Meshless Algorithms for Computational Biomechanics of the Brain

Abstract

In this chapter, we discuss meshless methods of computational biomechanics, which utilise computational grids in a form of clouds of points, in the context of computation of the brain deformations due to surgery and injury. We highlight that meshless discretisation may be regarded as a possible solution for overcoming the limitations of the finite element method. We advocate application of weak-form meshless methods that use background cells for spatial integration, explicit time stepping, and total Lagrangian formulation of continuum mechanics (where the derivatives with respect to the spatial coordinates can be pre-computed). We make specific recommendations regarding the following key aspects of the brain deformation computation using meshless methods: (1) shape functions that facilitate robust numerical solution for irregular node placement (a feature necessary to enable end-users who are not experts in computational mechanics to build patient-specific computational biomechanics models of the brain and other body organs), (2) ensuring the desired accuracy of Gaussian spatial integration for non-polynomial shape functions through adaptive integration schemes, and (3) determining a critical time step that ensures stability of the solution provided by explicit dynamics meshless algorithms. We also discuss soft tissue dissection simulation using a visibility criterion (where the model nodes located on the opposite sides of the dissection-induced crack cannot interact with each other) while leaving open the question about the method of choice for three-dimensional dissection simulation. We provide examples of verification of the discussed meshless algorithms against the reference solutions obtained using the well-established non-linear finite element procedures.