Estimation of multiple fiber orientations from diffusion tensor MRI

266 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 52, NO. 1, FEBRUARY 2005

Estimation of Multiple Fiber Orientations FromDiffusion Tensor MRI Using Independent

Component AnalysisSungheon Kim, Member, IEEE, Jeong-Won Jeong, Member, IEEE, and Manbir Singh, Senior Member, IEEE

Abstract—Determination of fiber orientation from diffusiontensor is problematic when the tensor is not linear anisotropic.Particularly planar anisotropy is often an indicator of multiplefibers in a voxel. A novel method has been developed to identify theorientations of multiple fiber tracts in a voxel of diffusion tensormagnetic resonance imaging (MRI) using independent componentanalysis (ICA). A computationally efficient algorithm to estimatethe independent sources has been derived by introducing a newadaptive nonlinear function to model the cumulative distributionof the sources. Monte Carlo simulation was used to evaluate themethod. Simulations suggest that the orientations of two tensors ina voxel can be estimated with mean error of less than 10 for mostinterfiber angles when the signal-to-noise ratio is higher than 30.A processing and source selection strategy has been proposed andsuccessfully tested with simulated tensor fields incorporating fibercrossing and with human data. Qualitative assessment of the resultfrom human data analysis demonstrated that the ICA methodreasonably estimated multiple fiber orientations corresponding toanatomically known white matter tracts.

Index Terms—Diffusion-tensor magnetic resonance imaging(MRI), independent component analysis, Monte Carlo simulation,multiple fiber orientations.

I. INTRODUCTION

D IFFUSIONtensor-magnetic resonance imaging(DT-MRI)has been widely used to investigate fibrous tissues, such as

white matter, spinal cord, and cardiac muscle [1], [2]. Amongthe many application areas, in vivo white matter tractography isone of the most exciting areas that have become feasible onlythrough using DT-MRI. Numerous attempts have been made tomap anatomical connections between different functional areasof the brain [3]–[5]. While there is a high potential to noninva-sively measure and visualize white matter tracts of individualbrains, there are still a number of problems, such as noise ef-fect, partial volume effect, tract quantification, and computa-

Manuscript received November 15, 2003; revised June 2, 2004. This workwas supported in part by Grants NIA-NIH P50 AG05142 and NIH-MH RO153213.

S. Kim was with the Department of Biomedical Engineering, the Universityof Southern California, Los Angeles, CA 90089 USA. He is now with the Labo-ratory for Molecular Imaging, Department of Radiology, the University of Penn-sylvania, Philadelphia, PA 19104 USA (e-mail: [email protected]).

J.-W. Jeong is with the Department of Biomedical Engineering, the Univer-sity of Southern California, Los Angeles, CA 90089 USA (e-mail: [email protected]).

M. Singh is with the Departments of Radiology and Biomedical Engineering,the University of Southern California, Los Angeles, CA 90089 USA (e-mail:[email protected]).

Digital Object Identifier 10.1109/TNS.2004.843137

tional time, that should be resolved before the technology canbe readily used for clinical and research purposes.

One of the most challenging problems in white matter trac-tography using DT-MRI is how to determine the fiber orienta-tions when there are multiple fiber tracts with different orien-tations in one voxel [1]–[5]. The conventional ways to processdiffusion tensor data are based on principal component analysis(PCA) that estimates three orthogonal components. The direc-tion of the major principal component, i.e., the eigenvector ofthe largest eigenvalue, has been assumed to reflect the fiber ori-entation. Unfortunately, it is valid only under the assumption ofone tract in one voxel, which cannot be guaranteed even by thebest spatial resolution of commercially available MRI machinesto date.

