Multi-fiber Reconstruction from DW-MRI using a Continuous Mixture of Hyperspherical von Mises-Fisher Distributions

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1 Multi-fiber Reconstruction from DW-MRI using a Continuous Mixture of Hyperspherical von Mises-Fisher Distributions Ritwik Kumar 1, Baba C. Vemuri 1 Fei Wang 2, Tanveer Syeda-Mahmood 2, Paul R. Carney 3, and Thomas H. Mareci 4 1 Dept. of CISE, University of Florida, Gainesville, FL, USA, 2 IBM Almaden Research Center, San Jose, CA, USA 3 Dept. of Pediatrics, University of Florida, Gainesville, FL, USA 4 Dept. and Molecular Biology, University of Florida, Gainesville, FL, USA Abstract. Multi-fiber reconstruction has attracted immense attention lately in the field of diffusion weighted MRI analysis. Several mathematical models have been proposed in literature but there is still scope for improvement. The key issues of importance in multi-fiber reconstruction are, fiber detection accuracy, robustness to noise and computational efficiency. To this end, we propose a novel mathematical model for representing the MR signal attenuation in the presence of multiple fibers at a single voxel and estimate the parameters of this model given the diffusion weighted MRI data. Our model for the diffusion MR signal consists of a continuous mixture of Hyperspherical von Mises-Fisher distributions. Being a continuous mixture, our model does not require the specification of the number of mixture components. We present a closed form expression for this continuous mixture that leads to a computationally efficient implementation. To validate our model we present extensive results on both synthetic and real data (human and rat brain) and demonstrate that even in presence of noise, our model clearly outperforms the stateof-the-art methods in fiber orientation estimation while maintaining a substantial computational advantage. 1 Introduction Diffusion Weighted Magnetic Resonance Imaging (DW-MRI), arguably one of the most important imaging inventions of the twentieth century, is the only existing non-invasive and in-vivo imaging method that allows examination of neural tissue architecture at a microscopic scale. By quantitatively capturing the diffusion of water molecules in the brain tissues one can determine the white matter fiber tracts even in the presence of complex local geometries such as fiber crossings ( [21, 17, 6]) and connectivity of different brain regions ([8]). DW-MRI works by measuring the loss of precession synchronization of hydrogen molecules when a pair of directional magnetic gradient pulses with opposing This research was in part supported by UF Alumni Fellowship to RK, NIH EB to BCV and NIH EB to PC and TM.

2 polarity are applied. This is quantitatively captured in the DW-MR image as the loss of signal required to pull the precession back in synchronization, defined for each gradient direction. From this raw data, water displacement probabilities (PDF) at each voxel can be obtained by computing the fourier transform of the corresponding signal attenuation function, reconstructed from its directional samples. The local fiber orientation can then be inferred by computing the maxima of the displacement probability function or the orientation distribution function (ODF), which can be obtained by radially integrating or by taking the radial iso-surface of the PDF. Further processing of this information, for instance, via tractography can reveal connections between different regions of the brain. From the above description it readily follows that the accurate reconstruction of the signal attenuation from its directional, often noisy, samples is of fundamental importance to the success of subsequent steps. In this paper we present a novel technique to accomplish this, which provides superior accuracy for fiber orientation detection while maintaining computational efficiency advantage over the state-of-the-art techniques. Our method captures the inherent antipodal symmetry of the DW-MR signal using the Knutsson mapping [16] and models the signal attenuation function using a continuous mixture of Hyperspherical von Mises-Fisher distributions, for which we have derived a closed form expression. Over the last two decades, various methods have been suggested for MR signal attenuation and ODF modeling. Pioneering work in this field was presented in [7] which modeled diffusion by rank 2 tensors (DTI). Though effective in characterizing the diffusivity function in most of white matter, DTI could not explain complex fiber geometry such as crossings. To address this, techniques using spherical harmonics [13] and higher order tensors [22], [4] were proposed using high resolution data (HARDI), but these produced diffusivity functions whose peaks are known not to correspond to the fiber orientations [21]. Bypassing the signal reconstruction, direct ODF modeling was proposed in [10] and [14] using Q-Ball imaging but this produced a displacement probability function which was corrupted via convolution with a zeroth order Bessel function. Diffusion Spectrum imaging [12] was proposed as an alternate to Q-Ball but it suffers from time intensive sampling requirements. Diffusion Orientation Transform, proposed in [21], analytically evaluated the radial component of the fourier integral but it yields a displacement probability that is also corrupted via convolution with a function that can not be specified in analytic form. In contrast to the above methods are the multi-compartmental models (e.g. [25, 3, 20, 9]) that use a discrete mixture of basis functions to approximate the MR signal attenuation. Though effective, these methods face the model selection problem of fixing the number of mixture components a priori, and this was addressed by the deconvolution based approaches ([24, 1, 2]), which assume a distribution of fibers at each voxel. More recently, continuous mixture models have been proposed in [15] and [17] which model MR signal attenuation as

