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3D XFEM-Based Modeling of Retraction for Preoperative Image Update Lara M. Vigneron 1 , Simon K. Warfield 2 , Pierre A. Robe 3 , and Jacques G. Verly 4 1 Department of Electrical Engineering and Computer Science, University of Li` ege, Li` ege, Belgium 2 Computational Radiology Laboratory, Children’s Hospital, Harvard Medical School, Boston, MA, USA 3 Department of Neurosurgery, University of Utrecht Medical Center, The Netherlands 4 Department of Electrical Engineering and Computer Science, University of Li` ege, Li` ege, Belgium Abstract The problematic of improving outcomes for neurosurgery patients by enhancing intraoper- ative navigation and guidance is considered. Current navigation systems do not accurately account for intraoperative brain deformation. So far, most studies of brain deformation have focused on brain shifts while this paper focuses on the brain deformation due to retraction. The heart of our system is a 3D nonrigid registration technique using a biomechanical model driven by the deformations of key surfaces tracked between two intraoperative images. The key surfaces, e.g the whole-brain region boundary and the lips of the retraction cut, thus deform due to the combination of gravity and retractor deployment. The tissue discontinu- ity due to retraction is handled via the eXtented Finite Element Method (XFEM), which has the appealing feature of being able to handle arbitrarily-shaped discontinuity without any remeshing. Our approach is shown to significantly improve alignment of intraoperative MRI. Keywords: Image registration, biomechanical model, retraction, XFEM Introduction The main goal of brain surgery is to remove as much as possible of lesion tissues, while avoiding contacts with eloquent areas and fiber tracts located in white-matter tissues. Surgery is planned on the basis of preoperative images of multiple modalities, such as Computed Tomography (CT), structural and functional Magnetic Resonance Imaging (sMRI and fMRI), Diffusion Tensor Imaging (DTI), Positron Emission Tomography (PET), and Magneto-encephalography (MEG). Surgery is generally performed using an image-guided navigation system that relates the 3D preoperative images to (3D) patient coordinates, thereby allowing the surgeon to position his instruments in the patient’s brain, while navigating on the preoperative images. However, throughout surgery, the brain deforms, mostly as a result of the leakage of the cerebrospinal fluid out of the skull cavity, modifications in cerebral perfusion, pharmacological modulation of the extracellular fluid, and surgical acts, such as cuts, retractions, and resections [27]. As surgery progresses, preoperative images become less representative of the actual brain, and nav- igation accuracy decreases. These navigation errors could be reduced if one could acquire, throughout surgery, fresh images of the same modalities and quality as the preoperative ones. These images are major chal- lenges, since intraoperative MR images are - with the exception of a handful surgical facilities - usually acquired using low-field MRI scanners that provide lower resolution and contrast than 1

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Page 1: 3D XFEM-Based Modeling of Retraction for Preoperative Image

3D XFEM-Based Modeling of Retraction

for Preoperative Image Update

Lara M. Vigneron1, Simon K. Warfield2, Pierre A. Robe3 , and Jacques G. Verly4

1Department of Electrical Engineering and Computer Science,University of Liege, Liege, Belgium

2Computational Radiology Laboratory, Children’s Hospital,Harvard Medical School, Boston, MA, USA

3Department of Neurosurgery, University of Utrecht Medical Center, The Netherlands4Department of Electrical Engineering and Computer Science,

University of Liege, Liege, Belgium

Abstract

The problematic of improving outcomes for neurosurgery patients by enhancing intraoper-ative navigation and guidance is considered. Current navigation systems do not accuratelyaccount for intraoperative brain deformation. So far, most studies of brain deformation havefocused on brain shifts while this paper focuses on the brain deformation due to retraction.The heart of our system is a 3D nonrigid registration technique using a biomechanical modeldriven by the deformations of key surfaces tracked between two intraoperative images. Thekey surfaces, e.g the whole-brain region boundary and the lips of the retraction cut, thusdeform due to the combination of gravity and retractor deployment. The tissue discontinu-ity due to retraction is handled via the eXtented Finite Element Method (XFEM), whichhas the appealing feature of being able to handle arbitrarily-shaped discontinuity withoutany remeshing. Our approach is shown to significantly improve alignment of intraoperativeMRI.

Keywords: Image registration, biomechanical model, retraction, XFEM

IntroductionThe main goal of brain surgery is to remove as much as possible of lesion tissues, while avoidingcontacts with eloquent areas and fiber tracts located in white-matter tissues. Surgery is plannedon the basis of preoperative images of multiple modalities, such as Computed Tomography (CT),structural and functional Magnetic Resonance Imaging (sMRI and fMRI), Diffusion TensorImaging (DTI), Positron Emission Tomography (PET), and Magneto-encephalography (MEG).Surgery is generally performed using an image-guided navigation system that relates the 3Dpreoperative images to (3D) patient coordinates, thereby allowing the surgeon to position hisinstruments in the patient’s brain, while navigating on the preoperative images. However,throughout surgery, the brain deforms, mostly as a result of the leakage of the cerebrospinalfluid out of the skull cavity, modifications in cerebral perfusion, pharmacological modulationof the extracellular fluid, and surgical acts, such as cuts, retractions, and resections [27]. Assurgery progresses, preoperative images become less representative of the actual brain, and nav-igation accuracy decreases.

These navigation errors could be reduced if one could acquire, throughout surgery, fresh imagesof the same modalities and quality as the preoperative ones. These images are major chal-lenges, since intraoperative MR images are - with the exception of a handful surgical facilities -usually acquired using low-field MRI scanners that provide lower resolution and contrast than

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their preoperative counterparts. Moreover, even in those rare places that possess high-fieldintraoperative MRI scanners, the acquisition of intraoperative DTI images remains rather time-consuming. Finally, to this date, several useful imaging modalities, such as PET and MEG,cannot be acquired intraoperatively.

The solution that is generally favored is to acquire reduced-quality intraoperative images atseveral critical points during surgery, to track the tissue deformation from one image to thenext, to update (i.e. to deform) the preoperative images accordingly, and to feed the resultingimages to the navigation system [30]. The deformation of preoperative images so they conformto the intraoperative images falls into the domain of nonrigid registration. One category of suchtechniques uses biomechanical models based on the Finite Element Method (FEM). Prior tosurgery, a biomechanical brain model specific to the patient is built from preoperative images:the model consists of a volume mesh of finite elements (FEs), and of one or more mechanicalbehavior laws assigned to them. During surgery, a number of key anatomical landmarks are ex-tracted and tracked through successive intraoperative images. When using the approach knownas the forced-displacement method, the estimated displacements of these landmarks are directlyapplied to the biomechanical model and drive its deformation. The resulting displacement fieldof the biomechanical model is used to deform the preoperative images. So far, most of themechanical conditions of the brain cannot be estimated in the operating room, such as thevolume of cerebrospinal fluid flowing out of the skull cavity, or the forces applied by a retractortool. The fact that an intraoperative image can provide the knowledge of the current state ofthe brain after some deformation partly eliminates the need for a complete evaluation of thesemechanical conditions. The nonrigid registration technique replaces them with the landmarkdisplacements evaluated from successive intraoperative images.

