A tumor growth model with autophagy: The reaction-(cross-)diffusion system and its free boundary limit

Figure 1.  Illustration of the ODE model. The two gray squares denote the densities of normal cells $ n_1 $ and autophagic cells $ n_2 $. And the green square denotes the nutrient concentration $ c $. Normal cells change into autophagic cells with a transition rate $ K_1 $. And autophagic cells change into normal cells with a transition rate $ K_2 $. Normal cells and autophagic cells both grow with a net growth rate $ G $. Dashed lines show the consumption or supply of nutrients by cells. Both normal cells and autophagic cells consume nutrients with a rate $ \psi $. The key assumption is that autophagic cells will "kill" themselves with an extra death rate $ D $ to provide nutrients with a supply rate $ a $. Nutrients are added with a rate $ \lambda c_B $ and discharged with a rate $ \lambda c $

Figure 2.  Densities of cells at time $ t = 0,1.5,3 $ with a heterogeneous initial density fraction. Blue dash-dotted line: the total density $ n $. Red solid line: the density of autophagic cells $ n_2 $. Orange dashed line: the theoretical value for the density of autophagic cells in the "well-mixed" limit. Parameters: $ G(c) = gc,\psi(c) = c,K_1 = K_2 = 1,g = 1,a = 0.4,D = 0.3 $

Figure 3.  Evolution of $ ||\mu(\cdot,t)-\mu^*||_{L^{2n}({\Omega(t)})} $ with respect to time under different parameter regimes. Parameters: $ \gamma = 80 $, $ G(c) = gc,\psi(c) = c $ with $ g = 1,a = 0.5 $ and $ c_B = 1 $. Here the threshold value of nutrient concentration $ c_0 = 0.5 $

Figure 4.  Graph of nutrient concentration $ c $ and moving speed of the boundary for different $ \mu $. When $ \mu = 1 $ there is no autophagic cells, when $ \mu = 0 $ all cells are autophagic cells. In the presence of autophagy, the nutrient is more sufficient and the growth rate tends to be linear in $ R $, which leads to an exponential growth of $ R $. Parameters: $ c_B = 1,a = 0.5,g = 1,D = 0.3 $

Figure 5.  Graph of pressure $ p $ for cell density model for $ \gamma = 80 $. Left: $ ga>D $. Right $ ga<D $, there is a necrotic core in the middle

Figure 6.  Plots of the total density $ n $ and the pressure $ p $ for different $ \gamma $ at $ t = 1 $. Left: $ n $. Right $ p $. $ \gamma = \infty $ stands for the analytical solution of the free boundary problem. As $ \gamma $ increases, the solution of the compressible model approximates the solution of the free boundary model. Parameters: $ g = 1,a = 0.5,D = 0.3,c_B = 1,K_1 = K_2 = 1 $

Figure 7.  Growth of tumor radius w.r.t time for different $ D $. Other parameters: $ g = 1,a = 0.5 $. Left: radius w.r.t. time. Solid line: $ D = 0.3 $ and thus $ ga>D $. Dashed line: $ D = 0.5 $ and thus $ ga = D $. Right: $ \log R $ w.r.t. time for $ D = 0.3 $ from $ t = 10 $ to $ t = 20 $

Figure 8.  Plots of the total cell density $ n $ and the autophagic cell density $ n_2 $, and the evolution of tumor radius in the case $ K_1,K_2 $ are not constants. Parameters $ a = 0.5,D = 0.3,\gamma = 80,K_1(c) = (\frac{1-c}{c+0.1})_+,K_2(c) = \frac{2c}{c+1} $