3 a continuous mixture of some appropriately chosen bases and recover model parameter by solving linear systems. Of all the methods presented above, multi-compartmental models and continuous mixture models are most closely related to our technique presented here. In particular, it is related to the work presented by McGraw et al. [20], Bhalerao et al. [9], Jian et al. [15] and Kumar et al. [17]. McGraw et al. [20] and Bhalerao et al. [9] present multi-compartmental models for MR signal attenuation modeling using discrete mixtures of von Mises- Fisher and Hyperspherical von Mises-Fisher distributions respectively. Both of these techniques require computationally intensive non-linear optimizations to recover the model parameters at each voxel. Furthermore, [20] suffers from the well known model selection problem, where the number of mixture components are arbitrarily set. Though the method in [9] tries to avoid this problem using Akaike information criterion, it however results in choosing between one or two components by exhaustively comparing the two choices, which puts this technique only on a slightly better footing than the one in [20]. Jian et al. [15] and Kumar et al. [17] formulate their models in the continuous mixture framework, which allows them to overcome the model selection problems associated with discrete mixtures mentioned above. Though Jian et al. [15] define the current state-of-the-art in terms of fiber orientation detection accuracy, their technique requires eigenvalues of the diffusion tensors in the continuous mixture to be fixed, in order to translate its discretization problem from P 3, space of symmetric positive definite matrices, to that of discretizing S 2,the sphere. Moreover, since it seeks non negative weights, it requires the use of the Non-Negative Least Squares [18] algorithm at each voxel of the DW-MRI data, which can be computationally intensive. The model proposed by Kumar et al. [17] avoids these problems at the expense of being less accurate than the method in [15]. Our method addresses all of of the above mentioned pitfalls by providing more accurate fiber orientation detection results than [15] while maintaining the computational advantage similar to [17]. 2 Theory 2.1 Knutsson Mapping DW-MR data is inherently antipodally symmetric. If a mixture model for its estimation uses basis functions which are not inherently antipodally symmetric, it would need additional constraints to enforce antipodal symmetry of the estimation. Existing mixture models like those described in [20] and [17] handle antipodal symmetry by combining pairs of basis functions with antipodal orientations into one basis function. Here we use a much more seamless method that uses higher dimensional mapping called Knutsson mapping [16] to account for antipodal symmetry. Knutsson [16] proposed a mapping ν : S 2 S 4 which was agnostic to directions, ν( x ) = ν( x ), locally preserved angular metric for constant magnitude

