﻿ Joe Eyles

Mathematics PhD

For my PhD I modelled tumour growth. To do this I first numerically solved a system of partial differential equations, and then performed a significant amount of Mathematics analysis on the system of equations as well as the numerical solutions. This was a Mathematics PhD, but involved approximately 50% programming. I was engaged in independent (supervised) research, including Mathematical analysis, scientific programming, and producing literature (a paper (arxiv link here) and a thesis (link to the publication here)). I was also involved in teaching undergraduate students, and marking work. The system of equations I numerically solved and analysed is shown below, along with a diagram detailing the notation for the interior and exterior of the tumour.

\begin{align*}&space;\Delta&space;u&space;=&space;1&space;\quad&space;\mbox{&space;on&space;}&space;\Omega(t)\\&space;\nabla&space;u&space;\cdot&space;\mathbf{n}&space;+&space;\frac{u}{\alpha}&space;=&space;Q&space;\quad&space;\mbox{&space;on&space;}&space;\Gamma(t)&space;\\&space;V&space;=&space;\beta&space;\kappa&space;+&space;\frac{u}{\alpha}&space;\quad&space;\mbox{&space;on&space;}&space;\Gamma(t)\\&space;\end{align*}

Since I wanted to use a sharp interface approach and finite element methods, I had to maintain an evolving mesh on this moving boundary. This is a highly complex problem, that has no standard solution. In order to generate a solution, I had to implement a number of algorithms from the literature. I settled on a combination of approaches. The first approach smooths the velocity of the boundary over the interior of the tumour, allowing the mesh to evolve with it. The second re-builds the mesh when it reaches a point at which it can no longer be smoothed. You can see this below.

Eye tracking

In the past I have worked with a robotics research group at the University of Portsmouth. I was researching eye tracking methods (finding the eyes in an image of a face, or a webcam feed). I then used this to aproximate the gaze location on a computer screen. I programmed in both C++ and Matlab to create my own method, which was as effective as many of the existing methods. We published a paper to an IEEE journal. Go to here for the paper. For more information see the informal blog that I wrote during the work.