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Journal of Neuroscience Methods 184 (2009) 375–379

Contents lists available at ScienceDirect

Journal of Neuroscience Methods

journa l homepage: www.e lsev ier .com/ locate / jneumeth

hort communication

enetic white matter fiber tractography with global optimization

i Wua,b,c, Qing Xuc,d, Lei Xuc,e, Jiliu Zhoua, Adam W. Andersonc,f, Zhaohua Dingc,d,f,g,∗

College of Electronics and Information Engineering, Sichuan University, PR ChinaDepartment of Electronic Engineering, Chengdu University of Information Technology, PR ChinaVanderbilt University Institute of Imaging Science, Vanderbilt University, USADepartment of Electrical Engineering and Computer Science, Vanderbilt University, USADepartment of Biostatistics, Vanderbilt University, USADepartment of Biomedical Engineering, Vanderbilt University, USAChemical and Physical Biology Program, Vanderbilt University, USA

r t i c l e i n f o

rticle history:eceived 27 May 2009eceived in revised form 30 July 2009ccepted 30 July 2009

a b s t r a c t

Diffusion tensor imaging (DTI) tractography is a novel technique that can delineate the trajectoriesbetween cortical region of the human brain non-invasively. In this paper, a novel DTI based white mat-ter fiber tractography using genetic algorithm is presented. Adapting the concepts from evolutionary

eywords:iffusion tensor imagingiber trackingenetic algorithmlobal optimization

biology which include selection, recombination and mutation, globally optimized fiber pathways aregenerated iteratively. Global optimality of the fiber tracts is evaluated using Bayes decision rule, whichsimultaneously considers both the fiber geometric smoothness and consistency with the tensor field.This global optimality assigns the tracking fibers great immunity to random image noise and other localimage artifacts, thus avoiding the detrimental effects of cumulative noise on fiber tracking. Experimentswith synthetic and in vivo human DTI data have demonstrated the feasibility and robustness of this new

and a

fiber tracking technique,

. Introduction

Diffusion tensor imaging (DTI) has become a primary tool foron-invasive exploration of the structure of living tissue in vivoBasser et al., 1994). Since its first introduction a decade ago, thisew imaging modality has been widely used to reconstruct neu-onal fiber pathways in the human brain (Mori and van Zijl, 2002).o date a variety of tracking algorithms have been proposed to inferber connections in the human brain (Lu et al., 2006; Friman et al.,006; Zhang et al., 2009), the basic principle of which is sequentially

ntegrating local fiber directions from pre-defined seed point(s)o generate fiber connection pathways. Typically, these trackinglgorithms “grow” fiber pathways by piecing a line segment tohe end of the preceding segment. These methods can be broadlyivided into two categories: deterministic fiber tractography (Lu etl., 2006) and probabilistic fiber tractography (Friman et al., 2006).

A common drawback of the above streamline-like trackingethods, either deterministic or probabilistic, is cumulative errors

rising from random image noise and/or partial volume averagingPVA) along the tracking path (Alexander et al., 2001; Anderson,

∗ Corresponding author at: Vanderbilt University Institute of Imaging Science,161 21st Avenue South, MCN AA-1105, Nashville, TN 37232-2310, USA.el.: +1 615 322 7889; fax: +1 615 322 0734.

E-mail address: zhaohua.ding@vanderbilt.edu (Z. Ding).

165-0270/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.jneumeth.2009.07.032

n improved performance over commonly used probabilistic fiber tracking.© 2009 Elsevier B.V. All rights reserved.

2001), even with certain regularizations on the basis of geometricor other constraints. Furthermore, as the direction information thatstream-like tracking methods rely on is only derived locally, thetracking results from these methods are not a globally optimizedsolution.

In this work, a novel fiber tracking technique based on wellestablished genetic algorithms was proposed. The proposed tech-nique allows globally optimized solutions to the fiber pathwaysto be obtained, and hence possesses superb immunity to localimaging artifacts. Additionally, the proposed technique providesoptimal solutions for fiber connection pathways between two des-ignated ROIs; this offers a great potential of applying it to studiesof structure–function relations in the human brain, in which thestructural connectivity between two functionally related regions isoften sought.

2. Method

A common practice of fiber tracking is to track fiber pathwaysthat connect to certain ROIs, among which a most useful applicationis to find connecting pathways between an ROI pair. Fiber tracking

between a pair of ROIs can be cast as a path finding problem towhich an optimal solution is found by genetic algorithm (GA). Inthis context, the goal is to evolve from initial solutions to produce aset of fibers with best fitness to the given data, which is presumablyan optimal solution to the fiber path. Details of the implementation

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76 X. Wu et al. / Journal of Neuros

rocedure for one iteration of the genetic tracking algorithm, calledeneTrack henceforth, are given below.

