Modular-type nonlinearity in the modeling of tumor spheroid growth

A space–time trigger model of tumor growth is considered depending on the concentration of hydrogen ions, oxygen, and tumor cell density at the initial stage of the tumor spheroid development. A system of parabolic equations with a piecewise linear right-hand side of modular type is used to solve this problem. Numerical implementation is carried out in a three-dimensional domain shaped as a cube with an edge of 0.1 mm on a uniform grid using the method of lines, the Rosenbrock scheme, and the factorization method in the spatial coordinates. The presented model quite well describes the dynamics of variation in the spheroid area at the initial stage of the tumor development depending on time. A distinctive feature of the model is that it reflects both the process of tumor growth into the external environment and the formation of a necrotic core at the tumor center. Based on the presented system of equations, it is possible to design a model that takes both the heterogeneity of the environment and more complex mechanisms of tumorigenesis into account.