Recently, it has been shown that the detailed three-dimen-sional (3-D) shape of the diffusion pattern in a voxel can bedetermined from a large number of angular measurements [6].However, this method requires considerably long scanning timeand strong gradients. It also requires a fitting process to a mul-tiple tensor model in order to determine the underlying fiberorientations. In this study, we developed a novel method usingindependent component analysis (ICA) [7]–[10] to estimate thenonorthogonal tracts in a voxel from conventional DT-MRI with6 gradient directions and multiple -values.

Arfanakis et al. [11] have applied ICA to raw images ofDT-MRI under the assumption that the raw images are linearcombinations of spatially independent component images. Thisapproach is similar to applying spatial ICA in source estimationof functional MRI [10]. It was demonstrated that ICA couldseparate noise, eddy current effects, some tracts, and diseasedareas as independent source images. The spatial ICA estimateswhich source each voxel belongs to, but does not estimatesthe individual fiber orientations within a voxel. Our methodis to apply ICA to the measurement data of an individualvoxel in order to estimate the underlying fiber orientations foreach voxel. This paper presents a specific formulation of ICAmethod to estimate fiber orientations from DT-MRI data. Themethod was first tested with Monte Carlo simulation data andthen applied to human data.

II. THEORY

A. Formulation of ICA for DT-MRI Data Analysis

ICA is a widely known method to separate independentsources without assuming orthogonality between sources [7].Assume that there is an -dimensional zero-mean vector

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KIM et al.: ESTIMATION OF MULTIPLE FIBER ORIENTATIONS FROM DIFFUSION TENSOR MRI 267

Fig. 1. ICA method to estimate the underlying individual tensors. (a)Ellipsoidal representation of two tensors corresponding to two crossing tractsin a voxel. (b) Pancake shaped tensor that may be estimated based on theDT-MRI data of the voxel in (a). (c) Two separate tensors to be estimated byICA.

, such that the componentsare mutually independent. A measurement data vector

is observed at each time point .It is assumed that there is a full rank scalar matrixsuch that

(1)

The goal of ICA is to find a linear transformation of thedependent sensor signal that makes the outputs as indepen-dent as possible. Then, the estimated sources , could be ob-tained by

(2)

where may include only permutation and scaling upon suc-cessful operation.

Let us assume a voxel with two fibers crossing as depictedin Fig. 1(a). It is assumed that each fiber is represented by alinear tensor and the DT-MRI signal from one fiber is not al-tered by the other one. It is also assumed that diffusion weightedMR scan measures the linearly combined signal from both fiberswith multiple -values and gradient directions. The conventionaldiffusion tensor estimation will yield a pancake shaped tensor asdepicted in Fig. 1(b). Our goal is to formulate a novel frameworkof ICA to estimate the two original linear tensors, as shown inFig. 1(c).

Conventional DT-MRI measures the diffusion weightedsignal multiple times with different combinations of diffusionweighting gradient direction and strength [1]. Using the avail-able multiple data per voxel, a characteristic pattern of thesignal representing information about the tensor can be defined.For instance, consider a vector

(3)

where is the number of nonzero -values, is the numberof diffusion weighting gradient directions, and is a dif-fusion weighted magnetic resonance signal measured with th-value and th gradient direction. This vector is unique to a

tensor. The pattern of the signal is closely related to the direc-tion and magnitude of a tensor. The number of available datais the number of -values multiplied by the number of gradientdirections used in the DT-MRI scanning.

The series of DT-MRI data for the th voxel is reparameter-ized as

(4)

and

(5)

where . When measurements areacquired, the measurement vector is defined as

. For example, let us assume a lineartensor that may represent a white matter fiber tract

(6)

where its unit is . DT-MRI signals can be calcu-lated for four -values (90, 363, 817, and 1453 ) andsix gradient directions

. When the measurement data are arrangedaccording to (3–5), it can be plotted as a function of ,as depicted in Fig. 2(a).

Once all the measurement data vectors are constructed as de-scribed, ICA can be applied to estimate a linear transformation

which estimates the original sources as follows:

(7)

The estimated sources can be rearranged as

(8)

A tensor can be calculated from each estimated source corre-sponding to a single tensor compartment.

