Biomechanics of Arterial Walls under Supra-Physiological Loading Conditions

在这个研究项目中，描述了动脉壁的超生理负荷情况，例如在球囊血管成形术期间发生的情况。在这样的加载条件下，纤维间水平的微观损伤在组织中被诱导，导致在应力-应变响应中观察到复杂的软化滞后。为了描述这种行为，本项目第一阶段在连续损伤力学的基础上建立了损伤模型。将模型与实验结果进行了比较，结果显示出较好的相关性。然而，在损伤力学公式中通常观察到的椭圆性损失可能导致网格依赖解和材料稳定性损失。为了解决这些实质性的基本问题，数学上安全且健壮的建模方法是第二阶段的中心任务。其基础是构建一个增量变分公式的损伤在软生物组织，使凸出。从而避免了椭圆度的损失，保证了与网格无关的解和材料的稳定性。本质上，有效的模型和算法是计划在松弛增量变分公式的基础上开发的，这反映了生物软组织损伤滞后的具体特征。首先要建立一种相当现象学但很有效的方法，从这种方法开始，必须建立一个考虑纤维色散的模型。因此，损伤的演变是通过单个纤维取向的逐渐增加来描述的。为了使该模型也能在合理的时间内实现最小二乘最小化意义上的优化参数调整，我们计划构造一个代理模型，并在参数优化时使用代理模型。开发的公式必须在有限元程序中实现。然后，在模拟病变动脉壁的边值问题的基础上，对它们的性能进行分析。

In this research project supra-physiological loading situations of arterial walls, e.g., occurring during balloon angioplasty, are described. Under such loading conditions microscopic damage on the interfibrillar level is induced in the tissue leading to a complex softening hysteresis observed in the stress-strain response. To describe this behavior, damage models were developed in the first phase of this project on the basis of continuum damage mechanics. These models were compared with experiments and a good correlation could be shown. However, the loss of ellipticity, which is typically observed for damage mechanics formulations, may lead to mesh-dependent solutions and the loss of material stability. In order to solve these substantial fundamental problems, mathematically secured and robust modeling approaches are central task in this second phase. The basis is the construction of an incremental variational formulation for damage in soft biological tissues which enables convexification. Thereby, the loss of ellipticity can be precluded and mesh-independent solutions as well as material stability are ensured. Essentially, efficient models and algorithms are planned to be developed on the basis of relaxed incremental variational formulations, which reflect the specific characteristics of the damage hysteresis in soft biological tissues. Starting from a rather phenomenological but efficient approach which is to be constructed first, a model has to be developed, which takes into account fiber dispersion. Thereby damage evolution is described by a progressive recruitment of individual fiber orientations. In order to enable an optimized parameter adjustment in the sense of a least-square minimization in reasonable time also for this model, a surrogate model is planned to be constructed and used during parameter optimization. The developed formulations have to be implemented in a finite-element program. Then they have to be analyzed with respect to their performance on the basis of boundary value problems where diseased arterial walls are simulated.