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Session S01 - Modeling and Computation for Control and Optimization of Biological and Physical Systems

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Discretization and Variational Formulation of a Continuous Model in PDE of Tumor-induced Angiogenesis

Ana Kristhel Esteban López

Universidad Juárez Autónoma de Tabasco, México   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The term angiogenesis literally means blood vessel formation. The concept that "tumor growth is dependent on angiogenesis" has prompted continued progress in the development of angiogenesis inhibitors toward the goal of future tumor therapy. This hypothesis, which was first proposed in 1971, can be expressed in its simplest terms: once the tumor is "taken", each increase in the tumor cell population must be preceded by an increase in new capillaries that converge towards the tumor. Thus, this work aims to show the discretization and variational formulation of a system of nonlinear partial differential equations, which describes the dynamics of the density of endothelial cells that migrate through a tumor and form neovascular structures in response to a chemical signal specific known as tumor angiogenic factor (TAF). The complete system of equations describing the interactions of endothelial cells, TAF and fibronectin is \begin{eqnarray*} \begin{split} \frac{\partial n}{\partial t} &=D_{n} \nabla^{2}n-\nabla \cdot \left( \frac{\chi_{0}k_1}{k_1+c}n \nabla c\right) -\nabla \cdot \left(\varrho _{0}n \nabla f\right), \\ \frac{\partial f}{\partial t}&= w n-\mu nf,\\ \frac{\partial c}{\partial t}&= -\lambda nc, \end{split} \end{eqnarray*}

where, $n$ is the endothelial cell density at time $t$, $f$ is the fibronectin concentration at time $t$, $c$ is the TAF concentration at time $t$, $D_{n}$ is the cell random-motility coefficient, $\chi_{0}$ is the chemotactic coefficient, $\varrho _{0}$ is the haptotactic coefficient and $k_1$, $\lambda$, $w$ y $\mu$ are positive constants.

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