# Approach-guided controlled resolution brain meshing for FE-based interactive neurosurgery simulation

Michel Audette, Denis Rivière, Matthew Ewend, Sébastien Valette

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Michel Audette, Denis Rivière, Matthew Ewend, Sébastien Valette. Approach-guided controlled resolution brain meshing for FE-based interactive neurosurgery simulation. Workshop on Mesh Processing in Medical Image Analysis, Rasmus R. Paulsen; Joshua A. Levine, Sep 2010, Toronto (CA), Canada. hal-05053842

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# Approach-guided controlled resolution brain meshing for FE-based interactive neurosurgery simulation

Michel Audette¹, Denis Rivière², Matthew Ewend³, Sébastien Valette⁴

¹ Kitware

² Service Hospitalier Frédéric Joliot

³ University of North Carolina, Dept. Neurosurgery

⁴ CREATIS; CNRS; Université de Lyon, France

**Abstract.** This paper presents an adaptation of a two-stage tetrahedralization, featuring boundary-faithful controlled-resolution surface and volumetric meshing, suitable for multi-grid finite-element based neurosurgery simulator, whose architecture reflects the choice of neurosurgical approach and the nature of the pathology. Moreover, the first application of our approach-guided methodology is a simulation of the pterional approach, which entails a brain mesh that is separable about the left or right Sylvian fissure, providing access to deep-seated lesions. In addition to this sulcal separability constraint, the emphasis of this procedure on lesions in the skull base also suggests that the meshing should model ventricles faithfully. However, another requirement that conflicts with the faithfulness to these sulcal and ventricular boundaries is the need to limit the number of tetrahedra, at the coarsest level, in order to produce an interactive yet materially and geometrically nonlinear response, which will be achieved by GPU-accelerated Total Lagrangian Explicit Dynamics. Finally, we view a controlled resolution as a cornerstone of a multi-grid approach to the simulation, which in turn we view as a necessity, given also the conflicting requirements of having to compute a deformation on a relatively large organ such as the brain, while also capturing finely detailed critical tissues, such as blood vessels, deleterious gestures to which must be penalized by the simulation.

## 1 Introduction: clinical need and conflicting requirements

CBTRUS has calculated a worldwide estimate of 186,678 newly diagnosed primary non-malignant brain and central nervous system tumors per annum for 2002 [4]. The five-year relative survival rate following diagnosis of a primary malignant brain tumor is 33.8% for males and 37.5% for females. It has been recognized that the main factor in long-term patient survival, other than the nature of the tumor, is completeness of the excision [5][22] underscoring the importance of surgical efficacy.

Surgical efficacy can be improved by simulation-based training, but despite advances in other areas and an important clinical need, simulation has not made major inroads into neurosurgery. Surgery simulation can be broadly categorized into predictive and interactive techniques, respectively used by expert surgeons for planning, through high-quality finite element (FE) computations, and by surgical residents for developing their skill, through interactive biomechanics and haptics. Correction of scoliosis of
the spine [1] and maxillofacial bone repositioning [13] exemplify predictive simulation. Meanwhile, validation studies have shown that residents having trained with an interactive simulator outperformed residents without such training in performing laparoscopic cholecystectomies [18]. The two types of simulation can be seen as poles of a spectrum characterized by the conflicting requirements of interactivity and faithfulness to tissue response, which research increasingly attempts to bridge through new efficiencies [10].

The clinical need that justifies pursuing interactive neurosurgery simulation with a strong predictive aspect lies in two tendencies in modern surgery: the compression of training schedules of surgical residents and the constant influx of new therapeutic technologies available to expert surgeons. Training schedule compression coincides with a recent movement to limit resident duty hours [14]. Interactive surgery simulation can provide a means for accelerating the training of residents, allowing them to take a more active role than in the traditional framework, whereby residents observe senior surgeons and gradually assume increased responsibility, and can result in measurable improvements to both skill and patient outcome. Meanwhile, the influx of new technology in neurosurgery is characterized by the increasing use of physics- and computer-based technology in modern practice [6]: stereotactic and minimally invasive neurosurgery, modalities such as ultrasound and digital atlases, as well as recent therapies such as the ultrasonic surgical aspirator, lasers, radiotherapy, and molecular therapies. The result is an armamentarium that is broad and technically complex, where simulation could play a pivotal role for expert surgeons assimilating or refining new treatments.