.1. Initialization

In a typical GA, solutions are encoded as strings of binary bits,hich are analogous to chromosomes in biology. Binary strings

an be extended to continuous strings, which are chosen as initialolutions in our design. More specifically, we represent an arbi-rary fiber pathway between a pair of ROIs with a space curve (f),nd express the curve f as Fourier series (of continuous values) inartesian system, i.e.

d(t) =N∑

n=1

[ad

n cos(nt) + bdn sin(nt)

], (1)

here superscript d denotes the x, y, or z direction in the Cartesianystem, n is the order of Fourier series, t is the number of points inurve f, the an and bn are coefficients of the cosine and sine compo-ents respectively, and N is the maximum order of Fourier series

or approximation with reasonable accuracy (t = 50 and N = 10 inhis work). According to the equation above, every fiber curve cane represented by 2N Fourier coefficients in each direction, whichonsists of N coefficients for both an and bn. The 6N coefficients inll three directions, which correspond to the genes of the chromo-omes, encode the solutions for the fiber pathways between an ROIair and will be used for recombination and mutation to produceffspring in later steps.

.2. Selection

In designing objective functions for selecting best fit solutionsmong the candidates, two primary considerations are taken intoccount. First, the solutions should possess best geometric smooth-ess along the entire fiber path. Second, the solutions should behose best fit the diffusion tensor field among all candidates. Simul-aneously considerations of these (with certain trade-off) may yieldolutions that are globally optimal in the sense that the curves areoth smooth and reasonably fit the data.

Globally optimal solutions may be obtained by using the clas-ical Bayesian theory (Duda and Hart, 1973). Let T denote theiffusion tensor field and C denote spatial curves that cover all pos-ible fiber pathways connecting the designated ROI pair. Accordingo the Bayes decision rule, the optimal solution Copt is the one with

aximum a posteriori (MAP) probability p(C|T):

(C|T) = p(C)p(T|C)p(T)

. (2)

Since p(T) is independent of C, maximizing p(C|T) reduces toaximizing the product of p(C) and p(T|C). The term p(C) is a prior

robability of curve C, which is the probability of the existence ofurve C without any measurement data; the term p(T|C) is a condi-ional probability, which defines the probability of the existence ofensor field T given curve C.

To find the MAP solution, p(C) and p(T|C) need to be modeled. Inur design, these probabilities are modeled so that constraints onurve smoothness and consistency with tensor field are imposed.irst, let us assume the tensor field to be a Markovian random fieldMRF), and vt be a unit vector representing the local tangentialirection at the tth point of the curve. According to the MRF theory,

t is a random realization of the vector field in the neighborhood of, which observes a Gibbs distribution (Geman and Geman, 1984):

(C) = 1Z1

∏

t

e−p(vt ) = 1Z1

e−∑

t

p(vt ) (3)

Methods 184 (2009) 375–379

where Z1 is a normalization constant. When only the precedingpoint along the curve is considered, a simple definition of P(vt) is

p(vt) = arccos(vt · vt−1) (4)

This definition of the prior probability gives preference to thecurves with low curvature, thus imposing a smoothness constraintto the fiber.

Second, assuming that the tensor measurement in voxel tdepends only on the local fascicle direction vt , the conditional prob-ability p(T|C) can be rewritten as a second Gibbs distribution below:

p(T|C) = 1Z2

∏

t

e−p(Tt |vt ) = 1Z2

e−∑

t

p(Tt |vt ), (5)

where Z2 is a normalizing constant.In order for the probability p(Tt|vt) to decrease with the discrep-

ancy between the local fiber direction vt and the major eigenvectorof the local diffusion tensor et, we propose the following conditionalprobability model:

p(Tt |vt) = arccos(vt · et) (6)

Combining all the previous models (Eqs. (3) and (5)) leads to anexpression for p(C|T) which turns out to be a new Gibbs distribu-tion:

p(C|T) = 1Z3

e−∑

t

p(vt )+˛p(Tt |vt ) (7)

where Z3 = Z1 × Z2.With this new Gibbs distribution, an optimal solution can be

reached by minimizing the following cost function:

fcos t =∑

t

p(vt) + ˛∑

t

p(Tt |vt). (8)

The first term in the above cost function imposes a smoothnessconstraint on the fiber pathway, and the second term encourages aconsistency between the fiber and tensor dominant directions. Therelative weights of these two terms are determined by the parame-ter ˛, which regulates the trade-off between the smoothness of thefiber and consistency with the data. In this work, the range of ˛ ischosen to be between 0 and 10.

