Recovering the Boundary of a Vessel Wall from Phase Contrast Magnetic Resonance Images In Low Resolutions

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1 Recovering the Boundary of a Vessel Wall from Phase Contrast Magnetic Resonance Images In Low Resolutions Kichun Lee a,, John D. Carew b, Ming Yuan a a Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 332-, USA b Biostatistics and Epidemiology, Carolinas Healthcare, Charlette, NC , USA Abstract The recovery of blood vessels from phase contrast magnetic resonance images is essential in various medical applications including the estimation of arterial wall shear stress and the monitoring of calcified plaques. In this paper, we propose a combined procedure for recovering blood vessels from images in low resolutions. The proposed procedure consists of segmentation based on a fast marching level set and an interpolation of segmented edges using support vector machines (SVMs). The decision boundary of the SVMs between the inner- and outer-most edges are used as the interpolation. The approach is computationally efficient, and its effectiveness is demonstrated by comparing it with previously suggested methods using phantom images and illustrating it using real-life data of carotid stenosis. Keywords: magnetic resonance imaging, segmentation, interpolation, level set, SVMs, spline 1. Introduction Atherosclerosis, the primary cause of myocardial infarction and most cerebral infarctions, is the leading cause of mortality and morbidity in the West- Corresponding author addresses: skylee@gmail.com (Kichun Lee), john.carew@carolinashealthcare.org (John D. Carew), myuan@isye.gatech.edu (Ming Yuan) Preprint submitted to Medical Image Analysis April 27,

2 ern world. Atherosclerosis preferentially forms in areas of reduced wall shear stress (WSS) [1, 2, 3]. A common problem when trying to recover the crosssectional boundary of an anatomical structure or object in an image arises from the estimation of arterial WSS. One method for noninvasively estimating WSS is through the post-processing of phase contrast magnetic resonance imaging (PC-MRI) [4, ]. PC-MRI is capable of measuring blood velocity along the direction of arterial blood flow. The WSS is proportional to the component normal to the wall of the gradient of the blood velocity at the arterial wall. An accurate estimation of the boundary of the artery from the PC-MRI is a prerequisite to estimating the WSS. Vessel segmentation algorithms using PC-MRI [6] are some of the critical components of circulatory blood vessel analysis systems. Identifying the pixels that intersect the boundary may, however, not be sufficient for some applications, particularly when the image is at a low pixel resolution and a sub-pixel resolution is necessary. The extraction of a closed curve to boundary pixels would provide an arbitrary resolution and allow the calculation of boundary normals, tangents, and areas. Particularly with regard to the WSS estimation, we need to shrink a boundary and translate discretized points along the normal direction to the boundary [7]. Recovering the shape of blood vessels on medical images is an essential step in solving several practical applications beyond estimating the WSS, such as the automated diagnosis of the vessels and the registration of patient images obtained at different times. Another specific application of the method is monitoring changes of calcified plaques in early atherosclerosis in terms of the areas of the vessles, which is linked to local hemodynamics at the carotid bifurcation in humans [8]. Measuring the arterial wall area, an indicator of plaque size, requires recovering the boundaries of outer and inner arterial walls from MR images Previous Work Various vessel extraction techniques have been previously implemented. Comprehensive summaries on vessel detection and medical image segmentation are found in Kirbas and Quek [9]. Thresholding the image, the most basic approach that works on very simple images, is often combined with subsequent approaches. Later approaches make use of moving interfaces such as active contours. In recent years, level set implementations have become popular since they can handle different shapes of the image and fast marching methods. Applications of the level set method in MRI vessel detection 2

