Tracking the LV in Velocity MR Images Using Fuzzy Clustering

Tracking the LV in cine MR cardiac images is a challenging computing application that is also relevant to the needs of clinicians. Using fuzzy clustering as the method of segmentation, this paper reports on whether velocity 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 images produced 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 is rendered as 3 images, vx, vy and vz , corresponding to the cartesian components of the velocity vector field V at each pixel. The reference coordinate system has the x-y plane lying on the plane of imaging (aligned with 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 time space between each frame and the next is approximately 40 ms. Figure 1 displays example frames from a sequence. Figure 2 displays only the first frame of each of the three velocity components. Figure 2. Examples of velocity images, frame 0 of vx, vy, and vz . Clustering algorithms have been used for image analysis, particularly segmentation, probably since the early seventies. The motivation for this use is that image intensity values tend to cluster in ways that e-mail address: [email protected] correspond to the physical description of the image. So, for example, in a picture of a dark-coloured aeroplane up in the sky, the sky’s colour would cluster around “bright blue”, while the aeroplane’s colour would cluster around “dark grey”. There are many types of clustering algorithms. In this paper, we use an objective-function-type algorithm called fuzzy c-means (FCM) that outputs fuzzy descriptions of each of the clusters. See [2] for a general review of FCM’s use in MR image segmentation. FCM has also been used for segmentation of Nuclear Medicine cardiac images [3]. A variant of FCM, called the fuzzy c-shells (FCS) algorithm, has been used to 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 algorithm is a description of the clusters it found. A suitably concise output is a set of prototype data points, where each of these prototypes represents a cluster of data points. In the simplest case, the prototype location is the geometric mean of the locations of data points in the cluster it represents. This way, the aim of the algorithm becomes finding the best placement of these prototypes. Mathematically, this can be formulated using an objective function. Assuming that we are looking for c prototypes, the objective function measures how 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 with each of the c prototypes. The value of this degree lies between 0 and 1; the closer it is to 0 the less the prototype is representative of the point, while the closer it is to 1 the more the prototype is representative of the point. The memberships can be arranged in a matrix, called the fuzzy partition matrix, which gives for 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 Xi=1 N Xk=1 umikd2ik subject to Pci=1 uik = 1 8k = 1::N where N is the number of points in the data set, c is the number of clusters, U = [uik] is a c N fuzzy c-partition matrix, and P = (p1; : : : ;pc) is the c-set of prototypes. d is the distance metric given by k xk pi k2A where k k is any inner product induced norm. The solution commonly 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 the prototypes and their associated memberships according to some derived update equations. As like other iterative optimisation methods, FCM may provide locally-optimal solutions of the objective function. Because of the possible presence of one or more locally-optimal solutions, the solution of an FCM run depends on the initialisation of that run. An important requirement for the application of the FCM algorithm and its derivatives is that c, the number of prototypes, must be provided by the user. There are various options we can take when using FCM and its extensions and derivatives. The first is the choice of metric with which points are compared to the 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 rather take other shape formations. As the cluster we seek (the LV) is more or less hyper-elliptical in shape we can safely use a point prototype. To simplify computational calculations, the metric with which to compare points to prototypes is chosen to be Euclidean distance.