Fractional derivatives to study cancer and its treatments

The bone is an organ in constant change. There are two main cells involved in the bone remodelling process: the osteoclasts and osteoblasts, responsible for the resorption and formation of bone, respectively. When tumour cells are present in the bone environment, they interfere with the bone remodelling process by promoting osteoclasts’ activity and causing the release of growth factors favorable to the survival and proliferation of tumour cells. This article presents differential equation models with a fractional order spatial derivative that include the emergence of drug resistance and the effect of therapy, taking the drug’s PK/PD into account. Fractional orders of differentiation, corresponding to anomalous diffusion, are considered in all models, and random mutations are included in a model where they were previously not considered. Simulations of the models are performed using a numerical approach.