B. Source Model and Learning Rule

Among several methods to estimate the linear transformationmatrix, we used the information-theoretic approach [9] to max-imize the separation with small number of samples. The mostimportant factor for successful ICA decomposition is to have anadequate probability density model of the sources. In the frame-work of the aforementioned ICA method, the source distributionis a function of MRI scanning parameters; -values and gradientdirections.

With the previous scanning parameters, for example, thecumulative distribution of data has a distinct shape shown inFig. 2(b). It shows the cumulative distributions of the simulatedmeasurement data from 100 tensors of the same size as (6),but randomly oriented in 3-D space. Repeated simulations withdifferent -values and gradient directions produced similarpatterns of the cumulative distribution curve with two regions:nonstiff and stiff. The shape of the cumulative distributionshown in Fig. 2(b) can be understood from the example plot inFig. 2(a). It has an upper envelop of the signal correspondingto smaller MR signal attenuation when the diffusion weighting


Fig. 2. Example of diffusion weighted measurement data normalized by signal without diffusion weighting, S(0). (a) Calculated measurement data of tensordefined by (6) is arranged according to (3–5) as a function of k. (b) Cumulative distributions of 100 randomly oriented tensors with same size as (6). (c) Comparisonbetween one cumulative distribution (thin line) and the least square fitted expo-exponential function (thick line).

gradient is perpendicular to the fiber orientation, which con-tributes to the stiff region of the cumulative distribution.Likewise, the lower envelop of the signal corresponds to largersignal attenuation when the diffusion weighting gradient isparallel to the fiber orientation. A similar shape of cumulativedistribution is expected if a highly linear tensor is measuredwith multiple -values.

The cumulative distribution curves shown in Fig. 2(b) areclose to neither a super-Gaussian function which has only onestiff slope in the middle nor a sub-Gaussian function which hastwo stiff slopes. Thus, it rules out the original [7] and extended[8] ICA methods as their source models are only super- andsub-Gaussian functions. A mixture of log-sigmoids should beable to represent this shape in principle [9]. However, the mix-ture would take a longer time to process and needs to estimatea larger number of parameters (i.e., the number of parametersto be estimated (the number of channels) (the number oflog-sigmoidals) 3).

We propose to use the following expo-exponential function asa model of the skewed sigmoidal shape observed in Fig. 2(b):

(9)

where and is a constant. The previous function isbounded between 0 and 1 so that it can be used as a cumula-tive distribution model. It is also used as a nonlinear function toestimate the cumulative density function of a source during ICAprocess. An example of this function is shown in Fig. 2(c) where

and , demonstrating that it can model theskewed shape of a sigmoidal function adequately.

Based on the probability density model of the source, analgorithm can be derived to minimize the mutual informationbetween the estimated sources. The mutual information is min-imized by maximizing the entropy of the nonlinear transformedvector of the estimated source vector. The density functionis obtained by taking the derivative of the cumulative densityfunction

(10)

where andFrom the natural gradient algorithm by Bell and Sejnowski

[7], the learning rule for the weighting matrix is given as

(11)


where is a learning rate, is

(12)

and

(13)

As shown by Xu et al. in [9], taking derivative of the negativeentropy of a monotonically increasing nonlinear transform, ,with respect to the unknown parameters yields learning rules forthe parameters as follows:

(14)

and

(15)

where and are the learning rates of and , respectively.This algorithm can be used to estimate a linear tensor source.

However, the proposed expo-exponential function is not ad-equate to model Gaussian noise in measurement data. Thus,this algorithm is to be used together with the algorithm for alog-sigmoidal function suggested by Xu et al. [9]. During theICA process, the estimation channel with the highest sphereanisotropy can be assigned to use a log-sigmoidal functioninstead of the expo-exponential function so that the Gaussiannoise can be separated from the actual signals effectively.

