Blood Vessel Visualization on CT Data

Transcription

1 WDS'12 Proceedings of Contributed Papers, Part I, 88 93, ISBN MATFYZPRESS Blood Vessel Visualization on CT Data J. Dupej Charles University Prague, Faculty of Mathematics and Physics, Prague, Czech Republic. Abstract. In this paper we researched some of the currently available methods for duct visualization on volume data. We then designed implemented and tested a new method based on Straightened Curved Planar Reformation (SCPR) that provides more spatial context for more intuitive orientation in the resulting image. Additionally, we used the Bishop frame to calculate the local coordinate system that minimizes its twist to further enhance the visualization. Introduction Computed tomography is a medical examination that can be used to diagnose a multitude of body parts or organs. This includes the cardiovascular system. Blood vessels, however, are somewhat challenging when it comes to proper visualization. Naturally, direct volume rendering by ray casting can be used, but due to the narrow tube-like shape of the arteries this may be inadequate for a proper diagnosis. Before the actual scanning, the patient is injected with a contrast agent. This must be done because the typical density of blood is very close to the density of muscle tissue, which would make the vascular system virtually invisible without any enhancement. In this paper, we will research pertinent work of other authors, propose an enhanced algorithm for blood vessel visualization and finally present and discuss the results of testing our new method. Previous Work There is an algorithm designed specifically for duct visualization by (Kanitsar et al. 2002). Actually, there have been three methods proposed in that paper. We are especially interested in a single method proposed there i.e. the Straightened Curved Planar Reformation (SCPR). This is due to its favorable properties, like complete linearization of the studied vessel and isometry preservation, which indicates that the visualization will match vessel s length. Furthermore, the cross-section will never appear distorted. The operation of this method is rather straight-forward. As illustrated in Figure 1, given the original volume image and the center line of the desired blood vessel segment, the algorithm uniformly samples the volume in short line segments, centered at the vessel s center line and perpendicular to the tangent at that point. The segments are called the lines-of-interest and are later reassembled by stacking next to each other to form the final linear image. Other authors, like (Xinrong et al. 2008) modified SCPR to account for the vessel s cross-section and to allow the local cross-section area estimation. The authors proposed sampling the line-of-interest at every point on the center line in several pre-defined directions. This creates a star-like image of the Figure 1. Straightened CPR samples the vessel in short lines-of-interest perpendicular to the center line and reassembles them into a linear image. 88

2 vessel s cross-section. Finally, they optimized a deformable model in each of these slices to estimate the shape of the vessel and, incidentally its cross-section area. Additionally, the authors of the CPR presented an enhancement in (Kanitsar et al. 2003) that automatically flattens the whole pre-tracked vascular system into a single planar image. Proposed Algorithm Before we start designing our algorithm, let us consider where the SCPR may fall short. First, (Kanitsar et al. 2002) is rather vague on calculating the actual direction of the line-of-interest, specifically its rotation about the vessel center line. We are going to address that issue in this chapter. Furthermore, it is easy to lose the sense of orientation in an image created by CPR due to the absence of spatial context, e.g. portions of nearby organs or blood vessels, as the line-of-interest only samples a small portion of the vessel surroundings. We will endeavor to remedy that problem as well. Let us briefly outline the proposed algorithm. We choose equidistant samples on the vessel center line and calculate the local coordinate system in each of them. We then sample a square-shaped area of the volume centered at the center line, as opposed to short line segments that SCPR extracts. These slices are then stacked onto each other and rendered as a block of volume data using maximum intensity projection (MIP) (Wallis & Miller 1991). To enable trading image quality for processing speed, we designed the algorithm to allow for reduced sampling of the vessel s surrounding area. Further details are discussed later in this chapter. Local Coordinate System It will be necessary to generate the local coordinate system, i.e. vessel normal u and binormal v in every sampling point along the vessel center line in a way that minimizes twisting as the vessel is traversed. We found two methods to calculate the local frame that may be suited for us in (McCreary 1998). They are designated the Frenet and Bishop frames. Both methods impose restrictions on the curve γ the local coordinate system will follow. The Bishop frame requires that γ be C 2 -continuous and that its first derivative γ be non-zero. On the other hand, the Frenet frame requires C 3 -continuity and nondegenerateness of γ. We therefore opted to use the Bishop frame. The Bishop frame, otherwise known as the parallel transport frame, is defined by the following equation: d t 0 k 1 k 2 t ds u = k u, (1) v k v where t, u, v are the local center line tangent, normal and binormal; as illustrated in Figure 2 and s denotes the arc length. Furthermore, the coefficients k 1 and k 2 satisfy (2). κ(t) = k k 2 2 τ(t) = θ θ(t) = arctan k 2 k 1 (2) Figure 2. The local coordinate system in every point P on the vessel center line. 89

