Growth of developing and regenerative biological tissues of different cell types is usually driven by stem cells and their local environment. Here, we present a computational framework for continuum tissue growth models consisting of stem cells, cell lineages, and diffusive molecules that regulate proliferation and differentiation through feedback. To deal with the moving boundaries of the models in both open geometries and closed geometries (through polar coordinates) in two dimensions, we transform the dynamic domains and governing equations to fixed domains, followed by solving for the transformation functions to track the interface explicitly. Clustering grid points in local regions for better efficiency and accuracy can be achieved by appropriate choices of the transformation. The equations resulting from the incompressibility of the tissue is approximated by high-order finite difference schemes and is solved using the multigrid algorithms. The numerical tests demonstrate an overall spatiotemporal second-order accuracy of the methods and their capability in capturing large deformations of the tissue boundaries. The methods are applied to two biological systems: stratified epithelia for studying the effects of two different types of stem cell niches and the scaling of a morphogen gradient with the size of the Drosophila imaginal wing disc during growth. Direct simulations of both systems suggest that that the computational framework is robust and accurate, and it can incorporate various biological processes critical to stem cell dynamics and tissue growth.

中文翻译：

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二维干细胞组织生长的数值方法。

不同细胞类型的发育和再生生物组织的生长通常由干细胞及其局部环境驱动。在这里，我们提出了由干细胞、细胞谱系和通过反馈调节增殖和分化的扩散分子组成的连续组织生长模型的计算框架。为了处理二维开放几何和封闭几何（通过极坐标）模型的移动边界，我们将动态域和控制方程转换为固定域，然后求解转换函数以明确跟踪界面。通过适当的转换选择，可以实现当地地区的聚类网格点，以获得更好的效率和准确性。由组织的不可压缩性产生的方程由高阶有限差分格式近似，并使用多重网格算法求解。数值测试证明了该方法的整体时空二阶精度及其捕获组织边界大变形的能力。这些方法适用于两个生物系统：分层上皮用于研究两种不同类型的干细胞壁龛的影响，以及生长过程中果蝇成虫翼盘大小的形态发生素梯度的缩放。对这两个系统的直接模拟表明，计算框架稳健且准确，并且可以整合对干细胞动力学和组织生长至关重要的各种生物过程。