Mathematical modeling and numerical simulation of tumor angiogenesis

Resumo

Cancer is nowadays the second leading cause of death in the world. Cancer starts with a single cell that has accumulated several mutations usually over a long period of time. One of the main characteristics of this cell is its ability to replicate unbounded producing, in most occasions, a mass of densely packed daughter cells that form a solid tumor. Cancerous cells depend on nutrients and oxygen supplied by pre-existent blood vessels to proliferate. This supply is not enough to maintain the growth and most tumors remain small and benign due to this constraint. Occasionally, however, some cells develop the ability to promote the growth of new capillaries towards them. This process is called tumor-induced angiogenesis and through it tumors acquire a constant supply of nutrients and oxygen and access to the whole body through the circulatory system. Tumors that are able to trigger angiogenesis become malignant as they can grow unbounded and metastasize. Malignant tumors can grow large enough to damage the functionality of the host organ and, eventually, lead to death. A few decades ago, scientists have proposed that blocking angiogenesis could be an effective treatment against cancer. This therapy, known as antiangiogenic therapy, has shown promising results in pre-clinical trials, but has not translated into the expected results in the clinic. A new emerging paradigm in Medicine, namely, Predictive Medicine, is expected to change the Oncology field and may be the tool to understand the problems with this therapy. Predictive Medicine is based on mathematical modeling and computation and has been so far successfully applied to several areas of Medicine. In the Oncology field it has only been started, but has already produced sound advances. In particular, in tumor-induced angiogenesis numerous researchers have proposed mathematical models to address the problem of antiangiogenic therapies, although there is still a need for improvement to attain this goal. Angiogenesis is a complex multiscale process that involves several key mechanisms, many of them not yet accounted for in the models. In addition, most models are simulated in two-dimensional simple geometries, while angiogenesis is a three-dimensional process and occurs in tissues with non-trivial geometries. In this thesis we develop mathematical models of tumor-induced angiogenesis that include key biological mechanisms and we simulate these models in relevant experimental setups and in three-dimensional, subject-specific geometries. To achieve our modeling and simulation goals, we develop adequate numerical algorithms. Every model in this thesis is hybrid and involves coupling averaged continuous theories and cellular-scale discrete agents. In addition, the models are grounded on the phase-field method which requires solving higher-order partial differential equations. To overcome these problems we developed a seamless coupling between the continuous variables and the discrete elements to permit an efficient numerical treatment of the coupled problem. For the resolution of the high-order partial differential equations involved in the formulation we used isogeometric analysis, which in addition provides accuracy, robustness, and the geometric flexibility that we require to perform simulations in real geometries. This manuscript presents three new mathematical models for tumor-induced angiogenesis. In the first one we started with a previous model to which we added a conceptual model for haptotaxis, one of the main mechanisms that governs capillary growth. The extended model permitted us to assess the role of haptotaxis in tumor angiogenesis. Furthermore, we developed for the first time a simulation of one of the most widely used in vivo assays, the mouse corneal micropocket angiogenesis assay, using a three-dimensional, subject-specific geometry. The results are in agreement with the experiments and predict well-known vascular structures in three dimensions. In addition, they suggest that, for mathematical models to achieve the topological complexity observed in in vivo angiogenesis experiments, two-dimensional simulations may not be enough. The second model focuses on the long-term dynamics of tumor-induced angiogenesis, that is, on the regression and regrowth events occurring after the first growth of blood vessels. The model was simulated both in a two-dimensional replication of the mouse corneal micropocket assay and in a reproduction of an in vivo experimental setup. Our simulations predict plasticity and dynamic evolution of angiogenesis at long time spans and are in quantitative agreement with experiments. Finally, we developed a fully continuum theory for fluid flow at the tissue scale, which we coupled with our model for tumor angiogenesis. The model shows how fluid flow alters tumor-induced vascular patterns through convection, which has been overlooked in mathematical modeling. Our model predicts a substantial impact of convection in angiogenesis and an increased malignancy of small solid tumors.