Virtual Cutting in Medical Data. Institute for Real-time Computer Systems and Robotics. University of Karlsruhe, Department for Computer Science

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1 Virtual Cutting in Medical Data A. Mazura and S. Seifert Institute for Real-time Computer Systems and Robotics Prof. Dr. U. Rembold, Prof. Dr. R. Dillmann University of Karlsruhe, Department for Computer Science 7618 Karlsruhe, Germany This paper has been published in the proceedings of \Medicine Meets Virtual Reality", IOS Press and Ohmsha 1997, edited by K.S. Morgan et. al. It has been presented on the conference \Medicine Meets Virtual Reality 5" in San Diego, CA, in January Abstract In this paper we present a method of virtual D cutting operations in D tomographic data. When cutting interactively the user species a series of D cutting points and correponding cut depths on the surface of the object. Each pair of succeeding cutting points then denes a freeform shape of the incision. A D meshing algorithm constructs a Finite Element mesh formed by tetrahedrons, representing the surgical intervention. 1 Introduction Planning of surgical interventions is based, up to now, almost exclusively on the experiences of the operating surgeon and on the interpretation of two-dimensional static sources of information. Our aim is to support the surgeon in his work by a D simulation of the consequences of the planned intervention, that is by the "playing" of alternative ways, their eects and risks. By providing the medical data with biophysical properties we think that the following craniofacial surgeries can be carried out more precisely in the future: tumor resection in bone and soft-tissue, surgical correction of pronounced maxillary and facial asymmetries, precision radiation therapy restricted to the tumor-area by computer-aided insertion of the radiation-source into the tumor. By using computer assisted simulation and surgical systems, we expect an advance in quality and a diminution of risks in surgery. More extensive and more radical interventions seem to be possible. Responsibility of the intervention, however, remains exclusively with the surgeon. In this paper a system of the simulation of virtual cutting operations is described. For this purpose a new D Finite Element approach is presented, where nonlinear elastic material and nonlinear geometric properties are included. 1

2 In earlier methods elliptic incisions can be applied on a three layer shell, approximating the surface of the medical object, from which range scanner data is available (s. []). Another method, modelling the virtual scalpel in a D manner is described in [5]. The method presented in this paper demonstrates, how a Finite Element mesh can be constructed, which represents an interactively specied free form incision, based on D matched MRI and CT data (s. []). The main advantages are: the geometry of the cut object can be approximated with subvoxel precision the resulting mesh contains compatible and almost regular node elements the renement of the mesh is controlled by the material properties of the underlying D data, the applied forces and the shape of the cutting area. Overview At rst a review of an automatic mesh generator (s. [6]) slightly modied for dealing with soft tissue is discussed. Then the modelling of an arbitrary incision is demonstrated (s. []). Its result is the local modication of the underlying Finite Element mesh induced by the interactively specied cutting operation. The paper ends by presenting the results, our further investigations and the conclusion. Octree Based Meshing When two corresponding CT and MRI data sets are matched eachvoxel can be classied as bone, muscle or remaining soft tissue (s. []). Before an incision can be specied, an automatic mesh generator based on the work of Yerri and Shepard (s. [6]) is used, which meshes the tomographic objects with tetrahedrons. The rst step consists of building an octree representation of the whole data which deviates from a standard octree in two ways. For reasons that will be explained later, a neighbour criteria concerning the size of octants on the same level must be guaranteed. Their edge ratio must not exceed a ratio of 1:. In gure 1 a standard octree representation can be seen, while in gure the ratio dependent structure is shown. Figure 1: Normal octree representation The second extension concerns the leaves of the tree, where supplementary cut octants are introduced. For satisfying the inter element compatibility criteria which is necessary for avoiding numerical instabilities and for a restriction of the distinction of cases, so called midedgepoints are introduced on the edges of an octant, where smaller octants are bordering on.

3 Figure : octree including the octant ratio condition interior exterior Figure : Cut octants at the object boundary The structure of the octree is not only determined by the geometric shape of the object, but also by the material of the data too. Homogeneous regions result in big octants, whereas heterogeneous areas will be divided until homogeneous partitions are found or the minimal octant size (at least voxel size) is reached. Meshing The Octants Having built the extended octree representation of the medical object each individual leave of the tree must be substituted by a set of tetrahedrons fullling the compatibility requirement wrt. the tetrahedron set inserted in the neighbour octants. For that reason two symmetric patterns are constructed for complete octants without midedgepoints which represent a region of homogeneous material properties if this octant is bigger than the minimal octant size. In case of minimal octants it may happen, that the bordering 8 nodes have dierent material properties. The only statement which can be assured, is, that these 8 nodes are in the interior of the object. If the pattern is selected on the basis of the relative position in the corresponding level (e.g. left pattern for position 1,,5,7 and right pattern for 0,,,6) compatibility is guaranteed (s. g. ). Figure : two patterns of complete octants In case of complete octants with midedgepoints the octant is divided into 6 pyramids. If such apyramid has not set any midegepoints on the edges of its quadrilateral surface, the pyramid is separated into two tetrahedrons. The direction of the diagonal which is splitting this surface, depends on the relative octant position (s. g. 5). If midedgepoints occur, two diagonals form a cross on the basic quadrilateral face of the corresponding pyramid. Figure 6 shows an example of a resulting tetrahedron subdivision.

