Allen-Cahn Equation for Modeling Temporal Evolution of Non-Conserved Field Variables in Cancer Cell Migration

This work explores the temporal evolution of non-conserved field variables through the application of the Allen-Cahn equation. The equation forms the basis for various phase-field models used in cell migration studies, particularly in the context of tumor cells and cancer metastasis. The model portrays cells as 2D soft bodies, integrating mechanical and biological aspects to simulate cell movement. The investigation delves into the mathematical representation of cell migration, vital in understanding cancer development and metastasis. The model employs an order parameter to characterize each cell, representing their presence within a cell cluster. By minimizing a specific free energy functional, the equilibrium shape of the soft cell bodies is determined, incorporating parameters that influence elasticity and energetic costs. Additionally, the interaction between cells is incorporated, contributing to a comprehensive portrayal of cell migration. The study yields insights into the complex dynamics of cell migration, enhancing our comprehension of biological processes and potentially informing cancer research strategies.