4 vectors, δν( x ) = α δ x with α being some constant and lead to direction independent magnitude in the mapped space, ν( x ) = β for some constant β. This mapping is defined as ν([r, θ, φ]) = [sin 2 (θ) cos 2φ, sin 2 (θ) sin 2φ, sin 2θ cos φ, sin 2θ sin φ, 3(cos 2 θ 1 3 )], (1) where [r, θ, φ] is the spherical parameterization of a vector in S 2. Knutsson mapping proves to be a handy tool for dealing with data which only depends on axes and not directions, like DW-MR data. In the context of continuous mixture model, this mapping alleviates the need to pair antipodal bases into one. But as a result of this mapping, the problem now is no longer defined in S 2 and would require analysis in S 4. It was employed in the past by [9] in context of processing MR signals. 2.2 Continuous Mixture of Hyperspherical von Mises-Fisher Distributions A continuous mixture model defined on the spherical domain S p 1 can be represented as f(x) = D(µ)K(x, µ)dµ, (2) S p 1 where K(x, µ) is called the basis function or the kernel and D(µ) is called the mixing density. Here x and µ are vectors in S p 1. This formulation can be looked at as a generalization of the discrete mixture of basis functions where the continuous mixing density has replaced the discrete mixing weights. Here, we seek a continuous mixture of Hyperspherical von Mises-Fisher distributions [19], which for the most general case (defined on S p 1 ), is given as M p (x; µ, κ) = ( κ 2 )p/2 1 exp (κµ T x) Γ (p/2)i p/2 1 (κ). (3) where µ, (with µ = 1) is the mean direction along which the distribution is oriented with a concentration given by parameter κ 0. I ν is the Bessel function of the first kind and order ν. As mentioned in the previous section, in order to handle antipodal symmetry we project the vectors from S 2 to S 4, and thus, the kernel function for our continuous mixture model should also be defined in S 4, hence it has the form M 5 (x; µ, κ) = 1 2 κ 3/2 3 π I 3/2 (κ) exp (κµt x). (4) Note that in this expression x and µ are defined as vectors in S 4 and not S 2. It can be noted in Eq. 2 that the mixing density is a density defined on the domain of integration. In our case, it needs to be defined on S 4. As the spherical analog of the Gaussian distribution, Hyperspherical von Mises-Fisher

5 density again presents itself as a natural choice. But since the formulation in Eq. 2 is convolutional in nature, if both the mixing density and the kernel are chosen to be single lobed functions, so would be the continuous mixture. In order to provide our model flexibility to accommodate intra-voxel orientation heterogeneity (IVOH), we use a discrete mixture of hyperspherical von Mises- Fisher distributions as the mixing density, given as D 5 (µ; {γ i }, κ ) = N i=1 1 2 κ 3/2 w i 3 π I 3/2 (κ ) exp (κ γ T i µ). (5) This density is parameterized by a discrete collection of N directions {γ i }, and constant concentration κ. We must point out that N here decides the resolution of spherical discretization and is in no way related to the number of expected fiber bundles. This kind of discretization is often used in continuous mixtures ([15], [17]) and should be looked at as the number of basis functions to be used while reconstructing the MR signal attenuation. Generally the quality of reconstruction improves as this resolution increases. For our method we obtain the set of direction {γ i } by a 4 th order subdivision of the icosahedral tessellation of the unit hemi-sphere. Note that we have not discretized the space for parameter κ as we will set it later based on numerical considerations. Substituting the expressions for the kernel and the mixing density in Eq.2 leads to the following expression for the MR signal attenuation S(q)/S 0 S(q) = S 0 S P N 2(κ κ) 3 2 w i 9πI 3 (κ )I 3 (κ) exp (κ γ T i µ) exp (κµ T q)dµ, (6) 2 2 i=1 where S(q) is the DW-MRI signal value associated with reciprocal space vector q, S 0 the zero gradient signal. Here we make the important observation that if κ is set to unity, this expression can be looked at as a Laplace transform of the Hyperspherical von Mises-Fisher distribution in S 4, which we have analytically evaluated to be S(q) S 0 = N i=1 3 2κ 2 /(3 πi 3 w (1)I [ 3 (κ )) 2 2 cosh( κ ] µ i + q ) i (κ cosh(κ ) sinh(κ )) κ µ i + q 2 sinh( κ µ i + q ) κ µ i + q 3. (7) This expression provides the value of signal attenuation when the magnetic gradient is applied in the direction q. Since the MR data is obtained for multiple such directions, unknown weights w i can be obtained by solving a linear system using the method described subsequently. 2.3 Numerical Solution When the data is available for various gradient directions q j, unknown weights from the expression in Eq. 7 can be obtained by solving a system of linear equations Aw = S, where w is the vector of N unknown weights, S is the vector

