Fractal geometry-governed oxygen diffusion: Tumors vs. Normal Tissues

{\bf Purpose}: 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. {\bf Methods}: 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 $\theta$. 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. {\bf Results}: The model reveals systematic suppression of long-range transport and enhanced localization as tissue structural complexity increases. Increasing $\theta$ 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 $\theta>0$, 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.