Ensemble averages of a stochastic model show that, after a formation stage, the tips of active blood vessels in an angiogenic network form a moving two dimensional stable diffusive soliton, which advances toward sources of growth factor. Here we use methods of multiple scales to find the diffusive soliton as a solution of a deterministic equation for the mean density of active endothelial cells tips. We characterize the diffusive soliton shape in a general geometry, and find that its vector velocity and the trajectory of its center of mass along curvilinear coordinates solve appropriate collective coordinate equations. The vessel tip density predicted by the soliton compares well with that obtained by ensemble averages of simulations of the stochastic model.

中文翻译：

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肿瘤诱导血管生成中的二维孤子

随机模型的集合平均值表明，在形成阶段之后，血管生成网络中活跃血管的尖端形成一个移动的二维稳定扩散孤子，向生长因子的来源前进。在这里，我们使用多尺度的方法来找到作为活性内皮细胞尖端平均密度确定性方程的解的扩散孤子。我们在一般几何学中表征了扩散孤子形状，并发现其矢量速度和其质心沿曲线坐标的轨迹可以求解适当的集体坐标方程。由孤子预测的血管尖端密度与通过随机模型模拟的集合平均值获得的密度相当。