Angiogenesis, the development of new blood capillaries, is crucial for the wound healing process. This biological process allows the proper blood supply to the tissue, essential for cell proliferation and viability. Several biological factors modulate angiogenesis, however the vascular endothelial growth factor (VEGF) is the main one. Given the complexity of angiogenesis, in the last years, computational modelling aroused the interest of scientists since it allows to model this process with different, more economic and faster methodologies, comparatively to experimental approaches. In this work, a mathematical model motivated by the analysis of the effect of VEGF diffusion gradient in endothelial cell migration is presented. This is the process that allows capillary formation and it is essential for angiogenesis. The proposed mathematical model is combined with the Radial Point Interpolation Method, being the area discretized considering an unorganized nodal cloud and a background mesh of integration points, without predefined relations. The nodal connectivity was achieved using the “influence‐domain” approach. The interpolation functions were constructed using the Radial Point Interpolators techniques. This method combines a radial basis functions with a polynomial functions to obtain the approximation. This preliminary work does not account for the whole complexity of cell and tissue biology, and numerical results are presented for an idealised two‐dimensional setting. Nevertheless, the developed RPIM software is a valid numerical tool that can be adjusted to biological problems and may also be able to complement the biological and medical subjects.

中文翻译：

---

使用无网格方法对血管生成过程中内皮细胞迁移的初步研究。

血管生成，即新毛细血管的发育，对伤口愈合过程至关重要。这种生物过程可以为组织提供适当的血液供应，这对细胞增殖和活力至关重要。几种生物因素调节血管生成，但血管内皮生长因子 (VEGF) 是主要的一种。鉴于血管生成的复杂性，在过去几年中，计算建模引起了科学家的兴趣，因为与实验方法相比，它允许使用不同的、更经济和更快的方法对这一过程进行建模。在这项工作中，提出了一个数学模型，其动机是分析 VEGF 扩散梯度对内皮细胞迁移的影响。这是允许毛细血管形成的过程，它对血管生成至关重要。所提出的数学模型与径向点插值法相结合，是考虑无组织节点云和积分点背景网格的离散区域，没有预定义的关系。节点连接是使用“影响域”方法实现的。插值函数是使用径向点插值器技术构建的。该方法将径向基函数与多项式函数结合以获得近似值。这项初步工作并没有说明细胞和组织生物学的整体复杂性，而是针对理想化的二维设置提供了数值结果。尽管如此，开发的 RPIM 软件是一种有效的数值工具，可以根据生物学问题进行调整，也可以补充生物学和医学学科。