Tracking the LV in Velocity MR Images Using Fuzzy Clustering

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Tracking the LV in Velocity MR Images Using Fuzzy Clustering
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  Tracking the LV in Velocity MR Images Using FuzzyClustering Ahmed Ismail Shihab    and Peter BurgerImperial College of Science, Technology and MedicineDepartment of Computing180 Queens Gate, London SW7 2BZ Abstract. Tracking the LV in cine MR cardiac images is a challenging computing application that is also relevant tothe needs of clinicians. Using fuzzy clustering as the method of segmentation, this paper reports on whethervelocity data can improve the accuracy of the results obtained through only tissue data. 1 PURPOSE Our application consists of analysing MR image cine sequences acquired at the mid-ventricular plane of the heart. We describe our use of the fuzzy c   -means clustering algorithm to track the LV area across a‘heartbeat’. The images we use are conventional MR tissue density images as well as velocity imagesproduced using a phase-sensitive MRI technique.The cine sequences of images are aligned with the short-axis of the left ventricle (LV). The velocity data isrendered as 3 images, v  x   , v  y   and v  z   , correspondingto the cartesian componentsof the velocity vector field V  at each pixel. The reference coordinate system has the x   - y   plane lying on the plane of imaging (alignedwith the short-axis of the left ventricle) and the z   axis perpendicular to it (aligned with the LV long-axis). Figure 1.  Examples of tissue density images: frames 0, 2, 6, 10 and 14 in an image sequence.The image sequences contain 16 frames. The sequences start at systole and end at early diastole. The timespace between each frame and the next is approximately 40 ms. Figure 1 displays example frames from asequence. Figure 2 displays only the first frame of each of the three velocity components. Figure 2.  Examples of velocity images, frame 0 of  v  x   , v  y   , and v  z   .Clustering algorithms have been used for image analysis, particularly segmentation, probably since theearly seventies. The motivation for this use is that image intensity values tend to cluster in ways that   e-mail address: ais@doc.ic.ac.uk   correspond to the physical description of the image. So, for example, in a picture of a dark-colouredaeroplane up in the sky, the sky’s colour would cluster around “bright blue”, while the aeroplane’s colourwould cluster around “dark grey”.There are many types of clustering algorithms. In this paper, we use an objective-function-typealgorithmcalled fuzzy c   -means (FCM) that outputs fuzzy descriptions of each of the clusters. See [2] for a generalreview of FCM’s use in MR image segmentation. FCM has also been used for segmentation of NuclearMedicine cardiac images [3]. A variant of FCM, called the fuzzy c   -shells (FCS) algorithm, has been usedto segment the myocardial wall in MR images [4]. 2 METHOD Fuzzy c   -Means (FCM) is an objective-function-based method of clustering. It is also sometimes called  fuzzy ISODATA . The monograph by James Bezdek [1] is the most widely cited reference for FCM.The input to any clustering algorithm is called the  data set  . There is no agreed-on name for the output;this is because it depends on the type of algorithm used. In the abstract sense, the output of the algorithmis a description of the clusters it found. A suitably concise output is a set of   prototype  data points, whereeach of these prototypes represents a cluster of data points. In the simplest case, the prototype locationis the geometric  mean  of the locations of data points in the cluster it represents. This way, the aim of thealgorithm becomes finding the best placement of these prototypes. Mathematically, this can be formulatedusinganobjectivefunction. Assumingthatwe are lookingfor c   prototypes,theobjectivefunctionmeasureshow good (or bad) the set of  c   prototypes describes the data set.The clustering suggested by FCM is a fuzzy one, i.e., each data point has a degree of membership witheach of the c   prototypes. The value of this degree lies between 0 and 1; the closer it is to 0 the less theprototype is representative of the point, while the closer it is to 1 the more the prototype is representativeof the point. The memberships can be arranged in a matrix, called the fuzzy partition matrix, which givesfor each data point, its memberships with each of the prototypes.Now we state the problem mathematically. FCM seeks to minimise the objective function: J  (  P; U  )=  c  X  i  =1  N  X  k  =1  u  m ik  d  2  ik  subject to P  c i  =1  u  ik  =1  8  k  =1  ::N   where N   is the number of points in the data set, c   is the number of clusters, U  =  u  ik  ]  is a c    N   fuzzy c   -partition matrix, and P  =(  p  1  ;:::; p  c  )   is the c   -set of prototypes. d  is the distance metric given by k  x  k  ?  