Tumor growth with nutrients: Regularity and stability
Matt Jacobs, Inwon Kim, Jiajun Tong
Abstract
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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -contraction estimates and establish the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript 1 comma alpha"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi> α </mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">C^{1,\alpha }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -boundary regularity of the tumor patch. Furthermore, whenever the initial nutrient <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n 0"> <mml:semantics> <mml:msub> <mml:mi>n</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">n_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> either lies entirely above or entirely below the critical value <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n 0 equals 1"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>n</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n_0=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we are able to give a complete characterization of the long-time behavior of the system. When <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n 0"> <mml:semantics> <mml:msub> <mml:mi>n</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">n_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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.