2.3. Recombination

The subset of optimal fibers selected above is recombined toform a new set of fibers. This is implemented by randomly select-ing a value for each of the 6N coefficients in Eq. (1) from the parentfibers, so that the coefficients for each new fiber come from differ-ent parents. This new set of fibers, along with a small number ofbest fit parents, constitute the offspring for the next generation.

2.4. Mutation

The mutation process perturbs the coefficients of the abovefibers so that each solution contains a new set of values for thecoefficients. To maintain the stability of this algorithm, the amountof perturbation for each coefficient is at the order of one standarddeviation of the coefficients in the existing fibers. This provides sta-bility to the solution, and in the meantime offers an opportunity togenerate new solutions that better fit the cost function.

3. Tracking experiments and results

To evaluate comprehensively the performance of the GeneTrackalgorithm proposed, we have carried out a series of fiber trackingexperiments on synthetic and in vivo human DTI datasets. Fiber

X. Wu et al. / Journal of Neuroscience Methods 184 (2009) 375–379 377

Fig. 1. Fibers tracking with synthetic tensor data. (a) A 3D view of one slice of the synthetic dataset. (b) Enlarged 2D view of the boxed region in (a). (c) Results from theGeneTrack algorithm. (d) Results from the Friman method. (e) Density map of connection probability from GeneTrack algorithm. (f) Density map of connection probabilityf ber ot FiberR ed to

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rom Friman method. (g) Variations of the mean value of cost function with the numhe GeneTrack and Friman method. (i) Fiber tracts resulting from the GeneTrack. (j)OI at the left; cyan denotes fibers tracked from the ROI at the right; fibers connect

racking on synthetic datasets was performed using both the Gen-Track and a probabilistic fiber tracking method by Friman et al.2006), and comparisons between them were made with respecto tracking accuracy and validation. The reproducibility of Gene-rack was demonstrated from the tracking result in five differentn vivo datasets.

.1. Fiber tracking with synthetic dataset

Evaluation of the performance of the algorithm began with aynthetic tensor field, which was designed to contain five curvesf different geometric complexity. To mimic closely physiologicalonditions in vivo, synthetic tensors in the curves were constructedo have a trace of 2.1 × 10−5 cm2/s and a fractional anisotropy (FA)f 0.8. These diffusion parameters were similar to those in nor-al brain parenchyma. Diffusion weighted imaging was simulated

long 32 non-collinear directions with a b value of 1000 s/mm2,

nd the diffusion weighted data were corrupted with zero meanaussian noise at a standard deviation of 0.05.

The synthetic dataset had one curve with two extrema and fourarabolic curves with increasing curvature, as shown in Fig. 1a. Toetter visualize the details of the synthetic fibers, the boxed region

f iterations. (h) A synthetic dataset for comparing the tracking efficiency betweentracts resulting from the Friman method. Magenta denotes fibers tracked from theboth ROIs are denoted in red or blue.

in Fig. 1a is shown in an enlarged view in Fig. 1b. Each voxel is visu-alized by an ellipsoid whose principal axes are the three orthogonaleigenvectors of the tensor and the radii of the ellipsoid along theaxes are proportional to their corresponding eigenvalues.

The proposed GeneTrack algorithm was first used to track fibersbetween two seed points in each curve (denoted as a diamond anda square in Fig. 1c). For each synthetic curve, a total of 1000 ran-dom curves were generated initially by assigning random valuesto the coefficients an and bn in Eq. (1), followed by 60 iterations ofselection, recombination and mutation. In each iteration, 100 par-ents were sampled, recombined and then mutated to generate 900children as offspring. These children along with 100 parents withbest cost function values were fed to the next generation as par-ents. Meanwhile, a probabilistic fiber tracking method by Frimanet al. (2006) was also implemented on the same dataset, with thesame seed point near one end of the curve (denoted as a diamond inFig. 1d). Parameter settings of this method were the same as those

described in Friman et al. (2006). A total of 1000 repeated trialswere performed with a step size of one voxel.

