A study of tumour development and effects of cytotoxic drugs using Ordinary Differential Equations and Integro-Differential Equations.
This project serves as my undergraduate dissertation, completed under the supervision or Dr. Tommaso Lorenzi at The University of St. Andrews.
Abstract:
In order to comprehend the behaviour of solid tumours, it is essential to analyse the dynamic evolution of cancer cell populations and the driving ecological and evolutionary mechanisms. Of equal importance is an understanding of the phenotypic variants commonly found in cancer cell populations and how they evolve with time. Mathematical models are of key importance in cancer research as they create important predictions that would otherwise require lengthy experimentation.
The results in this paper are achieved via two central models. The first is an ordinary differential equation model, which describes the time evolution of the size of a well-mixed population of cancer cells. The model is considered both with and without the effect of a cytotoxic drug acting on the population. An integro-differential equation model is then introduced to account for and further study the evolution of phenotypic heterogeneity found in cancer cell populations.
Results illustrate that tumours tend to maximum population sizes in the absence of treatment. When cytotoxic treatment is incorporated, the ordinary differential equation model highlights the efficacy of bolus injections over continuous infusions. Furthermore, it is shown that in heterogeneous tumours one phenotypic trait will dominate over others in the long-term.