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

  • *Corresponding author: Zhennan Zhou

    *Corresponding author: Zhennan Zhou
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  • In this paper, we propose a tumor growth model to incorporate and investigate the spatial effects of autophagy. The cells are classified into two phases: normal cells and autophagic cells, whose dynamics are also coupled with the nutrients. First, we construct a reaction-(cross-)diffusion system describing the evolution of cell densities, where the drift is determined by the negative gradient of the joint pressure, and the reaction terms manifest the unique mechanism of autophagy. Next, in the incompressible limit, such a cell density model naturally connects to a free boundary system, describing the geometric motion of the tumor region. Analyzing the free boundary model in a special case, we show that the ratio of the two phases of cells exponentially converges to a "well-mixed" limit. Within this "well-mixed" limit, we obtain an analytical solution of the free boundary system which indicates the exponential growth of the tumor size in the presence of autophagy in contrast to the linear growth without it. Numerical simulations are also provided to illustrate the analytical properties and to explore more scenarios.

    Mathematics Subject Classification: Primary: 35Q92, 92-10; Secondary: 35R35.


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  • 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} $

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