Finite Element Algorithms for Computational Biomechanics of the Brain

In this chapter, we advocate application of explicit dynamics (i.e. with explicit time stepping) finite element analysis as the finite element procedure of choice for computational biomechanics of the brain. Our recommendations regarding specific formulations, algorithms, and computing hardware for such analysis include: (1) application of total Lagrangian formulation of continuum mechanics (where the derivatives with respect to the spatial coordinates can be pre-computed); (2) dynamic relaxation for problems (such as image registration) that require steady-state solution; (3) low order (underintegrated eight-noded hexahedra with hourglass control and four-noded tetrahedra with averaged nodal pressure to alleviate volumetric locking) for construction of finite element meshes of the brain; (4) implementation of the finite element algorithms on graphics processing units (GPUs) to facilitate real-time computation (through parallel processing/multithreading) on commodity hardware. We provide examples of verification of the discussed finite element algorithms against the reference solutions obtained using the well-established non-linear static finite element procedures. We discuss time-consuming generation of patient-specific finite element meshes and deterioration of the solution accuracy when the elements undergo distortion induced by large deformations as the limitations of finite element method in the context of surgical simulation.