A structure-preserving computational method in the simulation of the dynamics of cancer growth with radiotherapy
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In this work, we consider a two-dimensional mathematical model that describes the growth dynamics of cancer when radiotherapy is considered. The mathematical model for the local density of the tumor is described by a parabolic partial differential equation with variable diffusion coefficient. The nonlinear reaction term considers both the logistic law of proliferation of tumor cells and the effect of a treatment against cancer. Suitable initial-boundary conditions are imposed on a bounded spatial domain, and a theorem on the existence and the uniqueness of solutions for the initial-boundary-value problem is proved. Motivated by this result, we design a finite-difference methodology to approximate the solutions of our mathematical model. The scheme is a linear method that is capable of preserving the positivity and the boundedness of the approximations. Some simulations are presented in order to illustrate the performance of the method. Among other conclusions, the numerical results show that the method is able to preserve the analytical features of the relevant solutions of the model.
KeywordsCancer growth modeling with therapy Diffusion–reaction equation Existence and uniqueness of solutions Structure-preserving finite-difference scheme Positivity and boundedness
Mathematics Subject Classification92-08 65M06 92C37 35K55 35K57
This manuscript is an extended version of a paper presented in July 2017 at the XVII International Conference “Computational and Mathematical Methods is Science and Engineering”, in Rota, Spain. The authors want to thank the comments on this work by various participants of that event.
- 3.S.W. McCue, D.G. Mallet, et al., A cellular automata model to investigate immune cell–tumor cell interactions in growing tumors in two spatial dimensions, in Mathematical models of tumor-immune system dynamics (Springer, Berlin, 2014), pp. 223–251Google Scholar
- 4.A. Streck, K. Thobe, H. Siebert, Analysing cell line specific EGFR signalling via optimized automata based model checking, in Computational methods in systems biology (Springer, Berlin, 2015), pp. 264–276Google Scholar
- 9.S. Xu, M. Huang, Global existence and uniqueness of solutions for a free boundary problem modeling the growth of tumors with a necrotic core and a time delay in process of proliferation. Math. Probl. Eng. 2014, 480147 (2014)Google Scholar
- 16.A.N. Kolmogorov, I. Petrovsky, N. Piskunov, Etude de léquation de la diffusion avec croissance de la quantité de matiere et son application a un probleme biologique. Mosc. Univ. Bull. Math. 1, 1–25 (1937)Google Scholar
- 17.A. Friedman, Partial differential equations of parabolic type, 1st edn. (Prentice Hall Inc, Upper Saddle River, 1964)Google Scholar