{"title": "Neuronal Fiber Delineation in Area of Edema from Diffusion Weighted MRI", "book": "Advances in Neural Information Processing Systems", "page_first": 1075, "page_last": 1080, "abstract": "", "full_text": "Neuronal Fiber Delineation in Area of Edema\n\nfrom Diffusion Weighted MRI\n\n\u2217\nOfer Pasternak\n\nSchool of Computer Science\n\nTel-Aviv University\n\nTel-Aviv, ISRAEL 69978\n\noferpas@post.tau.ac.il\n\nNathan Intrator\n\nSchool of Computer Science\n\nTel-Aviv University\n\nnin@post.tau.ac.il\n\nNir Sochen\n\nDepartment of Applied Mathematics\n\nTel-Aviv University\n\nsochen@post.tau.ac.il\n\nYaniv Assaf\n\nDepartment of Neurobiochemistry\n\nFaculty of Life Science\n\nTel-Aviv University\n\nassafyan@post.tau.ac.il\n\nAbstract\n\nDiffusion Tensor Magnetic Resonance Imaging (DT-MRI) is a non inva-\nsive method for brain neuronal \ufb01bers delineation. Here we show a mod-\ni\ufb01cation for DT-MRI that allows delineation of neuronal \ufb01bers which\nare in\ufb01ltrated by edema. We use the Muliple Tensor Variational (MTV)\nframework which replaces the diffusion model of DT-MRI with a mul-\ntiple component model and \ufb01ts it to the signal attenuation with a vari-\national regularization mechanism.\nIn order to reduce free water con-\ntamination we estimate the free water compartment volume fraction in\neach voxel, remove it, and then calculate the anisotropy of the remaining\ncompartment. The variational framework was applied on data collected\nwith conventional clinical parameters, containing only six diffusion di-\nrections. By using the variational framework we were able to overcome\nthe highly ill posed \ufb01tting. The results show that we were able to \ufb01nd\n\ufb01bers that were not found by DT-MRI.\n\n1 Introduction\n\nDiffusion weighted Magnetic Resonance Imaging (DT-MRI) enables the measurement of\nthe apparent water self-diffusion along a speci\ufb01ed direction [1]. Using a series of Diffusion\nWeighted Images (DWIs) DT-MRI can extract quantitative measures of water molecule\ndiffusion anisotropy which characterize tissue microstructure [2]. Such measures are in\nparticular useful for the segmentation of neuronal \ufb01bers from other brain tissue which then\nallows a noninvasive delineation and visualization of major brain neuronal \ufb01ber bundles in\nvivo [3]. Based on the assumptions that each voxel can be represented by a single diffusion\ncompartment and that the diffusion within this compartment has a Gaussian distribution\n\n\u2217\n\nhttp://www.cs.tau.ac.il/\u223coferpas\n\n\fDT-MRI states the relation between the signal attenuation, E, and the diffusion tensor, D,\nas follows [4, 5, 6]:\n\nE(qk) = A(qk)\nA(0)\n\n= exp(\u2212bqT\n\nk Dqk) ,\n\n(1)\n\nwhere A(qk) is the DWI for the k\u2019th applied diffusion gradient direction qk. The notation\nA(0) is for the non weighted image and b is a constant re\ufb02ecting the experimental diffu-\nsion weighting [2]. D is a second order tensor, i.e., a 3 \u00d7 3 positive semide\ufb01nite matrix,\nthat requires at least 6 DWIs from different non-collinear applied gradient directions to\nuniquely determine it. The symmetric diffusion tensor has a spectral decomposition for\nthree eigenvectors U a and three positive eigenvalues \u03bba. The relation between the eigen-\nvalues determines the diffusion anisotropy using measures such as Fractional Anisotropy\n(FA) [5]:\n\n(cid:1)\n\nF A =\n\n3((\u03bb1 \u2212 (cid:1)D(cid:2))2 + (\u03bb2 \u2212 (cid:1)D(cid:2))2 + (\u03bb3 \u2212 (cid:1)D(cid:2))2)\n\n2(\u03bb2\n\n1 + \u03bb2\n\n,\n\n2 + \u03bb2\n3)\n\n(2)\nwhere (cid:1)D(cid:2) = (\u03bb1 + \u03bb2 + \u03bb3)/3. FA is relatively high in neuronal \ufb01ber bundles (white\nmatter), where the cylindrical geometry of \ufb01bers causes the diffusion perpendicular to\nthe \ufb01bers be much smaller than parallel to them. Other brain tissues, such as gray mat-\nter and Cerebro-Spinal Fluid (CSF), are less con\ufb01ned with diffusion direction and exhibit\nisotropic diffusion. In cases of partial volume where neuronal \ufb01bers reside other tissue\ntype in the same voxel, or present complex architecture, the diffusion has no longer a sin-\ngle pronounced orientation and therefore the FA value of the \ufb01tted tensor is decreased. The\ndecreased FA values causes errors in segmentation and in any proceeding \ufb01ber analysis.\n\nIn this paper we focus on the case where partial volume occurs when \ufb01ber bundles are in\ufb01l-\ntrated with edema. Edema might occur in response to brain trauma, or surrounding a tumor.\nThe brain tissue accumulate water which creates pressure and might change the \ufb01ber archi-\ntecture, or in\ufb01ltrate it. Since the edema consists mostly of relatively free diffusing water\nmolecules, the diffusion attenuation increases and the anisotropy decreases. We chose to\nreduce the effect of edema by changing the diffusion model to a dual compartment model,\nassuming an isotropic compartment added to a tensor compartment.