Fractal geometry-governed oxygen diffusion: tumors vs. normal tissues.

To develop a geometry-governed diffusion framework that explains differential tissue response under FLASH ultra-high dose rate (UHDR) irradiation by explicitly accounting for structural heterogeneity and anomalous transport in biological tissues. Approach: We formulate a generalized diffusion-reaction model on fractal substrates to describe molecular transport in heterogeneous media. Tissue architecture is characterized by a fractal (Hausdorff) dimension D, while scale-dependent transport inefficiency and memory effects are captured by a fractional parameter θ. Analytical solutions for radially symmetric geometries are derived and compared with classical normal (Euclidean) diffusion and a Gaussian reference model under identical physical conditions. Transport behavior is quantified through transient probability distributions and steady-state spatial profiles. Main results: The model reveals systematic suppression of long-range transport and enhanced localization as tissue structural complexity increases. Increasing θ leads to subdiffusive dynamics, reduced effective diffusion lengths, and persistent non-Gaussian concentration profiles, even in the steady state. While increasing D alone enhances spatial accessibility, fractional dynamics dominate transport behavior when θ > 0, counteracting geometric connectivity. These effects produce a separation between regimes characterized by efficient inter-track overlap and rapid homogenization, and regimes marked by isolated, long-lived reactive domains. Conclusion: Fractal geometry provides a unifying physical framework for understanding tissue-dependent transport and differential response under FLASH UHDR irradiation. Normal tissues, characterized by near-Euclidean geometry and weak anomalous effects, permit greater inter-track interaction and recombination, whereas tumor-like tissues with elevated structural complexity exhibit localized transport and reduced collective chemical reactivity. This proof-of-principle study establishes tissue architecture as a fundamental determinant of transport efficiency and offers a mechanistic basis for experimentally observed FLASH tissue sparing, motivating geometry-aware modeling of radiobiological response. Significance: This work shifts the emphasis from purely chemical kinetics toward a geometry-governed description of oxygen and reactive oxygen species transport in cells and tissues.