American Institute of Mathematical Sciences

Advanced
September  2006, 1(3): 399-439. doi: 10.3934/nhm.2006.1.399

Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity

M.A.J Chaplain 1, and  G. Lolas 1,

1. 

The SIMBIOS Centre, Division of Mathematics, University of Dundee, Dundee DD1 4HN, United Kingdom, United Kingdom

Received  February 2006 Revised  June 2006 Published  July 2006

Solid tumours grow through two distinct phases: the avascular and the vascular phase. During the avascular growth phase, the size of the solid tumour is restricted largely by a diffusion-limited nutrient supply and the solid tumour remains localised and grows to a maximum of a few millimetres in diameter. However, during the vascular growth stage the process of cancer invasion of peritumoral tissue can and does take place. A crucial component of tissue invasion is the over-expression by the cancer cells of proteolytic enzyme activity, such as the urokinase-type plasminogen activator (uPA) and matrix metalloproteinases (MMPs). uPA itself initiates the activation of an enzymatic cascade that primarily involves the activation of plasminogen and subsequently its matrix degrading protein plasmin. Degradation of the matrix then enables the cancer cells to migrate through the tissue and subsequently to spread to secondary sites in the body.
    In this paper we consider a relatively simple mathematical model of cancer cell invasion of tissue (extracellular matrix) which focuses on the role of a generic matrix degrading enzyme such as uPA. The model consists of a system of reaction-diffusion-taxis partial differential equations describing the interactions between cancer cells, the matrix degrading enzyme and the host tissue. The results obtained from numerical computations carried out on the model equations produce dynamic, heterogeneous spatio-temporal solutions and demonstrate the ability of a rather simple model to produce complicated dynamics, all of which are associated with tumour heterogeneity and cancer cell progression and invasion.
Keywords: haptotaxis, spatio-temporal heterogeneity., cancer invasion of tissue, chemotaxis, matrix degrading enzyme.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: M.A.J Chaplain, G. Lolas. Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity. Networks & Heterogeneous Media, 2006, 1 (3) : 399-439. doi: 10.3934/nhm.2006.1.399
[1]

Thomas Hillen, Jeffery Zielinski, Kevin J. Painter. Merging-emerging systems can describe spatio-temporal patterning in a chemotaxis model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2513-2536. doi: 10.3934/dcdsb.2013.18.2513
[2]

Cicely K. Macnamara, Mark A. J. Chaplain. Spatio-temporal models of synthetic genetic oscillators. Mathematical Biosciences & Engineering, 2017, 14 (1) : 249-262. doi: 10.3934/mbe.2017016
[3]

Francesca Sapuppo, Elena Umana, Mattia Frasca, Manuela La Rosa, David Shannahoff-Khalsa, Luigi Fortuna, Maide Bucolo. Complex spatio-temporal features in meg data. Mathematical Biosciences & Engineering, 2006, 3 (4) : 697-716. doi: 10.3934/mbe.2006.3.697
[4]

Noura Azzabou, Nikos Paragios. Spatio-temporal speckle reduction in ultrasound sequences. Inverse Problems & Imaging, 2010, 4 (2) : 211-222. doi: 10.3934/ipi.2010.4.211
[5]

Xiaoying Chen, Chong Zhang, Zonglin Shi, Weidong Xiao. Spatio-temporal keywords queries in HBase. Big Data & Information Analytics, 2016, 1 (1) : 81-91. doi: 10.3934/bdia.2016.1.81
[6]

Hirofumi Izuhara, Shunsuke Kobayashi. Spatio-temporal coexistence in the cross-diffusion competition system. Discrete & Continuous Dynamical Systems - S, 2019  doi: 10.3934/dcdss.2020228
[7]

Pietro-Luciano Buono, Daniel C. Offin. Instability criterion for periodic solutions with spatio-temporal symmetries in Hamiltonian systems. Journal of Geometric Mechanics, 2017, 9 (4) : 439-457. doi: 10.3934/jgm.2017017
[8]

Buddhi Pantha, Judy Day, Suzanne Lenhart. Investigating the effects of intervention strategies in a spatio-temporal anthrax model. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1607-1622. doi: 10.3934/dcdsb.2019242
[9]

Jingdong Wei, Jiangbo Zhou, Wenxia Chen, Zaili Zhen, Lixin Tian. Traveling waves in a nonlocal dispersal epidemic model with spatio-temporal delay. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2853-2886. doi: 10.3934/cpaa.2020125
[10]

Niklas Kolbe, Nikolaos Sfakianakis, Christian Stinner, Christina Surulescu, Jonas Lenz. Modeling multiple taxis: Tumor invasion with phenotypic heterogeneity, haptotaxis, and unilateral interspecies repellence. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020284
[11]

Wenjia Jing, Panagiotis E. Souganidis, Hung V. Tran. Large time average of reachable sets and Applications to Homogenization of interfaces moving with oscillatory spatio-temporal velocity. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 915-939. doi: 10.3934/dcdss.2018055
[12]

Raimund BÜrger, Gerardo Chowell, Elvis GavilÁn, Pep Mulet, Luis M. Villada. Numerical solution of a spatio-temporal gender-structured model for hantavirus infection in rodents. Mathematical Biosciences & Engineering, 2018, 15 (1) : 95-123. doi: 10.3934/mbe.2018004
[13]

Lin Wang, James Watmough, Fang Yu. Bifurcation analysis and transient spatio-temporal dynamics for a diffusive plant-herbivore system with Dirichlet boundary conditions. Mathematical Biosciences & Engineering, 2015, 12 (4) : 699-715. doi: 10.3934/mbe.2015.12.699
[14]

Jinling Zhou, Yu Yang. Traveling waves for a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1719-1741. doi: 10.3934/dcdsb.2017082
[15]

Rui Xu. Global convergence of a predator-prey model with stage structure and spatio-temporal delay. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 273-291. doi: 10.3934/dcdsb.2011.15.273
[16]

Zhi-Xian Yu, Rong Yuan. Traveling wave fronts in reaction-diffusion systems with spatio-temporal delay and applications. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 709-728. doi: 10.3934/dcdsb.2010.13.709
[17]

Zelik S.. Formally gradient reaction-diffusion systems in Rn have zero spatio-temporal topological. Conference Publications, 2003, 2003 (Special) : 960-966. doi: 10.3934/proc.2003.2003.960
[18]

Rodrigo A. Garrido, Ivan Aguirre. Emergency logistics for disaster management under spatio-temporal demand correlation: The earthquakes case. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2369-2387. doi: 10.3934/jimo.2019058
[19]

Jan Giesselmann, Niklas Kolbe, Mária Lukáčová-Medvi${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over d} }}$ová, Nikolaos Sfakianakis. Existence and uniqueness of global classical solutions to a two dimensional two species cancer invasion haptotaxis model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4397-4431. doi: 10.3934/dcdsb.2018169
[20]

Janet Dyson, Eva Sánchez, Rosanna Villella-Bressan, Glenn F. Webb. An age and spatially structured model of tumor invasion with haptotaxis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 45-60. doi: 10.3934/dcdsb.2007.8.45

2019 Impact Factor: 1.053

Tools

Metrics

  • PDF downloads (97)
  • HTML views (0)
  • Cited by (116)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences