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In this paper, we study a tumor growth model with nutrients. The model presents dynamic patch solutions due to the incompressibility of the tumor cells. We show that when the nutrients do not diffuse and the cells do not die, the tumor density exhibits regularizing dynamics thanks to an unexpected comparison principle. Using the comparison principle, we provide quantitative $L^1$-contraction estimates and establish the $C^{1,\mathrm{\alpha <\#comment/>}}$-boundary regularity of the tumor patch. Furthermore, whenever the initial nutrient $n_0$either lies entirely above or entirely below the critical value $n_0=1$, we are able to give a complete characterization of the long-time behavior of the system. When $n_0$is constant, we can even describe the dynamics of the full system in terms of some simpler nutrient-free and parameter-free model problems. These results are in sharp contrast to the observed behavior of the models either with nutrient diffusion or with death rate in tumor cells.