Solving Tumor Growth Dynamic Using Runge-Kutta and Backward Euler Method

Abstract

The use of a model of tumor growth was a significant means of improving our understanding of cancer. The Gompertz model is the most common form of nonlinear growth behaviour of tumours that occurs with cancer growth. This study presented the author’s analysis of the numerical solution of the Gompertz model for malignant cancers using two numerical methods: the fourth-order Runge-Kutta (RK4) and the Backward Euler methods. The author conducted numerical simulations of the Gompertz model using predetermined initial conditions and parameters, and the numerical solutions were compared to the true analytical solution in order to assess their accuracy. The author performed an error analysis with a variety of different step sizes in order to assess the accuracy of the numerical methods. The analysis of these two method done using Python with the initial condition data take from Kaggle. The results of the study demonstrated that both numerical methods can be used to approximate the growth behaviour of tumours; however, the RK4 method consistently produces more accurate results as a result of its ability to produce higher-order elapsed time convergence, while the Backward Euler method produces larger amounts of numerical error, especially for larger step sizes. The author concluded that the RK4 method is more appropriate for the accurate simulation of the Gompertz tumour model and that this highlights the importance of selecting appropriate numerical methods and step sizes in order to produce reliable simulations of tumour growth.