Our main objective is a neurosurgery simulator that can effectively model skull base surgery. Its architecture is organized according to two nearly orthogonal parameters: i) the neurosurgical approach (e.g.: pterional, trans-nasal, and so on), which has implications for relevant critical tissues and the subvolume of clinical interest; and ii) the nature of the pathology, which dictates the choice of surgical tools and treatment. A requirement of skull base surgery simulation is depicted in figure 1 (a): one of the techniques available to the neurosurgeon for resecting deep-seated lesions involves parting the brain tissue about the Sylvian fissure. This technique characterizes pterional approach, which consists of approaching the brain and the anterior skull base through their lateral aspect after removing the frontal and temporal bones and the greater wing of the sphenoid [12]. The extensive spectrum of the neural and the vascular structures within reach includes insula, basal ganglia, lateral ventricle, middle cerebral artery, temporal operculum, frontal and parietal opercula, uncus, orbit, anterior cranial fossa, optic nerve, internal carotid artery and branches, lamina terminalis, and interpeduncular fossa.

This paper presents on-going research that features an adaptation of controlled-resolution clustering-based surface meshing, which is sulcal-separable if the surgical approach requires it of the anatomical model, followed by controlled-resolution tetrahedralization suitable for multi-grid FE [24] based neurosurgery simulator. Currently, the second stage is the Almost-regular Tetrahedralization method [9], which uses a mesh size objective consistent with that of the surface mesh. Moreover, simulation of the pterional approach entails a brain mesh that is separable about the left or right Sylvian fissure, providing access to deep-seated lesions. In addition to this sulcal separability constraint, the emphasis of this procedure on lesions in the skull base also suggests that the meshing should model ventricles faithfully. However, another requirement that con-
flicts with the faithfulness to these sulcal and ventricular boundaries is the need to limit
the number of tetrahedra, to roughly 2000 or fewer, at the coarsest level, in order to pro-
duce an interactive yet materially and geometrically nonlinear response, which will be
achieved by GPU-accelerated Total Lagrangian Explicit Dynamics [20][10]. We view a
multi-grid approach to the simulation as a necessity, given the conflicting requirements
of modeling a large organ such as the brain and finely detailed critical tissues, however
the scope of this paper is restricted to the computation of the coarse-level mesh. Last,
this model is used in conjunction with the SOFA platform, which implies that this tetra-
hedral mesh can be significantly coarser than the surface-based visual model, as long as
the main structures relevant to the biomechanics, e.g. ventricles, are represented.

<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Relevance and implementation of sulcal separability. (a) illustration of parting of Sylvian fissure to provide access to deep-seated tumor, and depiction of critical tissues; (b) illustration of BrainVISA results for labeling sulci.</figcaption></figure>

## 2 Preprocessing: segmentation and sulcal labeling

The first stage of our current processing method to produce descriptive anatomical mod-
els of the brain is the image analysis pipeline that segments T1-weighted MR data
according to white and grey matter (WM, GM) as well as corticospinal fluid (CSF),
based on scale-space analysis, which is also the first step of the BrainVISA image
analysis pipeline [3]. An alternate segmentation method is an atlas-based Expectation
Maximization [15], if sulcal labeling is not required. With both methods, a separate
tumor segmentation [16] will eventually be integrated for practical patient-specific neu-
rosurgery simulation. The latter method identifies tumors based on their statistical out-
lying characteristic in T₁-T₂ space, in conjunction with a probabilistic digital brain at-
las that incorporates T₁-T₂ features vectors of WM, GM and CSF, and with a level-set
based region growing algorithm. In the BrainVISA pipeline, preliminary classification
in terms of WM, GM and CSF is followed by morphological analysis to separate hemi-
spheres and detect sulci on the basis of a white matter skeleton, which are subsequently
<figure><img src="image_3.png" /><figcaption>**Fig. 2.** Post-segmentation sulcus-aware morphological processing of brain binary image: (a) raw brain binary image; (b) result of eliminating irrelevant sulci, while emphasizing clinically relevant left Sylvian and hemisphere fissures, while preserving the ventricles.</figcaption></figure>

expressed as nodes within relational graphs, which nodes coincide with simple surface primitives of various topological types [17]. The last computational stage of BrainVISA inputs this relational graph, within which each node is labeled according to its underlying surface type, to a multi-layer perceptron, which can deal with topological variability of sulci across individuals. The output of the sulcal labeling pipeline is shown in figure 1 (b), and is postprocessed to produce a suite of VTKPolydata files, which serves as input to the next stage, in addition to the brain tissue segmentation produced in the preprocessing.