Most studies of brain deformation based on biomechanical models have focused on shifts (thetopology of the brain is not modified) occurring after the opening of the skull and dura [3,11, 14, 19, 21, 31, 35, 36, 44, 45], and, thus, do not take explicitly into account any cut and sub-sequent deformation. A good review of these different studies can be found in [4, 17, 24, 44].In comparison, few studies have focused on brain deformations due to cuts, retractions, andresections (the last two necessarily involving a cut). One of the main difficulties associated witha cut is the discontinuity it implies in the tissues, which changes brain topology. Indeed, thefinite element method (FEM) cannot handle discontinuities directly and requires one to realignthe discontinuity with element boundaries based on mesh-adaptation [29, 34, 37] or remeshingtechniques [2, 12, 26, 28], which provide, from an initial mesh, a new mesh conform to the dis-continuity.

This paper presents a solution for updating multimodal preoperative images with respect tosurgical brain deformations due to retraction that can be seen and quantified using iMR im-ages. The landmarks tracked between the iMR images correspond to the whole-brain regionboundary and the lips of the retraction cut, both deforming due to the combination of gravityand retractor deployment. In the field of fracture mechanics, which studies the growth andpropagation of cracks in mechanical parts, some methods have been developed to avoid usingFEM in combination with mesh adaptation or remeshing techniques [8]. One of them is the eX-tended Finite Element Method (XFEM or X-FEM), which appeared in 1999 [25] and has beenthe object of considerable research since then [1]. XFEM works by allowing the displacementfield to be discontinuous within some FEs of the mesh. The mesh does not have to conform tothe discontinuities, so that these can be arbitrarily located with respect to the underlying FEmesh. Because XFEM allows for an accurate representation of the discontinuities while avoidingmesh adaption or remeshing, and because of the similarity between cracks in mechanical partsand cuts in tissue, we proposed the use of XFEM for handling cut, retraction, and resection in

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the updating of preoperative images. As explained earlier, the goal of this work is to incremen-tally update the preoperative images with each new pair of successive iMR images; as each iMRimage is acquired, this new image and the preceding one are used to estimate the deformationof the brain. Typically, after brain shift, the patient undergoes successive resections with priorretraction depending on the location of the tumour site. In the modeling of successive brain de-formations, XFEM is thus appropriate in order to avoid successive remeshing. In the past, some2D results were presented in [41] for retraction modeling and successive resection modeling, bothhandled separately. This paper is the first complete and detailed account of the generalisationto 3D of our 2D work on retraction modeling. In the future, 3D retraction followed by succes-sive resections will be modeled. Different constitutive laws have already been used for modelingbrain deformation. The simplest constitutive law is the linear elastic law [3, 11] and is used inthis work, while the brain shift was also modeled using a hyperviscoelastic law characterizedby a strain rate dependence [45], or a poroelastic law characterized by solid matrix and a fluidphase [19].

The structure of the paper is as follows. In next section, the state-of-the-art in retractionmodeling is presented. The basic principles of XFEM are then introduced. Our global systemfor updating preoperative images is described and details are given about our algorithms forretraction modeling. Afterwards, a patient case that illustrates our approach is considered, andour results are validated. Finally, conclusions are drawn and future work discussed.

State-of-the-artRetraction is usually performed when the target, e.g. the tumour, is deep inside the brain. Thesurgeon cuts through brain tissues and inserts the blades of a retractor to spread tissues outfrom the incision and to create a free path towards the tumour. Ferrant et al. [11] modeledretraction using a linear elastic biomechanical model. They deleted the FEs falling into the re-traction path, which is visible on the intraoperative MR (iMR) image acquired after retraction.This way of modeling retraction implied the removal of some tissues, which was not physicallycorrect, and did not model explicitly the motion of tissues as they were spread out.

Miga et al. [22] modeled retraction using a linear poroelastic biomechanical model. This modelwas subject to the effects of gravity, as well as to the deployment of the retractor. They usedoptical images from the operating microscope and cortical features, such as surface vasculature,to estimate the position of the retractor. They modeled the tissue discontinuity by splitting themesh along existing boundary edges, representing as best as possible the cut by a jagged dis-continuity. In order to model the opening of the path, they subsequently moved the duplicatednodes apart. The nodes located on the front side (i.e. facing the opening) of the retractor weremoved within a distance consistent with the optical images. The nodes on the back side werenot allowed to move orthogonally to the retractor blade.

Platenik et al. [32, 33] attempted to validate the retraction modeling of Miga et al. [22]. Theyperformed in vivo incremental interhemispheric retraction experiment (from 3 mm up to 10mm) on four porcine subjects, using a controlled retractor mounted on the stereotactic frameattached to the subject. They used intraoperative CT (iCT) scans to estimate the positionof the retractor. Validation was performed using beads, implanted in the porcine brains, andtracked between successive iCT scans. They also improved the model of Miga et al. by al-lowing tissue on the back side of the retractor to move with the retractor at the beginning ofthe retraction, and, then, to separate at larger retractions. This phenomenon was visible oniCT scans. They were able to capture up to 85% of brain motion. Lamprich et al. [18] alsoattempted to validate the retraction modeling of Miga et al. [22] on ex vivo interhemisphericretraction porcine experiment images using iMR images.

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Lunn et al. [20] used the data collected by Platenik et al. [33] from these porcine experiments totest a different way of modeling retraction. The volume displacement field of the biomechanicalmodel was computed as a weighted combination of basis solutions, computed prior to surgerywith a linear poroelastic biomechanical model. The weight coefficients were computed with aminimization procedure using the implanted beads as control points, in order to reduce the errorbetween the observed and computed displacements of the beads. They analysed different basissolutions, and evaluated the best combination of them. They obtained the best results witha combination of three basis solutions: gravity, retraction with normal displacements to theretractor blades, and pressure on the back of the retractor. They also analysed the amount andlocations of control points necessary to obtained such results. While the results showed equalor decreased mean vector error compared to the results of Platenik et al., the method still callsfor further study. Indeed, the method is of course sensitive to the type of basis solutions used,and does not necessarily provide better solutions with additional basis solutions. Furthermore,while it was feasible to use implanted beads for porcine subjects and to track them in iCT scans,the use and tracking of control points in real patients is expected to be more challenging.

