Abstract

Spatial transcriptomics enables the scoring of Cancer Hallmarks at a high resolution across a tissue biopsy, yet current analyses of these data remain largely descriptive, treating each spot as a static, independent observation. This thesis develops a coupled system of advection-diffusion-reaction partial differential equations to model the hallmark landscape as the steady-state solution of an underlying spatial process, and casts the recovery of its physical parameters as a PDE-constrained inverse problem. The framework is implemented in FEniCSx on tissue-specific finite element meshes derived from two endometrial cancer biopsies provided by the Huntsman Cancer Institute, with subdomains, boundary interfaces, and an advective velocity field constructed from ESTIMATE purity scores and nucleus-level cell density. Applying this framework, we determine the utility of such a model. Under a biologically motivated source term, which confines each hallmark's production to its expected tissue compartment, reconstruction quality is largely determined by how closely a hallmark's spatial organization aligns with either the stroma or tumor compartments, preventing several hallmarks from being well reconstructed. A principal component analysis shows the hallmark fields to be effectively low-dimensional, occupying approximately three independent spatial axes. Replacing the compartment source with a rank-three, leave-one-out data-driven source recovers precisely those hallmarks that the compartment source fails to reconstruct, in both samples, while preserving those it already captured. Subsequent optimization of the physical parameters does not improve upon this source-driven fit: the recovered diffusion and advection coefficients are driven to nearly zero, and the pairwise interaction coefficients are shown to be non-identifiable, a consequence of the strong collinearity of the hallmark fields and of the absence of temporal information in spatial transcriptomics data. These results are presented not as a failure of the mechanistic model but as a precise characterization of what single-timepoint spatial data can support and cannot support.

Degree

MS

College and Department

Computational, Mathematical, and Physical Sciences; Mathematics

Rights

https://lib.byu.edu/about/copyright/

Date Submitted

2026-08-06

Document Type

Thesis

Keywords

cancer hallmarks, endometrial cancer, spatial transcriptomics, tumor microenvironment, reaction-diffusion-advection, partial differential equations, data assimilation, inverse problems, PDE-constrained optimization, finite element method, mathematical oncology

Language

english

Download

Share

COinS