Integrating Mathematical Modeling and Oncology: An Improvised Collocation Approach for Prediction of Chemotherapy...

Cancer is one of the major global health challenges and predicting the response of malignant tumors to chemotherapy is particularly difficult due to the complex interaction between malignant cell growth and drug effects across time and space. Mathematical modeling plays an important role in understanding these dynamics, as it provides a systematic way to simulate malignant tumor behavior under treatment and to evaluate potential outcomes of different therapeutic strategies. In this work, a reaction-diffusion model is considered to analyze the dynamics of malignant tumors during chemotherapy treatment. The temporal derivative of the model equation is discretized using the Crank-Nicolson scheme, while the spatial derivatives are approximated using an improvised cubic B-spline collocation method. Nonlinearities in the governing model are treated with the Rubin-Graves method, which transforms the nonlinear terms into a tractable linear form. This approach maintains diagonal dominance of the system matrix while ensuring consistent enforcement of boundary conditions. To evaluate stability, we apply the Fourier spectral analysis to the proposed scheme. Four numerical experiments are conducted to test the effectiveness of the scheme under different treatment parameters. The computed numerical solutions and graphical representations are accurate and provide good agreement with previously reported results. Our approach is both easy to implement and computationally efficient, making it a reliable approach for biomedical simulations of tumor dynamics.