Mathematical Modeling of Spherical Shell-Type Pattern of Tumor Invasion

Abstract

Cancer cell migration, as the principal element of tumor invasion, involves different cellular mechanisms. Various modes of cell migration including single and collective motions contribute to the invasion patterns. The competition between adhesive cell–cell and cell–matrix forces is a key factor that determines such patterns. In this paper, we study a distinct shell-type mode of tumor invasion observed in brain and breast tumors. In this mode, cells at the outer layer of the tumor collectively move away from the core and form a shell-type shape. Both the core and the shell sustain a sharp interface between cells and the surrounding matrix. To model the preserved interface, we adopted a Cahn–Hilliard-type free energy relation with the contribution of the interfacial stress. This nonconvex form of free energy allows for cells to remain together and preserve the tumor core via adhesive cell–cell forces while separating the core from the surrounding matrix across a continuous sharp interface. In addition, the motion of the shell was modeled using the chemotactic migration of cells in response to the gradient of nutrients. The associated fluxes of cells were implemented in a general form of balance law. A non-Michaelis–Menten kinetics model was adopted for the proliferation rate of cells. The flux of nutrients was also modeled using a simple diffusion equation. The comparison between the model predictions and experimental observations indicates the ability of the model to manifest the salient features of the invasion pattern.

3D-Printed Tumor-on-a-Chip Model for Investigating the Effect of Matrix Stiffness on Glioblastoma Tumor Invasion

“…The generation of cell flux primarily occurs through the interplay of diffusive and adhesive forces, as well as the directional movement resulting from chemotaxis. We adopted a nonconvex form of free energy and a standard chemotactic model presented in [ 40 ] to model the collective motion of cells ( Figure 3 a). Additionally, the motion of single invasive cells, including random and directional motions, was predicted by a discrete model ( Figure 3 b).…”

“…Multiple modes of cell migration, including ual and collective motions, contribute to the overall patterns of invasion. These migration can potentially give rise to finger-type and ring-type invasion patterns tively [39,40]. The former accounts for the migration of single cells into the surro tissue and the latter corresponds to the collective motion of cells in the form of a single cell motion, cells gain the ability to move by adopting either a mesench amoeboid morphology [41].…”

Unveiling the role of metabolic rates variation in driving tumor heterogeneity and controlling growth and invasion: insights from an integrated multiscale in-vitro in-silico framework

“…Volumetric tumor growth is limited due to environmental restrictions, resulting in different patterns of invasion. Various stimuli such as mechanical stress, hypoxia, chemoattractants, and nutrient gradients can trigger such invasion [47]. For instance, the expression of genes that induce matrix degradative enzymes (MDE), metalloproteinases (MMP), and epithelial-mesenchymaltransition (EMP) proteins can be upregulated in hypoxia (Fig.…”