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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.

MSC:
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
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[1] Adalsteinsson, D.; Sethian, J.A., The fast construction of extension velocities in level set methods, J. comput. phys., 148, 1, 2-22, (1999) · Zbl 0919.65074
[2] Adam, J., General aspects of modeling tumor growth and immune response, (), 15-87
[3] Byrne, H.M.; Chaplain, M.A.J., Growth of necrotic tumors in the presence and absence of inhibitors, Math. biosci., 135, 187-216, (1996) · Zbl 0856.92010
[4] Byrne, H.M.; Chaplain, M.A.J., Modelling the role of cell-cell adhesion in the growth and development of carcinomas, Math. comput. model., 24, 1-17, (1996) · Zbl 0883.92014
[5] M.A.J. Chaplain, G.D. Singh, J.C. MacLachlan (Eds.) On Growth and Form: Spatio-temporal Pattern Formation in Biology, Wiley Series in Mathematical and Computational Biology, New York, NY, 1999 · Zbl 0932.92006
[6] Chen, S.; Merriman, B.; Osher, S.; Smereka, P., A simple level set method for solving Stefan problems, J. comput. phys., 135, 1, 8-29, (1997) · Zbl 0889.65133
[7] Cristini, V.; Lowengrub, J.S.; Nie, Q., Nonlinear simulation of tumor growth, J. math. biol., 46, 3, 191-224, (2003) · Zbl 1023.92013
[8] J.J. Dongarra, I.S. Duff, D.C. Sorensen, H.A. van der Vorst, Numerical Linear Algebra for High-Performance Computers, Philadelphia, PA (1998), ISBN 0-89871-428-1
[9] Fedkiw, R.P.; Aslam, T.; Merriman, B.; Osher, S., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. comput. phys., 152, 2, 457-492, (1999) · Zbl 0957.76052
[10] H. Frieboes, C.-H. Sun, B. Tromberg, V. Cristini, Diffusional instability as a mechanism for glioblastoma growth and invasion, 2004 (in preparation)
[11] F. Gibou, R. Fedkiw, A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem, J. Comput. Phys. 202 (2) (2005), in press · Zbl 1061.65079
[12] Gibou, F.; Fedkiw, R.; Caflisch, R.; Osher, S., A level set approach for the numerical simulation of dendritic growth, J. sci. comput., 19, 183-199, (2003) · Zbl 1081.74560
[13] Gibou, F.; Fedkiw, R.; Cheng, L.T.; Kang, M., A second order accurate symmetric discretization of the Poisson equation on irregular domains, J. comput. phys., 176, 1, 205-227, (2002) · Zbl 0996.65108
[14] Glimm, J.; Marchesin, D.; McBryan, O., A numerical-method for 2 phase flow with an unstable interface, J. comp. phys., 39, 179-200, (1981) · Zbl 0469.76079
[15] Gonzalez, R.; Woods, R., Digital image processing, ISBN: 0-201-18075-8, (1992), Addison-Wesley Reading, MA
[16] S. Gottlieb, C.W. Shu, Total variation diminishing Runge-Kutta schemes, Math. Comp. 67 (1998) 73-85 · Zbl 0897.65058
[17] Gottlieb, S.; Shu, C.W.; Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM rev., 43, 1, 89-112, (2001) · Zbl 0967.65098
[18] Jiang, G.S.; Peng, D., Weighted ENO schemes for multi-dimensional Hamilton-Jacobi equations, SIAM J. sci. comput., 21, 6, 2126-2143, (2000) · Zbl 0957.35014
[19] Jiang, G.S.; Shu, C.W., Efficient implementation of weighted ENO schemes, J. comput. phys., 126, 2, 202-228, (1996) · Zbl 0877.65065
[20] Kansal, A.A.; Torquato, S.; Harsh, G.R.; Chiocca, E.A.; Deisboeck, T.S., Simulated brain tumor growth dynamics using 3-D cellular automaton, J. theor. biol., 203, 4, 367-382, (2000)
[21] Liu, X.D.; Fedkiw, R.; Kang, M., A boundary condition capturing method for poisson’s equation on irregular domains, J. comput. phys., 160, 1, 151-178, (2000) · Zbl 0958.65105
[22] P. Macklin, Numerical simulation of tumor growth and chemotherapy, M.S. Thesis, University of Minnesota School of Mathematics, September 2003
[23] P. Macklin, J. Lowengrub, Nonlinear simulations of tumor response to chemotherapy, 2004 (in preparation) · Zbl 1067.65111
[24] Maher, E.A.; Furnari, F.B.; Bachoo, R.M.; Rowitch, D.H.; Louis, D.N.; Cavenee, W.K.; DelPinho, R.A., Malignant glioma: genetics and biology of a grave matter, Genes dev., 15, 1311-1333, (2001)
[25] Malladi, R.; Sethian, J.A.; Vemuri, B.C., A fast level set based algorithm for topology-independent shape modeling, J. math. imaging vision, 6, 269-289, (1996)
[26] Malladi, R.; Sethian, J.A.; Vemuri, B.C., Shape modeling with front propagation: a level set approach, IEEE trans. pattern anal. Mach. intell., 17, 2, (1995)
[27] McElwain, D.L.S.; Morris, L.E., Apoptosis as a volume loss mechanism in mathematical models of solid tumor growth, Math. biosci., 39, 147-157, (1978)
[28] Osher, S.; Sethian, J.A., Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. comput. phys., 79, 12-49, (1988) · Zbl 0659.65132
[29] Osher, S.; Fedkiw, R., Level set methods and dynamic implicit surfaces, ISBN: 0-387-95482-1, (2002), Springer New York, NY
[30] Peng, D.; Merriman, B.; Osher, S.; Zhao, H.K.; Kang, M., A PDE-based fast local level set method, J. comput. phys., 155, 410ff, (1999)
[31] Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P., Numerical recipes in C: the art of scientific computing, ISBN: 0-521-43108-5, (1992), Cambridge University Press Cambridge · Zbl 0845.65001
[32] L. Preziosi, Cancer Modeling and Simulation, Boca Raton, LA , 2003, ISBN 1-58488-361-8 · Zbl 1039.92022
[33] S. Ramakrishnan, Department of Pharmacology, University of Minnesota, Personal Communication
[34] Sethian, J.A., Level set methods and fast marching methods, ISBN: 0-521-64557-3, (1999), Cambridge University Press New York, NY · Zbl 0929.65066
[35] Sussman, M.; Fatemi, E., An efficient, interface preserving level set re-distancing algorithm and its application to interfacial incompressible fluid flow, SIAM J. sci. comput., 20, 4, 1165-1191, (1999) · Zbl 0958.76070
[36] Sussman, M.; Fatemi, E.; Smereka, P.; Osher, S., An improved level set method for incompressible two-phase flows, Comput. fluids, 27, 5-6, 663-680, (1998) · Zbl 0967.76078
[37] Zhao, H.; Chan, T.; Merriman, B.; Osher, S., A variational level set approach to multiphase motion, J. comput. phys., 127, 179ff, (1996)
[38] X. Zheng, S.M. Wise, V. Cristini, Nonlinear simulation of tumor necrosis, neo-vascularization and tissue invation via an adaptive finite-element/level-set method, B. Math. Biol. (in press) · Zbl 1334.92214
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