3 problems are found in Tek et al. [], Persson et al. [6], and Lingrand et al. [11], among others. Compared to other segmentation techniques [12, 13], the fast marching level set method can achieve the highest level of accuracy in a very short time. However, the accurate interpolation of segmented pixels for estimating the boundary at any resolution has received little attention in the literatures. Spline models have been applied in a context in which no issue of low resolution has arisen [14, 11]. In particular, periodic cubic smoothing splines were applied separately to two Cartesian coordinate components parameterized with the distance between two points [1]. Even though the technique is straightforward to apply, it is inaccurate when applied to segmented pixels that can be flat because it often yields flat smoothing for flat segmentation Contributions of the Paper Accurate and robust recovering of a vessel boundary is often a challenging task. Specifically, it has been found that (i) the presence of significant noise levels and the fluctuation of intensity in images often form multiple edges inside and outside vessels; (ii) the shape of the vessel cross-sectional boundary often deviates from a circular shape; (iii) segmented pixels at low resolution often lead to an inaccurate boundary. In this paper, we propose a combined method for accurately recovering vessel boundaries and robustly aiming to overcome the above mentioned difficulties. Our approach consists of two steps: In the first step, boundary pixels are identified with a fast marching level set method; in the second step, the segmented pixels are interpolated with SVMs. The fast marching level set method has merits in not only its computation time but also its effective removal of a homogeneous background. The SVMs produce such a classification, which serves as interpolation, between inner and outer cells that its decision boundary calculates at an arbitrary resolution, and it is smooth from a global structure viewpoint of the boundary Description of the Data The method is illustrated in a phantom data set. The goal of the phantom studies was to determine if our method is capable of estimating boundary in situations in which we have approximate knowledge of the truth. The data consist of a PC-MR image of constant flow of a blood substitute, which was matched to the viscosity of human blood. The flow rate was set to 6.9 ml/s, which is the typical flow rate during diastole in humans. The PC-MR 3

4 image comes as a perfectly co-registered magnitude image that has good contrast for resolving the walls of the phantom vessels. The magnitude image was used to identify the vessel wall pixels and segment the interior pixels in the PC-MR images. The software and algorithms used to carry out this analysis were developed for Matlab (Mathworks, Inc.). The data set is from a straight glass tube phantom that has a circular cross-section with a 3.9 mm inner diameter. The image field of view was mm mm and the image matrix was pixels. The PC-MR image was reduced to a rectangular region of approximately pixels that covered the entire glass tube. The images are shown in Figure (a) (b) (c) Figure 1: The phantom of PC-MRI; (a) the whole magnitude image with the vessel part of interest highlighted in the red box; (b) the vessel part of interest (32 by 32); (c) the vessel part of interest in 3D. 2. Methodology 2.1. Segmentation with Fast Marching Level Set Methods A fast marching level set method is a numerical technique that can follow the evolution of contours and surfaces [16, 17]. It is computationally advantageous and particularly useful for shape recovery of complex geometry without having to parameterize the object when speed F of the evolution, which describes the motion of each point, is strictly positive. In this paper, we used the fast marching level set method and suggest an image-based speed function that works well in a context of PC-MRI boundary recovery. Time dependent surface Γ(t) is implicitly represented as the zero level set of a function φ(x, t) : R 2 R R as Γ(t) = {x; φ(x, t) = }, where φ is 4

5 defined such that <, inside Γ, φ(x, t) = =, on Γ, >, outside Γ. With the definition above, by differentiating φ(x, t) = with respect to t, we track the motion of φ with the evolution equation φ t + x φ =, t in which φ denotes the gradient of φ: φ = ( φ, φ ). Since the outward x y unit normal n and speed F = F(x, y) of the surface in the direction of n are the evolution equation becomes n = φ x, and F = φ t n, φ t + F φ =. (2.1) This is the level-set method. When speed F is strictly positive, the equation in Eq. (2.1) resolves the stationary case, the well-known Eikonal equation F T = 1, where T is the crossing time of the propagating surface. It implies that the gradient of the arrival time is inversely proportional to the speed of the surface. We apply the following iterative algorithm to segmenting vessels from the PC-MR images. The goal is to construct T(x, y) numerically from T = 1/F from smaller values of T to larger values. Given brightness value (pixel intensity) I(x, y) at each pixel within an N by N grid, speed F at (x, y) is dependent on the image. For the sake of numerical convenience, the reciprocal of F(x, y) is suggested as follows. 1/F(x, y) = { γ, I(x, y) > τ,, I(x, y) τ. (2.2) Parameter τ can be set with the prior information of the image: In this paper, three times the standard deviation of intensities in the background plus its mean is used. Parameter γ can be set to N or N 2 : In this paper, N 2 was

