Journal of Scientific Computing

, Volume 35, Issue 2–3, pp 266–299 | Cite as

A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth

  • Paul Macklin
  • John S. LowengrubEmail author


In this paper, we present a ghost cell/level set method for the evolution of interfaces whose normal velocity depend upon the solutions of linear and nonlinear quasi-steady reaction-diffusion equations with curvature-dependent boundary conditions. Our technique includes a ghost cell method that accurately discretizes normal derivative jump boundary conditions without smearing jumps in the tangential derivative; a new iterative method for solving linear and nonlinear quasi-steady reaction-diffusion equations; an adaptive discretization to compute the curvature and normal vectors; and a new discrete approximation to the Heaviside function. We present numerical examples that demonstrate better than 1.5-order convergence for problems where traditional ghost cell methods either fail to converge or attain at best sub-linear accuracy. We apply our techniques to a model of tumor growth in complex, heterogeneous tissues that consists of a nonlinear nutrient equation and a pressure equation with geometry-dependent jump boundary conditions. We simulate the growth of glioblastoma (an aggressive brain tumor) into a large, 1 cm square of brain tissue that includes heterogeneous nutrient delivery and varied biomechanical characteristics (white matter, gray matter, cerebrospinal fluid, and bone), and we observe growth morphologies that are highly dependent upon the variations of the tissue characteristics—an effect observed in real tumor growth.


Ghost fluid method Ghost cell method Level set method Tumor growth NAGSI Nonlinear elliptic equations Heaviside function Heterogeneous media Heterogeneous tissue structure Adaptive normal vector calculation Normal derivative jump boundary condition Poisson equation 


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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.SHISU. of Texas Health Science CenterHoustonUSA
  2. 2.Dept. of MathematicsUniversity of CaliforniaIrvineUSA

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