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Generalized Cahn-Hilliard equation for biological applications.

Evgeniy Khain1, Leonard M Sander

  • 1Department of Physics, Oakland University, Rochester, Michigan 48309, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|July 23, 2008
PubMed
Summary

This study models cell invasion fronts using a generalized Cahn-Hilliard equation. It finds good agreement between continuum and discrete models for both subcritical and supercritical adhesion regimes.

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Area of Science:

  • Mathematical modeling
  • Biophysics
  • Statistical physics

Background:

  • Cell invasion dynamics are crucial in wound healing and disease.
  • Previous discrete models captured cell movement, proliferation, and adhesion.
  • A continuum approach is needed for a broader understanding.

Purpose of the Study:

  • To develop and analyze a continuum model for cell invasion fronts.
  • To investigate front velocity selection in subcritical adhesion.
  • To explore transient behaviors in supercritical adhesion.

Main Methods:

  • Utilized a generalized Cahn-Hilliard equation (GCH) with a proliferation term.
  • Analyzed propagating fronts in the subcritical adhesion regime.
  • Investigated relaxation dynamics of the Cahn-Hilliard equation without proliferation for supercritical adhesion.

Main Results:

  • The GCH model accurately predicts front velocity selection, matching numerical solutions.
  • Subcritical adhesion yields propagating 'pulled' fronts, akin to Fisher-Kolmogorov models.
  • Supercritical adhesion shows complex transient behavior with secondary peaks, exhibiting self-similar relaxation dynamics.

Conclusions:

  • The continuum GCH model effectively describes cell invasion fronts.
  • Both discrete and continuum models show good agreement across different adhesion regimes.
  • The study provides insights into the physics of collective cell motion and pattern formation.