Journal of Mathematical Biology

, Volume 63, Issue 1, pp 141–171 | Cite as

Mathematical modeling of cancer cell invasion of tissue: biological insight from mathematical analysis and computational simulation

  • Vivi AndasariEmail author
  • Alf Gerisch
  • Georgios Lolas
  • Andrew P. South
  • Mark A. J. Chaplain


The ability of cancer cells to break out of tissue compartments and invade locally gives solid tumours a defining deadly characteristic. One of the first steps of invasion is the remodelling of the surrounding tissue or extracellular matrix (ECM) and a major part of this process is the over-expression of proteolytic enzymes, such as the urokinase-type plasminogen activator (uPA) and matrix metalloproteinases (MMPs), by the cancer cells to break down ECM proteins. Degradation of the matrix enables the cancer cells to migrate through the tissue and subsequently to spread to secondary sites in the body, a process known as metastasis. In this paper we undertake an analysis of a mathematical model of cancer cell invasion of tissue, or ECM, which focuses on the role of the urokinase plasminogen activation system. The model consists of a system of five reaction-diffusion-taxis partial differential equations describing the interactions between cancer cells, uPA, uPA inhibitors, plasmin and the host tissue. Cancer cells react chemotactically and haptotactically to the spatio-temporal effects of the uPA system. The results obtained from computational simulations carried out on the model equations produce dynamic heterogeneous spatio-temporal solutions and using linear stability analysis we show that this is caused by a taxis-driven instability of a spatially homogeneous steady-state. Finally we consider the biological implications of the model results, draw parallels with clinical samples and laboratory based models of cancer cell invasion using three-dimensional invasion assay, and go on to discuss future development of the model.


Cancer invasion uPA system Haptotaxis Spatio-temporal heterogeneity Organotypic culture Invasion index 

Mathematics Subject Classification (2000)

92C17 92-08 92C15 35K57 65M20 


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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Vivi Andasari
    • 1
    Email author
  • Alf Gerisch
    • 2
  • Georgios Lolas
    • 3
  • Andrew P. South
    • 4
  • Mark A. J. Chaplain
    • 1
  1. 1.Division of MathematicsUniversity of DundeeDundeeScotland
  2. 2.Technische Universität Darmstadt, Fachbereich MathematikDarmstadtGermany
  3. 3.Department of Chemical EngineeringNational Technical University of AthensAigaleo, AthensGreece
  4. 4.Department of Surgery and Molecular OncologyNinewells Hospital, University of DundeeDundeeScotland

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