Taming of the tumour: a mathematical saga
A tumour is a heterogeneous material. A mathematical model of a tumour helps to make predictions about its growth and effects of drugs faster than real-time experiments. Construction of a good model that can accurately mimic tumour growth is a multi-step process. Tumour growth, like any biological phenomenon, is an extremely complicated event where hundreds of proteins, growth-regulating agents, and chemical signalling mechanisms are involved. A direct study of all these contributing factors might consume an unreasonable amount of time. Therefore, the first step in modelling tumour growth is to understand the tumour in a simple basic form. Fundamentally, a tumour can be conceived as a mixture of tumour cells and a nutrient-rich fluid that fills the void between the cells. The second step is to translate the interactions between the cells and the fluid into mathematical equations. The final model obtained from the second step is too complex to be exactly solved, and gaining insights into its solutions requires the usage of computer-based simulations, based on algorithms that, in mathematics, we call numerical methods. Designing numerical methods that capture the physical and biological properties of this heterogeneous model and justifying them mathematically is the third challenge. The project also consists of developing an optimised algorithm that can be programmed on a computer. This optimisation is crucial as it enables efficient and fast computation of approximate solutions, which contain information about the growing tumour. In future, more complexities, like the impact, on the model, of therapy, can be added to this basic model in a step-by-step manner.