In this paper, we present a ghost cell/level set method for the evolution of interfaces whose normal velocity depend upon the solutions of linear and nonlinear quasi-steady reaction-diffusion equations with curvature-dependent boundary conditions. Our technique includes a ghost cell method that accurately discretizes normal derivative jump boundary conditions without smearing jumps in the tangential derivative; a new iterative method for solving linear and nonlinear quasi-steady reaction-diffusion equations; an adaptive discretization to compute the curvature and normal vectors; and a new discrete approximation to the Heaviside function. We present numerical examples that demonstrate better than 1.5-order convergence for problems where traditional ghost cell methods either fail to converge or attain at best sub-linear accuracy. We apply our techniques to a model of tumor growth in complex, heterogeneous tissues that consists of a nonlinear nutrient equation and a pressure equation with geometry-dependent jump boundary conditions. We simulate the growth of glioblastoma (an aggressive brain tumor) into a large, 1 cm square of brain tissue that includes heterogeneous nutrient delivery and varied biomechanical characteristics (white matter, gray matter, cerebrospinal fluid, and bone), and we observe growth morphologies that are highly dependent upon the variations of the tissue characteristics-an effect observed in real tumor growth.

中文翻译：

---

移动边界问题的一种新的 Ghost Cell/Level Set 方法：在肿瘤生长中的应用。

在本文中，我们提出了一种用于界面演化的鬼单元/水平集方法，其法向速度取决于具有曲率相关边界条件的线性和非线性准稳态反应扩散方程的解。我们的技术包括一种鬼单元法，它可以准确地离散法向导数跳跃边界条件，而不会在切向导数中涂抹跳跃；求解线性和非线性准稳态反应扩散方程的一种新的迭代方法；用于计算曲率和法向量的自适应离散化；以及 Heaviside 函数的新离散近似。我们提供的数值示例证明了对于传统鬼单元方法无法收敛或达到最佳亚线性精度的问题，其收敛性优于 1.5 阶。我们将我们的技术应用于复杂、异质组织中的肿瘤生长模型，该模型由非线性营养方程和具有几何相关跳跃边界条件的压力方程组成。我们模拟胶质母细胞瘤（一种侵袭性脑肿瘤）在 1 平方厘米的大脑组织中的生长，该组织包括异质的营养输送和不同的生物力学特征（白质、灰质、脑脊液和骨骼），并观察生长形态高度依赖于组织特征的变化——这是在真实肿瘤生长中观察到的效果。