\n\n2 Theory\n\nThe method we offer is based on the dual compartment model which was already demon-\nstrated as able to reduce CSF contamination [7], where it required a large number of diffu-\nsion measurement with different diffusion times. Here we require the conventional DT-MRI\ndata of only six diffusion measurement, and apply it on the edema case.\n\n2.1 The Dual Compartment Model\n\nThe dual compartment model is described as follows:\n\nE(qk) = f exp(\u2212bqT\n\nk D1qk) + (1\u2212 f) exp(\u2212bD2) .\n\n(3)\n\nThe diffusion tensor for the tensor compartment is denoted by D1, and the diffusion coef\ufb01-\ncient of the isotropic water compartment is denoted by D2. The compartments have relative\nvolume of f and 1\u2212f. Finding the best \ufb01tting parameters D1, D2 and f is highly ill-posed,\nespecially in the case of six measurement, where for any arbitrarily chosen isotropic com-\npartment there could be found a tensor compartment which exactly \ufb01ts the data.\n\n\fFigure 1: The initialization scheme. In addition to the DWI data, MTV uses the T2 image\nto initialize f. The initial orientation for the tensor compartment are those that DT-MRI\ncalculated.\n\n2.2 The Variational Framework\n\nIn order to stabilize the \ufb01tting process we chose to use the Multiple Tensor Variational\n(MTV) framework [8] which was previously used to resolve partial volume caused by\ncomplex \ufb01ber architecture [9], and to reduce CSF contamination in cases of hydrocephalus\n[10]. We note that the dual compartment model is a special case of the more general mul-\ntiple tensor model, where the number of the compartments is restricted to 2 and one of the\ncompartments is restricted to equal eigenvalues (isotropy). Therefore the MTV framework\nadapted for separation of \ufb01ber compartments from edema is composed of the following\nfunctional, whose minima should provide the wanted diffusion parameters:\n\n(cid:5)\n\n(cid:2)\n\n(cid:3)\n\nd(cid:4)\n\nk=1\n\nS(f, D1, D2) =\n\n\u03b1\n\n\u2126\n\n(E(qk) \u2212 \u02c6E(qk))2 + \u03c6(|\u2207U 1\ni |)\n\nd\u2126 .\n\n(4)\n\n(cid:6)\n\n\u2202y )2 + ( \u2202I\n\n\u2202x)2 + ( \u2202I\n( \u2202I\n\nThe notation \u02c6E is for the observed diffusion signal attenuation and E is calculated using\n(3) for d different acquisition directions. \u2126 is the image domain with 3D axis (x, y, z),\n|\u2207I| =\n\u2202z )2 is de\ufb01ned as the vector gradient norm. The notation U 1\ni\nstands for the principal eigenvector of the i\u2019th diffusion tensor. The \ufb01xed parameters \u03b1 is\n(cid:6)\nset to keep the solution closer to the observed diffusion signal. The function \u03c6 is a diffusion\n\ufb02ow function, which controls the regularization behavior. Here we chose to use \u03c6i(s) =\nwhich lead to anisotropic diffusion-like \ufb02ow while preserving discontinuities\n[11]. The regularized \ufb01tting allows the identi\ufb01cation of smoothed \ufb01ber compartments and\nreduces noise. The minimum of (4) solves the Euler-Lagrange equations, and can be found\nby the gradient descent scheme.\n\n1 + s2\nK2\ni\n\n2.3 Initialization Scheme\n\nSince the functional space is highly irregular (not enough measurements), the minimization\nprocess requires initial guess (\ufb01gure 1), which is as close as possible to the global mini-\nmum. In order to apriori estimate the relative volume of the isotropic compartment we used\na normalized diffusion non-weighted image, where high contrast correlates to larger \ufb02uid\nvolume. In order to apriori estimate the parameters of D1 we used the result of conventional\nDT-MRI \ufb01tting on the original data. The DT-MRI results were spectrally decomposed and\nthe eigenvectors were used as initial guess for the eigenvectors of D1. The initial guess for\n\n\fthe eigenvalues of D1 were set to \u03bb1 = 1.5, \u03bb2 = \u03bb3 = 0.4.\n\n3 methods\n\nWe demonstrate how partial volume of neuronal \ufb01ber and edema can be reduced by apply-\ning the modi\ufb01ed MTV framework on a brain slice taken from a patient with sever edema\nsurrounding a brain tumor. MRI was performed on a 1.5T MRI scanner (GE, Milwau-\nkee). DT-MRI experiments were performed using a diffusion-weighted spin-echo echo-\nplanar-imaging (DWI-EPI) pulse sequence. The experimental parameters were as follows:\nT R/T E = 10000/98ms, \u2206/\u03b4 = 31/25ms, b = 1000s/mm2 with six diffusion gra-\ndient directions. 