SIAM Journal on Applied Mathematics, Volume 81, Issue 3, Page 1287-1302, January 2021.
We consider diffusion in a disk, representing a cell with a circular interior compartment. Using bipolar coordinates, we perform exact calculations, not restricted by the size or location of the intracellular compartment. We find Green functions, hitting densities and mean times to move from the compartment to the cellular surface and vice versa. For molecules with diffusivity $D$, mean times are proportional to $R^2/D$, where $R$ is the radius of the cell. We find explicit expressions for the dependence on $a^2$ (the fraction of the cell occupied by the intracellular compartment) and on the displacement of the compartment from the center of the cell. We consider distributions of initial conditions that are (i) uniform on the nuclear surface, (ii) uniform on the cellular surface, or (iii) given by the hitting density of particles diffusing from the nuclear to the cellular surface.

中文翻译：

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具有圆形夹杂物的圆盘中的扩散

SIAM 应用数学杂志，第 81 卷，第 3 期，第 1287-1302 页，2021 年 1 月。
我们考虑圆盘中的扩散，代表具有圆形内部隔室的细胞。使用双极坐标，我们执行精确计算，不受细胞内隔室的大小或位置的限制。我们发现格林函数、击中密度和从隔间移动到细胞表面的平均时间，反之亦然。对于扩散系数为 $D$ 的分子，平均时间与 $R^2/D$ 成正比，其中 $R$ 是细胞的半径。我们找到了对 $a^2$（细胞内隔室占据的细胞比例）和隔室从细胞中心的位移的依赖的明确表达式。我们考虑初始条件的分布（i）在核表面均匀，（ii）在细胞表面均匀，