Numerical analysis of a chemotaxis model for tumor invasion.

This paper is devoted to the study of a time-discrete scheme and its corresponding fully discretization approximating a d-dimensional chemotaxis model describing tumor invasion, d ≤ 3. This model describes the chemotactic attraction experienced by the tumor cells and induced by a so-called active extracellular matrix, which is a chemical signal produced by a biological reaction between the extracellular matrix and a matrix-degrading enzyme. In order to construct the numerical approximations and to control the chemo-attraction term in the tumor cells equation, we introduce an equivalent model with a new variable given by the gradient of the active extracellular matrix and use an inductive strategy. Then, we consider a first-order and non-linear time-discrete scheme which is mass-conservative and possesses the property of positivity for all the biological variables. After, we study the corresponding fully discrete finite element with “mass-lumping” approximation proving well-posedness, mass-conservation and the non-negativity of the extracellular matrix, the degrading enzyme, and the active extracellular matrix. In addition, we obtain uniform strong estimates required in the convergence analysis, and we prove optimal error estimates and convergence towards regular solutions. Finally, we provide some numerical results in agreement with our theoretical analysis with respect to the positivity and the error estimates. [ABSTRACT FROM AUTHOR]