Travelling Wave Analysis of a Nonlinear Diffusion Equation

  • Hanadi Alzubadi

Student thesis: Doctoral ThesisDoctor of Philosophy


Cell monolayers are a widely used tool in tumour and tissue repair studies. In many cell lines, cell proliferation is contact-inhibited such that the cell population reaches confluence at long times. Given suitable initial conditions, moving fronts of proliferation and migration can be observed prior to confluence, with two key measurable quantities being the width and propagation speed of the proliferating region. In this thesis we consider the continuum limit of a model of an off-lattice, cell-based simulation of a contact-inhibited cell monolayer. Numerical solutions of the continuum model, which can be formulated as a nonlinear diffusion free boundary (nonlinear Stefan) problem, indicate a travelling wave behaviour that is subsequently investigated using a travelling wave analysis. Considering first a simplified linear problem, we perform a travelling wave analysis that identifies a small parameter, the natural logarithm of the compression ratio of cells at which contact inhibition stops proliferation, from which asymptotic expressions for the wave speed and proliferating rim width are derived. Subsequently, we use perturbation theory to derive approximations for the wave speed and proliferating rim width in the case of nonlinear diffusion coefficients that represents the continuum limit of a discrete model in which cells interact via a nonlinear force law. The related results depend on which force law in which diffusion coefficient type is considered. We investigate the solution of nonlinear diffusion models for a number of different inter- cellular force laws (e.g. Hertz, linear, cubic) where the power of the force law has a profound effect on the qualitative behaviour.
Date of Award2019
Original languageEnglish
SupervisorPhilip Murray (Supervisor) & Fordyce Davidson (Supervisor)


  • Travelling wave
  • Contact inhibition
  • Nonlinear diffusion

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