Bio-Medical Materials and Engineering 24 (2014) 1253–1259 DOI 10.3233/BME-130927 IOS Press

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A statistical approach to segmentation of diffusion tensor imaging Ying Wen*,a, Lianghua Heb a

Department of Computer Science and Technology, East China Normal University, Shanghai 200241, China b

Key Laboratory of Embedded System and Service Computing, Ministry of Education, Tongji University, Shanghai 200092, China

Abstract. The aim of this study is to design a statistical segmentation technique to allow extraction of grey matter, white matter and cerebral spinal fluid volumes from diffusion tensor imaging. Four channel maps of the DTI are used as the input features, which provide more information for brain tissue segmentation compared with single channel map. An Improved Bayesian decision in the subspace spanned by the eigenvectors which are associated with the smaller eigenvalues in each class is adopted as the brain tissue segmentation criterion. Our method performed well, giving an average segmentation accuracy of about 0.88, 0.85 and 0.76 for white matter, gray matter and cerebrospinal fluid respectively in terms of volume overlap. Keywords: Diffusion tensor imaging, image segmentation, bayesian decision

1. Introduction Brain tissue segmentation is of great significance in studying the structure of the brain [1, 2] and an important preprocessing step in brain research and clinical applications because these contrasts define the boundaries of most brain structures. Different methods have been employed for segmentation of human brain imaging data in both magnetic resonance imaging (MRI) and DTI space in the past years, especially from k-means [3], statistical [4], fuzzy c-means [5], self-organizing map [6], and neural network [7]. White matter (WM), gray matter (GM) and cerebrospinal fluid (CSF) are three basic tissues in the brain. Intensity inhomogeneities imply intensity variations over the same class of tissue that are not caused by random noise in MRI, making it difficult to obtain accurate segmentation results. The intensities in different substructures, even in the same tissue class, are also more or less different due to the inherent regional differences in imaging related properties across substructures, like the composition, density and magnetic properties of different tissues at different positions. Due to the above mentioned adverse impact, it makes intensity distribution within a particular tissue class flatter, and results in overlapping intensity components among different tissues that are neighbors in the intensity histogram, especially in GM or cerebellum tissue segmentation. Many useful approaches have been proposed to *

Corresponding author. E-mail: [email protected]

0959-2989/14/$27.50 © 2014 – IOS Press and the authors. All rights reserved

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Y. Wen and L. He / A statistical approach to segmentation of diffusion tensor imaging

compensate the intensity inhomogeneities. Among them, a multi-channel fusion technique was employed to define accurate tissue maps when dealing with fused structural and diffusion information of DTI data [8-10]. Compared with the traditional segmentation methods, the multi-channel fusion technique performs estimation and segmentation in the fused way from multiple DTI data channels so that intermediate information gained from the current segmentation can be used to improve the estimation result, which can in turn lead to more accurate segmentation. This paper uses 4 channel maps of the DTI providing more information for brain tissue segmentation compared with the single channel map. For training samples, some regions denoting the labels of the WM, GM and CSF are marked manually. An improved Bayesian discriminant in the subspace spanned by the eigenvectors which are associated with the smaller eigenvalues in each class is adopted as the classification criterion. To solve the problem of the matrix inverse, the smaller eigenvalues are substituted by a small threshold which is decided by minimizing the training error in given database. The performance of the proposed method is tested for the segmentation of WM, GM and CSF. Extensive evaluations and comparison studies were provided to demonstrate the reasonable performance of our approach. This article is organized as follows: section 2 presents a statistical approach to segmentation of DTI; section 3 presents experimental results on real DTIs, which show effective and reliable segmentation, and the segmentation accuracy of the proposed algorithm is also evaluated; the final section is devoted to conclusions. 2. Statistical approach to segmentation of DTI 2.1. Preprocessing and multi-channel maps extraction Diffusion Tensor Images with low signal-to-noise ratio are usually very noisy, because DTIs are artificially reconstructed data based on diffusion-weighted imaging data, which are acquired through lengthy pulse sequences since image gradient has to be applied along many directions for reconstruction of diffusion tensors. Due to the motion artifacts and the noise in the data-channel of acquisition, outliers thus appear frequently in the acquired DWI datasets, which are deleterious to the accuracy of reconstruction of diffusion tensors. In this paper, four channel maps of DTI are employed as the input of segmentation algorithm: Fractional Anisotropy (FA), Apparent Diffusion Coefficient (ADC), Kullback- Leibler Anisotropy (KLA), Axial Diffusion (AD) of DTI, as shown in Fig.1. FA measures the fraction of the magnitude of the tensor that can be described by anisotropic diffusion. ADC describes the overall mean-squared displacements of water molecules and the existence of any obstacles to diffusion. KLA effectively detects the transitions between white and gray matter and shows a better discrimination in areas with great confluence of fibers. AD is the diffusivity value parallel to the fibers. In these maps, KLA and FA can be used to divide WM and GM+CSF. ADC and AD can be used to divide CSF and WM+GM. These two group parametric maps have complementary information for brain tissue segmentation [11, 12]. The calculation of FA, ADC, KLA and AD are listed: 2 2 2 3 (λ1 − λ ) + (λ2 − λ ) + (λ3 − λ ) FA = 2 λ 21 + λ 2 2 + λ 23

