Institute of Mechanics, Computational Mechanics Group, Technical University of Darmstadt, 64287 Darmstadt, Germany.
Institute of Applied Mathematics, Leibniz Universität Hannover, 30167 Hannover, Germany.
J R Soc Interface. 2022 Jun;19(191):20220087. doi: 10.1098/rsif.2022.0087. Epub 2022 Jun 15.
In this paper, we introduce a new framework for generating synthetic vascular trees, based on rigorous model-based mathematical optimization. Our main contribution is the reformulation of finding the optimal global tree geometry into a nonlinear optimization problem (NLP). This rigorous mathematical formulation accommodates efficient solution algorithms such as the interior point method and allows us to easily change boundary conditions and constraints applied to the tree. Moreover, it creates trifurcations in addition to bifurcations. A second contribution is the addition of an optimization stage for the tree topology. Here, we combine constrained constructive optimization (CCO) with a heuristic approach to search among possible tree topologies. We combine the NLP formulation and the topology optimization into a single algorithmic approach. Finally, we attempt the validation of our new model-based optimization framework using a detailed corrosion cast of a human liver, which allows a quantitative comparison of the synthetic tree structure with the tree structure determined experimentally down to the fifth generation. The results show that our new framework is capable of generating asymmetric synthetic trees that match the available physiological corrosion cast data better than trees generated by the standard CCO approach.
在本文中，我们引入了一种新的基于严格基于模型的数学优化的生成合成血管树的框架。我们的主要贡献是将寻找最优全局树几何形状重新表述为一个非线性优化问题（NLP）。这种严格的数学公式允许采用有效的求解算法，如内点法，并允许我们轻松地改变施加于树的边界条件和约束。此外，它还可以创建三分叉，而不仅仅是分叉。第二项贡献是增加了一个树拓扑的优化阶段。在这里，我们将约束构造优化（CCO）与启发式方法相结合，以搜索可能的树拓扑。我们将 NLP 公式和拓扑优化结合到一个单一的算法方法中。最后，我们尝试使用人体肝脏的详细腐蚀铸型来验证我们的新基于模型的优化框架，这允许对合成树结构与第五代实验确定的树结构进行定量比较。结果表明，我们的新框架能够生成与可用的生理腐蚀铸型数据更匹配的不对称合成树，比标准的 CCO 方法生成的树更好。
