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# Segmentation of Tumour
Using K-mean Clustering Algorithm

## INTRODUCTION

## METHEDOLOGY

### CONCLUSION

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Abstract In radiotherapy using 18-fluorodeoxyglucose positron emission tomography (18F-FDG-PET), the accurate delineation of the biological tumour volume (BTV) is a crucial step. In this study, new approach to segment the BTV in F-FDG-PET images is suggested. The technique is based on the k-means clustering algorithm incorporating automatic optimal cluster number estimation, using intrinsic positron emission tomography image information. Partitioning data into a finite number of k homogenous and separate clusters (groups) without use of prior knowledge is carried out by some unsupervised partitioning algorithm like the k-means clustering algorithm. To evaluate these resultant clusters for finding optimal number of clusters, properties such as cluster density, size, shape and separability are typically examined by some cluster validation methods. Mainly the aim of clustering analysis is to find the overall compactness of the clustering solution, for example variance within cluster should be a minimum and separation between the clusters should be a maximum.

There are several approaches to validate the segmentation techniques such as phantom studies and the macroscopic surgical specimen obtained from histology. The use of macroscopic samples for validation of segmentation techniques in positron emission tomography (PET) images is one of the most promising approaches reported so far in clinical studies, the procedure consists of the comparison of the tumour volumes defined on the PET data with actual tumour volumes measured on the macroscopic samples recorded from histology (where PET was performed prior to surgery). Segmentation using the cluster –based algorithms is very popular, but the main problem in this case is the determination of the optimal and desired number of clusters. In this, we have implemented an approach based on k-means algorithm with an automatic estimation[1][2] of the optimal number of clusters, based on the maximum intensity ratio in a given volume of interest (VOI).

Calculate the VE for a range of k (2–50 clusters), and the optimal number which corresponds to the minimum of SVEs. This method gives good results but consumes a significant computation time by performing the clustering for a large range of cluster values before selecting the optimal number of clusters. Several approaches have been proposed in the literature to identify the optimal cluster number to better fit the data, three of them are used. Unfortunately, the results are not promising because they are not adapted to PET image segmentation. So our goal in this study is to improve the k-means clustering method, by incorporating an automatic determination of the optimal number of clusters using a new criterion based on PET image features. After analysing the variation of the maximum activity (intensity) of the uptake in the VOI by scanning all slices, we conclude for all patients that the maximum intensity value decreases from middle to frontier slices, and the maximum intensity is often situated almost at the centre of the BTV.

The optimal cluster number has a minimum value at the centre of the BTV, and increases from central to frontier slices. This correlation between the optimal number of clusters and the maximum intensity motivates our choice of the following slice image feature: Where is the maximum activity (intensity) of the uptake in the corresponding slice, is the maximum activity in all slices that encompasses tumour volume BTV inside the, and is the difference between the maximum and the minimum values of (Imax (slice)/Imax (VOI)) in the Similarly to the new criterion, has a maximum value for the middle slices and decreases for the frontiers of the BTV.

**Note that the r values range from ‘0’ to ‘1’ for all patients.**

1. Modelling: This section is dedicated to finding a relationship between the optimal number of clusters k, and the new criterion r. This relationship could be used to determine the optimal cluster number for the segmentation of new PET images using only the new slice image feature r. After analysing the variation of k in function of r criterion (for all patients included in this study), we use two fitting models: an exponential and a power function given by (a) and (b), respectively, k = α ⋅ eβ. r + 1 (a) k = a + 1 (b) Where α, β, a, b are coefficients of fitting models and r is the proposed criterion. The fitting accuracy evaluation is based on the R-square criterion. Note that we added ‘1’ to the original fitting equation to avoid clustering the image with one cluster for the high values of r.

2. Generalisation: The aim of this step is to automate the choice of the optimal cluster number for all patients using one corresponding relationship function by defining a generalised model for all patients. For this reason, we have divided the database randomly into two parts of 50% each. The first part (validation set), contains three patients is used for optimising the model coefficients and fixing the optimal power and exponential generalised model. The second part (test set), contains four patients, is used for testing the accuracy of the fixed optimal model.

According to the R-square criterion, the optimal exponential and power generalised function can be rewritten as follows: k = 46.52e−5.918 × r + 1 k = 1.683r−1.264 + 1 k mean alogorithm [3] : Fig 1. K-mean clustering algorithm 3. Optimization by new technique: The underlying idea behind it is an analysis of the “movement” of objects between clusters, considered either forward from k to k+1 or backwards from k+1 to k groups. In other words we find the movement in membership or joint probability around k groups. The joint probability obtained from adjacent consecutive k numbers of groups will be used to produce a diagonally dominant probability matrix for optimal value of k homogenous and separable groups. The maximum normalised value of the trace as the greatest value for k within the range tested, will be defined as the optimal value of k clusters for the given dataset. Formally, we may describe our approach as follows.

For a given choice of k = number of clusters, a given choice of clustering technique U, and a given choice of V = set of parameters v1…vn used to control the clustering technique, we first construct a set of clusters (U,V) = { } with i=1..k. Next, we construct sets of clusters (U,V), and (U,V) using the same clustering technique. In the work reported here, we will not vary U and V so we may write these cluster sets more simply as, and Now these three and consecutive groups around k will be used to find the proportion of common objects from Ck to and to create a rectangular proportion matrix of size m x n, where m and n correspond to rows and columns of proportion matrix We denote the proportion of data elements in common between a particular pair of clusters, say cluster from and cluster from by, which can be abbreviated to Similarly, we can compute, k to create a rectangular matrix of size n x m where n correspond to columns and m to rows.

Note that in general is not equal to as they have different cardinality meaning k+1 not equal to k and vice versa. To investigate how much movement of objects occurs from Ck to Ck+1 and from to we will apply the dot product of matrix for size m x n to and n x m rectangular matrices ( to ) to get the joint information in square matrix m x m for clusters. Due to the row sum constant of 1 the resultant square matrix is also known as a row stochastic matrix [4] or probability and transition matrix.

The trace of resultant set of clusters will be normalised (average of the trace) to determine the set of more stable (optimal k) clusters as if the normalised trace is maximum and may change occur in the depending on the dataset for the range of adjacent set of k values. This matrix will be used to determine the maximum normalised trace value for determining set of more stable or consistent clusters , that will indicate set of clusters in are stable and completely separated from one another.

**The steps involves in determining the optimal value of k from the resultant clusters are follows as:**

1. Create the m x n forward proportion matrix from k to k+1 = (1)

2. Create the n x m backward proportion matrix from k+1 to k = (2) Where = 1,2,3 … and = 1,2,3 … . . The dot product of (1) by (2) will give us a m x m matrix as in (3) below with the entries showing the joint probabilities of the forward/back movement of the objects between the set of clusters from k to k+1 and k+1 to k. = (3)

The new index can be calculated from in (3) as follows: = (4) From (4) the normalised maximum trace value for will indicate the set of stable cluster at value. In an extreme situation the normalised trace is equal to 1, that is where the set of clusters in will keep splitting until the value of the trace is 1 and it may decrease or increase as we continue but will always stay less than 1.

A new unsupervised cluster-based approach for segmenting the BTV in FDG-PET images is introduced. The system is more reliable and has very less error. It can be improved by technique used in determining an optimal value of K in K-means clustering, for which k-means clustering it uses a method to find an optimal value of k number of clusters, using the features and variables inherited from datasets. The new proposed method is based on comparison of movement of objects forward/back from k to k+1 and k+1 to k set of clusters to find the joint probability, which is different from the other methods and indexes that are based on the distance.

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