# American Institute of Mathematical Sciences

• Previous Article
Characterization of the dynamic behavior of nonlinear biosystems in the presence of model uncertainty using singular invariance PDEs: Application to immobilized enzyme and cell bioreactors
• MBE Home
• This Issue
• Next Article
Mathematically modeling PCR: An asymptotic approximation with potential for optimization
2010, 7(2): 385-400. doi: 10.3934/mbe.2010.7.385

## Diffusion-limited tumour growth: Simulations and analysis

 1 Center for Models of Life, Niels Bohr Institute, Blegdamsvej 17, 2200 Copenhagen O, Denmark 2 H. Lee Moffitt Cancer Center & Research Institute, Integrated Mathematical Oncology, 12902 Magnolia Drive, Tampa, FL 33612, United States

Received  June 2009 Revised  September 2009 Published  April 2010

The morphology of solid tumours is known to be affected by the background oxygen concentration of the tissue in which the tumour grows, and both computational and experimental studies have suggested that branched tumour morphology in low oxygen concentration is caused by diffusion-limited growth. In this paper we present a simple hybrid cellular automaton model of solid tumour growth aimed at investigating this phenomenon. Simulation results show that for high consumption rates (or equivalently low oxygen concentrations) the tumours exhibit branched morphologies, but more importantly the simplicity of the model allows for an analytic approach to the problem. By applying a steady-state assumption we derive an approximate solution of the oxygen equation, which closely matches the simulation results. Further, we derive a dispersion relation which reveals that the average branch width in the tumour depends on the width of the active rim, and that a smaller active rim gives rise to thinner branches. Comparison between the prediction of the stability analysis and the results from the simulations shows good agreement between theory and simulation.
Citation: Philip Gerlee, Alexander R. A. Anderson. Diffusion-limited tumour growth: Simulations and analysis. Mathematical Biosciences & Engineering, 2010, 7 (2) : 385-400. doi: 10.3934/mbe.2010.7.385
 [1] Elena Izquierdo-Kulich, Margarita Amigó de Quesada, Carlos Manuel Pérez-Amor, Magda Lopes Texeira, José Manuel Nieto-Villar. The dynamics of tumor growth and cells pattern morphology. Mathematical Biosciences & Engineering, 2009, 6 (3) : 547-559. doi: 10.3934/mbe.2009.6.547 [2] Matthias Ebenbeck, Harald Garcke, Robert Nürnberg. Cahn–Hilliard–Brinkman systems for tumour growth. Discrete and Continuous Dynamical Systems - S, 2021, 14 (11) : 3989-4033. doi: 10.3934/dcdss.2021034 [3] Russell Betteridge, Markus R. Owen, H.M. Byrne, Tomás Alarcón, Philip K. Maini. The impact of cell crowding and active cell movement on vascular tumour growth. Networks and Heterogeneous Media, 2006, 1 (4) : 515-535. doi: 10.3934/nhm.2006.1.515 [4] Vicent Caselles. An existence and uniqueness result for flux limited diffusion equations. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1151-1195. doi: 10.3934/dcds.2011.31.1151 [5] Andrea Signori. Penalisation of long treatment time and optimal control of a tumour growth model of Cahn–Hilliard type with singular potential. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2519-2542. doi: 10.3934/dcds.2020373 [6] R. P. Gupta, Shristi Tiwari, Shivam Saxena. The qualitative behavior of a plankton-fish interaction model with food limited growth rate and non-constant fish harvesting. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2791-2815. doi: 10.3934/dcdsb.2021160 [7] Benjamin Söllner, Oliver Junge. A convergent Lagrangian discretization for $p$-Wasserstein and flux-limited diffusion equations. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4227-4256. doi: 10.3934/cpaa.2020190 [8] Sarah Bailey Frick. Limited scope adic transformations. Discrete and Continuous Dynamical Systems - S, 2009, 2 (2) : 269-285. doi: 10.3934/dcdss.2009.2.269 [9] Xiaofeng Ren. Shell structure as solution to a free boundary problem from block copolymer morphology. Discrete and Continuous Dynamical Systems, 2009, 24 (3) : 979-1003. doi: 10.3934/dcds.2009.24.979 [10] H.M. Byrne, S.M. Cox, C.E. Kelly. Macrophage-tumour interactions: In vivo dynamics. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 81-98. doi: 10.3934/dcdsb.2004.4.81 [11] John R. King, Judith Pérez-Velázquez, H.M. Byrne. Singular travelling waves in a model for tumour encapsulation. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 195-230. doi: 10.3934/dcds.2009.25.195 [12] Marek Bodnar, Monika Joanna Piotrowska, Urszula Foryś, Ewa Nizińska. Model of tumour angiogenesis -- analysis of stability with respect to delays. Mathematical Biosciences & Engineering, 2013, 10 (1) : 19-35. doi: 10.3934/mbe.2013.10.19 [13] Zhaosheng Feng, Goong Chen. Traveling wave solutions in parametric forms for a diffusion model with a nonlinear rate of growth. Discrete and Continuous Dynamical Systems, 2009, 24 (3) : 763-780. doi: 10.3934/dcds.2009.24.763 [14] Matthieu Alfaro, Thomas Giletti. When fast diffusion and reactive growth both induce accelerating invasions. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3011-3034. doi: 10.3934/cpaa.2019135 [15] Yilong Wang, Xuande Zhang. On a parabolic-elliptic chemotaxis-growth system with nonlinear diffusion. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 321-328. doi: 10.3934/dcdss.2020018 [16] Benedetto Bozzini, Deborah Lacitignola, Ivonne Sgura. Morphological spatial patterns in a reaction diffusion model for metal growth. Mathematical Biosciences & Engineering, 2010, 7 (2) : 237-258. doi: 10.3934/mbe.2010.7.237 [17] Anotida Madzvamuse, Raquel Barreira. Domain-growth-induced patterning for reaction-diffusion systems with linear cross-diffusion. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2775-2801. doi: 10.3934/dcdsb.2018163 [18] Kolade M. Owolabi, Kailash C. Patidar, Albert Shikongo. Efficient numerical method for a model arising in biological stoichiometry of tumour dynamics. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 591-613. doi: 10.3934/dcdss.2019038 [19] Hong Zhou, M. Gregory Forest. Anchoring distortions coupled with plane Couette & Poiseuille flows of nematic polymers in viscous solvents: Morphology in molecular orientation, stress & flow. Discrete and Continuous Dynamical Systems - B, 2006, 6 (2) : 407-425. doi: 10.3934/dcdsb.2006.6.407 [20] Macarena Boix, Begoña Cantó. Using wavelet denoising and mathematical morphology in the segmentation technique applied to blood cells images. Mathematical Biosciences & Engineering, 2013, 10 (2) : 279-294. doi: 10.3934/mbe.2013.10.279

2018 Impact Factor: 1.313