A study on brain tumor dynamics in two-dimensional irregular domain with variable-order time-fractional derivative

Understanding tumor growth in the brain is a crucial and complex challenge. This study aims to develop and analyze a brain tumor growth model that incorporates variable-order time-fractional derivatives within a two-dimensional irregular domain. The purpose is to explore the effects of time-fractional orders, mutation rates, and growth parameters on tumor dynamics.

The model employs the finite difference method for temporal discretization and Gaussian radial basis functions based on Kansa's method for spatial variables. Ulam-Hyers stability analysis is performed to ensure the model's stability and the existence and uniqueness of the solution are established. Additionally, the stability and convergence of the scheme are analyzed. Code verification is conducted to confirm the accuracy and reliability of the computational approach. Key parameters, such as the mutation rate K_c and growth parameters σ₁ and σ₂, are investigated under various time-fractional derivative orders, including variable orders.

The numerical simulations provide a detailed analysis of tumor cell dynamics, accounting for heterogeneity and fractional effects. Graphical representations reveal novel behaviors induced by variable-order time-fractional derivatives, including their impact on tumor cell population growth. Changes in the mutation rate and growth parameters significantly influence the results, demonstrating sensitivity to parameter variations.

This study demonstrates that the integration of variable-order time-fractional derivatives into brain tumor models introduces memory effects, revealing new insights into tumor behavior. The findings highlight the importance of fractional-order parameters in accurately modeling brain tumor growth, which could have potential implications for predicting tumor progression and developing targeted treatments.