OPTIMAL TRANSPORT IN BIOSCIENCE: MODELING TRANSPORT MASS AND COST FOR NEURAL AND CELLULAR SYSTEMS

<p dir="ltr">The optimal transport (OT) problem is a mathematical framework for determining the most efficient way to move and redistribute resources from one distribution to another while minimizing a specified cost. Originally formulated by Gaspard Monge in 1781 and later reformulated by Leonid Kantorovich in 1942, OT has become a powerful tool across numerous scientific and engineering disciplines, including biology, neuroscience, physics, economics, and environmental science, due to its ability to compare and transform distributions. It is also known as the Wasserstein Distance or Earth Mover's Distance (EMD) and is widely used in machine learning and computer vision.</p><p dir="ltr">This thesis builds on an entropy-regularized and time-efficient variant of OT, known as the Sinkhorn distance, to develop novel frameworks for analyzing complex biological systems, specifically neural and cellular systems. We focus on two fundamental components of OT, <i>mass of transport</i> and <i>cost of transport</i>, to develop these frameworks. In the first part of the thesis, we model the spatial organization of axons in peripheral nerves by treating nerve cross-sections as spatial objects extracted from microscopy images and defining their spatial statistics as the mass of transport. A linear cost function is initially used to quantify architectural similarity between nerve samples. We further incorporate non-linear cost functions that account for multi-class nerve structures to enhance biological relevance, enabling a more detailed and informed analysis. </p><p dir="ltr">We employ a spatial representation of nerve architecture to reveal its inhomogeneous and anisotropic nature, challenging the traditional view of randomness. We integrate spatial statistics into the OT framework, thereby enriching it with structural and functional insights that can inform targeted electro-modulation therapies and reduce off-target effects. Extending beyond the direct application of the optimal transport framework, we also explore spatial analysis to model spatial networks originating from the enteric nervous system, contributing to a broader understanding of the architecture of another part of the peripheral nervous system. Through preliminary experiments, we address the necessity of hybrid spatial models to represent neural systems more comprehensively. We demonstrate how morphometric features of peripheral nerve components relate to their spatial organization, laying the groundwork for future integrative models that combine structural and functional aspects of neural tissue.</p><p dir="ltr">In the second part, we extend the OT framework to high-parameter flow cytometry, which presents challenges for visualization and interpretation when current methods are used. We propose a graph-based visualization framework grounded in OT, where cell populations are defined by marker expression profiles and inter-population similarities are quantified using the Sinkhorn distance. A phenotype-aware Hamming distance is used to generate biologically consistent 2D graph layouts, while a customized graph edit distance captures structural differences between samples. Additionally, we introduce a learning-based model that projects cell populations into a latent space informed by OT distances, preserving single-cell resolution and enhancing interpretability. Our method provides robust, quantitative visual summaries, enabling the statistical analysis of disease progression and treatment response.</p><p dir="ltr">Taken together, this thesis demonstrates the versatility and power of the OT framework in modeling, quantifying, visualizing, and interpreting complex biological systems, bridging the gap between computational methods and biological insight.</p>