6 that contains the M acquired MR signal samples S(q j )/S 0, and A is the M N matrix where the entry A ji in the j th row and the i th column is given by 2κ 3 2 /(3 πi 3 2 A ji = (1)I [ 3 (κ )) 2 cosh( κ ] µ i + q j ) (κ cosh(κ ) sinh(κ )) κ µ i + q j 2 sinh( κ µ i + q j ) κ µ i + q j 3. (8) Since the data can be noisy and the number of fibers (bundles) at a voxel are sparse (as compared to N), we obtain the unknown weights by using a regularized least squares formulation leading to the minimization of the following expression w = argmin w Aw S 2 + λ w 2 (9) which leads to the following closed form solution for the unknown w w = A T (AA T + λi) 1 S. (10) This expression can be evaluated without inverting any matrix. This constrained least squares is also called Damped Least Squares and has been employed in the past in [15]. At this stage we set the parameters κ and λ so as to ensure better conditioning of the process in Eq. 10. We must point out that these parameters need to be set only once, as the matrix A remains the same even if the actual MR data changes from voxel to voxel. 2.4 Fiber Orientation Recovery The water molecule displacement probability function can be obtained from the signal attenuation function via a Fourier transform given as S(q) P (r) = e 2πiq T r dq, (11) S 0 where q is the reciprocal space vector, S(q) is the DW-MRI signal value associated with vector q, S 0 the zero gradient signal and r is the displacement vector. Through there are various methods available in literature for obtaining the water displacement probability, in our implementation we use the method proposed in [5] for its effectiveness. Once the water displacement probability has been obtained, fiber orientations can be recovered by finding the maxima of either a radial iso-surface of P (r) or the radial integral of P (r). We use the former of these techniques to evaluate the fiber orientation primarily on account of its simplicity and good accuracy. 3 Experimental Results & Discussion First, we try to quantitatively capture the performance of our technique relative to various state-of-the-art methods. Towards this, in Fig. 1(A) we present fiber

7 Fig. 1. (A). Fiber orientation detection accuracy comparison with DOT [21], MoW [15], Q-Ball ODF [10] and MovMF [17]. (B). Probability profiles for 2 fibers crossing at 90 as the amount of noise increases from left to right. orientation detection errors for Mixture of Wisharts (MoW) [15], continuous mixture of von Mises-Fisher distributions (MovMF) [17], Q-Ball Orientation Distribution Function [10], Diffusion Orientation Transform (DOT) [21] and the proposed method for various noise levels in the signal. For this experiment we simulated a 2-fiber crossing using the so called Söderman s model proposed in [23] which captures the MR signal attenuation from particles diffusing inside a cylindrical boundary. The signal was simulated for 81 gradient directions with a b value of 1250 s/mm 2. We added Rician noise to the data with increasing variance and obtained signal with the signal to noise ratio (SNR) ranging from (for no noise) to 3.6 (large noise). It can be readily noted from Fig. 1(A) that the proposed method provides noise robust performance and achieves higher accuracy as compared to MoW, Q-Ball ODF, MovMF and DOT. Next in Fig. 1(B) we visualize the probability profiles generated by our method for the various noise levels used in Fig. 1(A). We have presented 2 column of water displacement probability profiles for each case with the corresponding SNR mentioned below it. It can be noted that as the SNR decreases the computed probability profiles starts showing deviations from the expected 90 crossings with near perfect detection in the first column (noiseless case). We used the same 2-fiber crossing testbench to evaluate the computational efficiency of the proposed method. For this experiment we implemented MoW [15] and MovMF [17] and our method in MATLAB 7.4. Then we noted the time taken by each technique for 2 fiber crossing reconstruction for scan volumes of varying sizes. The experiment was conducted on an Intel Centrino 2.4 GHz machine with 3 GB memory and the obtained run times are reported in Fig. 2(A). It can be noted that our method maintains the same computational efficiency as MovMF and is substantially faster than MoW. The primary reason for this

8 Fig. 2. (A) Computational efficiency comparison as the scan volume size increases. It can be noted that our technique provides better performance than competing methods (MoW[15] and MovMF [17]). (B) Fiber orientation detection errors (in degrees) for 1, 2 and 3 fiber crossings. The second column indicates the maximum inter-fiber angular separation when multiple fibers are present. Column headings for next eight columns indicate the SNR in db. is that for MoW, the linear system is solved using non-negative least squares method while our method uses a simpler linear system solver. In Fig. 2(B) we present more comprehensive quantitative results for our method as both the number of fibers in the crossings and the intra-fiber angles are varied. For this experiment, we simulated MR signal attenuation with various different fiber crossing angles and with varying amount of noise. The angles in the second column are the maximum inter-fiber angle for the cases with multiple fibers. In the next eight columns fiber orientation detection errors (in degrees) are presented with the corresponding SNR mentioned in the column headings. Following observations can be made from these results. Firstly, as the number of fibers crossing at each voxel increases, so does the ambiguity in detection of fiber orientation and thus results for single fiber are better than 2 or 3 fibers. Secondly, for all the cases (1, 2 or 3 fibers), as the noise level increases, so does the orientation detection error rates. Thirdly, as the angle between the intersection fibers decreases, the error rates go up. Note that as long as the error rates are not larger than half the fiber separation angle, number of fibers can still be accurately detected and thus for all the cases presented, correct number of fibers can be resolved irrespective of the error. Thus far we presented results on simulated data for quantitative comparison of various methods, next we present comprehensive results of our method on real human and rat brain data. In these results we have shown water molecule displacement probability iso-surfaces estimated from the reconstructed signal using the proposed method. Each probability profiles is colored according to the dominant fiber direction and the color map is provided in the top left corner of