C. Processing and Source Selection Strategy

ICA requires the number of noncollinear measurement chan-nels to be larger than the number of sources to be estimated.The volume fractions of fibers in a voxel remain constant unlessthe subject moves. Thus, multiple measurements of the samevoxel are not adequate as inputs to ICA. It was assumed thatwhen a voxel has two fiber tracts, there are one or more neigh-boring voxels in the same imaging plane containing the samefiber tracts but with different volume fractions.

Let us consider a voxel where and are the row andcolumn positions of the voxel on the image plane, respectively.The rearranged measurement vector for the voxel according to(3–5) can be denoted as . Then, the following four mea-surement matrices were defined:

(16)

It can be noted that only four-point connected voxels are usedand they are grouped to make an L-shape in order to avoid anypossible bias in orientation estimation from using a set of voxels

in one direction. Each measurement matrix, , can be aninput to the ICA method to estimate two linear tensor sources atmost. Applying the ICA method to all four sets may yield up toeight sources. It is necessary to have a reasonable process to se-lect the sources for the voxel only. One plausible way is tofind a pair of sources out of at most 8 sources such that the com-bination of the two sources provides the best least squares fit tothe measurement data. Assuming there is no significantly fastexchange of water particles between two fiber compartments,the DT-MRI measurement can be modeled as a linear summa-tion of the signals from each tensor independently [12]

(17)

where represents the volume fraction of fiber 1 within the testvoxel, is -value matrix, and is the diffusion tensor of fiber

in the test voxel.

III. METHODS

A. Simulation Study I

Monte Carlo simulations were carried out to generateDT-MRI data with different fiber crossing configurations inorder to evaluate the performance of the ICA method to estimatethe underlying tensors. Simulation was conducted only for thecase where two fibers crossed each other. Each fiber was de-fined by a diffusion tensor with eigenvalues of 1.7, 0.3, and 0.3

[6]. The angle between the two fibers was increasedfrom 10 to 90 in steps of 10 . At each angle, maintaining theangle in between them, the two fibers were randomly orientedin 3-D space in order to avoid any bias related to the diffusionweighting gradient directions. The random orientation wasrepeated 200 times per fiber-crossing angle. Noise with Riciandistribution was added to simulate noisy MR data as describedin [13]. The noise level was adjusted to a signal-to-noise ratio(SNR) ranging from 20–50 for nondiffusion weighted data andthen applied to the entire data set. The MR imaging parameterswere same as those used for data acquisition described inSection III-C. For each case, three voxels were simulated withthe volume fraction of 0.25, 0.5, and 0.75 for one fiber, andthe remaining fraction for the other fiber. The proposed ICAmethod was applied to estimate the original tensor orientations.

B. Simulation Study II

Another simulation study was performed to test the pro-cessing and source selection strategy described in Section II-C.A tensor field was created as shown in Fig. 3(a) such thatthe voxels in the middle region contained both horizontal andvertical tracts. For the voxels with fiber crossing, the volumeratios of the two tracts were randomized between 0.4 and0.6. It was assumed that the DT-MRI measurement is a linearaddition of signals from two individual tensor compartmentsas expressed by (17). As shown in Fig. 3(b), these voxelswere estimated as pancake shaped tensors by the conventionaldiffusion tensor estimation. Except for voxels at the edge ofthe field, the aforementioned processing and source selectionstrategy was applied to each voxel. The test was repeated withvarious levels of Rician noise.


Fig. 3. Simulated tensor field to test the ICA method. (a) Original tensorfield where the voxels containing two crossing fibers are depicted with twoseparate ellipsoids. (b) Ellipsoidal representation of tensors estimated fromdata simulated with the tensor field in (a). Note that the voxels with two tensorsare represented by pancake shaped ellipsoids.

