# Finite Element Modeling of Cells Adhering to a Substrate: An Overview
## 2 Physical Modeling of Cells in Adhesion Phenomena
Under these hypotheses, the cell's spreading on the substrate leads to an increasing contact area, which can be approximately considered a circular area. Hence, the physical model developed in [71] describes at the macroscopic level the overall change in cellular morphology during adhesion by reproducing the increase in contact area experimentally observed for fibroblast spreading on glass substrates [70; 72]. Localized changes in the cell profile involving curvature variations were neglected, and two geometrical parameters were used to define at the macroscopic level the cell morphologies resembling spherical caps: the contact radius \(R\) and the contact angle \(\theta\) (Figure 3). The contact radius \(R\) is defined by the diameter of the contact area identified by the planar interaction region between the cell and the underlying substrate. The contact angle \(\theta\) is identified by the plane tangent to the substrate's top surface and the plane tangent to the cell's outer profile passing through a point that in the wetting theory is called a three-phase contact point (i.e., the point where the cell's free profile of meets substrate). In [72], contact radius \(R\) was estimated using the cell top view, from which the adhesion region can be evaluated by the projection of the cell surface area onto the substrate. Finally, the contact radius of the adhesion area was found as the root square of the projected cell area \(A_{p}\) divided by \(\pi\) (i.e., \(R=\sqrt{(A_{p}/\pi)}\)). The model developed by [PERSON] et al. [71] allows time progress curves of the contact radius to be plotted as the ratio between the parameters \(w_{\text{ad}}\) and \(\tau\) changes. By fitting the physical model to experimental data, the theoretical curve that best reproduces contact radius experimental values measured during spreading was obtained, thus estimating adhesion energy \(w_{\text{ad}}\) and cortical tension \(\tau\) entailed by cell-substrate adhesion mechanism. Modeling approaches based on continuum mechanics can be successfully used to analyze the mechanical response of cells characterized by the presence of a structured cytoskeletal architecture. Epithelial cells belong to this category and exhibit an elasticity-dominated mechanical response. In order to develop a simplified FE framework to mimic the adhesion behavior of airway epithelial cells, [PERSON] et al. [69] implemented a plane strain 2D homogeneous continuum linear elastic model simulating the cell's response to an external stimulus generated by magnetic tweezer micromanipulation and the corresponding induced stress-strain field. In the proposed FE framework, the external mechanical stimulus is represented by a force of 100 pN/\(\upmu\)m acting in the Y direction (i.e., the parallel direction to the substrate on which the cell adheres) applied to the center of a rigid bead with a 0.4 \(\upmu\)m diameter initially attached to the apical cellular region. Under the action of the external force, the bead is subjected to translation and rotation movements towards the right, while staying adherent to the cell surface. This generates a stress-strain field in the cell. The highest effective stress computed by the numerical framework (i.e., about 5 Pa) is localized at the left end of the contact interface between the bead and cell surface. Strains in the Y direction vary from 5% to 1% passing from the left to the right end of the contact region. Since increasing membrane stiffness too much would have led to an overestimated  
Figure 3: Side view of the cell that adheres and spreads on the substrate. The cell is schematized as a region consisting of a high-viscosity internal fluid domain (colored light blue) enveloped by the cortex (colored dark blue), which carries the cortical tension \(\tau\). The cell schematization in (**a**) refers to the initial phases of the adhesion process (i.e., after a few minutes). Cell schematization reported in (**b**) refers to the final phases of the adhesion process (i.e., after hours).