Mathematical and statistical methods for single cell data

The availability of single-cell data has increased rapidly in recent years and presents interesting new challenges in the analysis of such data and the modelling of the processes that generate it. In this thesis, we attempt to deal with some of those challenges by developing and exploring mathematical and statistical models for the evolution of population distributions over time, and methods for using aggregated single-cell data from individual patients in predictive diagnostic models of disease. In the first part of the thesis, we explore structured population models - a class of partial differential equations for describing the evolution of individual-level cell properties in a population over time. We begin by analysing an age-structured model of cell growth in which rates of proliferation and cell death are controlled by an external resource. We follow this with a method for extracting properties of a more general class of structured population models directly from single-cell data. In the final part of the thesis, we develop a flexible Bayesian statistical framework for building predictive models from possibly high-dimensional data collected from patients using single-cell technologies and find that the performance is promising compared to a number of existing methods.