6 adopted as γ. The speed form promotes the surface evolution among pixels that do not exceed threshold τ. By the approximation of T in Rouy and Tourin [18], we obtain a solution of the Eikonal equation in the unit box as follows: [max ( max(d x T, ), min(d +x T, ) )] 2 + [ max ( max(d y T, ), min(d +y T, ) )] 2 = 1/F 2 (x, y), (2.3) where the difference operator notation is used as follows: D +x T = T(x + 1, y) T(x, y) and so on. The numerical solution of Eq. (2.3) is in reference to Sethian [16]. We start with a point (x, y ) in the background. [1] Initializing. (a) Let A (set of alive points) be {(x, y )}. (b) Let Narrow Band be {(x 1, y ), (x +1, y ), (x, y 1), (x, y + 1)}; set T(x, y) = 1/F(x, y) for all points in the Narrow Band and T(x, y) =, otherwise. [2] Marching. (a) Let (x min, y min ) be the point in the Narrow Band with the smallest value for T. (b) Add the point (x min, y min ) to A; remove it from the Narrow Band. (c) Take as temporary neighbors any points (x min 1, y min ), (x min + 1, y min ), (x min, y min 1), (x min, y min +1) that are not in A; include them in the the Narrow Band. (d) Update the values of T for all points in the temporary neighbors according to Eq. (2.3). (e) Return to [2]-(a) until the number of iterations reaches N 2. The total work required to solve Eq. (2.3) is O(N 2 ), and the computational order in finding the minimum of the Narrow Band in [2]-(a) is O(logN) by using a heap data structure. An illustration of this algorithm is shown in Figure Interpolation with Support Vector Machine Methods SVMs find a hyperplane with a decision boundary that maximizes the gap between class labels in a transformed feature space [19, ]. It can be formulated as the minimization of hinge losses and penalty terms. For a given labeled data set (x i, c i ), c i = ±1, a soft margin SVM boundary is obtained by 6

7 (a) (b) (c) (d) Figure 2: An illustration of the fast marching level set method for the phantom image with the proposed image-based speed function in Figure 1; initial point (x, y ) was (3, 3); (a) 1th iteration; (b) 2th iteration; (c) 4th iteration; (d) the final stage. arg min w,ξ 1 2 w 2 +C n i=1 ξ i s.t. c i (w t x i + b) 1 ξ i, ξ i, i = 1,...n, (2.4) in which C is a constant that regulates the extent of miscalculation. To support non-linear boundaries, a non-linear kernel function K on the domain of x is introduced. The resulting algorithm to solve Eq. (2.4) is formally similar, except that every dot product between w and x i is replaced by the non-linear kernel function K. A Gaussian radial basis function is frequently used as the kernel, resulting in the corresponding feature space being a Hilbert space of an infinite dimension: K(x 1, x 2 ) = exp( x 1 x 2 2 ), σ with parameter σ explaining how wide the curvature of the boundary is. To formulate the problem of boundary estimation into SVM, labeled instances c i for all points in the segmentation are obtained in the following manner. Given the segmentation of vessels from PC-MR images from the previous stage, the pixels of the inner-most and outer-most edges are pixels that are inside and outside the segmentation, respectively, and adjacent to the angular points of the segmentation as are shown (in gray) in Figure 3. 7

8 We assign c i = 1 to the pixels of the inner-most edges and c i = 1 to those of the outer-most edges. The assignment of the two values can be arbitrary as long as they differ. Both the inner-most and outer-most edges are x i. (a) (b) (c) Figure 3: Extraction of inner-most or outer-most edges (in gray) from angular points of segmentation (in black). To extract inner-most and outer-most edges from the segmentation, the following algorithm is used. First, segmented points are sorted by increase in the angle of polar coordinates: the ordered points are (x (t), y (t) ). Then the procedure commences: [1] Set t =. Let E I be the set of the inner-most edges; E O be the set of the outer-most edges. [2] If x (t+1) x (t) = 1 and y (t+1) y (t) = 1, (a) add the point ( x (t+1) sgn (y (t+1) y (t) ), y (t+1) +sgn (x (t+1) x (t) ) ) to E I ; (b) add the point ( x (t) + sgn (y (t+1) y (t) ), y (t) sgn (x (t+1) x (t) ) ) to E O. [3] Increase t by 1 and repeat [2] until it reaches the length of the segmentation. The algorithm above is of O(N) since it basically examines the segmentation. Figure 1(a) shows the segmentation of the phantom image and Figure 1(b) shows E I (in dark black) and E O (in dark gray), extracted from the segmentation. Then the support vectors and the decision boundary with a Gaussian radial basis kernel of σ = 128 are shown in Figure 4(c), and the estimated boundary overlaps with the original image in Figure 4(d). 8