Sun et al. [40] studied the capacity of capturing cortical motion during retraction, using stereocameras mounted on the operating microscope. They registered stereo images acquired aftercraniotomy to a linear poroelastic biomechanical model of the patient’s brain, and tagged thenodes of the external surface of the biomechanical model that corresponded to the visible sur-face of the brain. From these nodes, they defined a subset that were visually hidden by theretractor on the stereo images acquired after the insertion of the retractor, and tagged themas the retractor nodes. They estimated the motion of the retractor based on stereo imagesacquired before and after the deployment of the retractor, using an iterative closest point (ICP)algorithm. They transposed the displacement of the retractor to displacements applied to theretractor nodes of the biomechanical model. They validated their method by computing, basedon the ICP algorithm, the displacements of the tagged nodes minus the retractor nodes fromstereo images, and compared these displacements to the displacements of nodes resulting fromthe deformation of the biomechanical model. For the patient case studied, they found that theycould capture approximately 75% of the cortical motion.

The methods described above have been all developed using a FEM-based biomechanical modelfor intraoperative image registration. Surgical simulation is another research field that broadlyuses FEM-based biomechanical model. The objective of a surgical simulator is to provide aninteractive manipulation with force feedback of the anatomical part to be operated using varioussurgical instruments. In order to model a large range surgical procedures, a real-time interactivecutting method should be included in the simulator. Jerabkova et al. [15] have applied nonlinearXFEM for simulating cut, and have shown that XFEM is successfully efficient for such purpose.

XFEM PrinciplesBefore presenting XFEM principles, a few key properties of FEM are introduced that make theunderstanding of XFEM easier. FEM [47] discretises the solid of interest into a mesh, i.e. intoa set of finite elements (FEs) interconnected by nodes, and approximates the displacement fieldu(x) by the FEM displacement field uFEM (x) defined as

uFEM (x) =∑

i∈I

ϕi(x)ui, (1)

where x is the position of a point of the solid, I is the set of nodes, the ϕi(x)’s are the nodalshape functions (NSFs), and the ui’s are the nodal degrees of freedom (DOFs). Each ϕi(x) isdefined as being continuous on its compact support ωi, which corresponds to the union of the

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domains of the FEs connected to node i, with the property

ϕi(xj) = δij , (2)

where xj is the position of node j and δij the usual Kronecker signal. In our approach, linearNSFs are used.

FEM requires its displacement field uFEM (x) to be continuous over each FE. In contrast,XFEM handles a discontinuity by allowing the displacement field to be discontinuous withinFEs [5,25,38,39]. Arbitrarily-shaped discontinuities can then be modeled without any remesh-ing. The XFEM displacement field generalises the FEM displacement field (1) with

uXFEM (x) =∑

i∈I

ϕi(x)ui +∑

i∈J

ϕi(x)nEi∑

j=1

gj(x)aji. (3)

The first term corresponds to the FEM displacement field (1), where I is the set of nodes, theϕi(x)’s are the FEM NSFs, and the ui’s are the nodal FEM DOFs. The heart of XFEM is the“enrichment” that adds a number, nEi , of DOFs aji to each node i of the set J , which is thesubset of nodes of I whose support is intersected by the discontinuity of interest. These DOFsare multiplied by the NSFs ϕi(x) and the discontinuous functions gj(x).

Evaluating the function uXFEM (x) at the position location xk of some node k that is enriched,i.e. for k ∈ J , one finds

uXFEM (xk) =∑

i∈I

ϕi(xk)ui +∑

i∈J

ϕi(xk)nEi∑

j=1

gj(xk)aji

= uk +nEi∑

j=1

gj(xk)ajk.

(4)

Since the last sum is generally not zero, one has

uXFEM (xk) 6= uk. (5)

This means that, for nodes that are enriched, the FEM DOF uk does not represent the dis-placement of node k, and thus, the XFEM displacement field uXFEM (x) does not interpolatethe nodal displacements. This should be contrasted to the case of FEM, where the FEM DOFuk represents the displacement of node k. Equation (4) also implies that nodal displacementscannot be imposed by eliminating the DOFs uk, such as it is done with FEM, but must beapplied as a linear combination between several DOFs with, for instance, Lagrange multipliersfor nodes that are enriched [9].

The use of specific XFEM enrichment functions gj(x) for a node i ∈ J depends on the typeof discontinuity, e.g. crack, hole, material interface, etc., to be modeled. Suppose that ourgoal is to model a crack, characterised by a discontinuity in the displacement field (as opposedto a material interface for instance, characterized by a discontinuity in the derivative of thedisplacement field). When the crack fully intersects the support of the node, a simple choice isa piecewise-constant unit function that changes sign at the boundary across the crack, i.e. theHeaviside function

H(x) ={

1 for (x− x∗).en > 0−1 for (x− x∗).en < 0,

(6)

where x is again the position of a point of the solid, x∗ is the position of the point on thecrack that is the closest to x, and en is the outward normal to the crack at x∗, as shown in

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Fig. 1(a) [25].

The support of any node in J is cut into two disjoint pieces by the crack. If, for some node,one of the two pieces is very small compared to the other, then the Heaviside function is almostconstant over the nodal support, which causes the XFEM problem to be ill-conditioned. Conse-quently, the node has to be removed from J , which is equivalent to moving the crack location tothe boundary of the support. A criterion should be used to remove from J a node for which theratio of the volume of the smallest piece to that of the full support is less than some tolerancevalue, e.g. 10−4 [39].

The XFEM equations that result from the minimization of the total potential energy remainsparse and symmetric as for FEM. Whereas FEM requires a remeshing and the duplication ofthe nodes along the crack to take into account any discontinuity, XFEM requires the identi-fication of the nodes belonging to the set J and the addition of DOFs: (1) any node whosesupport is not intersected by the discontinuity remains unaffected and thus possesses threeDOFs; (2) any node whose support is intersected by the discontinuity is enriched with threeHeaviside DOFs and thus possesses six DOFs. Because of the additional DOFs in XFEM, thenumber of equations to be solved is larger than for FEM.

