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 paper 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±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±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.

https://www.dealii.org/, 9.5.0-pre