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