3 In these formulas, κ(t) denotes the local curvature in the point t, which can be evaluated as the reciprocal of the osculating circle radius, while τ(t) stands for the local curve torsion, which can be understood as the amount, by which the curve s osculating plane s normal changes. It is trivial to observe that the coefficients k 1 and k 2 are in fact the Cartesian equivalents for the polar coordinates κ and θ that define the local osculating circle of the curve. As a result, we only need to project the centerline tangent onto the previous normal and binormal to get those coefficients: k 1 = t(i + 1) v(i) k 2 = t(i + 1) u(i) The initial normal and binormal vectors still need to be calculated. The initial tangent t 0 in p 0 is known. Let a be the axial direction vector (one of {(1,0,0), (0,1,0), (0,0,1)}) whose dot product with the initial tangent is the lowest. That vector is then projected onto the vessel s first cutting plane P = {p t 0 (p p 0 ) = 0} and normalized. This yields our initial curve normal vector. The binormal is calculated as the cross product of the normal and t 0. After this initial step, the coordinate frame vectors t, u, v are calculated using the differential equation in (1). (3) Figure 3. Our algorithm samples whole cross-section patches and generates a list of volume slices. Resampling to Cross-Sections and Visualization When the normal and binormal u, v vectors in p on the curve are known, we sample the volume at these locations: q = p + au + bv (4) Where a, b r, r are the coordinates on the current vessel cross-section slice. The argument r 2 2 controls the size of the surrounding area to be captured. Trilinear interpolation is used to extract smooth samples from the volume. Due to heavy random memory access, this part is likely to be the bottleneck. The algorithm can therefore be configured to generate slices with height of h pixels and a width of h d pixels, where d 1 is the subsampling ratio. Note that the slice still spans the area described in (4), however it is sparser by a factor of d in one direction. This is not a problem as in the subsequent visualization the denser side will be facing the camera as depicted in Figure 4. Let us reiterate that the slices generated in the previous step are stacked together to form a block of volume data that will be rendered using MIP. Due to the elongated shape of the block we opted to use an orthogonal projection (as shown in Figure 4) instead of a perspective projection. In our tests we will designate this new method MIP-CPR. Results Before we present the results, it should be noted that one of our proposed method s inputs is the vessel center line. We calculated that line using the method presented in (Dupej 2011). For the testing purposes we used a voxels large CT scan of a portion of a human cranium, shown in Figure 5. The subject was injected with a contrast agent, of course. 90

