Herbert Tang | Applied Math, University of Waterloo
The Effects of Extrinsic and Intrinsic Noise on a Tumour and a Proposed Metastasis Model.
This thesis examines the effects of both extrinsic and intrinsic noise on a standard model for tumour growth and a proposed new model for metastasis. Traditional partial differential equation (PDE) models only produce solutions that are smooth and deterministic (given the same initial condition we always reach the same solution) and thus only examines the average behaviour of the system. Extrinsic noise is a simple mathematical device that ostensibly considers any external influences on the system. Intrinsic noise, on the other hand, examines the inherent randomness of chemical reactions and takes into account discrete number effects (e.g. integer numbers of births and deaths) which are smoothed over by a PDE model. We know from real life that most experiments will not have the same outcome even if the initial conditions are exactly the same. Thus, a stochastic model produced by including noise in an already established PDE model will add an extra element of fidelity to the system by recreating this natural variability.
The tumour model that we will consider is a simple reaction-diffusion model that was actually first used by R. A. Fisher in 1937. The addition of noise in this model allows us to investigate the effects of patient to patient variability predicted by this model, and give error bounds on estimated tumour size and survival time. Our proposed metastasis model is based on an excitable system used to model signal propagation in a cell undergoing mitosis. Excitable systems have the ability to switch between to stable steady states (a high ``on'' state and a low ``off'' state) under the influence of a sufficiently large stimulus, and we make use of this property to model how a normal cancer cell can turn into a metastatic one. Noise in this case allows the system to enter into its ``on'' state under a broader range of conditions than allowed under the deterministic setting, and explains how this metastatic signal can be transmitted across the entire tumour.