Needle insertion simulations play an important role in medical training and surgical planning. Most simulations require boundary conforming meshes, while the diffuse domain approach, currently limited to stiff needles, eliminates the need for meshing geometries. In this article the diffuse domain approach for needle insertion simulations is first extended to the use of flexible needles with bevel needle tips, which are represented by an Euler-Bernoulli beam. The model parameters are tuned and the model is evaluated on a real-world phantom experiment. Second, a new method for the relaxation of the needle-tissue system after the user releases the needle is introduced. The equilibrium state of the system is determined by minimizing the potential energy. The convergence rate of the coupled Laplace equations for solving the Euler-Bernoulli beam is 1.92 $\pm$ 0.14 for decreasing cell size. The diffuse penalty method for the application of Dirichlet boundary conditions results in a convergence rate of 0.73 $\pm$ 0.21 for decreasing phase field width. The simulated needle deviates on average by 0.29 mm compared to the phantom experiment. The error of the tissue deformation is below 1 mm for 97.5% of the attached markers. Two additional experiments demonstrate the feasibility of the relaxation process. The simulation method presented here is a valuable tool for patient-specific medical simulations using flexible needles without the need for boundary conforming meshing. To the best of the authors' knowledge this is the first work to introduce a relaxation model, which is a major step for simulating accurate needle-tissue positioning during realistic medical interventions.

中文翻译：

---

用于灵活插入和放松针的扩散域方法

进针模拟在医疗培训和手术计划中发挥着重要作用。大多数模拟需要符合边界的网格，而目前仅限于刚性针的扩散域方法消除了对网格几何形状的需要。在本文中，针插入模拟的扩散域方法首先扩展到使用具有斜角针尖的柔性针，其由欧拉-伯努利梁表示。调整模型参数并在现实世界的模型实验中评估模型。其次，介绍了一种在使用者释放针后放松针组织系统的新方法。系统的平衡状态是通过最小化势能来确定的。求解欧拉-伯努利梁的耦合拉普拉斯方程的收敛率为 1.92$\pm$0.14 用于减小单元尺寸。应用狄利克雷边界条件的扩散惩罚方法导致收敛率为 0.73$\pm$0.21 用于减小相场宽度。与模型实验相比，模拟针平均偏差 0.29 mm。97.5% 的附着标记的组织变形误差低于 1 mm。另外两个实验证明了松弛过程的可行性。这里介绍的模拟方法是使用柔性针进行针对特定患者的医学模拟的宝贵工具，无需边界一致的网格划分。据作者所知，这是第一个引入松弛模型的工作，这是在现实医疗干预期间模拟精确针组织定位的重要一步。