Integrative mathematical oncology

The role of extracellular matrix in glioma invasion: a cellular Potts model approach

In this work, a cellular Potts model based on the differential adhesion hypothesis is employed to analyze the relative importance of select cell-cell and cell-extracellular matrix (ECM) contacts in glioma invasion. To perform these simulations, three types of cells and two ECM components are included. The inclusion of explicit ECM with an inhomogeneous fibrous component and a homogeneously dispersed afibrous component allows exploration of the importance of relative energies of cell-cell and cell-ECM contacts in a variety of environments relevant to in vitro and in vivo experimental investigations of glioma invasion. Simulations performed here focus chiefly on reproducing findings of in vitro experiments on glioma spheroids embedded in collagen I gels. For a given range and set ordering of energies associated with key cell-cell and cell-ECM interactions, our model qualitatively reproduces the dispersed glioma invasion patterns found for most glioma cell lines embedded as spheroids in collagen I gels of moderate concentration. In our model, we find that invasion is maximized at intermediate collagen concentrations, as occurs experimentally. This effect is seen more strongly in model gels composed of short collagen fibers than in those composed of long fibers, which retain significant connectivity even at low density. Additional simulations in aligned model matrices further elucidate how matrix structure dictates invasive patterns. Finally, simulations that allow invading cells to both dissolve and deposit ECM components demonstrate how Q-Potts models may be elaborated to allow active cell alteration of their surroundings. The model employed here provides a quantitative framework with which to bound the relative values of cell-cell and cell-ECM interactions and investigate how varying the magnitude and type of these interactions, as well as ECM structure, could potentially curtail glioma invasion.

A fully continuous individual-based model of tumor cell evolution

The aim of this work is to develop and study a fully continuous individual-based model (IBM) for cancer tumor invasion into a spatial environment of surrounding tissue. The IBM improves previous spatially discrete models, because it is continuous in all variables (including spatial variables), and thus not constrained to lattice frameworks. The IBM includes four types of individual elements: tumor cells, extracellular macromolecules (MM), a matrix degradative enzyme (MDE), and oxygen. The algorithm underlying the IBM is based on the dynamic interaction of these four elements in the spatial environment, with special consideration of mutation phenotypes. A set of stochastic differential equations is formulated to describe the evolution of the IBM in an equivalent way. The IBM is scaled up to a system of partial differential equations (PDE) representing the limiting behavior of the IBM as the number of cells and molecules approaches infinity. Both models (IBM and PDE) are numerically simulated with two kinds of initial conditions: homogeneous MM distribution and heterogeneous MM distribution. With both kinds of initial MM distributions spatial fingering patterns appear in the tumor growth. The output of both simulations is quite similar.

Invasion emerges from cancer cell adaptation to competitive microenvironments: quantitative predictions from multiscale mathematical models

In this review we summarize our recent efforts using mathematical modeling and computation to simulate cancer invasion, with a special emphasis on the tumor microenvironment. We consider cancer progression as a complex multiscale process and approach it with three single-cell-based mathematical models that examine the interactions between tumor microenvironment and cancer cells at several scales. The models exploit distinct mathematical and computational techniques, yet they share core elements and can be compared and/or related to each other. The overall aim of using mathematical models is to uncover the fundamental mechanisms that lend cancer progression its direction towards invasion and metastasis. The models effectively simulate various modes of cancer cell adaptation to the microenvironment in a growing tumor. All three point to a general mechanism underlying cancer invasion: competition for adaptation between distinct cancer cell phenotypes, driven by a tumor microenvironment with scarce resources. These theoretical predictions pose an intriguing experimental challenge: test the hypothesis that invasion is an emergent property of cancer cell populations adapting to selective microenvironment pressure, rather than culmination of cancer progression producing cells with the "invasive phenotype". In broader terms, we propose that fundamental insights into cancer can be achieved by experimentation interacting with theoretical frameworks provided by computational and mathematical modeling.