p  i  k  2  A   where kk   is any inner product induced norm. The solutioncommonly used is an iterative one based on the gradient descent technique.The algorithm iteratively minimises the value of the objective function by changing the location of theprototypes and their associated memberships according to some derived update equations. As like otheriterative optimisation methods, FCM may provide locally-optimal solutions of the objective function. Be-causeofthepossiblepresenceofoneormorelocally-optimalsolutions,thesolutionofanFCMrundependson the initialisation of that run.An important requirementfor the applicationof the FCM algorithm and its derivativesis that c   , the numberof prototypes, must be provided by the user. There are various options we can take when using FCMand its extensions and derivatives. The first is the choice of metric with which points are compared tothe prototype. Another choice to make is the prototype description. Normally this is taken to be a point,however clusters may not always be shaped like point clouds (hyper-elliptically-shaped), but can rathertake other shape formations. As the cluster we seek (the LV) is more or less hyper-elliptical in shape wecan safely use a point prototype. To simplify computationalcalculations, the metric with which to comparepoints to prototypes is chosen to be Euclidean distance.  3 RESULTS We assessed the impact of velocity data by clustering first with- V θϕ  y zvxv zv yx Figure 3.    and    define the directionof the velocity vector at a given point.out it, and then with combinations of it. The input in the firstinstance was three-dimensional: x   and y   , for the x   and y   co-ordinates of a pixel; and d   , for the tissue density at that pixel.In the second experiment we added a fourth dimension, V   , con-sisting of the combined magnitude of the v  x   , v  y   , and v  z   velocitycomponents at each pixel, as shown in figure 3 and described by: V  =  p  v  x  2  +  v  y  2  +  v  z  2  : In the third experiment, we removed V  and replaced it with    and    . These angles describe the directionof the velocity field at a given pixel (see figure 3).The m   fuzziness factor of the FCM algorithm was chosen to be2.0, and c   was set to four as this gave the most intuitive segmen-tation of the images. The output was in the form of cluster pro-totypes and the membership matrix. For each frame after the firstone, we initialised FCM with the prototype locations of the previ-ous frame. We then selected the cluster corresponding to the LVblood pool area and tracked its membership across the sequence.To count the pixels in the LV, we considered a pixel part of the LV cluster if its membership with thatcluster was higherthan its membershipswith the otherclusters. This rule is commonlycalled the  max rule .Then we then ran a simple region growing routine to count the pixels in the LV area.In figure 4, we compare the calculated areas of the left-ventricular blood pool using the three routes wetook with a ’ground truth’ established by a clinician. The cine sequence is that of a normal patient. 50010001500200025003000350040000246810121416    L   V    P   l  a  n  a  r   A  r  e  a   (  p   i  x  e   l  s   ) Frame NumberGround TruthMax rule: Density data onlyMax rule: Density + VMax rule: Density + directions Figure 4.  Comparison of calculated LV area for the three data sets used.We note that clustering with velocity magnitude tends to produce quite erratic results at frames 12 and13. We find that the density and density-and-directional data produced almost identical results in terms of LV area (but not in terms of clustering). We also find that, in general, the clustering results are all on theconservative side of estimating LV area.  4 CONCLUSIONS As the aim of our research is to empower the clinician with a simple-to-use tool like FCM, we did notattempt to derive a new algorithm to take account of the special nature of the velocity data. Our resultsprove that including the velocity magnitude data in the data set can actually give misleading results, butthat the directional velocity information is useful in verifying density-data results.The FCM algorithm has proven itself to be quite robust. Given the fact that it starts with almost no priorinformation, it is quite accurate. However, we note the shortcoming of having to experiment in order tofind a suitable value for c   . We remindthe readerthat we avoided the problemof non-repeatabilityof resultsby using the result of one frame to start off the next. However, the problem of non-repeatability still existswhen we cluster the first frame.Ourtechniqueiseasytouseandveryintuitive;cliniciansimmediatelygraspeditsapplication. Theaccuracyof our results show a lot of promise. Acknowledgements We would like to thank Dr Philip Kilner of the MR Unit, Royal Brompton Hospital for his comments onthe results of our research. References 1. J. Bezdek.  Pattern Recognition with Fuzzy Objective Function Algorithms . Plenum Press, 1981.2. M. C. Clark, L. O. Hall, D. B. Goldgof et al. “MRI segmentation using fuzzy clustering techniques.”  IEEE  Engineering in Medicine and Biology Magazine  13(5) , pp. 730–742, 1994.3. A. Boudraa, J.-J. Mallet, J.-E. Besson et al. “Left ventricle automated detection method in gated isotopic ventricu-lography using fuzzy clustering.”  IEEE Transactions on Medical Imaging  12(3) , September 1993.4. H.-K. Tu, D. B. Goldgof & E. Backer. “Utilizing fuzzy c   -shells for automatic approximate lv location for initializa-tion of myocardial structure and functions analysis algorithm.” In  Medical Imaging 1994: Physiology and Function from Multidimdensional Images . 1994.
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