Description of Baseline SystemOur retraction modeling is part of a global framework for updating preoperative images usingsuccessive intraoperative MR (iMR) images acquired at different critical points during surgery.(iMR images were acquired with the 0.5 Tesla intraoperative GE Signa scanner of the Brighamand Women’s Hospital, Boston, USA. iMR image size is 256× 256× 60 voxels, and voxel size is0.859×0.859×2.5 mm.) In this paper, two simplifying assumptions are made. First, retractiondeformation only is considered, which takes place between the second and third iMR images forthe patient case treated. Therefore, brain shift deformation is not taken into account, whichtakes place between the first and second iMR images. However, the modeling of retraction is notaffected by this simplification. (The modeling brain shift deformation - for a different patientcase - has been treated and illustrated in [42].) Second, the second iMR image is used as asubstitute for preoperative images. The biomechanical model is thus built based on structuresvisible in the second iMR image, instead of in the preoperative images, and the structures usedin the model are limited to the ones visible in the intraoperative image. Except for the rigidregistration between the preoperative images, the biomechanical model, and the second iMRimage, this simplified approach allows us to discuss, illustrate, and test all key aspects of thesystem.

The successive steps for evaluating brain retraction deformation consist of rigidly registering theiMR images, and of segmenting them. The biomechanical model is then built based on struc-tures segmented in the second iMR image. A set of common anatomical surface landmarks, thewhole-brain region boundary and the two lips of the cut, are segmented and tracked betweenthe two iMR images. The biomechanical model is deformed, based on XFEM, in accordancewith the displacement fields of these surface landmarks. The resulting volume displacementfield of the biomechanical model is then used to warp the second iMR image. Finally, themethod is validated by comparing images (original and deformed second iMR image with thirdiMR image) that are of same modality and quality, and, thus, that show the same anatomicalfeatures. In the following sections, the successive processing steps just mentioned are described.

Rigid Registration of Intraoperative ImagesAlthough the patient’s head is supposedly fixed during the operation, one cannot totally rule outthe possibility of slight head motion. To compensate for this potential source of rigid motion,

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the iMR images are rigidly registered using the point-based landmark transform available inVTK(http://www.vtk.org/), where the corresponding landmark points are manually selectedin the two iMR images.

Segmentation of Intraoperative ImagesThe segmentation of iMR images is necessary for both the building of the biomechanical modeland the capture of intraoperative deformation between the two iMR images (Figs. 2(1a)-2(2a)).In the present application, the only region of interest is the whole-brain region, i.e. with the skulland external cerebrospinal fluid masked out. The segmentation is performed in two steps. First,the whole-brain region is manually segmented out using 3D Slicer (http://www.slicer.org/)(Figs. 2(1b)-2(2b)) from the two iMR images. Second, the two segmented regions are smoothedout in order to minimize the dependance of the evaluation of surface landmark displacementfields on roughness of manual segmentation. This smoothing is performed by building a sur-face mesh of the regions segmented manually in each iMR image, and by subsequently fillingall voxels that fall inside the surface mesh with the numerical label of the segmented region(Figs. 2(1c)-2(2c)). By comparing Figs. 2(2b) and 2(2c), one sees that the smoothing causesthe depth and the opening width of the path created by retraction to decrease. In the future,this problem will need to be fixed.

Building of the Biomechanical ModelAs mentioned above, in the present work, the biomechanical model is built from the secondiMR image rather than from the preoperative images. The biomechanical model is built intwo steps. First, the volume mesh is built, and appropriate constitutive law is assigned to it.Second, the cut to be included in the biomechanical model is defined. This cut represents thediscontinuity that is handled via XFEM, i.e by enriching the appropriate nodes of the volumemesh with XFEM Heaviside DOFs.

To build the volume mesh, Isosurf (http://www-math.mit.edu/˜persson/mesh/) is first usedto produce a surface mesh of the boundary of the segmented whole-brain region. This softwaretakes as input a binary image, provided as stacks of slices, and representing segmented regions,and meshes the boundary of these regions into surfaces of triangles. Gmsh (http://www.geuz.org/gmsh/) [13],a 3D mesh generator, is then used to produce, based on the triangle surface mesh, a tetrahe-dron volume mesh of the whole-brain region. A linear elastic law is assigned to the whole-brainregion, with Young modulus E = 3000 Pa and Poisson ratio ν = 0.45 [11]. Note that, sincedisplacements only (and no forces) are applied to the homogeneous biomechanical model, itsdeformation is independent of the value of Young modulus [23].

For modeling retraction, a representation of the cut in the biomechanical model is needed,in order to enrich the appropriate nodes of the volume mesh with XFEM Heaviside DOFs. Thecut is defined as a segmented region, and ultimately represented by the surface mesh that is theintersection of the 3D volume mesh with this segmented region. For the patient case of interest,the mesh representing the cut is obtained as follows. First, the cut is assumed to be planar,and the cut plane, representing the location where the retractor was inserted, is taken as theplane between the two brain hemispheres (Fig. 3(a)). Second, the 2D region that is the dif-ference between the whole-brain regions segmented out from the two iMR images in the planeof the cut is determined (Figs 3(b)-3(c)), where additional image processing has beforehandbeen performed on Fig. 3(c) to remove small, isolated islands. The result is a 2D cut region,shown in Fig. 3(d). Third, the intersection of the cut with the volume mesh is computed. Thisis performed by first computing a distance map of the cut region, and then, using the filterVTKCutter, which allows one to compute the intersections of a volume mesh with an implicitfunction, such as a distance map. VTKCutter produces a triangle surface mesh, representing

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the cut mesh, composed of the set of points intersecting the volume mesh (Fig. 3(e)). Finally,the nodes whose supports are intersected by the discontinuity, i.e. the cut mesh, are enrichedwith XFEM Heaviside DOFs. The volume mesh consists of 2, 198 nodes which corresponds to6, 594 FEM DOFs. Enrichment adds 561 XFEM Heaviside DOFs, leading to a total of 7, 155DOFs.

Evaluation of Surface Landmark Displacement FieldsIn order to drive the deformation of the biomechanical model, the displacement field of severalsurface landmarks is evaluated: the whole-brain region boundary and the two lips of the cut.The lips of the cut are assumed to be superimposed before the deployment of the retractor, andthen spread out during the deployment of the retractor.

Displacement Field of Whole-Brain Region Boundary. To evaluate the displacement field ofwhole-brain region boundary between two iMR images, an active surface algorithm [16, 46] isused. An active surface, i.e. a surface mesh with mechanical constraints such as elasticity,is initialised from the region boundary segmented in the first iMR image, and then deformedunder the influence of external forces computed from the corresponding region boundary in thesecond iMR image. The external forces attract the surface during an iterative process, in sucha way that this surface deforms to cling to the region boundary to match. In our work, theexternal forces F (x) are computed using a gradient descent on a distance map of the regionboundary, i.e.

F (x) = Smin∇D(x) with Smin ={

1 if D(x) > D(x +∇D(x))−1 if D(x) < D(x +∇D(x)),

(7)

where D(x) is the distance map of the region boundary to match and Smin is chosen to be+1 or −1 to ensure that the external forces point towards a position on the distance map withsmaller distance value. Further details on this specific active surface algorithm can be foundin [10,11].

