A new mathematical model and numerical approach are proposed for the simulation of fluid and chemical exchanges between a growing capillary network and the surrounding tissue, in the context of tumor-induced angiogenesis. Thanks to proper modeling assumptions the capillaries are reduced to their centerline: a well posed mathematical model is hence worked out, based on the coupling between a three-dimensional and a one-dimensional equation (3D-1D coupled problem). Also the application of a PDE-constrained optimization formulation is here proposed for the first time for angiogenesis simulations. Under this approach no mesh conformity is required, thus making the method particularly suitable for this kind of application, since no remeshing is required as the capillary network grows. In order to handle both the evolution of the quantities of interest and the changes in the geometry, a discrete-hybrid strategy is adopted, combining a continuous modeling of the tissue and of the chemicals with a discrete tip-tracking model to account for the vascular network growth. The tip-tracking strategy, together with some proper rules for branching and anastomosis, is able to provide a realistic representation of the capillary network.
翻译:在肿瘤引起的血管形成的背景下,为模拟不断增长的毛细网络和周围组织之间的液体和化学交流,提出了一种新的数学模型和数字方法。由于适当的模型假设,毛细线将缩小到其中心线:因此,根据三维和一维方程式(3D-1D加在一起的问题)的组合,形成了一个完善的数学模型。在此还首次提议应用一个受PDE限制的优化配方,以进行血管生成模拟。在此方法下,不需要网状符合,从而使这种方法特别适合这种应用,因为随着毛细网络的成长,不需要再铺设。为了处理兴趣量的演变和几何学的变化,采用了一种离散的环绕式战略,将组织和化学品连续的模型与离散的小跟踪模型结合起来,以计算血管网络的成长。顶级跟踪战略,加上一些适当的分流和孔化规则,能够提供一个真实的圆柱形网络图示。