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Article Contents

# Cahn–Hilliard–Brinkman systems for tumour growth

• * Corresponding author: Harald Garcke
• A phase field model for tumour growth is introduced that is based on a Brinkman law for convective velocity fields. The model couples a convective Cahn–Hilliard equation for the evolution of the tumour to a reaction-diffusion-advection equation for a nutrient and to a Brinkman–Stokes type law for the fluid velocity. The model is derived from basic thermodynamical principles, sharp interface limits are derived by matched asymptotics and an existence theory is presented for the case of a mobility which degenerates in one phase leading to a degenerate parabolic equation of fourth order. Finally numerical results describe qualitative features of the solutions and illustrate instabilities in certain situations.

Mathematics Subject Classification: Primary: 35K35; Secondary: 35K57, 35Q92, 35R35, 35C20, 65M60, 92C42.

 Citation:

• Figure 1.  Typical situation for the formal asymptotic analysis

Figure 2.  The tumour and healthy regions $\Omega_T$ and $\Omega_H$

Figure 3.  Schematic sketch of the inner region close to $\Sigma(0)$

Figure 4.  Initial tumour size for initial data $r$: A slightly perturbed sphere

Figure 5.  Comparison of Cahn--Hilliard--Darcy and Cahn--Hilliard--Brinkman models: Tumour at time $t = 12$ for $\beta = 0.1$, left side for the CHD model, right side for the CHB model with $\eta = 10^{-5}$

Figure 6.  Influence of different mobilities: Tumour at time $t = 9$ for $\eta = 10^{-5}$, $\beta = 0.1$ and $\alpha = \rho_S = 2$, but with different mobilities, left $m(\varphi) = \frac{1}{2}(1+\varphi)^2$, middle $m(\varphi) = \epsilon$, right $m(\varphi) = 10^{-3}\epsilon$

Figure 7.  Influence of the adhesion parameter $\beta$: Evolution of the tumour with $m(\varphi) = \tfrac{1}{2}(1+\varphi)^2$ and $\eta = 0.1$, above for $\beta = 0.1$ at time $t = 1, 3, 6, 10$, below for $\beta = 0.01$ at time $t = 1, 1.5, 2, 2.5$

Figure 8.  Influence of viscosiy I: Tumour and velocity for $\beta = 0.01$ at time $t = 2.5$, left for $\eta = 0.1$, right for $\eta = 100$, on top the tumour and below the velocity magnitude

Figure 9.  Influence of viscosity II: Evolution of the tumour at time $t = 1, 3, 6, 10$ with $\beta = 0.1$, $\nu = 0$ and a no-slip boundary condition on the left boundary, on top for $\eta = 0.1$ and below for $\eta = 10$

Figure 10.  Velocity profiles for different viscosities: The velocity magnitude at time $t = 10$ with $\beta = 0.1$, $\nu = 0$ and a no-slip boundary condition on the left boundary, left for $\eta = 0.1$, right for $\eta = 10$

Figure 11.  Influence of viscosity contrast: Tumour at time $t = 10$ with $\beta = 0.1$, $\nu = 0$ and a no-slip b. c. on the left boundary, with $\eta_- = 0.01$, $\eta_+ = 1$; $\eta_- = 1$, $\eta_+ = 0.01$; $\eta_- = 0.01$, $\eta_+ = 10$; $\eta_- = 10$, $\eta_+ = 0.01$

Figure 12.  Influence of initial profile I: Tumour at time $t = 0, 0.3, 1, 1.6$ with $\eta = 100$ and with the initial profile corresponding to $r_1$

Figure 13.  Influence of initial profile II: Tumour at time $t = 0, 0.7, 1.2, 2.6$ with $\eta = 100$ and with the initial profile corresponding to $r_2$

Figure 14.  Influence of initial profile III: Tumour at time $t = 0, 0.5, 1.2, 2.4$ with $\eta = 100$ and with the initial profile corresponding to $r_3$

Figure 15.  Influence of initial profile IV: Tumour at time $t = 0, 0.3, 1.3, 2.3$ with $\eta = 0.01$ and with the initial profile corresponding to $r_4$

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