Detection and modelling of contacts in explicit finite-element simulation of soft tissue biomechanics

Realistic modelling of soft tissue biomechanics and mechanical interactions between tissues is an important part of biomechanically-informed surgical image-guidance and surgical simulation. This submission details a contact-modelling pipeline suitable for implementation in explicit matrix-free FEM solvers. While these FEM algorithms have been shown to be very suitable for simulation of soft tissue biomechanics and successfully used in a number of image-guidance systems, contact modelling specifically for these solvers is rarely addressed, partly because the typically large number of time steps required with this class of FEM solvers has led to a perception of them being a poor choice for simulations requiring complex contact modelling.

The presented algorithm is capable of handling most scenarios typically encountered in image-guidance. The contact forces are computed with an evolution of the Lagrange-multiplier method first used by Taylor and Flanagan in PRONTO 3D extended with spatio-temporal smoothing heuristics for improved stability and edge-edge collision handling, and a new friction model. For contact search, a bounding-volume hierarchy (BVH) is employed, which is capable of identifying self-collisions by means of the surface-normal bounding cone of Volino and Magnenat-Thalmann, in turn computed with a novel formula. The BVH is further optimised for the small time steps by reducing the number of bounding-volume refittings between iterations through identification of regions with mostly rigid motion and negligible deformation. Further optimisation is achieved by integrating the self-collision criterion in the BVH creation and updating algorithms.

The effectiveness of the algorithm is demonstrated on a number of artificial test cases and meshes derived from medical image data. It is shown that the proposed algorithm reduces the cost of BVH refitting to the point where it becomes a negligible part of the overall computation time of the simulation. It is also shown that the proposed surface-normal cone computation formula leads to about 40 % fewer BVH subtrees that must be checked for self-collisions compared with the widely used method of Provot. The proposed contact-force formulation and friction model are evaluated on artificial test cases that allow for a comparison with a ground truth. The quality of the proposed contact forces is assessed in terms of trajectories and energy conservation; a [Formula: see text]0.4 % drop off in total energy and highly plausible trajectories are found in the experiments. The friction model is evaluated through a benchmark problem with an analytical solution and a maximum displacement error of 8.2 %, and excellent agreement in terms of the stick/slip boundary is found. Finally, we show with realistic image-guidance examples that the entire contact-modelling pipeline can be executed within a timeframe that is of the same order of magnitude as that required for standard FEM computations.