SUMMARYAn algorithm which couples the level set method (LSM) with the extended ÿnite element method (X-FEM) to model crack growth is described. The level set method is used to represent the crack location, including the location of crack tips. The extended ÿnite element method is used to compute the stress and displacement ÿelds necessary for determining the rate of crack growth. This combined method requires no remeshing as the crack progresses, making the algorithm very e cient. The combination of these methods has a tremendous potential for a wide range of applications. Numerical examples are presented to demonstrate the accuracy of the combined methods.

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.

Mathematical modeling and computational analysis are essential for understanding the dynamics of the complex gene networks that control normal development and homeostasis, and can help to understand how circumvention of that control leads to abnormal outcomes such as cancer. Our objectives here are to discuss the different mechanisms by which the local biochemical and mechanical microenvironment, which is comprised of various signaling molecules, cell types and the extracellular matrix (ECM), affects the progression of potentially-cancerous cells, and to present new results on two aspects of these effects. We first deal with the major processes involved in the progression from a normal cell to a cancerous cell at a level accessible to a general scientific readership, and we then outline a number of mathematical and computational issues that arise in cancer modeling. In Section 2 we present results from a model that deals with the effects of the mechanical properties of the environment on tumor growth, and in Section 3 we report results from a model of the signaling pathways and the tumor microenvironment (TME), and how their interactions affect the development of breast cancer. The results emphasize anew the complexities of the interactions within the TME and their effect on tumor growth, and show that tumor progression is not solely determined by the presence of a clone of mutated immortal cells, but rather that it can be ‘community-controlled’. It Takes a Village – Hilary Clinton

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.

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