ADC =

λ1 + λ2 + λ3 3

(1) (2)

Y. Wen and L. He / A statistical approach to segmentation of diffusion tensor imaging

KLA =

λ 3 1 3 λ 1 3 ln(( ¦ i =1 i )( ¦ i =1 ln( )) λ 3 λi 2 3

AD = λ1

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(3) (4)

where λ1 , λ2 and λ3 are the three eigenvalues of the diffusion tensor ( λ1 > λ2 > λ3 ), and λ is the average of three eigenvalues. In this paper, 4 maps are chosen as multi-channel features due to their effectiveness for tissue segmentation. The reasons why other parametric maps are not chosen are as follows: 1) redundant information is decreased; 2) it is time-saving; 3) more feature maps do not mean a good segmentation result.

Fig.1 DTI data and selected parametric maps

Some points of the WM, GM and CSF are selected manually to mark the labels. In Fig.2, red, blue and green labels are marked on the WM, GM and CSF region, respectively. Thus, three groups of samples of the WM, GM and CSF are obtained. Then, points of 4 channel maps corresponding to the above marked position are extracted to construct a training set.

Fig.2 Manual labels for WM, GM and CSF

2.2. Improved Bayesian decision Bayesian decision theory is theoretically optimal when the distribution of samples has a given probability. Suppose that x is a 4-dimensional variable, x = {xFA , x ADC , xKLA , xAD } . The discriminant function is the logarithm of the product between prior and conditional probability [13]: gi ( x) = ln P(ωi ) + ln( p( x | ωi ), i = 1, 2,3 (5) where i denotes the number of tissue class. When the samples distribution is Gaussian distribution, Bayesian discriminant function can be written as: 1 1 gi ( x) = − ( x − μi )t ¦i−1 ( x − μi ) − ln ¦i + ln P(ωi ) (6) 2 2

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where ¦i−1 denotes the inverse matrix of the covariance matrix obtained from ¦i , the mean vector μi and covariance matrix ¦ i of the ith class are: 1 M (7) μi = ¦ x j n j =1 1 M (8) ¦i = ¦ ( x j − μi )t ( x j − μi ), j = 1,..., M , i = 1, 2,3 M j =1 There exists the orthonormal matrix Ui , so that ¦i = U it ΛiU i , where Λi is the eigenvalues matrix of ¦i and denoted as: Λi = diag (λi1...λiki λiki +1...λid ) , λi1 > " > λiki > λik +1 > " > λid (9) where λi is the one of eigenvalue of ¦i . Then, the Eq. (8) can be rewritten as: 1 1 gi ( x) = − ( x − μi )t (U it ΛiU i )−1 ( x − μi ) − ln Λi + ln P(ωi ) (10) 2 2 If the data distribution is Gaussian distribution, the classification capability is optimal when the total eigenvectors are calculated. The smaller the eigenvalue is, the better the classification performance will be. Because the smaller eigenvalue reflects the convergence of within-class, the corresponding eigenvector is more important. Therefore, the eigenvectors corresponding to zero eigenvalues are more important for the classification than others. But if eigenvalue is very close to zero, the inverse matrix of the covariance matrix Λ i−1 will have a problem because g ( x) cannot be calculated. Bayesian classifier is then invalid. To solve this problem, a threshold λ0 is utilized to replace all of eigenvalues which are less than λ0 , i.e., λij = λ0 , Λi = diag (λi1...λiki λ0 ...λ0 ) , if λij ≤ λ0 , j = ki +1 , ki + 2 ,", d . Although the value of λ0 is very small, ¦i−1 could be inversed so that the downsteaming process can be continued. The classification criterion: Given a test sample x , if r satisfies: g ( r ) ( x) = min{g (i ) ( x)} (11) i = 0,1,2

