Mathematical Modeling Of Tumor Invasion And Treatment

Abstract

Using mathematical models to simulate dynamic biological processes has a long history. Over the past couple of decades or so, many approaches have also made their way into cancer research. An increasing number of mathematical, physical, computational and engineering techniques have been applied to various aspects of tumor growth, with the ultimate goal of understanding the response of the cancer population to clinical intervention. Herein I present mathematical model which describe the invasion of host tissue by tumor cells and treatment. In the models, I focus on three key variables implicated in the invasion process and treatment; namely, tumor cells denoted by c, host tissue or extracellular matrix denoted by v and urokinase plasminogen activator (uPA) protease secreted by tumor cells and denoted by u. This model focuses on the macro-scale structure (cell population level) and considers the tumor as a single mass. The mathematical model consists of a system of partial differential equations describing the production and/or activation of degradative enzymes by the tumor cells, the degradation of the matrix and the migratory response of the tumor cells. I describe the stability of the equilibrium points which are locally and globally stable. As rate of proliferation tumor cell increases the rate of production of uPA also increases. After treatment this rate decreases and the tumor is in quiescent state. This is shown by mathematical simulation at the last.