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In the last decade along with classical cytotoxic agents, antiangiogenic drugs have been actively used in cancer chemotherapy. They are not aimed at killing malignant cells, but at blocking the process of angiogenesis, i.e., the growth of new vessels in the tumor and its surrounding tissues. Agents that stimulate angiogenesis, in particular, vascular endothelial growth factor, are actively produced by tumor cells in the state of metabolic stress. It is believed that blocking of tumor neovascularization should lead to a shortage of nutrients flow to the tumor, and thus can stop, or at least significantly slow down its growth. Clinical practice on the use of first antiangiogenic drug bevacizumab has shown that in some cases such therapy does not influence the growth rate of the tumor, whereas for other types of malignant neoplasms antiangiogenic therapy has a high antitumor effect. However, it has been shown that along with successful slowing of tumor growth, therapy with bevacizumab can induce directed tumor progression to a more invasive, and therefore more lethal, type. These data require theoretical analysis and rationale for the evolutionary factors that lead to the observation of epithelial-mesenchymal transition. For this purpose we have developed a spatially distributed mathematical model of growth and antiangiogenic therapy of heterogeneous tumor consisting of two subpopulations of malignant cells. One of subpopulations possesses inherent characteristics of epithelial phenotype, i.e., low motility and high proliferation rate, the other one corresponds to mesenchymal phenotype having high motility and low proliferation rate. We have performed the investigation of competition between these subpopulations of heterogeneous tumor in the cases of tumor growth without therapy and under bevacizumab monotherapy. It is shown that constant use of antiangiogenic drug leads to an increase of the region in parameter space, where the dominance of mesenchymal phenotype takes place, i.e., within a certain range of parameters in the absence of therapy epithelial phenotype is dominant but during bevacizumab administration mesenchymal phenotype begins to dominate. This result provides a theoretical basis of the clinically observed directed tumor progression to more invasive type under antiangiogenic therapy.

In this paper, we proposed a two-dimensional chemo-mechanical model of the growth of invasive carcinoma in epithelial tissue. Each cell is modeled by an elastic polygon, changing its shape and size under the influence of pressure forces acting from the tissue. The average size and shape of the cells have been calibrated on the basis of experimental data. The model allows to describe the dynamic deformations in epithelial tissue as a collective evolution of cells interacting through the exchange of mechanical and chemical signals. The general direction of tumor growth is controlled by a pre-established linear gradient of nutrient concentration. Growth and deformation of the tissue occurs due to the mechanisms of cell division and intercalation. We assume that carcinoma has a heterogeneous structure made up of cells of different phenotypes that perform various functions in the tumor. The main parameter that determines the phenotype of a cell is the degree of its adhesion to the adjacent cells. Three main phenotypes of cancer cells are distinguished: the epithelial (E) phenotype is represented by internal tumor cells, the mesenchymal (M) phenotype is represented by single cells and the intermediate phenotype is represented by the frontal tumor cells. We assume also that the phenotype of each cell under certain conditions can change dynamically due to epithelial-mesenchymal (EM) and inverse (ME) transitions. As for normal cells, we define the main E-phenotype, which is represented by ordinary cells with strong adhesion to each other. In addition, the normal cells that are adjacent to the tumor undergo a forced EM-transition and form an M-phenotype of healthy cells. Numerical simulations have shown that, depending on the values of the control parameters as well as a combination of possible phenotypes of healthy and cancer cells, the evolution of the tumor can result in a variety of cancer structures reflecting the self-organization of tumor cells of different phenotypes. We compare the structures obtained numerically with the morphological structures revealed in clinical studies of breast carcinoma: trabecular, solid, tubular, alveolar and discrete tumor structures with ameboid migration. The possible scenario of morphogenesis for each structure is discussed. We describe also the metastatic process during which a single cancer cell of ameboid phenotype moves due to intercalation in healthy epithelial tissue, then divides and undergoes a ME transition with the appearance of a secondary tumor.

Chemotaxis plays an important role in morphogenesis and processes of structure formation in nature. Both unicellular organisms and single cells in tissue demonstrate this property. In vitro experiments show that many types of transformed cell, especially metastatic competent, are capable for directed motion in response usually to chemical signal. There is a number of theoretical papers on mathematical modeling of tumour growth and invasion using Keller-Segel model for the chemotactic motility of cancer cells. One of the crucial questions for using the chemotactic term in modelling of tumour growth is a lack of reliable quantitative estimation of its parameters. The 2-D mathematical model of tumour growth and invasion, which takes into account only random cell motility and convective fluxes in compact tissue, has showed that due to competitive mechanism tumour can grow toward sources of nutrients in absence of chemotactic cell motility.

A mathematical model of tumor growth in tissue taking into account angiogenesis and antiangiogenic therapy is developed. In the model the convective flows in tissue are considered as well as individual motility of tumor cells. It is considered that a cell starts to migrate if the nutrient concentration falls lower than the critical level and returns into proliferation in the region with high nutrient concentration. Malignant cells in the state of metabolic stress produce vascular endothelial growth factor (VEGF), stimulating tumor angiogenesis, which increases the nutrient supply. In this work an antiangiogenic drug which bounds irreversibly to VEGF, converting it to inactive form, is modeled. Numerical analysis of influence of antiangiogenic drug concentration and efficiency on tumor rate of growth and structure is performed. It is shown that antiangiogenic therapy can decrease the growth of low-invasive tumor, but is not able to stop it completely.