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Modeling and simulation of a low grade urinary bladder carcinoma

In this work, we present a mathematical model of the initiation and progression of a low-grade urinary bladder carcinoma. We simulate the crucial processes involved in tumor growth, such as oxygen diffusion, carcinogen penetration, and angiogenesis, within the framework of the urothelial cell dynamics. The cell dynamics are modeled using the discrete technique of Cellular Automata, while the continuous processes of carcinogen penetration and oxygen diffusion are described by nonlinear diffusion-absorption equations. As the availability of oxygen is necessary for tumor progression, processes of oxygen transport to the tumor growth site seem most important. Our model yields a theoretical insight into the main stages of development and growth of urinary bladder carcinoma with emphasis on two most common types: bladder polyps and carcinoma {\it in situ}. Analysis of histological structure of bladder tumor is important to avoid misdiagnosis and wrong treatment and we expect our model to be a valuable tool in the prediction of tumor grade and progression patterns, based on the exposure to carcinogens and an oxygen dependent expression of genes promoting tumor growth. Our numerical simulations have good qualitative agreement with {\it in vivo} results reported in the corresponding medical literature.