A Cahn‐Hilliard–type equation with application to tumor growth dynamics

Abstract

We consider a Cahn-Hilliard-type equation with degenerate mobility and single-well potential of Lennard-Jones type. This equation models the evolution and growth of biological cells such as solid tumors. The degeneracy set of the mobility and the singularity set of the cellular potential do not coincide, and the absence of cells is an unstable equilibrium configuration of the potential. This feature introduces a nontrivial difference with respect to the Cahn-Hilliard equation analyzed in the literature. We give existence results for different classes of weak solutions. Moreover, we formulate a continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality. We prove the existence and uniqueness of the discrete solution for any spatial dimension together with the convergence to the weak solution for spatial dimension d = 1. We present simulation results in 1 and 2 space dimensions. We also study the dynamics of the spinodal decomposition and the growth and scaling laws of phase ordering dynamics. In this case, we find similar results to the ones obtained in standard phase ordering dynamics and we highlight the fact that the asymptotic behavior of the solution is dominated by the mechanism of growth by bulk diffusion. KEYWORDScontinuous Galerkin finite element approximation, degenerate Cahn-Hilliard equation, single well potential, tumor growth models 7598

Relaxation of the Cahn–Hilliard equation with singular single-well potential and degenerate mobility

“…Elliott and Garcke [18] prove this result using the definition of the regularized entropy and by a contradiction argument. For single-well potential, Agosti et al [2] used a reasoning on the measure of the set of solutions outside the set 0 ≤ n < 1 and find contradictions with the boundedness of the entropy. This is the route we follow here.…”

“…∂n ∂ν = ∂ (−γ∆n + ψ ′ (n)) ∂ν = 0 on ∂Ω × (0, +∞), (2) where ν is the outward normal vector to the boundary ∂Ω.…”

“…The typical expression in the applications we have in mind is b(n) = n(1 − n) 2 . Consequently, when considered as transport equations, both (1) and (3) impose formally the property that 0 ≤ n ≤ 1.…”

Bounded solutions and their asymptotics for a doubly nonlinear Cahn–Hilliard system

“…Besides being a fundamental contribution to Materials Science, the C-H system has had considerable success in many other branches of Science and Engineering where segregation of a diffusant leads to pattern formation, such as population dynamics [20], image processing [6], dynamics for mixtures of fluids [16], tumor modelling [1,12,13], to name a few.…”

Numerical error analysis for nonsymmetric interior penalty discontinuous Galerkin method of Cahn–Hilliard equation

“…Since then, the spectrum of applications for this model has been extended to a variety of scientific and engineering fields. The Cahn-Hilliard equation is popular in modeling spinodal decomposition [2], wettability [3], diblock copolymer [4], tumor growth [2,5], and image inpainting [6].…”