An adaptive multigrid algorithm for simulating solid tumor growth using mixture models

In this paper we give the details of the numerical solution of a three-dimensional multispecies diffuse interface model of tumor growth, which was derived in [S.M. Wise, J.S. Lowengrub, H.B. Frieboes, V. Cristini, Three-dimensional multispecies nonlinear tumor growth–I: model and numerical method, J. Theoret. Biol. 253 (2008) 524–543] and used to study the development of glioma in [H.B. Frieboes, J.S. Lowengrub, S.M. Wise, X. Zheng, P. Macklin, E.L. Bearer, V. Cristini, Computer simulation of glioma growth and morphology, NeuroImage 37 (2007) S59–S70] and tumor invasion in [E.L. Bearer, J.S. Lowengrub, Y.L. Chuang, H.B. Frieboes, F. Jin, S.M. Wise, M. Ferrari, D.B. Agus, V. Cristini, Multiparameter computational modeling of tumor invasion, Cancer Res. 69 (2009) 4493–4501; H.B. Frieboes, F. Jin, Y.L. Chuang, S.M. Wise, J.S. Lowengrub, V. Cristini, Three-dimensional multispecies nonlinear tumor growth—II: tissue invasion and angiogenesis, J. Theoret. Biol. 264 (2010) 1254–1278]. The model has a thermodynamic basis, is related to recently developed mixture models, and is capable of providing a detailed description of tumor progression. It utilizes a diffuse interface approach, whereby sharp tumor boundaries are replaced by narrow transition layers that arise due to differential adhesive forces among the cell-species. The model consists of fourth-order nonlinear advection–reaction–diffusion equations (of Cahn–Hilliard-type) for the cell-species coupled with reaction–diffusion equations for the substrate components. Computing numerical solutions of the model is challenging because the equations are coupled, highly nonlinear, and numerically stiff. In this paper we describe a fully adaptive, nonlinear multigrid/finite difference method for efficiently solving the equations. We demonstrate the convergence of the algorithm and we present simulations of tumor growth in 2D and 3D that demonstrate the capabilities of the algorithm in accurately and efficiently simulating the progression of tumors with complex morphologies.