Derivation and travelling wave analysis of partial integro-differential equation models for EMT-mediated immunosuppression in cancer

This thesis deals with the derivation and travelling wave analysis of partial integro-differential equation models for EMT-mediated immunosuppression in cancer with phenotypic heterogeneity, including also the dynamics of killer and regulatory immune cells. Experimental evidence shows that mesenchymal cancer cells recruit regulatory cells, which suppress killer cell activity, creating a protective environment for the tumour. On the other hand, epithelial cancer cells experience higher immune activity due to the greater presence of killer cells. Cancer cells could have a phenotypic state ranging from totally epithelial to completely mesenchymal, modelled by the structure variable y. Specifically, a mesenchymal phenotype is characterized by high mobility and low proliferative ability, while an epithelial phenotype prioritizes reproduction over movement, reflecting the biological principle of “go or grow”. Mesenchymal cells also show a high production of chemical attractant, which is responsible for recruiting regulatory cells which migrate due to chemotaxis. On the other hand, killer cells move up the density gradient of cancer cells and their activity is inhibited by regulatory cells. These biological considerations are expressed mathematically by choosing suitable functions to model the phenotype-dependent, mobility, production, and growth of cancer cells, as well as the chemical-dependent growth of regulatory and killer cells. After this, we can define jump probabilities in branching random walks on a lattice in physical and phenotypic space, formulating three coupled discrete agent-based models for the densities of cancer, regulatory and killer cells that are also coupled with a discrete finite difference equation for the concentration of the attractant. Through limit procedures, we formally derive the corresponding continuous model, composed of two partial integro-differential equations (IPDE) for the density of cancer and killer cells, coupled with two partial differential equations for the concentration of the chemical attractant and the density of regulatory cells. Despite the system’s complexity, we show through formal asymptotic techniques that a detailed analysis within the framework of the travelling wave can be performed, obtaining invasion fronts with phenotypic structure. As a preliminary step, we rescale the continuous model with appropriate powers of a small parameter ε, according to the biological relevance of the terms within the equations. Combining WKB ansatz for the density of cancer cells with asymptotic expansions, we derive an equation and a constraint for the new function u as ε → 0. Through several mathematical steps, we obtain a transport equation for ¯y, which is the dominant phenotypic trait at (t, x). Shifting our focus to the traveling wave framework, we investigate the monotonicity of traveling front solutions, the wavefront position and the minimum propagation speed. We conclude that the maximum phenotypic trait ¯y presents a monotonically increasing profile along the wave variable z while the density of cancer cells ρ has a decreasing behaviour. Finally, we can derive the profiles of the chemical concentration and densities of killer and regulatory cells using these results. As expected, mesenchymal cells dominate the wavefront while epithelial cells reside at the rear, following leader cell dynamics. Various scenarios are proposed to investigate how this spatial organisation influences the degree of infiltration of killer cells into the tumor.