48 slices with thickness of 3mm and no gap were acquired covering the\nwhole brain with FOV of 240mm2 and matrix of 128x128. Number of averages was 4, and\nthe total experimental time was about 6 minutes. Head movement and image distortions\nwere corrected using a mutual information based registration algorithm [12]. The corrected\nDWIs were \ufb01tted to the dual compartment model via the modi\ufb01ed MTV framework, then\nthe isotropic compartment was omitted. FA was calculated for the remaining tensor for\nwhich FA higher than 0.25 was considered as white matter. We compared these results to\nsingle component DT-MRI with no regularization, which was also used for initialization of\nthe MTV \ufb01tting.\n\n4 Results and Discussion\n\nFigure 2: A single slice of a patient with edema.\n(A) a non diffusion weighted image\nwith ROI marked. Showing the tumor in black surrounded by sever edema which appear\nbright. (B) Normalized T2 of the ROI, used for f initialization. (C) FA map from DT-MRI\n(threshold of FA> 0.25). Large parts of the corpus callosum are obscured. (D) FA map of\nD1 from MTV (thresholds f> 0.35, FA> 0.25). A much larger part of the corpus callosum\nis revealed\n\nFigure (2) shows the Edema case, where DTI was unable to delineate large parts of the\ncorpus callosum. Since the corpus callosum is one of the largest \ufb01ber bundles in the brain\n\n\fit was highly unlikely that the \ufb01bers were disconnected or disappeared. The expected FA\nshould have been on the same order as on the opposite side of the brain, where the corpus\ncallosum shows high FA values. Applying the MTV on the slice and mapping the FA value\nof the tensor compartment reveals considerably much more pixels of higher FA in the area\nof the corpus callosum. In general the FA values of most pixels were increased, which was\npredicted, since by removing any size of a sphere (isotropic compartment) we should be left\nwith a shape which is less spherical, and therefore with increased FA. The bene\ufb01t of using\nthe MTV framework over an overall reduce of FA threshold in recognizing neuronal \ufb01ber\nvoxels is that the amount of FA increase is not uniform in all tissue types. In areas where\nthe partial volume was not big due to the edema, the increase was much lower than in areas\ncontaminated with edema. This keeps the nice contrast re\ufb02ected by FA values between\nneuronal \ufb01bers and other tissue types. Reducing the FA threshold on original DT-MRI\nresults would cause a less clear separation between the \ufb01ber bundles and other tissue types.\nThis tool could be used for \ufb01ber tracking in the vicinity of brain tumors, or with stroke,\nwhere edema contaminates the \ufb01bers and prevents \ufb01ber delineation with the conventional\nDT-MRI.\n\n5 Conclusions\n\nWe show that by modifying the MTV framework to \ufb01t the dual compartment model we\ncan reduce the contamination of edema, and delineate much larger \ufb01ber bundle areas. By\nusing the MTV framework we stabilize the \ufb01tting process, and also include some biological\nconstraints, such as the piece-wise smoothness nature of neuronal \ufb01bers in the brain. There\nis no doubt that using a much larger number of diffusion measurements should increase the\nstabilization of the process, and will increase its accuracy. However, more measurement\nrequire much more scan time, which might not be available in some cases. The variational\nframework is a powerful tool for the modeling and regularization of various mappings. It\nis applied, with great success, to scalar and vector \ufb01elds in image processing and computer\nvision. Recently it has been generalized to deal with tensor \ufb01elds which are of great interest\nto brain research via the analysis of DWIs and DT-MRI. We show that the more realistic\nmodel of multi-compartment voxels conjugated with the variational framework provides\nmuch improved results.\n\nAcknowledgments\n\nWe acknowledge the support of the Edersheim - Levi - Gitter Institute for Functional Hu-\nman Brain Mapping of Tel-Aviv Sourasky Medical Center and Tel-Aviv University, the\nAdams super-center for brain research of Tel-Aviv University, the Israel Academy of Sci-\nences, Israel Ministry of Science, and the Tel-Aviv University research fund.\n\nReferences\n\n[1] E Stejskal and JE Tanner. Spin diffusion measurements: Spin echoes in the presence\n\nof a time-dependant \ufb01eld gradient. J. Chem. Phys., 42:288\u2013292, 1965.\n\n[2] D. Le-Bihan, J.-F. Mangin, C. Poupon, C.A. Clark, S. Pappata, N. Molko, and\nH. Chabriat. Diffusion tensor imaging: concepts and applications. Journal of Mag-\nnetic Resonance Imaging, 13:534\u2013546, 2001.\n\n[3] S. Mori and P.C. van Zijl. 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Mag-\nnetic Resonance in Medicine, 51:103\u2013114, 2004.\n\n\f", "award": [], "sourceid": 2917, "authors": [{"given_name": "Ofer", "family_name": "Pasternak", "institution": null}, {"given_name": "Nathan", "family_name": "Intrator", "institution": null}, {"given_name": "Nir", "family_name": "Sochen", "institution": null}, {"given_name": "Yaniv", "family_name": "Assaf", "institution": null}]}