4 cutting point Figure 5: subdivision of complete octants with at least one midedgepoint cutting point Figure 6: decomposition of the pyramids A cut octant ischaracterized by a minimal octant, where at least one node belongs to the exterior of the object. Normally 5 congurations of cut octants must be considered. Because of symmetry and ane transformations it is possible to reduce the total number of 5 to patterns. An example, where 6 nodes are in the interior and the remaining two octant nodes are in the exterior, is shown in gure 7 Figure 7: 6 nodes in the interior in the exterior of the object The resulting mesh serves as a basic mesh for applying interactive incisions. If only pushing and pulling of the object should be simulated, the user given forces invoke a local renement of the mesh. In this manner a dense information structure is supplied for nonlinear geometric considerations. Because of the fact, that the nodes of one tetrahedron may have totally dierent elastic material properties, there is a problem in relating an unambiguous material property to the corresponding tetrahedron. In our implementations the problem is solved by calculating the center of gravity and assigning its material parameters to the tetrahedron. If the center does not belong to the object, the median of the nodes will be selected.

5 5 The Cutting Model For applying an arbitrary cut at the Finite Element mesh, where the \virtual scalpel" changes place in a piecewise C 1 manner, bilinear reference planes are used. If the user chooses two cutting points at the surface of the object and species interactively the cutting direction and the corresponding depth, points will be given, which needn't to be complanar. An intuitive way of interpolating the shape of the incision in between consists of forming a bilinear reference plane (s. g. 8). It is called reference plane, cut end points cut entry points Figure 8: bilinear shape of the incision because additionally an adaption to the boundary of the object must be planned. As a result, the voxels which lie inside the object and have a distance to the object boundary are collected until the interpolated depth is reached. We call them cut voxels. It must be mentionned that the cut voxels are extracted from the originally matched cubic data. Clearly there is a dierence in volume between the cubic input data and the constructed Finite Element mesh constructed above. Therefore voxels outside the object are inserted too, for representing the cutting area by an approximation (the cut voxel set). The construction of the cut voxel set can be done very easily by an recursive algorithm. We are going to explain it in a vivid way. Let the two specied cutting surface points be p 0 and p 1 and let the corresponding cutting vectors be c 0 and c 1. Imagine that the vectors p i + c i with [0 1] are \piercing" the voxel set V i.now establish the midpoint m between p 0 and p 1. Because of the bilinear cutting shape it is easy to see that the vector m+c m with [0 1] and c m =0:5(c 0 +c 1 ) lie in the reference plane, too. Moving the interpolated cutting direction in m towards the boundary along the line m+c m with <, an adaption to the object shape can be realized. Now let the translated midpoint on the object surface be m s. Piercing again the voxels at m s in the direction c m produces the cut voxel set v s.ifv s is not a subset of v 0 [ v 1, the cut voxel set of the incision will not be complete. Therefore the problem extract(p 0 p 1 c 0 c 1 ) is divided into extract(p 0 p 1 c 0 c 1 ) and extract(p 0 p 1 c 0 c 1 ). Consequently the problem of extracting the cut voxels representing the incision is solved by a classical divide and conquer strategy. It is worth being noted that the translation of the vector m s to the object boundary is done in a preprocessing step, by which the possible boundary voxels between the interactively given incision points p 0 and p 1 are calculated. The segmentation process is done in the parameter space, regarding that a bilinear reference plane in D can be 5

