This paper presents and investigates the tumor growth based on a phase model. The tumor core is necrotic and inhibitor chemical species are considered. The interface of tumor and health tissue is tracked using a phase field equation. The reformulation of a classical model, accounting for cell-proliferation, apoptosis, cell-to-cell and cell-to-matrix adhesion, is derived. The advantages of the finite difference methodology employed are generality and relative simplicity implication. We present simulations of the nonlinear evolution of growing tumors morphology and discuss the effects of tumor microenvironment. Mechanisms reflecting the tumor growth and development behavior was preliminarily interpreted. Recently numerous mathematical have been developed to investigate the growth dynamics of tumor [1–8]. One of most significant model developed by Wise [8] is based on Cahn-Hilliard equation, which is conservation phase field method. Allen-Chan nonconservation phase field has been developed to track the moving interface for multiphase simulation by Sun [9]. Allen-Chan equation is second order, while Cahn-Hilliard equation is fourth order in space. Thus, we introduce the Allen-Chan phase method [9–10] to simulate the tumor growth, which is very simple for numerical simulation The computation domain is illustrated in Fig. 1, where ΩH denotes host tissue, the tumor domains is comprised of viable tumor cell ΩV and dead tumor cell ΩD. The numerical results are presented at Fig. (2–4). One can find that the growth of tumor strongly depend on the nutrients and nonlinear unstable growth may lead to finger shaped pattern, which is in agreement with recent experimental observations [7] of in vivo tumor. In summary, a phase method has been developed to study diffusion and consumption of the nutrients and tumor cell proliferation, necrosis and migration, which discloses the evolution of complex shape of tumor.
Upscaled phase-field models for interfacial dynamics in strongly heterogeneous domains
