A graph theoretical approach to modelling growing cellular tissue

Abstract

All organisms are built out of cellular tissue. Being able to recognise abnormalities in these tissues could be useful in recognizing cancerous cells. In this thesis we construct a mathematical model for cellular tissue based on its spatial structure. We consider cells as elements of the network. Touching cells are considered connected. Cells grow at different growth speeds. We determine the point in time when this network is fully connected, meaning there is a path between every pair of cells through touching cells. This point indicates the start of the last phase of the cellular growth, where friction restricts cell movement. We first use a Poisson point process to generate the locations of the cells. To make the model more similar to cellular tissue, we introduce determinantal point processes which have short-ranged repulsion, meaning points repel each other and thus spread. We compare the repulsion of Poisson and Determinantal point processes with real cellular tissue. We conclude that determinantal point processes have significantly higher repulsion than Poisson point processes. We also conclude that the repulsion in cellular tissue is higher than both point processes. Using simulations, we show that the model with determinantal point processes reaches connectivity significantly earlier than the Poisson model. We conclude that the analytically derived connectivity time point from the Poisson model can be used as an upper bound for the determinantal model.