Decoding Clinical Complexity: Mathematical Frameworks for Interpretable Precision Medicine

Precision medicine aims to decode disease at the level of individual patients. However, the data driving this field—complex imaging and high-dimensional molecular profiles—often exceed the limits of traditional interpretation. My research leverages mathematics as a principled language to navigate this complexity, transforming heterogeneous biomedical data into structured, interpretable representations of patient-specific biology. I first demonstrate this approach through the lens of immune organization in neuroblastoma. While immune infiltration is a known prognostic factor, current clinical assessments rely on coarse summaries that ignore tissue architecture. By integrating deep learning with topological data analysis (TDA), we can map whole-slide histopathology onto quantitative representations that capture both cell density and the higher-order spatial geometry of the tumor microenvironment. This framework extracts features that are robust to technical noise yet highly sensitive to biological variations in clinical outcomes. Expanding from tissue architecture to systemic cellular communication, I present a multi-tissue model of immune states in the human brain and blood. Using paired single-cell transcriptomic data, we apply tensor decomposition to model simultaneous interactions across individuals, cell types, and ligand–receptor pairs. This approach reveals structured, bidirectional signaling programs linking peripheral immune cells to microglia and neurons. Our findings demonstrate that immune cells entering the brain undergo a distinct "brain-adaptation" process, providing a new mathematical window into how systemic immunity influences neurological health. Formal mathematical frameworks do more than just manage large datasets; they reveal hidden biological structures that are invisible to standard analysis. By bridging the gap between abstract theory and clinical observation, these models provide a foundation for more precise diagnostics and the discovery of novel, patient-specific therapeutic targets.