6 represented by a function f : < ; > <. The idea is to realize ights into the object with a camera where a virtual cutting tool is mounted. 6 Cutting The Mesh First of all the octree will be locally modied. For each voxel of the cut voxel set it is checked by which octant it is included. The so marked octants are rened to the minimal octant size and the 1: ratio condition is reestablished. Then only a remeshing of these \hull octants" must be carried out. If material loss can be tolerated, e.g cutting only bones, an easy expansion of the cut voxel set suces, to simulate the incision. Pumping up the cut voxel set by the 6 neighbour voxels of each cut voxels guarantees that the approximated cutting area is D connected. Therefor only a local pruning operation to the octree is applied, where the octants with minimal octant size, containing the cut voxel set, vanish. If cutting without material loss is required, the following steps achieve the objective. Now the edges of the hull octants must be selected, which willbeinsertedby the tetrahedron pattern. Each edge of the hull octants will be tested if it has a point in common with the cut voxel set. This can be done by the D Bresenham algorithm (s. [1]) in combination with an implicit representation of the bilinear reference plane. Inserting the nodes of each tetrahedron of the hull octants into the implicit function restricts the sum of the tetrahedrons which are possible to be a cut tetrahedron. Afterwards the reduced set of tetrahedrons is tested by the Bresenham algorithm, if the elements lie in the cut area. An outline can be seen in gure 9. Now those tetrahedrons V0 Cut Voxel Set Minimal Octant V1 Figure 9: determination of the cut edges which have at least one cut edge are the remaining cut tetrahedrons. If a single reference plane is given the cutted tetrahedrons can be classied wrt. the number of its cutted edges which cannot exceed the total sum of. Inspecting the dierent classes it can be found that no more than 6 dierent cutting patterns must be distinguished. They can be seen in gure 10, 11 and 1. At this stage it's important to nd an admissible partitionning of cutted tetrahedrons in a way that the mentionned compatibility requirements are fullled. If only one or two cutting points exist if the symmetric patterns with three cutting points are present, the global identiers of the cutting points will be kept, because the concerned tetrahedrons form the border of the incision. In this way the corncerning edges are kept close during the wound opening. The decomposition scheme is shown in the gures 1 and 1. Otherwise it is necessary to \double" the cutting points, to make the opening of the wound. This means that a geometrical position gets two global identiers. The partitionning can be extracted from the gures 15 and 16. 6

7 normalisiert Figure 10: normalization of tetrahedrons with cutting points , 1-, - 0-, 0-1, 0-0-1, 1-, 1-0-, 1-, , 0-, - 0-, 1-, 0-0-1, 1-, , 0-, - 0-, 1-, 1-0-, 1-, - 7 Results normalized version Figure 11: normalization of tetrahedrons with cutting points As input data for our mesh generator we take matched CT- and MRI data (s. []). Dependent on the material properties associated with each class (bone, muscle and soft tissue), a nonlinear elastic stress-strain curve is determined, which approximates the material specic behaviour under loading. In gure 17 on the right the region of interest is selected. As boundary conditions the border of the selected region is xed. With the recongured mesh as input data the Finite Element method produces an output which can be seen in gure 17 on the right. Following the algorithm described above the originally specied shape of the incision is approximated by a triangular surface mesh. The CPU time on a HP 9000 workstation for the shown data with about elements lasts 6 minutes and seconds. After the calculation of the deformation, only the visible triangular patches were selected to speed up rendering time. 7

8 1 0 Figure 1: normalization of tetrahedrons with cutting points Figure 1: decomposition with one and two cutting points Figure 1: 1. decomposition in case of three cutting points cutting point center of gravity of the truncated tetrahedron Figure 15:. decomposition in case of three cutting points 8

9 Figure 16: decomposition in case of cutting points Figure 17: selection and exploration of the region of interest 9

10 8 Conclusion And Future Work In this paper a new method of realizing the virtual scalpel was presented. In contradiction to earlier papers D cutting with freeform incision shapes is now possible. After the incision has been interactively specied, the net modifying algorithm works locally limited to the region of interest. Therefore we reach an optimal performance wrt. to run time. The presented mesh generator causes a Finite Element mesh, where the renement depends on the geometry, the material properties, the applied incision and the user given forces. First of all we will substitute the D input device by a D tracking system (data glove). The movement of the user will be sampled which results in a series of piecewise bilinear cutting patches. Besides we are going to check out if the algorithm is practical for the simulation of sewing operations. In the future, we want also consider prestresses in our material model, which is present in living soft tissue. References [1] J. E. Bresenham. Algorithm for computer control of a digital plotter. IBM Systems Journal, (1):5{0, [] X. J. Deng. A Finite Element Analysis of Surgery of the Human Facial Tissues. PhD thesis, Columbia University, [] A. Mazura, T. Weingartner, and S. Seifert. Cutting nite element meshes. In ECMBM96, Oct [] P. Pokrandt. Fast non-supervised matching: A probabilistic approach. In CAR '96, pages 06{10, [5] G.J. Song and N.P. Reddy. Tissue cutting in virtual environments. In Interactive Technology and the New Paradigm for Healthcare, pages 59{6. IOS Press and Omsha, [6] M. A. Yerry and M. S. Shepard. Automatic three-dimensional mesh generation by the modied-octree technique. volume 0 of International Journal of Numerical Methods in Engineering,