Shapes with complex morphologies are ubiquitous in biology and materials science. Patterns in shapes hold valuable information on disease progression in living organisms and physical properties in materials. Quantifying and modeling shape textures are therefore essential questions, but they pose real challenges due to the disparity of porous shapes. Our approach, deeply rooted in geometry and topology, adopts a unifying view encompassing all kinds of shape patterns. We first propose a phase-field model, curvatubes, that generates a wide continuum of porous shapes, based on the optimization of curvature functionals generalizing classical models of biomembranes. This results in an expressive, efficient and customizable framework. Signed distance persistent homology (SDPH) quantifies shape textures with a summary of their multiscale topological features, based on the sublevel set filtration of their signed distance field. A major question in persistent homology is the interpretation of these barcodes. We bridge the gap by generalizing Morse theory to distance functions generated by smooth compact boundaries in the Euclidean setting, which underpins a rigorous interpretation in terms of critical points of the signed distance. We also accelerate the cycle matching method that detects salient features in data and finds topological correspondences between networks, by adapting state-of-the-art algorithms for persistent homology powered by cohomology. Finally, we combine these approaches into our driving biological application, the study of vascular remodeling by acute myeloid leukaemia in the bone marrow. The potential of these methods however goes far beyond vascular analysis, and can lead to impactful results in biomedical engineering, materials science, imaging and shape analysis, while motivating new mathematical questions.