C. DT-MRI Data Acquisition

Multislice diffusion weighted MR data were acquired fromone healthy subject using a 1.5 T Signa Echo-speed systemversion 5.7 (GE, Milwaukee, WI). A cardiac-gated pulsed-gradient spin-echo sequence with echo planar read-out wasused (TR of 4 R-R intervals). The field of view was 240 mm

240 mm (128 128 pixels). The slice thickness was 5 mmwith a gap of 1.5 mm between slices. The duration of thediffusion gradients was 38.2 m, and TE 72 msec. The -valueswere 90, 363, 817, and 1453 . Diffusion gradientswere applied along six directions

. The measurement was repeatedfour times with 16 slices. The total duration of diffusionweighted imaging was about 14 min.

D. Application of the ICA Method to Human Data

The multiple raw images were coregistered and averagedto minimize artifacts. Diffusion tensors were estimated byweighted multivariate linear regression [14]. From the eigen-values, linear, planar, and fractional anisotropy were calculated[1], [15]. The ICA method was applied to regions of interest(ROI) whose fractional anisotropy was relatively high, butlinear anisotropy was lower than planar anisotropy. Assumingthat there can be only up to two major tracts in a voxel, theprocessing and source selection strategy described earlier inSection II-C were used.

IV. RESULTS

A. Simulation Study I

The performance of ICA for estimating two fibers crossingat different angles was evaluated from the Monte Carlo simula-tion data. Results are summarized in Fig. 4, demonstrating thatICA can be used to estimate the orientations of two fibers fromconventional DT-MRI with six diffusion weighting gradient di-rections and multiple -values. The estimation error increasesas the SNR decreases. When SNR is higher than 30, the meanerror is smaller than 10 for most angular separations.

The previous results are comparable to those obtained by fit-ting a two-fiber model to simulated high-angular resolution dif-fusion (HARD) imaging data [6] where an SNR of about 45was required to achieve the mean angular error below 10 forall angular separations. HARD imaging requires measurementwith a large number of gradient directions, for instance 126. Atwo-fiber model is fitted to the measured data in least square

Fig. 4. Result of fiber direction estimation using the ICA method for thesimulation data with six noncollinear gradient directions. The estimation error(mean � standard deviation) is the angular difference between the simulatedfiber direction and the estimated eigenvector. The study was repeated forvarious SNRs; 20 (dotted/triangle-up), 30 (dash-dotted/triangle-down), 40(dashed/square), and 50 (solid/circle). The simulation data were generated for200 random orientations per interfiber angle with each SNR.

Fig. 5. Comparison between conventional tensor estimation and the ICAmethod. (a) and (c) Results of conventional single tensor estimation. (b) and(d) Results of the ICA method. SNRs were 30 for (a) and (b), and 20 for (c)and (d). The ellipsoid sizes are scaled by their fractional anisotropies.

sense. This type of fitting assumes certain characteristics of un-derlying individual fibers to reduce the number of parameters tobe estimated, and the estimation can be degraded significantlywhen the data are noisy.

On the other hand, the proposed ICA method does not makeany assumption about the size or anisotropy of individual fibers.Minimizing mutual information between the sources appears toestimate the source tensors with minimum planar anisotropy andmaximum linear anisotropy. It was observed that the estimatedsources usually had higher linear anisotropy than the measuredtensors.

B. Simulation Study II

The results of the second simulation study are shown in Fig. 5.As shown by Fig. 4, there was noticeable performance degrada-


Fig. 6. Result of the ICA method applied to human data. Ellipsoidal representations in (b) and (c) correspond to the ROI shown by white box in (a), and thosein (e) and (f) to the ROI in (d). In (a) and (d), the ROIs are shown on images without diffusion weighting (upper left), fractional anisotropy (upper right), linearanisotropy (lower left), and planar anisotropy (lower right) images. In (b) and (e), a voxel is displayed by an ellipsoid corresponding to the conventionally estimatedtensor scaled by its fractional anisotropy. In (c) and (f), the results of the ICA method are displayed by combinations of two ellipsoids for voxels. Images in (b) and(c) are tilted because some of ellipsoids are difficult to observe without tilting. All ellipsoids are scaled by their fractional anisotropies.

