Finite Element Modeling of Biological Systems

Mechanical properties have a decisive role in the fundamental functions of biological systems, including migration of cells, cell apoptosis, and proliferation of cells and bacteria. This is also true for cancer metastasis and morphogenetic processes during embryonic development. It isn’t easy, however, to study biological systems due to their complex behavior, such as their activity and nonlinear material properties. Note that while the individual mechanical properties of specific biological systems, such as biopolymers, have been well established, the collective behavior of these elements has a different response, as the comparative studies of the mechanical properties of single cancer cells and cancerous tissue demonstrate. Thus, numerous experimental instruments have been developed over the years to investigate biological systems’ mechanical properties, individually or collectively. These experimental techniques can evaluate mechanical properties at multiple scales. Theycan target individual biological entities, like single cells, or assess the collective mechanical properties of more complex biological systems, such as tissues or organoids. The resolution of these studies ranges from single-cell analyses to those concerning embryonic morphogenesis. Simulating biological systems’ individual or collective behavior using a discretized approach (i.e., Molecular Dynamics) or a continuum approach (i.e., Finite Element) is an adjunct to experimental studies. This thesis explores the collective behavior observed in individual cells and embryonic tissue. This exploration was carried out through the development of experimental protocols and the application of continuum mechanics models. In the initial two chapters of this thesis, we delve into the fundamental mechanical concepts essential for understanding the mechanical properties of cells and tissues. We also discuss prior studies that employed shell mechanics to model cellular and embryonic deformations. In the third chapter, we detail our collaborative work with Dr. Samaneh Rezvani focuses on the role of the actin cortex in the deformation of individual suspended spherical cells. For this purpose, we utilized double-trap optical tweezers in conjunction with a viscoelastic pressurized-thick-shell model. Using our simulation approach, we determined the mechanical properties of the actin cortex from the experimental results. The elastic shear modulus of the actin cortex ranged between 4.5 kPa and 7.5 kPa. In modeling the steady deformation of single cells with the shell model, we observed that cell volume remains conserved during deformation. Instead of reducing volume, cells extend the actin cortex to accommodate the increased surface area. We also introduced a multilayer viscoelastic shell model to examine the time-dependent mechanical behaviors of cells, focusing on hysteresis due to dissipative processes. Our model incorporated a fluid core within a viscoelastic shell, offering a more thorough understanding of cell mechanics. Our findings indicate that the damping response in cells is predominantly influenced by the viscosity of the cytosol rather than that of the actin cortex. The fourth chapter describes the modeling of experiments conducted by Dr. Renata Garces on gram-negative E. coli bacteria uniaxially compressed between parallel plates. We used Finite Element Modeling (FEM) to examine the collective mechanical behavior of the peptidoglycan layer (PG) in the bacterial cell wall, modeled as a thin, pressurized rod-shaped shell. Finally, in chapter five, we investigated, in collaboration with Dr. Chonglin Guan, the cells’ collective behavior in epithelial tissue during dorsal closure (DC) in developing Drosophila melanogaster embryos (DME). Utilizing glass microprobes, we deformed various tissue types, specifically amnioserosa (AS) and lateral epidermis (LE), and subsequently recorded their responses to assess the impact of tissue mechanical properties on embryonic development. We simulated a viscoelastic flat shell, replicating the geometry of individual embryos, using the Finite Element Method (FEM) to model tissue deformations. Through this methodology, we quantified the mechanical characteristics of amnioserosa and lateral epidermis, encompassing both their viscosity and elasticity. Our analyses determined the elasticity of AS to be approximately (110 to 180 kPa) and its viscosity to be (0.86 to 1.05 Pa.s). Additionally, we executed step-function experiments to ascertain tissue mechanical properties and evaluate tissue relaxation time. Our findings are in line with our previous results obtained from hysteresis studies.