UC Davis

Precision medicine is an emergent paradigm for health care research and delivery. Its primary goal is to improve clinical outcomes by developing treatments on the basis of each patient's individual characteristics. One focus within precision medicine is the problem of patient stratification, which is the method by which patients are partitioned into clinically meaningful subgroups. This is both a medical and mathematical problem, as it requires the identification of structures (i.e., clusters) in clinical data. Topological data analysis (TDA) is an emergent approach that uses topology --- a field of mathematics that studies shapes and the relationships between them --- to examine structure in data. The TDA algorithm Mapper converts data into graphs in order to create a sparse representation of the data. In Chapter 1, we provide an overview of precision medicine, Big Data, and the role that TDA plays within it. Chapter 2 is a scoping review on Mapper and its applications to patient stratification. We identify the successes of Mapper in this field, as well as three key shortcomings. Chapter 3 presents MapperPlus, an extension of Mapper that converts Mapper graphs into disjoint clusters for the purpose of patient stratification. We demonstrate the efficacy of MapperPlus and introduce a guideline for the selection of stable parameters. Chapter 4 is a case study that applies MapperPlus to the problem of risk-stratifying patients suffering from myocardial infarction (MI). Finally, in Chapter 5 we summarize the significance of this work, its contributions to both mathematics and precision medicine, and future directions for research.