| Cancer is a leading cause of premature death. Cancer starts with one tumour cell dividing uncontrollably and forming a growing tumour. This growth process can actually be captured and simulated by mathematical models. Since the 1970?s mathematicians have refined these models, being largely guided by experimental discoveries of the various growth mechanisms of tumours. The mathematical models are currently in a stage where the equations can be derived from first principles through continuum mechanics. Two types of models have arisen: sharp-interface models and diffuse-interface models. Both of these models have demonstrated great potential in the prediction of various growth behavior such as: aggressive growth resulting in the invasion of tissue and benign growth towards dormant states. However, it is unclear how the outcomes of simulations of both models are related. The aim of the proposed research is to assess and understand the differences between sharp-interface and diffuse-interface models in their prediction of key growth behaviours of tumours. To reach this goal, we first aim to provide a rigorous connection between the models. Based on this connection, we perform various analyses on geometrically simple and complex tumours. To aid in these analyses, we aim to develop efficient goal-oriented adaptive numerical methods for both models. The proposed research hopes to provide a first step into determining which of the models is the most appropriate for capturing the evolution of tumours, thereby bringing us a small mathematical step forward in the fight against cancer. |
