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Evolving interfaces via gradients of geometry-dependent interior Poisson problems: application to tumor growth. (English) Zbl 1067.65111
Summary: We develop an algorithm for the evolution of interfaces whose normal velocity is given by the normal derivative of a solution to an interior Poisson equation with curvature-dependent boundary conditions. We improve upon existing techniques and develop new finite difference, ghost fluid/level set methods to attain full second-order accuracy for the first time in the context of a fully coupled, nonlinear moving boundary problem with geometric boundary conditions (curvature). The algorithm is capable of describing complex morphologies, including pinchoff and merger of interfaces.
Our new methods include a robust, high-order boundary condition-capturing Poisson solver tailored to the interior problem, improved discretizations of the normal vector and curvature, a new technique for extending variables beyond the zero level set, a new orthogonal velocity extension technique that is both faster and more accurate than traditional partial differential equation-based approaches, and a new application of Gaussian filter technology ordinarily associated with image processing. While our discussion focuses on two-dimensional problems, the techniques presented can be readily extended to three dimensions.
We apply our techniques to a model for tumor growth and present several 2D simulations. Our algorithm is validated by comparison to an exact solution, by resolution studies, and by comparison to the results of a spectrally accurate method boundary integral method (BIM). We go beyond morphologies that can be described by the BIM and present accurate simulations of complex, evolving tumor morphologies that demonstrate the repeated encapsulation of healthy tissue in the primary tumor domain - an effect seen in the growth of real tumors.

65N06 Finite difference methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35R35 Free boundary problems for PDEs
92C55 Biomedical imaging and signal processing
Full Text: DOI
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