Mathematical and Numerical Modelling of Interference of Immune Cells in the Tumour Environment - Sweta Sinha; Paramjeet Singh; Mehmet Emir Koksal (Discrete Dynamics in Nature and Society)
In this article, the behaviour of tumour growth and its interaction with the immune system have been studied using a mathematical model in the form of partial differential equations. However, the development of tumours and how they interact with the immune system make up an extremely complex and little-understood system. A new mathematical model has been proposed to gain insight into the role of immune response in the tumour microenvironment when no treatment is applied. The resulting model is a set of partial differential equations made up of four variables: the population density of tumour cells, two different types of immune cells (CD4+ helper T cells and CD8+ cytotoxic T cells), and nutrition content. Such kinds of systems also occur frequently in science and engineering. The interaction of tumour and immune cells is exemplified by predator-prey models in ecology, in which tumour cells act as prey and immune cells act as predators. The tumour-immune cell interaction is expressed via Holling’s Type-III and Beddington-DeAngelis functional responses. The combination of finite volume and finite element method is used to approximate the system numerically because these approximations are more suitable for time-dependent systems having diffusion. Finally, numerical simulations show that the methods perform well and depict the behaviour of the model.