An easy to implement numerical framework for a cancer invasion mathematical model with two distinct cancer sub-populations

This work presents a finite element scheme for solving a model that involves sub-populations of cancer cells. The model is formulated by four coupled partial differential equations, which represent the evolution of tumor growth, the density of cancer cell sub-populations arising from mutations, the density of the extracellular matrix (ECM), and the concentration of matrix-degrading enzymes (MDE). A semi-implicit backward Euler finite element framework has been developed for this model. Unconditional error estimates have been established for all variables, and the unconditional stability of the solutions has also been demonstrated. To validate the proposed numerical scheme, we have performed numerical simulations, including an assessment of the convergence rate and comparisons between the numerical solutions and analytical solutions.