Prior to the computation of triangular surface mesh, the sulci are used to focus the morphological filtering of the binary brain volume, so as to prevent clinically irrelevant sulci from appearing in the anatomical model, while also imposing separability where the clinically relevant sulcus appears, namely the left Sylvian fissure. The basic idea is to ensure that the mesh does not bridge the sulcus about which separability is required, while limiting the element count in the rest of the brain. This stage is implemented with a series of morphological operations on each hemisphere:

- a small morphological dilation of a voxel-based representation of all sulci;

- a small morphological closing of the brain binary volume OR'ed with the dilated irrelevant sulci;

- a small morphological opening of the closed brain binary volume from step ii on the basis of the clinically relevant sulcus;
- a blurring with a small kernel to produce brain hemispheres and subcortical structures from which stable results can be obtained by the application of Marching Cubes;

- a MAX operation of the blurred hemispheres and subcortical structure.

In addition, our experience suggests that rather than producing surface and volume meshes that are of constant mesh size, it is more feasible to achieve a mesh size objective that is smaller at surfaces of high curvature, such as the ventricles and the relevant Sylvian fissure. To this effect, the ventricles are isolated by morphological filtering, and a distance map is computed from them as well as from the relevant Sylvian fissure. Last, this surface model will also emphasize the space between hemispheres, and the extraction of this space is improved by exploiting prior labeling of left and right hemispheres produced by BrainVISA. This process is illustrated in figure 2.

# 3 Surface mesh simplification with topological guarantees

## 3.1 Motivation for a two-stage approach

We adopt a two-stage method for achieving a controlled-resolution, boundary-faithful mesh, in the spirit of the method developed by us in [2]. We have found by experience that a number of one-stage tetrahedralization methods tend to fuse two boundaries that are close together, which inevitably annihilates the Sylvian fissure imbedded in the mesh. The surface meshing stage begins with a Marching Cubes computation on the preprocessed binary brain volume of image in figure 2 (b). This Marching Cubes result is shown in figure 3.1 (b).

<figure><img src="image_4.png" /><figcaption>Fig. 3. Preliminary steps in surface meshing: (a) Marching Cubes; (b) first iteration of clustering.</figcaption></figure>
## 3.2 Adaptive coarsening

The simplification of the surface resulting from the Marching Cubes algorithm is carried out using a modified version of the ACVD software. This algorithm, based on [23], constructs an approximate Centroidal Voronoi Tesselation (CVT) [8] on the input mesh by clustering its vertices $v_j$ in n clusters $C_i$, in order to minimize the following energy:

$$E_{CVD} = \sum_{i=1}^{n} \left( \sum_{v_j \in C_i} \rho(v_j) \|v_j - c_i\|^2 \right) \quad (1)$$

where $\rho(v)$ is a density function defined for each vertex $v_j$, and allows adaptive clustering. In our experiments, we used a linear combination of the previously computed distance map to generate more vertices near the ventricles and the Sylvian fissure. Once the clustering is constructed, the set of cluster sites $c_i$ defines a point set that can be easily triangulated by analysing the clusters connectivity. The resulting triangulation is a good candidate for numerical simulations.

Unfortunately, this algorithm does not guarantee that the resulting mesh will be a manifold surface, as some regions might locally require more clusters to be represented by a manifold surface in the simplified mesh. To overcome this issue, after the triangulation is constructed, we check that each simplified mesh vertex ring is equivalent to a topological disk. If some vertices do not respect this constraint, we insert one cluster near every failing vertex, and re-perform the clustering step. For this clustering step, we freeze all clusters except those that are in the 1-neighbourhood of the problem vertices. Freezing those clusters prevents the clusters to redistribute too much on the mesh, otherwise the clusters insertion would have little impact on the topology correction. This process is repeated until the resulting mesh is watertight. Since the input mesh is a watertight mesh, the convergence of this algorithm is guaranteed.

