UC Merced

Mathematical modeling of biological systems provides a structured framework for exploring and understanding intricate biological phenomena that can be key to unlocking the underlying principles that govern life. Even relatively elementary mathematical models applied to highly complex biological systems have exhibited a capacity to unveil significant patterns and insights, underscoring the substantial utility of computational models in scientific inquiries. In a simplified perspective, mathematical models of biological phenomena can be categorized into three groups: The first group assumes known governing equations and employs mathematical descriptions (e.g. Ordinary or Partial Differential Equations) for modeling dynamics and variations. The second group relies on complete observations to learn some or all system components, often utilizing Machine Learning to uncover useful patterns and relationships in the data. The third group aims to use incomplete observations or assumptions to learn from the system, often seeking to generalize findings to unknown agents in the same or different systems.

The goal of this dissertation is to propose novel mathematical techniques to address some of the intricacies inherent in modeling complex biological systems. This dissertation is structured into three parts, each corresponding to the modeling paradigms mentioned earlier, with individual chapters devoted to various methods and applications. The first part of this dissertation introduces a novel numerical method for modeling Partial Differential Equations on deforming geometries. Subsequently, the dissertation transitions to data-driven modeling of biological tissues in humans and animals, leveraging deep learning to capture meaningful insights that can improve the robustness and accuracy of biological analyses. In the final part of this dissertation, novel deep-learning algorithms are formulated to address data or label scarcity, offering substantial improvements in the analysis of systems where prior knowledge or the amount of observations is limited.

Similar to mathematical biology itself, the methods and systems explored in this dissertation encompass a wide spectrum, spanning various scales (from a single cell to tissues) and species (yeast, mice, humans) within the realm of biological sciences. However, they are unified in their pursuit to enable more robust and accurate analysis of complex biological systems. Despite their distinctions, the frameworks presented in this work underscore my dissertation’s overarching goal: to provide a comprehensive toolkit for modeling complex biological systems across different modeling regimes and limitations. Through these efforts, I hope to contribute to gaining a deeper under- standing of life and its underlying mathematical principles.