Then the sample belongs to the rth class. The algorithm to DTI segmentation is as follow: Step 1: Four maps for DT images are selected and a set of training samples are manually segmented by experienced neuroradiologists. Step 2: Calculate each class tissue’s mean and covariance based on integrated features (Eqs.(7-8)). Step 3: Calculate each class Bayesian discriminant function as the classification criterion according to Eq. (10). Step 4: the distance between a test sample and each class’ Bayesian decision results is compared to obtain segmentation results according to Eq.(11). 3. Experiments The segmentation was evaluated on brain DTI data from 15 subjects in all. In this study, the standard CSF and WM maps were manually segmented from FA and λ3 maps in each of the data sets. Manual segmentation was performed on a combination of the FA and λ3 maps (CSF appears hyperintense

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and WM appears hypointense on λ3 maps) and a co-registered (although not perfectly) T1-W map was used as a rough guide. Segmentation accuracy was assessed by comparing the automatically obtained results with manual segmentations. In this paper, the experimental results were reported using four measurements. The true positive fraction (TPF) and extra fraction (EF) were reported to express sensitivity and oversegmentation, respectively: TPF = TP (TP + FN ) (12) EF = FP (TP + FN ) (13) where TP, FN and FP denoted true positives, false negatives, and false positives, respectively. The similarity index (SI), or Dice coefficient [14,15], was used to express overlap between segmentations: SI = (T1  T2 ) ((T1 + T2 ) / 2) (14) where T1 and T2 denoted the segmented volumes and (T1ҏT2) was the overlap of T1 and T2. In addition, the overlap measure conformity [16] was used: C = 1 − ( FP + FN ) (TP) (15) The following section experimentally evaluated the accuracy of our method. Fig.3 showed a representative segmentation result of 8 slices of DTI. The yellow, blue and white areas were WM, GM and CSF tissue, respectively. Since a good segmentation was to match ground truth as well as possible, the boundary overlap could reflect the match standard. Fig.4 showed the overlap of WM, CSF and GM boundaries. The images in the first row were the maps derived by DTI data. The images in the second row were WM, GM and CSF segmentation results obtained by our method. The images in the third row were the boundary overlaps of segmentation results and the maps. The boundary overlaps showed a good match between segmentation results and the maps.

Fig.3 Segmentation result of a DTI based on our method.

In this paper, fuzzy c-means (FCM), Kernel FCM (KFCM) and K-Means clustering were adopted to compare the performance of our method on 15 DTI datasets with all normalized multi-channel maps as the input of these 4 methods. Table 1 presented accuracy comparison of the segmentation methods using an expert segmentation as ground truth. Overall, the different segmentation methods showed small differences in accuracy. As conformity and SI were closely related, they showed the same trends. However, for tissue types with less overlap, the conformity measurement showed a significant distinc-

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tion between the segmentation methods. The volume overlap from our method was roughly 0.88 in case of WM, 0.85 for GM and 0.76 for CSF. These values were slightly higher than the results obtained using other algorithms. In addition, TPF and C of our method were higher than those of others except EF. Four measurements showed that the performance of our method was better than other methods.