Note that even if the resulting mesh is watertight, its genus might be lower than the genus of the original surface. As an example when a small handle fits entirely in one cluster, it will be repaced by a single vertex in the simplified mesh. This allows our approach to filter out topological noise that would forbid good coarse simplifications.

Figure 4 shows a closeup view of a simplification example. The mesh shown in (a) contains relatively thin tubular regions where the clusters density is not big enough. The resulting triangulation is highly non-manifold, and vertices need to be inserted. After two insertion steps, the resulting mesh is manifold (e).

## 3.3 Further simplifications

Although the resulting mesh is watertight, it can happen that many vertices were inserted in thin regions, which can penalize the performance of the simulation toolchain. In order to further reduce the number of vertices in the mesh, we perform a topology-preserving simplification step, in spirit with [11]: we perform successive edge collapse operations on the mesh until the desired number of vertices is reached. Similarly to [11], the ordering of the edge collapses is computed using Quadric Error Metrics (QEM), and we forbid any edge collapse that would alter the mesh topology. Final results are shown in section 5.
<figure><img src="image_5.png" /><figcaption>**Fig. 4.** Enforcing the correct topology. (a) : close-up view of a tubular input surface. (b) initial clustering. (c) resulting triangulation. This triangulation is not manifold, hence we introduce more clusters in the vicinity of non-manifold vertices. (d) after one iteration. (e) after two iterations, the result is manifold</figcaption></figure>

# 4 Surface-mesh constrained Tetrahedral Meshing

The tetrahedralization stage uses as input the controlled-resolution surface meshing results from the previous stage. Currently this stage is still underway. We are experimenting with open-source tetrahedralization software such as TetGen [21] that is faithful to boundaries and with the Almost-regular Tetrahedralization (ART) [9] used by us in past implementations where resolution control and boundary faithfulness were emphasized. The latter method admits an explicit mesh size objective, which is used to govern insertion of Steiner points. The final tetrahedral mesh is converted to an input format supported by the Simulation Open Framework Architecture (SOFA) platform [19], namely the VTK legacy file format.

Preliminary results illustrating tissue parting on SOFA have so far been achieved with the Co-rotational FEM, which demonstrate small-deformation parting of the brain tissue about the left Sylvian fissure. The deformation tends to be localized, which motivates our on-going integration of haptic interactivity with Total Lagrangian Explicit Dynamics finite elements, which we feel will better resolve the large deformation required in this type of simulation.

# 5 Results

Figure 5 illustrates typical results of the surface meshing, whose wireframe rendering clearly shows the left Sylvian fissure and ventricles, as well as extremely coarse elements away from these structures. Figure 6 displays an exploded view of the tetrahedralization, and a preliminary mouse-based interaction via SOFA's Co-Rotational FEM biomechanics engine. These results also illustrate that a suitably sparse and coarse surface mesh can be achieved, while the development of the tetrahedral mesh that mirrors this edge size function is still work in progress. However we feel that the application of the ART method, which affords precise control over internal edge size, and the adaptation of the tetrahedral clustering method [7] in a manner that maintains the existing boundary, show promise of producing volumetric mesh with that adheres to the density function of the surface mesh.
<figure><img src="image_6.png" /><figcaption>**Fig. 5.** Surface mesh simplification results: (a) wireframe and (b) surface and edge-based renderings. The mesh displayed here contains fewer than 600 elements.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**Fig. 6.** Volume meshing results: (a) exploded view and (b) preliminary interactive biomechanics results via SOFA co-rotational tetrahedral FEM class.</figcaption></figure>
# 6 Conclusions

This paper presented a two-stage method for producing controlled-resolution brain mesh for neurosurgical simulation applications, where in particular the meshing strategy reflects an approach-specific adaptation. Moreover, we have shown that a mesh that is separable about a Sylvian fissure is both desirable and achievable for the specific case of the pterional approach. This method was restricted to the computation of the coarse-level representation of the brain in a multi-grid framework, but the same techniques can be applied to computing medium and fine representations of a subvolume of the brain.

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