Date of Award

December 2022

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Peter Hinow

Committee Members

Jeb Willenbring, Istvan Lauko, Gabriella Pinter, Lijing Sun

Keywords

Biomarkers, Electrophysiology, Machine Learning, Near Earth Space, Population Dynamics

Abstract

We present two applications of mathematics to relevant real-world situations.

In the first chapter, we discuss an automated method for the extraction of useful data from large file-size readings of cardiac data. We begin by describing the history of electrophysiology and the background of the work's setting, wherein a new multi-electrode array-based application for the long-term recording of action potentials from electrogenic cells makes large-scale readings of relevant data possible, opening the way for exciting cardiac electrophysiology studies in health and disease. With hundreds of simultaneous electrode recordings being acquired over a period of days, the main challenge becomes achieving reliable signal identification and quantification. In the context of this method of data collection, we set out to develop an algorithm capable of automatically extracting regions of high-quality action potentials from terabyte size experimental results and to map the trains of action potentials into a low-dimensional feature space for analysis. We establish that our automatic segmentation algorithm finds regions of acceptable action potentials in large data sets of electrophysiological readings. We use spectral methods and support vector machines to classify our readings and to extract relevant features. We are able to show that action potentials from the same cell site can be recorded over days without detrimental effects to the cell membrane. The variability between measurements 24 h apart is comparable to the natural variability of the features at a single time point. this work contributes towards a non-invasive approach for cardiomyocyte functional maturation, as well as developmental, pathological and pharmacological studies. As the human-derived cardiac model tissue has the genetic makeup of its donor, a powerful tool for individual drug toxicity screening emerges.

In the second chapter we consider the population of objects, largely considered debris, in the region of outer space close to Earth. the presence of this debris in Earth's orbit poses a significant risk to human activity in outer space. This debris population continues to grow due to ground launches, loss of external parts from space ships, and uncontrollable collisions between objects. We examine the background of human space launch, the current methods of tracking objects, and modelling work done to date. We propose a diffusion-collision model for the evolution of debris density in Low-Earth Orbit (LEO) and its dependence on ground-launch policy, to arrive at a computationally feasible continuum-based model. We parametrize this model and test it against data from publicly available object catalogs to examine timescales for uncontrolled growth. Finally, we consider sensible launch policies and cleanup strategies and how they reduce the future risk of collisions with active satellites or space ships, along with considering extensions of the model to directly account for launch policy determination through minimization of certain functionals.

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