Fig.4. Boundary overlaps of WM, CSF and GM and ADC and FA maps (left: WM; middle: CSF; right: GM). Table1. Accuracy comparison of the segmentation methods with an expert segmentation. Tissue WM

GM

CSF

Method FCM KFCM K-Means Our method FCM KFCM K-Means Our method FCM KFCM K-Means Our method

SI 0.84 0.86 0.83 0.88 0.83 0.84 0.82 0.85 0.73 0.74 0.73 0.76

TPF 0.80 0.82 0.81 0.91 0.72 0.73 0.73 0.78 0.83 0.84 0.80 0.83

EF 0.28 0.24 0.35 0.15 0.37 0.32 0.35 0.12 0.47 0.37 0.48 0.37

C 0.80 0.82 0.78 0.83 0.78 0.78 0.76 0.81 0.68 0.69 0.66 0.72

4. Conclusion This paper presents a brain tissue segmentation method based on DTI data. This method adopts four channel maps of DTI and an improved Bayesian decision is used for the criteria of brain tissue segmentation. Reasonable tissue segmentation results are obtained in the 15 cases, which demonstrate the

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feasibility and potential of the proposed method. Moreover, this method will be further evaluated and validated in larger datasets. Acknowlegement This work is supported by The National Natural Science Foundation of China (No. 61273261, 61272267) and Shanghai Pujiang Program (No.12PJ1402800), SRF for ROCS, SEM, Shanghai Knowledge Service Platform Project (No. ZF1213) and Program for New Century Excellent Talents in University (No. NCET-11-0381). References [1] R. Stahl, O. Dietrich, et al, Assessment of axonal degeneration on Alzheimer’s disease with diffusion tensor MRI, Radiologe, 43 (7), pp.566-575, 2003. [2] M. Kubicki, H. Park, et al. DTI and MTR abnormalities in schizophrenia: analysis of white matter integrity, Neuroimage, 2005 Jul 15; 26(4):1109-18. [3] J. Macqueen, Some methods for classification and analysis of multivariate observations, Proceedings of 5th Berkeley Symposium on mathematical statistics and probability, university of California press. pp.281-297, 1967. [4] B. Tanoori, Z. Azimifar et al, Brain volumetry: An active contour model-based segmentation followed by SVM-based classification, Computer in Biology and Medicine, 41, pp. 619-632, 2011. [5] G. Lin, C. Wang et al, Automated classification of multispectral MR images using unsupervised constrained energy minimization based on fuzzy logic, Magnetic Resonance Imaging 28, pp.721-738, 2010. [6] E. Berglund and J. Sitte, The parameterless self-organizing map algorithm, IEEE transactions on neural networks, vol.17, no. 2, pp.305-316, 2006. [7] M. Daliri, H. A. Moghaddam, et al, Skull segmentation in 3D neonatal MRI using hybrid Hopfield Neural Network, IEEE international conference on EMBC, pp.4060-4063, 2010. [8] H. Li, T. Liu, et al., Brain tissue segmentation based on DWI/DTI data, 3rd IEEE international Symposium on Biomedical Imaging, pp.57-60, 2006. [9] R. Bammer, Basic principles of diffusion-weighted imaging. Eur. J. Radiol. 45, 2003, 169-184. [10] T. Liu, H. Li, K. Wong, et al, Brain tissue segmentation based on DTI data, NeuroImage 38 , pp.114-123, 2007. [11] D.L. Bihan, J.F. Mangin, C. Poupon, C.A. Clark, S. Pappata, N. Molko, H. Chabriat. Diffusion tensor imaging: concepts and applications. Journal of Magnetic Resonance Imaging. 2001. 13(4): p. 534–46. [12] C.F. Westin, S. Peled, H. Gudbjartsson, R. Kikinis, F.A. Jolesz, Geometrical Diffusion Measures for MRI from Tensor Basis Analysis. ISMRM '97, Vancouver Canada, 1997. [13] Y. Wen, L. H. He, A classifier for Bangla Handwritten Numeral Recognition, Expert Systems with Applications 39 (2012) 948-953. [14] L.R. Dice, Measures of the amount of ecologic association between species. Ecology 26, 297-302, 1945. [15] A.P. Zijdenbos, B.M. Dawant, R.A. Margolin, A.C. Palmer, 1994. Morphometric analysis of white matter lesions in MR images: method and validation. IEEE Trans.Med. Imaging 13, 716-724. [16] H.H. Chang, A.H. Zhuang et al., Performance measure characterization for evaluating neuroimage segmentation algorithms. Neuroimage, 47, 122-135, 2009.

A statistical approach to segmentation of diffusion tensor imaging.

The aim of this study is to design a statistical segmentation technique to allow extraction of grey matter, white matter and cerebral spinal fluid vol...
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