Personalized prognosis of glioblastoma via fractional-order/biomechanical modeling: Bridging anomalous diffusion, continuum mechanics, and MRI data

• Novel fractional-order biomechanical model for GBM using Caputo tempered fractional derivative, showing 50% faster growth at α =0.7 vs. classical ( α =1). • Meshless Petrov-Galerkin method with Moving Kriging interpolation solves equations efficiently ( ≤ 5% error vs. clinical data). • findings highlight the restraining effect of tissue stiffness on tumor expansion and the diagnostic potential of stress-induced thermal effects. • Stress-induced thermal effects identified as potential diagnostic markers. • Validated with MRI data, supporting use in personalized treatment planning. Background and Objective: Glioblastoma multiforme (GBM) is one of the most aggressive and treatment-resistant brain tumors, with a poor prognosis despite therapeutic advancements. This study aims to develop a novel fractional-order mathematical model coupled with biomechanical interactions to predict GBM growth dynamics, providing insights into tumor progression and potential diagnostic markers. Methods: The study utilizes the Caputo tempered fractional derivative to capture memory effects and anomalous diffusion in tumor growth. A meshless local Petrov-Galerkin method based on Moving Kriging interpolation is employed to solve the resulting equations efficiently. The model incorporates heterogeneous tissue properties (gray/white matter, cerebrospinal fluid) and mechanical stress-strain responses within a continuum mechanics framework. Comparative analysis with a biomechanical finite element model is conducted to validate the results. Results : Computational simulations reveal that the fractional order ( α ) and tempering parameter ( λ ) significantly influence tumor growth rates, with lower α values correlating with faster proliferation. The model shows strong agreement ( ≤ %5 discrepancy) with classical biomechanical finite element results when α = 1 . Key findings highlight the restraining effect of tissue stiffness on tumor expansion and the diagnostic potential of stress-induced thermal effects. Conclusions: This study advances personalized GBM prognosis by integrating fractional dynamics, biomechanics, and computational efficiency. The proposed framework offers a promising approach for optimizing treatment strategies and improving understanding of GBM progression. The findings underscore the importance of biomechanical interactions in tumor growth and their potential clinical applications.