9 both Fig. 3 and Fig. 4 respectively. The peaks of the probability profile indicate the fiber orientation. We present results on human brain data in Fig. 3. This data was acquired using 46 gradient directions with repetition time of 8.5 sec and b value of 800 s/mm 2 on a 3 Tesla Phillips scanner. The matrix size was and the data was acquired at 2 mm 2 mm 2 mm resolution. In Fig. 3, images (A) and (C) show the coming together of the corpus callosum splenium and corpus callosum genu bundles with the inferior fronto-occipital tract respectively. At the join of these two bundles, fiber crossings can be seen and the plotted spherical functions depict the multiple lobes whose peaks point to the direction of the fibers, using which we can track fibers ([8]) as shown in Fig. 3(A)(iv) and Fig. 3(C)(iv). Fig. 3(B) shows another slice with single fiber bundles as well as fiber crossings. In Fig. 4 we present results obtained on rat brain data acquired using a 17.6 Tesla Bruker scanner. The data used in Fig. 4(A) and 4(C) was obtained from an excised perfusion-fixed rat brain. Thirty-two images were acquired using a spin-echo, pulsed-field-gradient sequence with repetition time 1.4 s, echo time 28 ms, field of view 30 mm 15 mm, matrix with 32 continuous 0.3 mm thick slices measured (oriented parallel to the long-axis of the brain). 46 diffusion weighted images were collected with 5 signal averages with approximate b values of 1250 s/mm 2, whose orientations were determined by the tessellation on a hemisphere. The data used in Fig. 4(B) was obtained from excised, perfusionfixed rat optic chiasm in 46 gradient directions with the b value as 1250 s/mm 2. Fig. 4(A) shows fiber orientations in the hippocampus region of the rat brain. The zoomed-in region of interest shows fiber crossings and single fiber structures consistent with already published reports on hippocampal structure [11]. In Fig. 4(B) we present estimated fiber orientations from various regions of myelinated axons from the two optic nerve bundles crossing each other to reach their respective contra-lateral optic tracts. Inset images (i) and (ii) show the optic tract where voxels are composed of single fibers. Inset (iii) shows the crossing fibers while the inset (iv) shows the fibers going their separate ways after the crossing. Fig. 4(C) shows fibers of cingulum and corpus callosum (upper right) intersecting each other. Single fibers and fiber crossings can be seen to be correctly rendered using our method. 4 Conclusion In this paper we introduced a novel mathematical model for modeling the MR signal attenuation leading to high fiber orientation detection accuracy and computational efficiency. We have experimentally demonstrated that our method is particularly robust to the presence of noise and outperforms existing methods in terms of fiber orientation detection accuracy. Further, since our techniques leads to a linear system which we solve using damped least squares, it is more computationally efficient, as compared to existing methods that employ non-linear techniques and constrained linear least-squares techniques. Through extensive