tion when the SNR was decreased from 30 to 20. A similar degra-dation was also observed for the second simulation study. There-fore, only the results of two SNRs, 30 and 20, are shown in Fig. 5.The ellipsoidal representations of the noisy tensor fields alongwith the results of applying the previously described processingand source selection strategy of the ICA method are presented inFig. 5. It is clearly demonstrated that all the planar anisotropicvoxels in the middle of the fields are successfully separated intotwo linear tensors. The result in Fig. 5(b) is quite comparable tothe original tensor field shown in Fig. 3(a). The estimated sourcesin Fig. 5(b) seem to have less linear anisotropies than the truevalues. However, it can be observed that their orientations arewell-matched to those of the true tensors.

When SNR is 20, the result was degraded noticeably as shownin Fig. 5(d). The estimation of two tensors was not always suc-cessful at this level of noise. This is largely due to the source se-lection step following the ICA procedure because the measure-ment data with too much noise can mislead the source selectionprocess despite the capability of the ICA method to separatesignals from noise. This simulation study demonstrates that theICA method should be used for data with SNR of 30 or higherfor the best result.

C. Human Data

The ICA method was also applied to human data. Two exam-ples are shown in Fig. 6. Two ROIs in Fig. 6(a) and (d) are lo-cated in white matter regions with high fractional anisotropies.However, both ROIs have low linear anisotropies and higherplanar anisotropies. The first ROI shown in Fig. 6(a) is a partof the area with a number of fibers running in different direc-tions. It includes the superior longitudinal fasciculus in ante-rior-to-posterior direction, the internal capsule in inferior-to-su-perior direction, and the coronal radiations. Presence of thesetracts in one area results in many voxels containing more thanone tract so that their estimated tensors are highly planar asshown by the lower right image in Fig. 6(a). Since these fibertracts are oriented not only in one plane, but also through theimage plane, it is difficult to show their full 3-D shapes prop-erly. Fig. 6(b) shows the conventionally estimated tensors usingellipsoids. Application of the ICA method yielded the resultshown in Fig. 6(c). Although it is difficult to name specific tractsfor all the estimated tensor sources, it is possible to observethat their orientations are not totally random, but rather struc-tured in several groups. One of them is a group of vertically


oriented ellipsoids that can be attributed to the internal capsule.The groups oriented in anterior-to-posterior and left-to-right di-rections can be attributed to the superior longitudinal fasciculusand the corona radiata, respectively.

The second ROI shown in Fig. 6(d) is another interestingarea as an intersection of fiber tracts. This area includes theanterior thalamic radiation from posterior side to anterior, theinferior fronto-occipital fasciculus propagating diagonally fromthe lower right side, and some lower part of the corpus callosaltract from left side of the ROI. Since these tracts run almostparallel to the image plane, a top view of the ellipsoidal rep-resentation in Fig. 6(e) can easily demonstrate the degree ofplanar anisotropy due to fiber crossing. The result of using theICA method for the second ROI is shown in Fig. 6(f). Similarto Fig. 6(c), it is possible to observe that the estimated sourcetensors are well organized in terms of their orientations. Theseorientations also correspond to the directions of known fibertracts in this region.

Unfortunately, it is not practically possible at this time tocross validate these results of in vivo human data analysis. How-ever, the results appear to match well with the anatomicallyknown white matter tracts.

V. DISCUSSION

A novel method based on ICA was developed to estimatemultiple fiber orientations in one voxel using conventionaldiffusion tensor imaging with 6 gradient directions and fournonzero -values. The method was tested successfully withMonte Carlo simulation data and human data. The resultsof human data appear to be consistent with known tracts inthe human brain. Quantitative evaluation of the human dataanalysis remains problematic because there is no noninvasivemethod currently available to cross-validate the result. Moreextensive qualitative tests could be performed together withwhite matter tractography.