4 Figure 4. viewport. The individual slices with reduced depth sampling, projected orthogonally onto the Figure 5. The cranium section we used for testing. Left direct volume rendering; Right automatically detected blood vessel center lines. Images from (Dupej 2011). As Figure 6 shows, MIP-CPR s sampling technique captures the vessel surroundings even if SCPR omits important parts in the vessel s proximity when the line-of-interest fails to intersect them. This gives the user much more spatial context e.g. blood vessel junctions or portions of nearby vessels, that make orientation in the visualization easier. To demonstrate the results in a case of diseased blood vessels, Figure 8 Stenosis in the human carotid artery. Left Straightened CPR, Right MIP-CPRshows the visualization of a stenosis in the human carotid artery. The left image is calculated with SCPR and while it does appear clearer, the right image, created with MIP-CPR captures another blood vessel diverging from the visualized duct, thus revealing a junction. Furthermore, if the vessel is afflicted with but a small calcification, it is possible that the anomaly will not be captured by ordinary SCPR. Again, due to the fact that MIP-CPR captures the complete surroundings of the blood vessel, it is unlikely that even tiny calcifications will be missed. The algorithm allows for reduced depth sampling, which balances the rendering speed and image quality. MIP-CPR has proven considerably slower than SCPR. This is, however not surprising considering MIP-CPR samples lr 2 /d pixels, where l is the number of extracted slices. The SPCR, on the other hand only extracts lr pixels. Figure 7 shows that the image still captures most of the blood vessel that runs almost parallel to the visualized one even at 4 depth subsampling. Note that for the speed tests we used the following parameters: h = 48 line-of-interest 48 pixels long 414 slices (lines-of-interest) sampled and visualized 91

5 Figure 6. A blood vessel segment. Top Straightened CPR; Bottom MIP-CPR. a b c d e f Figure 7. A blood vessel segment, effect of depth subsampling on quality. a MIP-CPR full sampling; b 2 subsampling; c 4 subsampling; d 8 subsampling; e 16 subsampling; f SCPR. Figure 8. Stenosis in the human carotid artery. Left Straightened CPR, Right MIP-CPR. The speed tests were conducted on a computer with an Intel Core i7-920 processor and 6 GB of RAM in triple-channel configuration. With the aforementioned configuration, rendering a vessel segment with MIP-CPR took 354 milliseconds at full resolution, whereas SCPR only took 19 ms. At a subsampling ratio of 4 the visualization took on average 99 ms, which is better than sufficient for interactivity. At the same time, the resulting image (Figure 7c) still contained most of the detail captured in the fully-sampled image (Figure 7a). If lower processing times of very long visualized strands are desired, the algorithm can be easily rewritten to run on multiple processors or on a GPGPU as both vessel cross-section sampling and MIP are easily parallelizable. 92

6 Conclusion We modified the straightened CPR algorithm for linearizing visualization of ducts in volume data. Our modified algorithm, MIP-CPR extracts multiple layers of voxels in the proximity of the visualized vessel as opposed to a single line-of-interest in SCPR. This captures much more spatial context in the visualization making orientation in the final image easier. The resulting processing speed deterioration is addressed by configurable reduction of depth sampling density. The results have shown that even at higher subsampling ratios, the results capture much of the vessel s surrounding mass. Furthermore, we used the Bishop frame to minimize the twist of the centerline-local coordinate system when the vessel cross-sections are generated to further improve the visualization. There is considerable room for improvement in the speed of the visualization. The sampling of the individual slices and subsequent volume rendering are easily parallelizable tasks that can be rewritten to run on multiple threads or on a GPGPU. References Dupej, J., Vessel segmentation. Master thesis, Faculty of Mathematics and Physics, Charles University in Prague. Kanitsar, A. et al., CPR - Curved Planar Reformation. VIS 2002, IEEE, 1. Kanitsar, A., Wegenkittl, R. & Fleischmann, D., Advanced curved planar reformation: Flattening of vascular structures. Proceedings of the 14th. DOI: /VISUAL McCreary, P.R., Visualizing Riemann Surfaces, Teichmueller Spaces, and Transformation Groups in Hyperbolic Manifolds Using Real-Time Interactive Computer Animator (RTICA) Graphics. PhD thesis, University of Illinois. Wallis, J.W. & Miller, T.R., Three-Dimensional Display in Nuclear Medicine and Radiology. The Journal of Nuclear Medicine, 32(3). PMID: Xinrong, L., Xinbo, G. & Hua, Z., Interactive curved planar reformation based on snake model. Computerized Medical Imaging and Graphics, 32. DOI: /j.compmedimag