MBI Publications

MBI Publications for Hybrid Model (2)

  • Y. Kim and H. Othmer
    A hybrid model for tumor spheroid grown in vitro I: Theoretical development and early results
    Math. Models Methods in Appl ScisVol. 17 (2007) pp. 1773-1798


    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.
  • Y. Kim, M. Stolarska and H. Othmer
    The Role of the Microenvironment in Tumor Growth and Invasion
    Progress in Biophysics and Molecular BiologyVol. 106 (2011) pp. 353-379 (Submitted)


    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 under- stand 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 â??community-controlledâ??.

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