Fractal Geometry and Fractional Calculus for Integrative Morphological Mapping of Breast Cancer Complexity

TL;DR

This review explores how fractal geometry and fractional calculus can be integrated to better model and understand the complex, heterogeneous morphology and dynamics of breast cancer, emphasizing the need for standardized methods.

Contribution

It synthesizes existing approaches, clarifies their methodological links, and proposes reproducibility standards for integrating fractal geometry and fractional calculus in breast cancer modeling.

Findings

Fractal analysis quantifies spatial and temporal irregularities in tumor morphology.

Fractional calculus models tumor growth with non-local, memory-dependent formulations.

Standardized protocols are essential for reproducible and interpretable modeling results.

Abstract

Breast cancer 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 and fractional calculus 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 breast cancer complexity. Fractal geometry-based analyses quantify spatial and temporal irregularities along with spatiotemporal morphodynamics, while fractional calculus 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 inconsistent methodologies and a lack of reproducible standards. This review consolidates existing evidence, delineates methodological interrelations between fractal geometry and fractional calculus, and outlines reproducibility requirements, including standardized preprocessing, parameter reporting, and benchmark datasets. Collectively, the findings emphasize that reproducible and biologically interpretable integration of these two approaches is fundamental to achieving clinically relevant modeling of breast cancer morphology and dynamics.