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Biphasic behaviour in malignant invasion.

Ben P Marchant1, John Norbury, Helen M Byrne

  • 1Rothamsted Research, Harpenden, Hertfordshire AL5 2JQ, UK. ben.marchant@bbsrc.ac.uk

Mathematical Medicine and Biology : a Journal of the IMA
|April 22, 2006
PubMed
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This study develops a mathematical model for malignant invasion, revealing how tumor cells spread into surrounding tissues. The model explains the complex relationship between invasion speed and tissue density, crucial for understanding cancer progression.

Area of Science:

  • Mathematical Biology
  • Cancer Research
  • Theoretical Oncology

Background:

  • Malignant invasion is key to tumor growth, enabling cancer cells to spread.
  • Tumor cell invasion is influenced by protease production and surrounding tissue matrix composition.
  • Experimental studies show a biphasic invasion speed dependence on tissue density and protease levels.

Purpose of the Study:

  • To develop a theoretical model of malignant invasion.
  • To analyze the behavior of tumor cells invading surrounding normal tissue.
  • To explain the biphasic invasion speed dependence observed in experimental studies.

Main Methods:

  • Developed a theoretical model using two coupled partial differential equations.
  • Employed numerical methods to analyze steady travelling wave solutions.

Related Experiment Videos

  • Utilized non-standard phase plane methods to study ordinary differential equations derived from wave solutions.

    • Main Results:

    • The model exhibits stable, steady travelling wave solutions, both smooth and discontinuous.
    • Focuses on biologically relevant discontinuous solutions with a jump in tumor cell concentration.
    • Successfully reproduces the biphasic invasion speed dependence on surrounding tissue density.

    Conclusions:

    • The theoretical model provides insights into the mechanisms of malignant invasion.
    • Discontinuous solutions, characterized by a shock, are crucial for understanding invasion dynamics.
    • The model explains how tissue density influences tumor cell invasion speed through phase plane analysis.