An active surface for the whole-brain region boundary built from the second iMR image woulddeform on the whole-brain region boundary of the third iMR image by entering into the freepath created by the retraction. The resulting displacement field of the active surface algorithmwould thus model a deformation that does not occur in reality. The deformation of the whole-brain region boundary is thus decoupled into two independent applications of the active surfacealgorithm, one for each of the two hemispheres. One can clearly see in Figs. 2(1a)-2(2a) thatthe two hemispheres deform differently. Figures 4(a)-4(b) show the result of the active surfacedisplacement fields of the two hemispheres. As expected, they deform differently; in particular,the magnitude of the displacement is much larger for the right hemisphere. (In accordance withradiological convention, the right side of the iMR images corresponds to the left side of thebrain. Consequently, what we refer to as the left (right) brain hemisphere actually appears onthe right (left) of the iMR images. The same remark can be made for the left and right lips ofthe cut.)

Displacement Field of Cut Lips. The displacement field of the cut lips is obtained by eval-uating the displacement of each node of the cut mesh, shown in Fig. 3(e), which represents theintersections of the cut with the volume mesh. In the following, these nodes are named cutintersections.

First the lips of the cut are assumed to be moved by the retractor along a direction that formsan angle, shown in Fig. 5(a), with the plane of the cut. Then a surface mesh of the whole-brainregion segmented out from the third iMR image is built. Finally, for each cut intersection, a

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vector is defined, which starts at this intersection, forms the pre-defined angle (Fig. 5(a)) withthe cut plane, and stops at its intersection with the surface mesh of the whole-brain region justmentioned. The length and direction of the vector represent the magnitude and direction of thedisplacement to apply to the cut intersection of the lip. Figures 5(b)-5(c) show the displacementfield for the set of cut intersections for the right lip.

The displacement d retr,rightn of each cut intersection n of the right lip is applied by imposing,

based on Equations (3) and (6), the conditions

ϕAn(xretrn )uAn + H(xretr

n )ϕAn(xretrn )aAn

+ ϕBn(xretrn )uBn

+ H(xretrn )ϕBn(xretr

n )aBn

= ϕAn(xretrn )uAn + ϕAn(xretr

n )aAn

+ ϕBn(xretrn )uBn

+ ϕBn(xretrn )aBn

= dretr,rightn

(8)

where xretrn is the position of cut intersection n, An and Bn are the two nodes of the FE edge

that cut intersection n is located on, ϕAm(x) and ϕBm(x) are the FEM NSFs of nodes Am andBm, uAn and uBn are the nodal FEM DOFs of nodes An and Bn, and aAn and aBn are thenodal XFEM Heaviside DOFs of nodes An and Bn. The Heaviside function H(x) is equal to+1 at each xretr

n , although it could also be taken as −1. The assignments of the values for H(x)+1 and −1 can indeed be exchanged, depending which value is assigned on the left and rightside of the cut. Conditions (8) are applied with Lagrange multipliers. A similar procedure isused for the left part of the brain.

Once the displacement fields of both brain hemispheres and lips of the cut have been estimated,they are applied to the biomechanical model to drive its deformation. These displacementfields come from independent computations, and are neither necessarily consistent with eachother, nor compatible with the volume mesh. Figures 6(1a)-6(1c) jointly show the displacementfields of the right brain hemisphere and of the right lip of the cut. One clearly sees that, inthe neighborhood of the external surface of the brain, the two displacement fields cross eachother. The application of these two displacement fields to the biomechanical model would leadto element flipping. Before applying these displacement fields, they have to be smoothed. Fig-ures 6(2a)-6(2c) show the two displacements fields that have been jointly smoothed based ona weighted-distance average using ten neighbor nodes. The displacement fields are now con-sistent with each other, and compatible with the volume mesh. Furthermore, the direction ofthe displacement field of the right hemisphere (Fig. 6(1c)) is clearly modified by the smoothing(Fig. 6(2c)). This modification of direction allows one to capture in a better way the opening ofthe path created by retraction. One thus concludes that the use of an active surface algorithmfor evaluating the displacement fields of the brain hemispheres is not sufficient, and that theevaluation of the displacement fields of the cut lips is required for correctly modeling retraction.A similar procedure is used for the left part of the brain. Note that left and right displacementfields are smoothed separately because the deformation of both sides of the brain are differentand relatively uncoupled.

XFEM-Based Biomechanical Model DeformationIn order to deform the biomechanical model, one must transpose the smoothed displacementfields of the brain hemispheres and lips of the cut into displacements to be applied to thebiomechanical model. The smoothed displacement field of the lips are directly applied to cutintersections. The displacement of each cut intersection is applied by imposing the conditions

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described in Equation (8) with Lagrange multipliers. The displacement of each node of theexternal surface of the biomechanical model is given by the displacement of the closest node ofboth initial active surfaces, representing the brain hemispheres, or of the surface mesh represent-ing the intersections of the cut with the volume mesh. The biomechanical model then deforms,based on XFEM. The FEM-software tool Metafor (http://metafor.ltas.ulg.ac.be) developedin our mechanical-engineering department is used, which we have added a XFEM module to.The initial stress state of the brain is unknown and is thus set to zero [11, 43]. The resultingvolume displacement field of the biomechanical model is shown in Fig. 7.

Results of Image Warping and ValidationThe resulting volume displacement field of the biomechanical model (Figs. 7(a)-7(b)) is used towarp the part of the second iMR image corresponding to the whole-brain region, i.e. with theskull and external cerebrospinal fluid masked out. Figure 8(1a) shows the second iMR image,and Fig. 8(2a) shows the result of rigidly registering the third iMR image to the second iMRimage. Figure 8(1b) shows the whole-brain region extracted from Fig. 8(1a), while Fig. 8(2b)shows the result of the warping of Fig. 8(1b).

To evaluate the accuracy of our 3D nonrigid registration technique, the image similarity iscompared between the two iMR images rigidly registered with the similarity between the imageresulting from the warping of the second iMR image and the third iMR image. This gives us anestimate of how well one is able to capture, and compensate for, the local deformations betweenthe two iMR images. To qualitatively estimate the similarity between two images, the edgesextracted from these images using the Canny edge detector itkCannyEdgeDetectionImageFilter,available in ITK (http://www.itk.org/), are compared. This allows us to visually, and also lo-cally, estimate the alignment of the iMR images. Figure 8(1c) shows the juxtaposition of edgesextracted from the whole-brain region of the two iMR images rigidly registered, while Fig. 8(2c)shows the juxtaposition of the edges extracted from the result of the warping of the second iMRimage and the whole-brain region of the third iMR image. The visual comparison of Figs. 8(1c)and 8(2c) shows that edges match much better in (2c) than in (1c), which qualitatively indicatesthe benefit of the nonrigid registration.

