In this paper, we study a tumor growth equation along with various models for the nutrient component, including a in vitro model and a in vivo model. At the cell density level, the spatial availability of the tumor density $ n$ is governed by the Darcy law via the pressure $ p(n) = n^{γ}$. For finite $ γ$, we prove some a priori estimates of the tumor growth model, such as boundedness of the nutrient density, and non-negativity and growth estimate of the tumor density. As $ γ \to ∞$, the cell density models formally converge to Hele-Shaw flow models, which determine the free boundary dynamics of the tumor tissue in the incompressible limit. We derive several analytical solutions to the Hele-Shaw flow models, which serve as benchmark solutions to the geometric motion of tumor front propagation. Finally, we apply a conservative and positivity preserving numerical scheme to the cell density models, with numerical results verifying the link between cell density models and the free boundary dynamical models.
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Figure 1.
Example 1: expanding disk with constant nutrient and initial data (5.55). Left: plot of solution at time
Figure 2.
Example 2: a single annulus with constant nutrient and initial data (5.57). Left: plot of solution at time
Figure 3.
Example 3: a double annulus with constant nutrient and initial data (5.59). Here we compare the numerical solution (black solid curve) and analytical solution (red dashed curve) at time
Figure 7.
A comparison of the front propagation speed in the 2D radial symmetric in vitro model and in vivo model. The dots represent the the position of the right boundary at each time, and the curve are computed via (3.43) and (3.44). Here
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