Graph Theory and the Pulmonary Arterial System
Supervisor: Dr Jay Mackenzie
School: Mathematics and Statistics
Description:
Blood vessel networks are comprised of many thousands or hundreds of thousands of individual tubes that are joined together in diverging and converging trees. For various mathematical modelling purposes, we want blood vessel networks that meet certain criteria. These criteria might be easy to write down, but hard to find a network that meets them. This is further complicated by the fact that the data (from medical images) of blood vessel networks don’t have a series of tubes joined together, but rather large point clouds where each spatial point has certain attributes.
The challenge, and what we explore in this project, is how to go from a seemingly formless cloud of data to a sorted, filtered, and useable network of vessels. Finally, we explore how blood flow through a vessel network is impacted by the shape of the network, including overall volume, number of vessels, and the size of the vessels involved.
In this project, using mathematical and computational techniques, you will develop algorithms for the visualising, sorting, and filtering the pulmonary arterial networks of mice. This will start by visualising the data sets before implementing Dijkstra’s algorithm and trying to reduce the size of the search space using graph theory. After this, we can begin to filter and sort the data, and analyse the differences in the pulmonary arterial structure between individuals.
This project dips into discrete maths and graph theory, topology, computer vision, and fluid dynamics. A good grasp of linear algebra is recommended, but not essential.