9 (a) (b) (c) (d) Figure 4: The application of SVM interpolation to the phantom image; (a) segmentation from the fast marching level set; (b) the inner-most edges in black and the outer-most edges in dark gray; (c) support vectors in red and the decision boundary in dotted black; the outer-most edges are rectangles; the inner-most edges are diamonds; (d) the boundary recovery; the yellow circles represent segmentation; the blue line represents the interpolated boundary Results To illustrate the effectiveness of our approach, we introduce another data set in which the vessel shape is not circular. The second data set, as shown in Figure (a), was from a modified version of the straight glass tube phantom in which the tube was laterally indented on one side to yield a non-convex cross section. Pre-processing was identical to that of the first data set and 9

10 N was 86. The image and results are shown in Figure. The segmentation, done by the fast marching level set method with τ = 1 and λ = 86 2, is shown in Eq. (2.2). The outer- and inner-most edges that were extracted are represented by a rectangle and a diamond, respectively, in Figure (b). The support vectors derived from the Gaussian radial basis function of σ = 128 are highlighted in red. The decision boundary in dashed cyan between the outer- and inner-most edges, which interpolates the boundary in such a way that it considers the global structure from the segmentation, provides estimations at any resolution. The boundary in blue overlaps with the original image, as in Figure (a) (a) (b) Figure : The application of SVM interpolation to the segmentation of the second phantom; (a) the boundary recovery; the yellow rectangles represent segmentation; the blue lines represent the interpolated boundary; (b) the support vectors in red and the decision boundary (dashed cyan line); the outer-most edges are rectangles; inner-most edges are diamonds. 3. Experiments and Comparisons To show the effectiveness of the combined approach of the fast marching level set and SVMs in recovering the boundary, we compared our proposed method with a few selected methods that have been previously suggested using simulated phantom PC-MR images. The three phantoms, shown in

11 Figure 6, simulate blood vessels in 3 3. After we applied an averaging low-pass filter and downsampling to the images, images of the blood vessels with a low resolution in were generated (N = 32) and perturbed with a signal-to-noise ratio (SNR) of, as shown in the bottom row of Figure 6. Phantom 3 is a compromise between a circle and an ellipsoid; Phantom 4 is circular; and Phantom is ellipsoidal (a) Phantom 3. (b) Phantom 4. (c) Phantom. (d) Phantom 3 in 3D (e) Noised Phantom 3. (f) Noised Phantom 4. (g) Noised Phantom. (h) Noised Phantom 3 in 3D. Figure 6: The upper rows are simulated phantom PC-MR images of blood vessels; the noised phantoms in the lower row correspond to those in the upper row with an SNR of ; (d) and (h) are 3D representations of (a) and (e), respectively. Four segmentation methods were tested: the fast marching level set method which was explained in this paper; the Canny method, the Sobel method, and the Laplacian of Gaussian. Details of the methods can be found in Canny [21] and Parker [22]. The segmentation results are shown in Figure 7. The Sobel method not only produced points inside the vessel, but the points were disconnected. Outputs of the Canny and Laplacian of Gaussian yielded connected boundaries, but the boundaries were mostly inside the true boundary. The segmentation results of the level set method overlaps the true boundary. We adopted the level set and Canny methods as segmentation 11

