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In this paper, we investigate the tumor instability by employing bothanalytical and numerical techniques to validate previous results and extend theanalytical findings presented in a prior study by Feng et al 2023. Buildingupon the insights derived from the analytical reconstruction of key results inthe aforementioned work in one dimension (1D) and two dimensions (2D), weextend our analysis to three dimensions (3D). Specifically, we focus on thedetermination of boundary instability using perturbation and asymptoticanalysis along with spherical harmonics. Additionally, we have validated ouranalytical results in a two-dimensional framework by implementing theAlternating Directional Implicit (ADI) method, as detailed in Witelski andBowen (2003). Our primary focus has been on ensuring that the numericalsimulation of the propagation speed aligns accurately with the analyticalfindings. Furthermore, we have matched the simulated boundary stability withthe analytical predictions derived from the evolution function, which will bedefined in subsequent sections of our paper. These alignment is essential foraccurately determining the stability or instability of tumor boundaries.

A Three-dimensional tumor growth model and its boundary instability