NONLOCAL FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS FOR MODELING ANOMALOUS DIFFUSION IN BIOLOGICAL TISSUES: A UNIFIED THEORETICAL FRAMEWORK

Résumé

Anomalous diffusion is a hallmark of complex biologically tissues where heterogeneity, microstructural barriers, and long-range correlations lead to significant deviations from classical Fickian behavior. These tissues include brain white matter, extracellular matrix networks, and tumor microenvironments where experimental data show nonlinear mean square displacement, memory effects, nonlocal interactions, and anisotropic transport patterns. Hence, traditional integer-order diffusion equations are insufficient to explain biological diffusion. This research builds a unified theoretical framework utilizing nonlocal fractional partial differential equations (FPDEs) that combine three essential aspects: generalized fractional temporal derivatives for modeling memory-driven subdiffusion, anisotropic fractional spatial operators for capturing the direction-dependent tissue microstructure, and kernel-based nonlocal interactions for explaining the long-range spatial coupling. The nonlocal boundary conditions integrated into the resultant FPDE model offer a mathematically rigorous description of the anomalous transport phenomena. Besides, the model analytical results such as existence, uniqueness, positivity preservation, energy estimates, and scaling laws, to name a few, provide evidence of the model’s physical consistency and well-posedness. To complete the picture, the authors have developed numerical schemes that approximate the full dynamics based on L1 time discretization, Fourier spectral methods, and nonlocal quadrature. Simulation results on drug transport, diffusion-weighted MRI, extracellular matrix diffusion, and tumor microenvironment modeling are representative of the framework's biological relevance and versatility. The current work serves as a solid basis for forthcoming multiscale, data-driven, and inverse modeling research on biological anomalous diffusion.