To quantitatively estimate the similarity of the two edge maps, the modified Hausdorff dis-tance is computed between the sets of edge points, i.e. voxels representing the edges, in thesetwo images. The modified Hausdorff distance H(A,B) [6] between two sets of points A and Bis defined as

H(A,B) = max(h(A,B), h(B,A)), (9)

where h(A,B) is the directed Hausdorff distance. The directed Hausdorff distance is a measureof the distance of the point set A to the point set B, and is defined as

h(A,B) =1

Na

a∈A

d(a,B), (10)

where Na is the number of points in set A, and d(a,B) is the distance of point a ∈ A to theclosest point in B, i.e. d(a,B) = minb∈B ‖a− b‖, where ‖a− b‖ is the Euclidean distance. Thedirected Hausdorff distance h(A,B) thus computes the average distance of points of A to pointsof B. The averaging minimizes the effects of outlier points, e.g. due to image noise. The valueof the modified Hausdorff distance H(A,B) increases with the amount of difference betweenthe two sets of edges points. The advantage of using a criterion such as the modified Haus-dorff distance is that it quantifies (numerically) the alignment of the iMR images. However,the quantification remains global, and cannot reflect that fact that some local regions could bebetter nonrigidly registered than other ones. The qualitative and quantitative evaluations of

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the nonrigid registration, based on the visual comparison of edges and the modified Hausdorffdistance, are thus complementary.

The modified Hausdorff distance distance decreases from 1.36 mm for the set of edges extractedfrom the two iMR images rigidly registered (Fig. 8(1c)), down to 1.10 mm for the iMR imagesnonrigidly registered (Fig. 8(2c)). This indicates that the nonrigid registration improves thealignment of the two iMR images, although the opening width of the path created by retractionis not totally captured, as shown in Fig. 8(2b) in comparison to Fig. 8(2a). We expect thisis due to the segmentation, as explained above. In addition, the need for jointly smoothing(respectively on each side of the brain) the displacement fields of the hemispheres and the lipsof the cut, as explained above, tends to globally decrease the magnitude of the displacementfields, and, thus, to decrease the opening width of the path.

ConclusionsWe developed a method using intraoperative images for evaluating the brain deformation dueto retraction. The nonrigid registration technique was based on a linear elastic biomechanicalmodel driven by the deformations of key surfaces, extracted and tracked from the two succes-sive iMR images. The cut was defined directly based on the two iMR images, and, then, wasincluded in the biomechanical model using XFEM. The iMR images used in our experimentswere acquired with a 0.5 Tesla intraoperative GE Signa scanner. To facilitate the evaluationof our results, the iMR image acquired just before the retraction was updated, as a substitutepreoperative image.

In this work, all performed in 3D, the whole-brain region boundary and the cut were cho-sen as surface landmarks. The deformation of the whole-brain region boundary was decoupledin two independent computations of the active surface algorithm for both hemispheres, andshowed that it was beneficial since they deformed very differently. The importance of evaluat-ing the displacement fields of the cut lips for correctly capturing the opening of the path createdby retraction was also shown. While the opening width of the path was not totally captured,the nonrigid registration was shown to improve the alignment of the two iMR images.

To further improve our retraction modeling, future work could be considered in fourth mainareas. First, a way of preventing the opening width of the path from decreasing should be deter-mined. The segmentation method, and the evaluation of the surface displacement fields shouldthus be improved to ensure that they do not imply this effect. Second, a linear formulation ofXFEM was used, which applies to small deformations. Nevertheless, a nonlinear formulationof XFEM [15] should be implemented for the surgery cases involving large deformations of thebrain. Third, the fact that iMR images are used could be further exploited. Indeed, theseimages provide volume information (rather than surface information only), are of good qualityin comparison to other intraoperative modalities, and possess a field of view that includes thefull volume of brain tissues (for the 0.5 Tesla GE Signa scanner). These images thus allow oneto evaluate what, and how, new structures of the brain could be used. Some regions, e.g. thelateral ventricles’ region, could be extracted from the two iMR images, and used as surfacelandmarks to drive the deformation of the biomechanical model. Fourth, as mentioned in theintroduction, retraction followed by successive resections should be modeled in order to be ableto update preoperative images for any kind of deformation encountered by the patient.

AcknowledgementsThis research was supported in part by the “Fonds Leon Fredericq”, Liege, Belgium, by the“Fonds National de la Recherche Scientifique (F.N.R.S.)”, Brussels, Belgium, by a researchgrant from CIMIT, and by NIH grants R01 RR021885, R01 GM074068 and R01 EB008015.

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References

[1] Abdelaziz Y, Hamouine A. A survey of the extended finite element. Computers andStructures 2008;86(11-12):1141–1151.

[2] Bielser D, Glardon P, Teschner M, Gross MH. A state machine for real-time cutting oftetrahedral meshes. Graphical Models 2004;66(6):398–417.

[3] Clatz O, Delingette H, Talos I-F, Golby AJ, Kikinis R, Jolesz FA, Ayache N, WarfieldSK. Robust non-rigid registration to capture brain shift from intra-operative MRI. IEEETransactions on Medical Imaging 2005;24(11):1417–1427.

[4] Cohen-Adad J, Paul P, Morandi X, and Jannin P. Knowledge modeling in imageguided neurosurgery: application in understanding intra-operative brain shift. Proceed-ings of the Medical Imaging: Visualization, Image-Guided Procedures and Display, SPIE2006;6141:709–716.

[5] Dolbow JE. An Extended Finite Element Method with Discontinuous Enrichment forApplied Mechanics. PhD thesis, Northwestern University; 1999.

[6] Dubuisson M-P, Jain AK. A modified Hausdorff distance for object matching. In Proceed-ings of the 12th IAPR International Conference on Pattern Recognition 1994;566–568.

[7] Duflot M. A meshless method with enriched weight functions for three-dimensional crack.International Journal for Numerical Methods in Engineering 2006;65(12):1970–2006.

[8] Duflot M, Nguyen-Dang H. A meshless method with enriched weight functions for fatiguecrack growth. International Journal for Numerical Methods in Engineering 2004;59:1945–1961.

[9] Felippa CA. Introduction to Finite Element Method. Available online, 2001.

[10] Ferrant M. Physics-based Deformable Modeling of Volumes and Surfaces for Medical Im-age Registration, Segmentation and Visualization. PhD thesis, Universite catholique deLouvain, Telecommunications Laboratory, Louvain-la-Neuve, Belgium, April 2001.

