Fractal geometry and fractional calculus for integrative morphological mapping of breast cancer complexity

Breast cancer (BC) exhibits intricate morphological and dynamical heterogeneity across cellular, tissue, and tumor scales, posing challenges to conventional modeling approaches that fail to capture its nonlinear, self-similar or self-affine, and memory-dependent behavior. Despite increasing applications of fractal geometry (FG) and fractional calculus (FC) in cancer modeling, their methodological integration and biological interpretation remain insufficiently consolidated. This review aims to synthesize these frameworks within an integrative morphological perspective to elucidate their collective potential for quantitative characterization of BC complexity. FG-based analyses quantify spatial and temporal irregularities along with spatiotemporal morphodynamics, while FC introduces non-local and memory-dependent formulations describing tumor growth. Together, these frameworks establish a mathematical link between fractal structure and fractional dynamics. Nevertheless, their application remains hindered by a lack of consistent methodologies and reproducible standards. Here, we consolidate existing evidence, delineate methodological interrelations between FG and FC, and outline reproducibility requirements, including standardized preprocessing, parameter reporting, and benchmark datasets. We emphasize that reproducible and biologically interpretable integration of these two approaches is fundamental to achieving clinically relevant modeling of BC morphology and dynamics.