Magdalena Stolarska

A HYBRID MODEL FOR TUMOR SPHEROID GROWTH IN VITRO I: THEORETICAL DEVELOPMENT AND EARLY RESULTS

Tumor spheroids grown in vitro have been widely used as models of in vivo tumor growth because they display many of the characteristics of in vivo growth, including the effects of nutrient limitations and perhaps the effect of stress on growth. In either case there are numerous biochemical and biophysical processes involved whose interactions can only be understood via a detailed mathematical model. Previous models have focused on either a continuum description or a cell-based description, but both have limitations. In this paper we propose a new mathematical model of tumor spheroid growth that incorporates both continuum and cell-level descriptions, and thereby retains the advantages of each while circumventing some of their disadvantages. In this model the cell-based description is used in the region where the majority of growth and cell division occurs, at the periphery of a tumor, while a continuum description is used for the quiescent and necrotic zones of the tumor and for the extracellular matrix. Reaction-diffusion equations describe the transport and consumption of two important nutrients, oxygen and glucose, throughout the entire domain. The cell-based component of this hybrid model allows us to examine the effects of cell–cell adhesion and variable growth rates at the cellular level rather than at the continuum level. We show that the model can predict a number of cellular behaviors that have been observed experimentally.

Multi-scale models of cell and tissue dynamics.

Cell and tissue movement are essential processes at various stages in the life cycle of most organisms. The early development of multi-cellular organisms involves individual and collective cell movement; leukocytes must migrate towards sites of infection as part of the immune response; and in cancer, directed movement is involved in invasion and metastasis. The forces needed to drive movement arise from actin polymerization, molecular motors and other processes, but understanding the cell- or tissue-level organization of these processes that is needed to produce the forces necessary for directed movement at the appropriate point in the cell or tissue is a major challenge. In this paper, we present three models that deal with the mechanics of cells and tissues: a model of an arbitrarily deformable single cell, a discrete model of the onset of tumour growth in which each cell is treated individually, and a hybrid continuum–discrete model of the later stages of tumour growth. While the models are different in scope, their underlying mechanical and mathematical principles are similar and can be applied to a variety of biological systems.

Center or periphery? Modeling the effects of focal adhesion placement during cell spreading

Focal adhesions are often observed at the cell’s periphery. We provide an explanation for this observation using a system-level mathematical model of a cell interacting with a two-dimensional substrate. The model describes the biological cell as a hypoelastic continuum material whose behavior is coupled to a deformable, linear elastic substrate via focal adhesions that are represented by collections of linear elastic attachments between the cell and the substrate. The evolution of the focal adhesions is coupled to local intracellular stresses which arise from mechanical cell-substrate interactions. Using this model we show that the cell has at least three mechanisms through which it can control its intracellular stresses: focal adhesion position, size, and attachment strength. We also propose that one reason why focal adhesions are typically located on the cell periphery instead of its center is because peripheral focal adhesions allow the cell to be more sensitive to changes in the microenvironment. This increased sensitivity is caused by the fact that peripherally located focal adhesions allow the cells to modulate its intracellular properties over a much larger portion of the cell area.