ABSTRACT We present a sharp‐interface partial differential equation (PDE) framework for modeling tumor invasion and basement membrane dynamics in colorectal cancer, focusing on the spatial organization of colonic crypts. The model describes tumor cell density evolution using a reaction–diffusion equation with a growth term, posed on a domain partitioned into lumen, epithelium, and extracellular matrix, with interfaces representing the basement membrane and luminal boundaries. The interface motion is governed by a Stefan‐type condition, directly coupling tumor growth to interface dynamics. To address the computational challenges posed by the evolving domain and moving boundaries, we employ the Cut Finite Element Method (CutFEM) with stabilization techniques, which enable the accurate resolution of sharp interfaces on unfitted meshes. This approach allows for robust simulation of tumor progression and basement membrane deformation, providing a mathematically precise and computationally efficient tool for investigating early‐stage colorectal cancer invasion. Simulation results demonstrate qualitative agreement with experimental ex vivo results, illustrating how a PDE model can effectively capture the spatial progression of tumor development and structural interactions with the basement membrane in early tumorigenesis. The results emphasize the critical role of crypt geometry and epithelial organization in shaping cancer dynamics.
Schmid et al. (Thu,) studied this question.