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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
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3.5. XFEM pros and cons
3.3. Choice of enrichment functions
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

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.

• 1) Discontinuity cuts nodal support into 2
  disjoint pieces
• Node enriched by Heaviside function H(x)

2. Inspiration fracture mechanics
4. Results for a 2D modelling of retraction

• 2) Discontinuity ends inside nodal support
• Node enriched by basis functions Fl(r,?)
  (l1,,4)
• corresponding to the behavior of the
  crack-tip
• displacement field for a linear elastic
  material

Material discontinuities
2.1. Goal
Predicting appearance and evolution of cracks in
mechanical parts
2.2. Methods for modelling discontinuity

1. Preop MRI slice

(b) Intraop MRI slice with modelled cut
modelled retraction
Object modelled by a mesh discontinuity
Current method FEM
Solution XFEM
? 2
? 2
F1(r,?) vr sin(-)
F2(r,?) vr cos(-)
Problem Expensive remeshing
3. XFEM
(c) Mesh from preop brain
(d) Deformed preop brain
3.1. Goal
5. Future work
Accounting for discontinuities without remeshing
? 2
? 2
3.2. Key enrichment of FEM displacement
approximation

• Dealing with intersecting, arbitrarily-shaped
  discontinuities
• Dealing with resection
• Generalization to 3D
• Validation
• Application to surgical simulation and
  image-guided surgery

F3(r,?) vr sin(-) sin(?)
F4(r,?) vr cos(-) sin(?)
Addition of discontinuous functions to the FEM
displacement approximation for nodes along the
discontinuity
3.4. XFEM displacement approximation
u(x) ? fi(x) ui ? fj(x) H(x) aj ? fk(x) (
? Fl(x) ck )
4 l1
l
number of enrichment functions for node i
FEM shape functions
i ? I
k ? K
j ? J
FEM displacement
XFEM Heaviside enrichment
XFEM tip enrichment
6. References
discontinuous enrichment functions
FEM unknowns

• 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.

All nodes
XFEM unkowns
u(x) ? fi(x) ui ? fi(x) ? gj(x) aji
Ei
n
i ? I
j1
i ? J
FEM
XFEM
set of all nodes
subset of enriched nodes