10 experiments on rat and human brain data, we showed that the proposed technique can provide excellent fiber detection results for real data as well. References 1. Daniel C. Alexander. Maximum entropy spherical deconvolution for diffusion MRI. In IPMI, Adam W. Anderson. Measurement of fiber orientation distributions using high angular resolution diffusion imaging. MRM, 54(5), Y. Assaf, R. Z. Freidlin, G. K. Rohde, and P. J. Basser. New modeling and experimental framework to characterize hindered and restricted water diffusion in brain white matter. MRM, A. Barmpoutis, B. Jian, B. C. Vemuri, and T. M. Shepherd. Symmetric positive 4th order tensors and their estimation from diffusion weighted mri. In IPMI, A. Barmpoutis, B. C. Vemuri, and J. R. Forder. Fast displacement probability profile approximation from hardi using 4th-order tensors. ISBI, A. Barmpoutis, B. C Vemuri, D. Howland, and J. R. Forder. Extracting tractosemas from a displacement probability field for tractography in dw-mri. In MICCAI, P. J. Basser, J. Mattiello, and D. Lebihan. Estimation of the Effective Self-Diffusion Tensor from the NMR Spin Echo. J. Magn. Reson. B, 103, P.J. Basser, S. Pajevic, C. Pierpaoli, J. Duda, and A. Aldroubi. In vivo fiber tractography using dt-mri data. Magnetic Resonance in Medicine, 44(4), A. Bhalerao and C.-F. Westin. Hyperspherical von mises fisher mixture (hvmf) modelling of high angular resolution diffusion mri. In MICCAI, M. Descoteaux, E. Angelino, S. Fitzgibbons, and R. Deriche. Regularized, fast and robust analytical q-ball imaging. Magnetic Resonance in Medicine, 58, T. M. Shepherd et al. Structural insights from high-resolution diffusion tensor imaging and tractography of the isolated rat hippocampus. NeuroImage, V. J. Wedeen et al. Mapping complex tissue architecture with diffusion spectrum magnetic resonance imaging. MRM, 54(6), L. R. Frank. Characterization of anisotropy in high angular resolution diffusionweighted MRI. MRM, 47(6), C. P. Hess, P. Mukherjee, E. T. Han, D. Xu, and D. B. Vigneron. Q-ball reconstruction of multimodal fiber orientations using the spherical harmonic basis. MRM, B. Jian, B. C. Vemuri, E. Özarslan, P. R. Carney, and T. H. Mareci. A novel tensor distribution model for the diffusion weighted mr signal. NeuroImage, H. Knutsson. Producing a continuous and distance preserving 5-d vector repesentation of 3-d orientation. In IEEE Computer Society Workshop on Computer Architecture for Pattern Analysis and Image Database Management, R. Kumar, A. Barmpoutis, B. C. Vemuri, P. R. Carney, and T. H. Mareci. Multifiber reconstruction from dw-mri using a continuous mixture of von mises-fisher distributions. MMBIA, C. Lawson and R. J. Hanson. Solving Least Squares Problems. Prentice-Hall, K. V. Mardia and P. Jupp. Directional Statistics. John Wiley and Sins Ltd., Newyork, 2nd Edition, T. E. McGraw, B. C. Vemuri, R. Yezierski, and T. H. Mareci. Von Mises-Fisher mixture model of the diffusion ODF. In ISBI, E. Özarslan, T. M. Shepherd, B. C. Vemuri, S. J. Blackband, and T. H. Mareci. Resolution of complex tissue microarchitecture using the diffusion orientation transform (DOT). NeuroImage, 31: , Evren Özarslan and Thomas H. Mareci. Generalized diffusion tensor imaging and analytical relationships between diffusion tensor imaging and high angular resolution diffusion imaging. MRM, 50(5), O. Söderman and B. Jönsson. Restricted diffusion in cylindirical geometry. J. Magn. Reson. B, 117(1), J.-D. Tournier, F. Calamante, D. G. Gadian, and A. Connelly. Direct estimation of the fiber orientation density function from diffusion-weighted MRI data using spherical deconvolution. NeuroImage, D. S. Tuch, T. G. Reese, M. R. Wiegell, N. Makris, J. W. Belliveau, and V. J. Wedeen. High angular resolution diffusion imaging reveals intravoxel white matter fiber heterogeneity. MRM, 2002.

11 Fig. 3. Results using human brain data. Detected water displacement probability profiles for various human brain regions including regions showing merging of the A. corpus callosum splenium and the inferior fronto-occipital tract and C. corpus callosum genu and the inferior fronto-occipital tract are presented. Part B shows single fiber bundles as well as fiber crossings.

12 Fig. 4. Results using rat brain data. Detected fiber orientations in A. Rat brain hippocampus, B. Optic chiasm from a rat brain and C. Cingulum and corpus callosum crossing.