12 methods to be further tested in interpolating their segmentation results (a) Level Set (b) Canny (c) Sobel (d) Laplacian Gaussian of Figure 7: The segmentation results, shown by black box markers with the various methods, overlap with Phantom 3. The segmentation methods were applied to the noised images. Next, we compared the SVM interpolation of this paper with the periodic cubic spline interpolation [1]. The segmented points are sorted by increasing the angle in the polar coordinates as in Section 2.2. Then, to assign a t i to each (x (i), y (i) ), t i was selected such that t i+1 t i was proportional to the distance between (x (i), y (i) ) and (x (i+1), y (i+1) ). Subsequently, periodic cubic smoothing splines were fitted to the (t i, x (i) ) and (t i, y (i) ) pairs. The weights of the penalizing regularization term in the spline were set at 7% and 9%: the weights of goodness-of-fit were % and %, respectively. The results are shown in Figure 8. The interpolation of the spline method is sensitive to weight and easily misguided by a few jumps, but the output of our method is very regular in the interpolation of the segmentation. To find the best method of recovering the boundary, we combined the level-set segmentation with both the SVM interpolation and the cubic spline interpolation and the Canny segmentation with the same two interpolation methods. Then we compared the performance of the four combinations. For each of the four combinations of methods, we generated 1, noised samples for each of the three phantoms with an SNR of. The mean squared error (MSE) was obtained by summing the squared difference between the true and estimated boundaries in polar coordinates. The boxplots of the MSEs are shown in Figure 9. Our approach, the combination of the level set and SVM interpolation method, outperformed the other combinations. 12

13 (a) (b) (c) Figure 8: The interpolation results using the various methods after the level set segmentation for Phantom 3; the red line is the true boundary and the black line is the interpolation; (a) SVMs; (b) spline with a 7% regularization term; (c) spline with a 9% regularization term MSE LevelSet SVMs LevelSet Spline Canny SVMs Canny Spline Figure 9: The boxplots of 1, empirical MSEs for the four methods. The three boxplots for each method represent Phantom 3, Phantom 4, and Phantom. The combination of level-set and SVM interpolation shows the best performance An Example in Carotid Stenosis As an ongoing prognostic study on HIgh-Resolution magnetic resonance Imaging in atherosclerotic Stenosis of the Carotid artery (HIRISC), Touze et al. [23] focused on the reproducibility of MRI in characterizing plaque components and measuring plaque volumes. They assessed the reproducibil- 13

14 ity of high-resolution MRI for the identification and quantification of carotid atherosclerotic plaque components in 8 patients: specifics of the experiments are in reference to Touze et al. [23]. Figure (a) shows a T2-weighted image with a patient of moderate carotid stenosis in the region of interest highlighted in the red box: a 3D representation of the region is shown in Figure (b). During their quantitative measurements, they obtained area measurements of vessels, lumens, plaques, and others for each location by manually tracing the boundaries of the images. We demonstrated our approach in automatically extracting the boundary of the lumen. Figure (c) shows the inner-most edges (in black dots) and the outer-most edges (in gray dots). The extracted boundary in blue, superimposed on the original image, is shown in Figure (d). We ran our method on one image of a patient that was made available to the public, and the algorithm was able to extract the boundary of the lumen. 4. Conclusion In this paper, we proposed a two-step approach to recover the boundaries of blood vessels from images in low resolutions: in the first step of segmentation, we use a fast marching level set and propose an SVM-based interpolation. The implemented level set method suggests an image-based speed function and computationally efficient marching algorithm for the segmentation. The interpolation of the segmented boundary is done by SVMs with two groups of inner- and outer-most edges. By taking the Gaussian radial kernel of an infinite dimension, the decision line smoothly embraces all edges and provides estimations at an arbitrary resolution. The accuracy of the two-step approach is compared with that of several other methods and illustrated. In simulations of noised phantom images, the accuracy of our proposed approach was compared with that of a few selected methods and the feasibility of our approach was illustrated in real-life data coming from carotid stenosis. We adhere to the concept of reproducible research. The m-files used for the calculations and the figures in this paper can be downloaded from the following web page [1] M. Friedman, C. Bargeron, O. Deters, G. Hutchins, F. Mark, Atherosclerosis 68 (1987)

15 (a) (b) (c) (d) Figure : Demonstration of the combined method; (a) a T2-weighted image of a patient with moderate carotid stenosis; (b) a 3D representation of the region of interest; (c) the inner-most edges (in black dots) and the outer-most edges (in dark gray dots): the inbetween line (in light gray) is the boundary segmentation; (d) the extracted boundary of the lumen part. [2] D. Ku, D. Giddens, C. Zarins, S. Glagov, Arteriosclerosis, Thrombosis, and Vascular Biology (198) 293. [3] C. Zarins, D. Giddens, B. Bharadvaj, V. Sottiurai, R. Mabon, S. Glagov, 1

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