure. The presence of these preferential connecting loci between beads and intracellular supporting architecture allows us to exert the external load almost completely on the cytoskeletal network, unlike what was assumed in the FE framework developed by [PERSON] et al. [69] where the mechanical resistance exhibited by the cell is strongly determined by membranes. Therefore, the higher stiffness exhibited by the cell surface in response to a simulated mechanical load acting in correspondence with receptor sites results from the linking of receptor sites with the underlying intracellular load-bearing architecture, which opposes external stimulus [86]. When the simulated mechanical load is exerted on a receptor site neighborhood at a certain distance from the receptor site, a drop in cell surface stiffness can be observed due to the absence of these preferential gateways (i.e., receptor sites) through which an external mechanical stimulus can be canalized towards cytoskeleton load-bearing architecture, which mainly concurs to determine the cell resistance to deformation. The same approximation in the bead-cell surface interaction mechanism can also be found in more complex 3D finite element frameworks developed for studying cell stiffness in response to micromanipulation techniques involving probing beads in contact with the cell surface [73]. The main reason for simplifying cell-bead interaction, by distributing the stress generated by bead-cell membrane surface contact across the outer layers of cellular material, is to circumvent the complex FE modeling of the bead-integrin-cytoskeletal filaments interlinking chain. As a summary for the reader, Table 1 lists the most representative studies surveyed in this review. The studies are categorized according to the types of cells analyzed.