[11] Ferrant M, Nabavi A, Macq B, Kikinis R, Warfield SK. Serial registration of intra-operativeMR images of the brain. Medical Image Analysis 2002;6(4):337–359.

[12] Ganovelli F, Cignoni P, Montani C, Scopigno R. A multiresolution model for soft objectssupporting interactive cuts and lacerations. Computer Graphics Forum 2000;19(3):271–282.

[13] Geuzaine C, Remacle J-F. Gmsh: a three-dimensional finite element mesh generator withbuilt-in pre- and post-processing facilities. International Journal for Numerical Methodsin Engineering 2009;79(11):1309–1331.

[14] Hagemann A, Rohr K, Stiehl HS. Coupling of fluid and elastic models for biomechanicalsimulations of brain deformations using FEM. Medical Image Analysis 2002;6(4):375–388.

[15] Jerabkova L, Kuhlen T. Stable cutting of deformable objects in virtual environments usingxfem. IEEE Computer Graphics and Applications 2009;29(2):61–71.

[16] Kass M, Witkin A, and Terzopoulos D. Snakes: Active contour models. InternationalJournal of Computer Vision 1988;1(4):321–331.

[17] Kyriacou SK, Mohamed A, Miller K, Neff S. Brain mechanics for neurosurgery: Modelingissues. Biomechanics and Modeling in Mechanobiology 2002;1(2):151–164.

12

Page 13: 3D XFEM-Based Modeling of Retraction for Preoperative Image

[18] Lamprich BK and Miga MI. Analysis of model-updated MR images to correct for braindeformation due to tissue retraction. Proceedings of the Medical Imaging: Visualization,Image-Guided Procedures and Display, SPIE 2003;5029:552–560.

[19] Lunn KE, Paulsen KD, Roberts DW, Kennedy FE, Hartov A, West JD. Displacementestimation with co-registered ultrasound for image guided neurosurgery: a quantitative invivo porcine study. IEEE Transactions on Medical Imaging 2003; 22(11):1358–1368.

[20] Lunn KE, Paulsen KD, Roberts DW, Kennedy FE, Hartov A, Platenik LA. Nonrigid brainregistration: synthesizing full volume deformation fields from model basis solutions con-strained by partial volume intraoperative data. Computer Vision and Image Understanding2003;89:299–317.

[21] Miga MI, Paulsen KD, Hoopes PJ, Kennedy FE, Hartov A, Roberts DW. In vivo quantifi-cation of a homogeneous brain deformation model for updating preoperative images duringsurgery. IEEE Transactions on Biomedical Engineering 2000;47(2):266–273.

[22] Miga MI, Roberts DW, Kennedy FE, Platenik LA, Hartov A, Lunn KE, Paulsen KD.Modeling of retraction and resection for intraoperative updating of images. Neurosurgery2001;49(1):75–84.

[23] Miller K, Wittek A. Neuroimage registration as displacement - zero traction problem ofsolid mechanics. In Workshop ”Computational Biomechanics for Medicine” at the Inter-national Conference on Medical Image Computing and Computer-Assisted Intervention2006.

[24] Miller K, Wittek A, Joldes G, Horton A, Dutta-Roy T, Berger J, Morriss L. Modelling braindeformations for computer-integrated neurosurgery. International Journal for NumericalMethods in Biomedical Engineering 2009;26(1):117–138.

[25] Moes N, Dolbow J, Belytschko T. A finite element method for crack growth withoutremeshing. International Journal for Numerical Methods in Engineering 1999;46:131–150.

[26] Mor A, Kanade T. Modifying soft tissue models: Progressive cutting with minimal newelement creation. In Proceedings of the International Conference on Medical Image Com-puting and Computer-Assisted Intervention 2000;1935:598–607.

[27] Nabavi A, Black PM, Gering DT, Westin CF, Mehta V, Pergolizzi RS, Ferrant M, WarfieldSK, Hata N, Schwartz RB, Wells WM, Kikinis R, and Jolesz FA. Serial intraoperativemagnetic resonance imaging of brain shift. Neurosurgery 2001;48(4):787–798.

[28] Nienhuys HW. Cutting in deformable objects. PhD thesis, Institute for Information andComputing Sciences, Utrecht University; 2003.

[29] Nienhuys HW, van der Stappen FA. A surgery simulation supporting cuts and finiteelement deformation. In Proceedings of the International Conference on Medical ImageComputing and Computer-Assisted Intervention 2001;2208:153–160.

[30] Nimsky C, Ganslandt O, Hastreiter P, Fahlbusch R. Intraoperative compensation for brainshift. Surgical Neurology 2001;56(6):357–365.

[31] Paulsen KD, Miga MI, Kennedy FE, Hoopens PJ, Hartov A, Roberts DW. A computationalmodel for tracking subsurface tissue deformation during stereotactic neurosurgery. IEEETransactions on Biomedical Engineering 1999;46(2):213–225.

13

Page 14: 3D XFEM-Based Modeling of Retraction for Preoperative Image

[32] Platenik LA, Miga MI, Roberts DW, Kennedy FE, Hartov A, Lunn KE, Paulsen KD.Comparison of an incremental versus single-step retraction model for intraoperative com-pensation. Proceedings of the Medical Imaging: Visualization, Image-Guided Proceduresand Display, SPIE 2001;4319:358–365.

[33] Platenik LA, Miga MI, Roberts DW, Lunn KE, Kennedy FE, Hartov A, Paulsen KD. Invivo quantification of retraction deformation modeling for updated image-guidance duringneurosurgery. IEEE Transactions on Biomedical Engineering 2002;49(8):823–835.

[34] Serby D, Harders M, Szekely G. A new approach to cutting into finite element models.Proceedings of the International Conference on Medical Image Computing and Computer-Assisted Intervention 2001;2208:425–433.

[35] Skrinjar O, Nabavi A, Duncan J. Model-driven brain shift compensation. Medical ImageAnalysis 2002;6(4):361–373.

[36] Songbai J, Hartov A, Robert D, and Paulsen K. Data assimilation using a gradient de-scent method for estimation of intraoperative brain deformation. Medical Image Analysis,2009;13(5):744–756.

[37] Steinemann D, Harders M.A, Gross M, Szekely G. Hybrid cutting of deformable solids.In Proceedings of the IEEE Computer Society Conference on Virtual Reality, Alexandria,Virginia, USA, March 25–29, 2006;35–42.

[38] Sukumar N, Moes N, Belytschko T, Moran B. Extended finite element method for three-dimensional crack modelling. International Journal for Numerical Methods in Engineering2000;48(11):1549–1570.

