A mathematical model of tumor hypoxia targeting in cancer treatment and its numerical simulation

Solid tumor includes the areas where oxygen concentration is very low called hypoxia, often in the surrounding areas of necrosis. Hypoxic cells in these areas are resistant to chemotherapy and radiation therapy. The presence of hypoxia and necrosis enables tumor-selective treatment, including hypoxia-activated prodrugs, tumor hypoxia-specific gene therapy and tumor-targeting bacterial therapy. This article deals with the mathematical formulation of tumor hypoxia-targeting by introducing a decay parameter of oxygen in the model given by Kolobov et al. (2009) and Avila et al. (2013). The well-posedness of the governing partial differential equations and numerical simulation are provided. For the purpose of numerical simulations, the conforming Q1 finite element method for space discretization and second-order diagonally implicit fractional step θ-scheme for temporal discretization are used. The effect of oxygen on hypoxia, necrotic region decay and the maximum age of growing tumor cells are computed and illustrated graphically. It is observed that the distribution of nutrients in tissues have substantial effect on tumor growth rate and structure.