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In this survey article, a variety of systems modeling tumor growth are discussed. In accordance with the hallmarks of cancer, the described models incorporate the primary characteristics of cancer evolution. Specifically, we focus on…
This paper concerns a model for tumor cell migration through the surrounding extracellular matrix by considering mass balance phenomena involving the chemical interactions produced on the cell surface. The well-posedness of this model is…
The hydrodynamic theory of active nematics has been often used to describe the spatio-temporal dynamics of cell flows and motile topological defects within soft confluent tissues. Those theories, however, often rely on the assumption that…
Intra-operative identification of malignant versus benign or healthy tissue is a major challenge in fluorescence guided cancer surgery. We propose a perfusion quantification method for computer-aided interpretation of subtle differences in…
In the field of biomedicine magnetic beads are used for drug delivery and to treat hyperthermia. Here we propose to use self-organized bead structures to isolate circulating tumor cells using lab-on-chip technologies. Typically blood flows…
We present a theoretical framework based on an extension of dynamical density functional theory (DDFT) for describing the structure and dynamics of cells in living tissues and tumours. DDFT is a microscopic statistical mechanical theory for…
In this paper we study the rigorous sharp interface limit of a diffuse interface model related to the dynamics of tumor growth, when a parameter $\epsilon$, representing the interface thickness between the tumorous and non tumorous cells,…
We conduct a theoretical study of a two-phase-fluid-structure interaction problem in which air is driven at constant volume flux into a liquid-filled Hele-Shaw channel whose upper boundary is an elastic sheet. A depth-averaged model in the…
Theoretical and computational tools that can be used in the clinic to predict neoplastic progression and propose individualized optimal treatment strategies to control cancer growth is desired. To develop such a predictive model, one must…
Biological tissues have been observed to display emergent fluid-like properties, owing to physical interactions between cells. However, it remains unclear in general how these fluid-like properties affect tissue structure and function.…
Viscous fingering occurs in the flow of two immiscible, viscous fluids between the plates of a Hele-Shaw cell. Due to pressure gradients or gravity, the initially planar interface separating the two fluids undergoes a Saffman-Taylor…
Primary tumors infrequently lead to demise of cancer patients; instead, mortality and a significant degree of morbidity result from the growth of secondary tumors in distant organs (metastasis). It is well-known that malignant tumors induce…
It is well established that the mechanical environment influences cell functions in health and disease. Here, we address how the mechanical environment influences tumor growth, in particular, the shape of solid tumors. In an in vitro tumor…
Angiogenesis is a key process in the tumoral growth which allows the cancerous tissue to impact on its vasculature in order to improve the nutrient's supply and the metastatic process. In this paper, we introduce a model for the density of…
Cell migration is important in many biological processes, including embryonic development, cancer metastasis, and wound healing. In these tissues, a cell's motion is often strongly constrained by its neighbors, leading to glassy dynamics.…
We consider the steady-state fingering instability of an elastic membrane separating two fluids of different density under external pressure in a rotating Hele-Shaw cell. Both inextensible and highly extensible membranes are considered, and…
Epithelia form protective barriers in multicellular organisms. To maintain homeostasis, they must be able to regenerate and heal damaged areas. This occurs through collective cell migration, during which finger-like protrusions commonly…
A theoretical model based on the molecular interactions between a growing tumor and a dynamically evolving blood vessel network describes the transformation of the regular vasculature in normal tissues into a highly inhomogeneous tumor…
This paper provides a unified mathematical analysis of a family of non-local diffuse interface models for tumor growth describing evolutions driven by long-range interactions. These integro-partial differential equations model cell-to-cell…
Unstable dynamics characterizes the evolution of most solid tumors. Because of an increased failure of maintaining genome integrity, a cumulative increase in the levels of gene mutation and loss is observed. Previous work suggests that…