[39] Sukumar N, Prevost JH. Modeling quasi-static crack growth with the extended finiteelement method. Part I: Computer implementation. International Journal of Solids andStructures 2003;40(26):7513–7537.

[40] Sun H, Kennedy FE, Carlson EJ, Hartov A, Roberts DW, Paulsen KD. Modeling of braintissue retraction using intraoperative data. Proceedings of the International Conference onMedical Image Computing and Computer-Assisted Intervention 2004;225–233.

[41] Vigneron LM, Duflot MP, Robe PA, Warfield SK, Verly JG. 2D XFEM-based modelingof retraction and successive resections for preoperative image update. Computer AidedSurgery 2009;14(1-3):1–20.

[42] Vigneron LM, Boman RC, Ponthot J-P, Robe PA, Warfield SK, Verly JG. EnhancedFEM-based modeling of brain shift deformation in image-guided neurosurgery. Journal ofComputational and Applied Mathematics 2010;234:2046–2053.

[43] Warfield SK, Talos F, Kemper C, Cosman E, Tei A, Ferrant M, Macq B, Wells WM,Black PM, Jolesz FA, Kikinis R. Augmenting intraoperative MRI with preoperative fMRIand DTI by biomechanical simulation of brain deformation. Proceedings of the MedicalImaging: Visualization, Image-Guided Procedures and Display, SPIE 2003;5029:79–86.

[44] Warfield SK, Haker SJ, Talos IF, Kemper CA, Weisenfeld N, Mewes AU, Goldberg-ZimringD, Zou KH, Westin CF, Wells WM, Tempany CM, Golby A, Black PM, Jolesz FA, KikinisR. Capturing intraoperative deformations: research experience at Brigham and Women’sHospital. Medical Image Analysis 2005;9(2):145–162.

[45] Wittek A, Miller K, Kikinis R, Warfield SK. Patient-specific model of brain deformation:Application to medical image registration. Journal of Biomechanics 2007;40(4):919–929.

14

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[46] Xu C, Pham DL, Prince JL. Handbook of Medical Imaging–Volume 2: Medical ImageProcessing and Analysis, chapter Medical Image Segmentation Using Deformable Models,2000;129–174.

[47] Zienkiewicz OC, Taylor RL. The Finite Element Method. Butterworth-Heinemann; 2000.

s1

es2(x*)

en(x*)

x

x*

Crack

surface

es1(x*)

s2

Figure 1: Parameters related to the XFEM enrichment Heaviside function corresponding to a3D crack surface.

Figure 2: Segmentation of intraoperative images. (1a) Original second iMR image, acquiredbefore retraction. (2a) Original third iMR image, acquired after retraction. (1b) Whole-brainregion segmented out manually from (1a) with 3D Slicer. (2b) Ditto for (2a). (1c) Smoothingof the segmented whole-brain region from (1b). (2c) Ditto for (2b). (1c) and (2) are performedin order to minimize the dependance of the active surface algorithm on segmentation roughnessand surface mesh resolution.

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Figure 3: Representation of the planar cut to be included in the biomechanical model. (a) Sur-face mesh of the biomechanical model with the location of the plane where the retractor wasinserted. This plane corresponds to the plane separating the two hemispheres. (b) Whole-brainregion segmented out from the second iMR image, acquired before retraction, in the selectedplane shown in (a). (c) Ditto for the third iMR image, acquired after retraction. (d) Cutdefined as the difference between (b) and (c), where (c) has beforehand been processed to re-move small, isolated islands. (e) Surface mesh representing the intersections of the cut withthe volume mesh of the whole-brain region brain.

(a) (b)

max: 3.44 min: 0.00

9.33

7.00

4.67

2.33

0.00

Displ. mag. mm

max: 9.33 min: 0.00

Figure 4: Results of the active surface algorithm for the two hemispheres of the brain. (a) Initialactive surface for right hemisphere with color levels corresponding to the magnitude of thedisplacement field. (b) Initial active surface for left hemisphere with same color coding. (Inaccordance with radiological convention, the right side of the iMR images corresponds to theleft side of the brain, and conversely.)

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(a) (b) (c)

10.7

8.07

5.43

2.78

0.14

Displ. mag. mm

Figure 5: Evaluation of the displacement field of the cut for 3D retraction modeling. (a) Whole-brain region segmented out from the third iMR image, with segment (in red) defining the angleof the displacement field of the right lip. (b) Surface mesh built from the segmented thirdiMR image, cut mesh (Fig. 3(e)) (in red), and displacements of cut intersections (in green).(Displacements go from right to left in figure.) (c) Cut mesh (Fig. 3(e)) with color levelscorresponding to the magnitude of the displacement field of the right lip.

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(1a) (1c)(1b)

(2a) (2c)(2b)

displacement field of

brain hemisphere

displacement field of

cut lip

displacement field of

brain hemisphere

displacement field of

cut lip

Figure 6: Comparison of unsmoothed (row 1) and jointly smoothed (row2) displacement fieldsof right hemisphere and right lip of the cut. (1a) Surface meshes of right hemisphere and rightlip of cut, and their surface displacement fields in green. (1b) Same as (1a) except that thesurface meshes are left transparent. (1c) Zoom of (1b) around the external surface of thebrain. (2a), (2b), and (2c) are respectively the same as (1a), (1b), and (1c), except that thedisplacement fields are jointly smoothed. (In each subfigure, the direction of the displacementvectors is defined such that the vectors start at the surface mesh nodes.)

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(a) (b)

12.0

9.03

6.02

3.02

0.02

Displ. mag. mm

x

y

z

(c)

x y

z

Figure 7: Deformation of the biomechanical model based on XFEM for retraction modeling.(a) External surface mesh of the biomechanical model with color levels corresponding to themagnitude of the displacement field. (b) Slice of the biomechanical model with same color cod-ing. (c) Final mesh resulting from the modeling of the retraction using XFEM. The tetrahedrathat were added to display the opening of the cut lips are there only for visualisation purposes.The edges of FEs cut by the discontinuity have actually been made discontinuous and theirnodes moved apart.

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Figure 8: Results of retraction modeling. (1a) Original second iMR image. (2a) Third iMRimage, rigidly registered to (1a). (1b) Whole-brain region extracted from (1a). (2b) Defor-mation of (1b) using the volume displacement field of the biomechanical model, computed viaXFEM. (1c) Juxtaposition of edges of whole-brain region of second iMR image (1a) in green,and of edges of whole-brain region of third iMR image (2a) in red. (2c) Juxtaposition of edgesof whole-brain region of second iMR image deformed for modeling brain shift (2b) in green, andof edges of whole-brain region of third iMR image (2a) in red. (The red edges in (1c) and (2c)are identical.)

20