1. EiEi New Approach for Efficient Prediction of Brain Deformation and Updating of Preoperative Images Based on the Extended Finite Element Method Lara M. Vigneron, Jacques G. Verly, Pierre A. Robe, Simon K. Warfield Signal Processing Group, Department of Electrical Engineering and Computer Science, University of Liège, Belgium Department of Neurosurgery, Centre Hospitalier Universitaire, University of Liège, Belgium Computational Radiology Laboratory, Surgical Planning Laboratory, Brigham and Women's Hospital and Harvard Medical School, Boston, USA 1 2 1 1 2 1. Abstract Pros: No remeshing required Arbitrarily-shaped discontinuities Arbitrary number of discontinuities FEM framework preserved, including symmetry and sparsity Cons: Increase of the number of unknowns and size of stiffness matrix Need for integration (by Gauss quadrature) of crack-tip function derivatives 1) Discontinuity cuts nodal support into 2 disjoint pieces  Node enriched by Heaviside function H(x) 2) Discontinuity ends inside nodal support  Node enriched by basis functions {F l (r,θ)} (l=1,…,4) corresponding to the behavior of the crack-tip displacement field for a linear elastic material: u(x) = ∑ φ i (x) u i + ∑ φ j (x) H(x) a j + ∑ φ k (x) ( ∑ F l (x) c k ) i Є I j Є J k Є K 4 l=1 l 4. Results for a 2D modelling of retraction 5. Future work 6. References (1) L. Vigneron, J. Verly, and S. Warfield. Modelling Surgical Cuts, Retractions, and Resections via Extended Finite Element Method. Proceedings MICCAI 2004, 311-318, 2004. (2) N. Moës, J. Dolbow, and T. Belytschko. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering, 46: 131-150, 1999. (3) N. Sukumar, N. Moës, T. Belytschko, and B. Moran. Extended Finite Element Method for three-dimensional crack modelling. International Journal for Numerical Methods in Engineering, 48 (11): 1549-1570, 2000. We introduce a new, efficient approach for modelling the deformation of organs following surgical cuts, retractions, and resections. It uses the extended finite element method (XFEM), recently developed in "fracture mechanics" for dealing with cracks in mechanical parts. A key feature of XFEM is that material discontinuities through meshes can be handled without remeshing, as would be required with the regular finite element method (FEM). This opens the possibility of using a biomechanical model to estimate intraoperative deformations accurately in real-time. To show the feasibility of the approach, we present a 2D modelling of a retraction. Dealing with intersecting, arbitrarily-shaped discontinuities Dealing with resection Generalization to 3D Validation Application to surgical simulation and image-guided surgery FEM displacement XFEM Heaviside enrichment XFEM tip enrichment nodes All nodes 3.5. XFEM pros and cons u(x) = ∑ φ i (x) u i + ∑ φ i (x) ∑ g j (x) a ji Accounting for discontinuities without remeshing i Є I i Є J j=1 n FEM shape functions FEM unknowns discontinuous enrichment functions set of all nodes subset of enriched nodes FEMXFEM number of enrichment functions for node i 3. XFEM XFEM unkowns 3.1. Goal 3.4. XFEM displacement approximation 2. Inspiration: fracture mechanics 2.1. Goal Predicting appearance and evolution of cracks in mechanical parts 2.2. Methods for modelling discontinuity 3.2. Key: ‘‘enrichment’’ of FEM displacement approximation Addition of discontinuous functions to the FEM displacement approximation for nodes along the discontinuity 3.3. Choice of enrichment functions (a) Preop MRI slice(b) Intraop MRI slice with modelled cut & modelled retraction (c) Mesh from preop brain(d) Deformed preop brain Object modelled by a mesh + discontinuity Current method: FEMSolution: XFEM F 2 (r,θ) = √r cos(−) θ2θ2 r sin(θ) r cos(θ) θ2θ2 F 3 (r,θ) = √r sin(−) sin(θ) r sin(θ) r cos(θ) F 4 (r,θ) = √r cos(−) sin(θ) θ2θ2 r sin(θ)r cos(θ) θ2θ2 F 1 (r,θ) = √r sin(−) r sin(θ)r cos(θ) Problem: Expensive remeshing Material discontinuities 3 3