A sharp-front moving boundary model for malignant invasion

Abstract

We analyse a novel mathematical model of malignant invasion which takes the form of a two-phase moving boundary problem describing the invasion of a population of malignant cells into a population of background tissue, such as skin. Cells in both populations undergo diffusive migration and logistic proliferation. The interface between the two populations moves according to a two-phase Stefan condition.Unlike many reaction-diffusion models of malignant invasion, the moving boundary model explicitly describes the motion of the sharp front between the cancer and surrounding tissues without needing to introduce degenerate nonlinear diffusion.Numerical simulations suggest the model gives rise to very interesting travelling wave solutions that move with speed c, and the model supports both malignant invasion and malignant retreat, where the travelling wave can move in either the positive or negative x-directions. Unlike the well-studied Fisher-Kolmogorov and Porous-Fisher models where travelling waves move with a minimum wave speed c ≥ c * > 0, the moving boundary model leads to travelling wave solutions with |c| < c * * . We interpret these travelling wave solutions in the phase plane and show

Semi-infinite travelling waves arising in a general reaction–diffusion Stefan model

“…While other methods have been proposed to mitigate these issues in related reaction-diffusion models [29], it is unclear how these techniques can be used in the RDS modelling framework. In addition to negative reaction kinetics, other extensions can be incorporated into the RDS modelling framework, including nonlocal reactions [31] and multiple-species reaction-diffusion equations [32,33]. Finally, it should be noted that a formal stability analysis of the travelling wave solutions presented in this work (cf.…”

Travelling waves in a free boundary mechanobiological model of an epithelial tissue

“…The second, φ = βη/(4k), is the ratio of the proliferation rate, β, and mechanical relaxation rate, that depends on the cell stiffness k and mobility coefficient η. Eq. ( 4) governs the evolution of the free boundary due to mechanical relaxation and mass conservation but can be thought of as a nonlinear analogue of a Stefan condition [14,15,13]. Eqs.…”

“…One way of overcoming the lack of a well-defined front is to incorporate degenerate nonlinear diffusion, as in the Porous-Fisher equation [9,10,11,12]. An alternative approach to obtain travelling wave solutions with a well-defined front is to re-formulate the Fisher-KPP and Porous-Fisher models as moving boundary problems with a Stefan condition at the moving boundary [13,14,15,16].…”