May  2007, 7(3): 581-604. doi: 10.3934/dcdsb.2007.7.581

Nonlinear three-dimensional simulation of solid tumor growth

1. 

Department of Mathematics, University of California, Irvine, CA 92697, United States

2. 

School of Health Information Sciences, University of Texas Health Science Center, Houston, TX 77030, United States

3. 

Center for Mathematical and Computational Biology, Department of Mathematics, University of California, Irvine, CA 92697-3875

4. 

Department of Mathematics, Center for Mathematical and Computational Biology, University of California, Irvine, CA 92697, United States

Received  September 2006 Revised  January 2007 Published  February 2007

We present a new, adaptive boundary integral method to simulate solid tumor growth in 3-d. We use a reformulation of a classical model that accounts for cell-proliferation, apoptosis, cell-to-cell and cell-to-matrix adhesion. The 3-d method relies on accurate discretizations of singular surface integrals, a spatial rescaling and the use of an adaptive surface mesh. The discretized boundary integral equations are solved iteratively using GMRES and a discretized version of the Dirichlet-Neumann map, formulated in terms of a vector potential, is used to determine the normal velocity of the tumor surface. Explicit time stepping is used to update the tumor surface. We present simulations of the nonlinear evolution of growing tumors. At early times, good agreement is obtained between the results of a linear stability analysis and nonlinear simulations. At later times, linear theory is found to overpredict the growth of perturbations. Nonlinearity results in mode creation and interaction that leads to the formation of dimples and the tumor surface buckles inwards. The morphologic instability allows the tumor to increase its surface area, relative to its volume, thereby allowing the cells in the tumor bulk greater access to nutrient. This in turn allows the tumor to overcome the diffusional limitations on growth and to grow to larger sizes than would be possible if the tumor were spherical. Consequently, instability provides a means for avascular tumor invasion.
Citation: Xiangrong Li, Vittorio Cristini, Qing Nie, John S. Lowengrub. Nonlinear three-dimensional simulation of solid tumor growth. Discrete and Continuous Dynamical Systems - B, 2007, 7 (3) : 581-604. doi: 10.3934/dcdsb.2007.7.581
[1]

Thomas Y. Hou, Pingwen Zhang. Convergence of a boundary integral method for 3-D water waves. Discrete and Continuous Dynamical Systems - B, 2002, 2 (1) : 1-34. doi: 10.3934/dcdsb.2002.2.1

[2]

Guoliang Ju, Can Chen, Rongliang Chen, Jingzhi Li, Kaitai Li, Shaohui Zhang. Numerical simulation for 3D flow in flow channel of aeroengine turbine fan based on dimension splitting method. Electronic Research Archive, 2020, 28 (2) : 837-851. doi: 10.3934/era.2020043

[3]

Yi Shi, Kai Bao, Xiao-Ping Wang. 3D adaptive finite element method for a phase field model for the moving contact line problems. Inverse Problems and Imaging, 2013, 7 (3) : 947-959. doi: 10.3934/ipi.2013.7.947

[4]

Thomas Y. Hou, Zuoqiang Shi. Dynamic growth estimates of maximum vorticity for 3D incompressible Euler equations and the SQG model. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1449-1463. doi: 10.3934/dcds.2012.32.1449

[5]

Nikodem J. Poplawski, Abbas Shirinifard, Maciej Swat, James A. Glazier. Simulation of single-species bacterial-biofilm growth using the Glazier-Graner-Hogeweg model and the CompuCell3D modeling environment. Mathematical Biosciences & Engineering, 2008, 5 (2) : 355-388. doi: 10.3934/mbe.2008.5.355

[6]

Michael V. Klibanov, Dinh-Liem Nguyen, Loc H. Nguyen, Hui Liu. A globally convergent numerical method for a 3D coefficient inverse problem with a single measurement of multi-frequency data. Inverse Problems and Imaging, 2018, 12 (2) : 493-523. doi: 10.3934/ipi.2018021

[7]

Shijin Ding, Zhilin Lin, Dongjuan Niu. Boundary layer for 3D plane parallel channel flows of nonhomogeneous incompressible Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4579-4596. doi: 10.3934/dcds.2020193

[8]

A. V. Fursikov. Stabilization for the 3D Navier-Stokes system by feedback boundary control. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 289-314. doi: 10.3934/dcds.2004.10.289

[9]

Anna Kostianko, Sergey Zelik. Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions. Communications on Pure and Applied Analysis, 2015, 14 (5) : 2069-2094. doi: 10.3934/cpaa.2015.14.2069

[10]

Mei Wang, Zilai Li, Zhenhua Guo. Global weak solution to 3D compressible flows with density-dependent viscosity and free boundary. Communications on Pure and Applied Analysis, 2017, 16 (1) : 1-24. doi: 10.3934/cpaa.2017001

[11]

Marcelo M. Disconzi, Igor Kukavica. A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid. Evolution Equations and Control Theory, 2019, 8 (3) : 503-542. doi: 10.3934/eect.2019025

[12]

Alessio Falocchi, Filippo Gazzola. Regularity for the 3D evolution Navier-Stokes equations under Navier boundary conditions in some Lipschitz domains. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1185-1200. doi: 10.3934/dcds.2021151

[13]

Fabrice Delbary, Kim Knudsen. Numerical nonlinear complex geometrical optics algorithm for the 3D Calderón problem. Inverse Problems and Imaging, 2014, 8 (4) : 991-1012. doi: 10.3934/ipi.2014.8.991

[14]

Boyan Jonov, Thomas C. Sideris. Global and almost global existence of small solutions to a dissipative wave equation in 3D with nearly null nonlinear terms. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1407-1442. doi: 10.3934/cpaa.2015.14.1407

[15]

Xiaoyu Chen, Jijie Zhao, Qian Zhang. Global existence of weak solutions for the 3D axisymmetric chemotaxis-Navier-Stokes equations with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2022, 42 (9) : 4489-4522. doi: 10.3934/dcds.2022062

[16]

Niklas Hartung. Efficient resolution of metastatic tumor growth models by reformulation into integral equations. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 445-467. doi: 10.3934/dcdsb.2015.20.445

[17]

Shihe Xu. Analysis of a delayed free boundary problem for tumor growth. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 293-308. doi: 10.3934/dcdsb.2011.15.293

[18]

Yaodan Huang, Zhengce Zhang, Bei Hu. Bifurcation from stability to instability for a free boundary tumor model with angiogenesis. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2473-2510. doi: 10.3934/dcds.2019105

[19]

Yong Zhou. Remarks on regularities for the 3D MHD equations. Discrete and Continuous Dynamical Systems, 2005, 12 (5) : 881-886. doi: 10.3934/dcds.2005.12.881

[20]

Hyeong-Ohk Bae, Bum Ja Jin. Estimates of the wake for the 3D Oseen equations. Discrete and Continuous Dynamical Systems - B, 2008, 10 (1) : 1-18. doi: 10.3934/dcdsb.2008.10.1

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (134)
  • HTML views (0)
  • Cited by (40)

[Back to Top]