Diffusion-limited tumour growth: Simulations and analysis

Previous Article
Mathematically modeling PCR: An asymptotic approximation with potential for optimization
MBE Home
This Issue
Next 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

2010, 7(2): 385-400. doi: 10.3934/mbe.2010.7.385

Diffusion-limited tumour growth: Simulations and analysis

Philip Gerlee ^1, and Alexander R. A. Anderson ^2,

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

Abstract
Related Papers

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.

Keywords: Tumour morphology., Diffusion-limited growth.

Mathematics Subject Classification: Primary: 82B24; Secondary: 37B1.

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]	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 & Heterogeneous Media, 2006, 1 (4) : 515-535. doi: 10.3934/nhm.2006.1.515
[3]	Vicent Caselles. An existence and uniqueness result for flux limited diffusion equations. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1151-1195. doi: 10.3934/dcds.2011.31.1151
[4]	H.M. Byrne, S.M. Cox, C.E. Kelly. Macrophage-tumour interactions: In vivo dynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 81-98. doi: 10.3934/dcdsb.2004.4.81
[5]	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
[6]	John R. King, Judith Pérez-Velázquez, H.M. Byrne. Singular travelling waves in a model for tumour encapsulation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 195-230. doi: 10.3934/dcds.2009.25.195
[7]	Sarah Bailey Frick. Limited scope adic transformations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 269-285. doi: 10.3934/dcdss.2009.2.269
[8]	Zhaosheng Feng, Goong Chen. Traveling wave solutions in parametric forms for a diffusion model with a nonlinear rate of growth. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 763-780. doi: 10.3934/dcds.2009.24.763
[9]	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
[10]	Yilong Wang, Xuande Zhang. On a parabolic-elliptic chemotaxis-growth system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 321-328. doi: 10.3934/dcdss.2020018
[11]	Matthieu Alfaro, Thomas Giletti. When fast diffusion and reactive growth both induce accelerating invasions. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3011-3034. doi: 10.3934/cpaa.2019135
[12]	Xiaofeng Ren. Shell structure as solution to a free boundary problem from block copolymer morphology. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 979-1003. doi: 10.3934/dcds.2009.24.979
[13]	Anotida Madzvamuse, Raquel Barreira. Domain-growth-induced patterning for reaction-diffusion systems with linear cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2775-2801. doi: 10.3934/dcdsb.2018163
[14]	Kolade M. Owolabi, Kailash C. Patidar, Albert Shikongo. Efficient numerical method for a model arising in biological stoichiometry of tumour dynamics. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 591-613. doi: 10.3934/dcdss.2019038
[15]	Tracy L. Stepien, Erica M. Rutter, Yang Kuang. A data-motivated density-dependent diffusion model of in vitro glioblastoma growth. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1157-1172. doi: 10.3934/mbe.2015.12.1157
[16]	Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Regularity of global attractors for reaction-diffusion systems with no more than quadratic growth. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1899-1908. doi: 10.3934/dcdsb.2017113
[17]	Tiberiu Harko, Man Kwong Mak. Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach. Mathematical Biosciences & Engineering, 2015, 12 (1) : 41-69. doi: 10.3934/mbe.2015.12.41
[18]	Linfeng Mei, Xiaoyan Zhang. On a nonlocal reaction-diffusion-advection system modeling phytoplankton growth with light and nutrients. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 221-243. doi: 10.3934/dcdsb.2012.17.221
[19]	Junping Shi, Jimin Zhang, Xiaoyan Zhang. Stability and asymptotic profile of steady state solutions to a reaction-diffusion pelagic-benthic algae growth model. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2325-2347. doi: 10.3934/cpaa.2019105
[20]	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 & Continuous Dynamical Systems - B, 2006, 6 (2) : 407-425. doi: 10.3934/dcdsb.2006.6.407

2018 Impact Factor: 1.313

American Institute of Mathematical Sciences