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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. WarfieldSignal 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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1. AbstractPros:

• 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 {Fl(r,θ)} (l=1,…,4) corresponding to the behavior of the crack-tip displacement field for a linear elastic material:

u(x) = ∑ φi(x) ui + ∑ φj(x) H(x) aj + ∑ φk(x) ( ∑ Fl(x) ck )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 Heavisideenrichment

XFEM tipenrichment

nodes nodesAll nodes

3.5. XFEM pros and cons

u(x) = ∑ φi(x) ui + ∑ φi(x) ∑ gj(x) aji

Accounting for discontinuities without remeshing

i Є I i Є J j=1

n

FEM shape functions

FEM unknowns

discontinuousenrichment functions

set ofall nodes

subset of enriched nodes

FEM XFEM

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 withmodelled cut & modelled retraction

(c) Mesh from preop brain (d) Deformed preop brain

Object modelled by a mesh

+ discontinuity

Current method: FEM Solution: XFEMF2(r,θ) = √r cos(−)

θ2

r sin(θ) r cos(θ)

θ2

F3(r,θ) = √r sin(−) sin(θ)

r sin(θ) r cos(θ)

F4(r,θ) = √r cos(−) sin(θ)θ2

r sin(θ) r cos(θ)

θ2

F1(r,θ) = √r sin(−)

r sin(θ) r cos(θ)

Problem: Expensive remeshing

Materialdiscontinuities

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