Estimation of multiple fiber orientations in a single voxel isa critical task in brain white matter tractography as previouslypointed out by many groups [1]–[6]. For most fiber trackingmethods, the primary eigenvector direction of the diffusiontensor is used to reconstruct tracts. The results of those methodsbecome less reliable where the primary eigenvector does notmatch the fiber orientation due to the presence of multiplefibers within a voxel. Fiber tracking could be significantlyimproved by incorporating the ICA method. To minimize theprocessing time, the ICA method can be applied only to voxelswith high planar anisotropy. Criteria to select a tensor forcontinuous tracking of a tract through a voxel from multipletensors estimated by the ICA method will require furtherwork. One simple, but intuitive solution could be to use theestimated source that makes the minimum angular change inthe trajectory.

This study was limited to the case of two-fiber crossing forsimplicity. However, the developed method is not limited to twofibers. Additional fibers within a voxel can be detected as longas the number of sources to be estimated by the ICA method isless than the number of input channels. In our study, the numberof input channels was three as measurement data from three

Fig. 7. Result of fiber direction estimation using the ICA method for thesimulation data with the six gradient vectors optimized for estimating thediffusion tensor [16]. The estimation error (mean � standard deviation) is theangular difference between the simulated fiber direction and the estimatedeigenvector. The study was repeated for various SNRs; 20 (dotted/triangle-up),30 (dash-dotted/triangle-down), 40 (dashed/square), and 50 (solid/circle). Thesimulation data were generated for 300 random orientations per interfiber anglewith each SNR.

neighboring voxels were used. There are at least two consid-erations for increasing the number of sources. First, increasingthe number of sources requires an increased number of voxelsto form the input data because the number of input channelsshould be larger than the number of sources for ICA to work.The proposed processing strategy is based on the assumptionthat neighboring voxels contain the same fiber tracts withoutsignificant changes in their orientation and size. This assump-tion may not hold when the total area covered by the selectedvoxels becomes larger. Thus, the processing strategy should bemodified in conjunction with image resolution and any avail-able information about the underlying fiber tracts. Second, themeasurement data length should be long enough compared tothe number of sources to be estimated. The length of measure-ment data is determined by the number of gradient vectors andnonzero -values used for the measurement. In general, it isdesirable to have more measurement data by using additional-values and gradient directions even without any increase in

the number of input channels in order to estimate the sourcedistribution more accurately. If the number of input channelsis increased, the available data length should also be increasedproportionately.

The performance of the ICA method may be affected by notonly how many but also the specific gradient directions and-values used for data acquisition. We used a commonly used

set of six noncollinear gradient directions only. There are otherways of defining the gradient directions, such as the optimizedgradient vectors by Jones et al. [16] or using icosahedral hemi-sphere [6]. A specific gradient scheme will determine the signalpattern shown in Fig. 2(a) as well as the cumulative distribu-tion shown in Fig. 2(b). It may also affect the whole estima-tion process by altering local minima and maxima. For instance,Fig. 7 shows the result of applying the ICA method to a simu-lated data set with the six gradient vectors optimized for esti-mating the diffusion tensor by Jones et al. [16]. Comparing thisresult with one in Fig. 4 with orthogonal gradient vectors, the


performance of the ICA method is noticeably improved whenthe inter fiber angle is 40 or larger. In a similar way, the dis-tribution of -values could affect the estimation performance.An optimal strategy of selecting gradient vectors and -valuesto maximize the performance of the ICA method is yet to beinvestigated.

ACKNOWLEDGMENT

The authors would like to thank Dr. S. Sinha, Radiology,UCLA, for providing the DT-MRI data used in this study aswell as for helpful discussions.

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Estimation of multiple fiber orientations from diffusion tensor MRI

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