THE NON-INTEGER SYNTHESIS: A REVIEW OF FRACTAL-FRACTIONAL MODELING FOR GEOMETRY AND MEMORY IN COMPLEX BIOLOGICAL SYSTEMS

Abstract

Classical mathematical models based on Euclidean geometry and integer-order calculus often fail to capture the pervasive irregularity (scale-invariance) and temporal memory (hereditary effects) characteristic of complex biological systems. This review synthesizes the theoretical and applied advancements of the Fractal-Fractional (FF) paradigm in biomedicine. We begin by detailing the mathematical merger of the fractal dimension (α) and non-integer order differentiation (β), highlighting the critical role of the Liouville-Caputo fractional derivative in incorporating physically meaningful memory effects into dynamic models. We then explore the leading application in mathematical oncology, where FF models simulate tumor-immune cell interactions to optimize radiotherapy protocols, demonstrating enhanced numerical stability and predictive accuracy. Finally, we review biophysical applications, including the modeling of anomalous diffusion in advanced magnetic resonance imaging (MRI), viscoelastic modeling of tissues, and the crucial connection between distributed fractional-order derivatives and the multi-factuality observed in electrophysiological time series. This review confirms that the FF framework provides the necessary mathematical tools for accurately and comprehensively modeling non-local and geometrically complex biological phenomena.