Mathematical Analysis of Taxis in Angiogenesis with Application to the Study of Tumor Growth

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Levine 9803992 The investigator and his collaborators develop models of angiogenesis. Angiogenesis is a process whereby capillary sprouts from a pre-existing vasculature are formed in response to externally supplied chemical stimuli. The sprouts, driven by endothelial cell migration and mitosis, develop and organize themselves into a dendritic structure. Angiogenesis has been observed for example during embryogenesis, wound healing, arthritic development and during the growth of solid tumors. This process is thought to occur in three steps: the degradation of the vascular membrane and interstitial matrix by endothelial cells, the migration and proliferation of the latter, and finally tubulogenesis. This project is specifically directed at constructing and analyzing mathematical models of tumor angiogenesis. These models are based on the idea of reinforced random walks combined with nonstandard reaction-diffusion mechanisms to describe how capillary sprouts are formed and how endothelial cells migrate into the interstitial cellular matrix. In particular, the modeling includes chemo-attraction via certain tumor angiogenesis factors emitted from the tumors and haptotaxis via secretion of fibronectin from the endothelial cells. Fundamental objectives of the study are to suggest testable hypotheses for the capillary sprout formation and to understand the mechanisms for anastomosis and the branching of capillaries. The goal is to use the models to suggest strategies whereby capillary sprout growth can be impeded and consequently inhibit the spread of capillary networks. By preventing the capillary dendritic structure from reaching the tumor, the tumor will be starved of a blood supply. Consequently tumor cells will be prevented from metastasizing to other parts of the body. Several years ago, Dr. Judah Folkman of Children's Hospital, Boston, proposed a model for malignant tumor growth. It is known that an avascular tumor (a tumor without a blood sup ply) can only grow to a certain size before it begins to die because of oxygen insufficiency. Folkman suggested that as it dies, it sends out a chemical signal that induces the body to produce a second chemical (called a tumor angiogenic factor, TAF). TAF, in turn, diffuses through the surrounding body tissue to nearby capillary vessels where it creates openings in the capillary walls. The (endothelial) cells that line all capillary walls are then able to leak through these openings and follow the chemical trail left by the TAF back to the tumor, forming new capillaries as they go. These secondary capillaries then deliver oxygen directly to the tumor, enabling it to grow more rapidly than it could in the avascular state. In order to inhibit the tumor growth, it has been suggested that it might be possible to use certain anti-angiogenic factors (antigens) to inhibit the action of the TAF and thus not only starve the tumor by preventing the growth of these secondary capillaries but also prevent the spread of tumor cells to other parts of the body through the circulatory system. The investigator and his collaborators aim to put Folkman's mechanism on a quantitative footing by developing a mathematical model that is both descriptive and predictive. That is, the model will not only aid in the understanding of Folkman's ideas at a fundamental scientific level but can also be used to suggest strategies whereby capillary growth can be impeded and consequently inhibit the spread of capillary networks. The model should ultimately suggest optimal dosages of antigens needed to prevent capillaries from reaching the tumor, thus minimizing risk to the patient as the tumor is deprived of a blood supply. On a more positive note, the same models could perhaps also be used to effectively model a mechanism by which capillary sprout growth can be encouraged, such as in embryo development or wound healing, by isolating those factors that encourage such growth.