Biomechanics of Arterial Walls under Supra-Physiological Loading Conditions

在这项研究项目中，描述了动脉壁的超生理负荷情况，例如在球囊血管成形术中发生的情况。在这种加载条件下，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。为了描述这一行为，在本项目的第一阶段，在连续损伤力学的基础上建立了损伤模型。将这些模型与实验进行了比较，发现它们之间有很好的关联性。然而，通常在损伤力学公式中观察到的椭圆度的损失可能导致依赖于网格的解和材料稳定性的损失。为了解决这些实质性的基本问题，数学上安全的和健壮的建模方法是第二阶段的中心任务。其基础是建立了软生物组织损伤的增量变分公式，从而使凸化成为可能。从而避免了椭圆度的损失，保证了解的网格无关性和材料的稳定性。本质上，我们计划在松弛增量变分公式的基础上开发高效的模型和算法，以反映软组织损伤滞后的具体特征。从一个相当现象学但有效的方法开始，首先要建立一个考虑光纤色散的模型。因此，损伤演化通过对单个纤维取向的渐进招募来描述。为了能够在合理的时间内对该模型进行最小二乘意义下的参数优化调整，计划构建一个代理模型，并在参数优化过程中使用该模型。所开发的公式必须在有限元程序中实现。然后，必须根据模拟病变动脉壁的边值问题来分析它们的表现。

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.