In addition, the number of fibers going through each voxel wasrecorded. These numbers were normalized against the total num-ber of trials to yield a density map that indicates the probability of

378 X. Wu et al. / Journal of Neuroscience Methods 184 (2009) 375–379

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ig. 2. GeneTrack resulting fibers between left/right thalamus (blue dots) and left/rind the pre-motor region (blue dots) in five different in vivo datasets.

ber connection along the fiber path between the two seed pointsfor the GeneTrack) or to the starting seed point (for the Friman

ethod).Fig. 1c and d shows the tracking results obtained using our

ethod and the Friman method, respectively. For each figure, 100fibers” (out of 1000) were randomly selected for demonstration.t can be seen that, at zero mean Gaussian noise with a standardeviation of 0.05, fibers reconstructed by both methods grossly fol-

ow the synthetic curves. However, close inspections reveal thathe Friman method tends to generate fibers that diverge distallylong the path, which results in some fibers with geometric struc-ure that is inconsistent with that of the “ground truth”; also, somebers show premature termination, and as such, there is a largeifference in the length of the fibers. These problems render sub-equent quantitative analysis rather difficult. On the other hand,bers from our method exhibit good convergence along the fiberath and no premature termination at all. This is attributable tohe fact that our method finds optimal fiber connections betweenwo designated regions, rather than tracking fibers from one seedegion as with the Friman method.

Density maps of the middle slice that depict the connectionrobability are shown in Fig. 1e and f for our method and the Frimanethod, respectively. As can be observed, the probability of con-

ection is relatively uniform along the entire curves for our method,lbeit there is a slight drop-off around the middle portion of someurves. However, the connection probability for the Friman methodxhibits gradual decreases along the curves, which is caused byeviations of some fibers from the curves along the path. This sit-ation is most pronounced for the 1st bundle whose density mapuickly drops to 0.3 or less.

Mean values of the cost functions of the tracked fibers andheir variations with the number of iterations are shown in Fig. 1g.or each curve, the mean value of the cost functions of 1000bers is calculated for each of the 60 iterations performed. It ishown that the mean values of the cost functions are high for allurves at the beginning, but decrease rapidly during the first a fewterations. They tend to stabilize to low values around 20 itera-ions, beyond which further declines are quite small. This indicateshat the algorithm starts to converge around the 20th iteration,t which the tracked fibers are close to the final optimal path-ays. This also demonstrates, although indirectly, the validity of

he GeneTrack since if there exist genuine fibers between twoOIs, the cost function will decrease and converge to a very lowalue.

To demonstrate the advantage of the GeneTrack algorithm overhe Friman method in terms of tracking efficiency, a synthetichantom was constructed that contained two thick parallel “fiberundles” bridged by a thin “fiber bundle” (see Fig. 1h). Fibers were

racked using the GeneTrack algorithm and the Friman methodith the same pair of ROIs (denoted by a diamond and square in

ig. 1i and j). For the GeneTrack, the same parameters as before weresed; for the Friman method, 5000 trials with the same parameterettings as in Friman et al. (2006) were launched from each of the

17 (green dots) in the occipital lobe, and between the left Broca’s area (green dots)

ROIs. The results of the two tracking methods are shown in Fig. 1iand j respectively. It can be appreciated from Fig. 1i that the fibersfrom the GeneTrack algorithm converge nicely to the intended fiberpathways. On the other hand, it took the Friman method a consid-erably larger number of trials to reach the results desired (Fig. 1j).This discrepancy in tracking efficiency comes from the fact thatthe GeneTrack algorithm takes the ROI pair as prior knowledge,whereas the Friman method tracks fibers without any targetedregions, and hence only resorts to post hoc filtering with a secondROI to reach intended results. Evidently, the lack of considerationof designated targets in the tracking process makes the Frimanmethod much more computationally expensive, and also puts it atthe risk of missing small fiber structures that have low connectionprobability.

3.2. Fiber tracking with in vivo human DTI data

To assess the performance of the GeneTrack algorithm on invivo data, diffusion weighted images (DWI) were acquired fromfive healthy human volunteers with a 3T Philips Intera AchievaMR scanner (Best, The Netherlands) and an eight-element SENSEcoil. A single shot, echo-planar pulsed gradient spin-echo imag-ing sequence was used, and diffusion weighting was performedalong 32 non-collinear directions with a b value of 1000 s/mm2.A total of 64 contiguous, 2-mm-thick slices with a matrix size of128 × 128 were acquired from a field of view of 256 mm × 256 mm,yielding an in-plane pixel size of 2 mm × 2 mm. Three repeatedscans were obtained from each subject, which were motion anddistortion corrected and then averaged using Philips diffusion reg-istration PRIDE tool (Release 0.4). Diffusion tensors were estimatedfrom the averaged DWI data using a linear least-square fittingprocedure (Basser et al., 1994), from which fractional anisotropy(FA) maps were computed. T1-weighted images were acquired aswell during the same session for the purpose of image registra-tion. Prior to the study, the subject gave informed consent forthe study protocol that was approved by the local ethics commit-tee.

To define the regions of interest for the in vivo experiments,two different approaches were used. First, the left and rightthalami were defined using the WFU PickAtlas tool in SPM2(http://www.fil.ion.ucl.ac.uk/spm/software/spm2/). To do so, theleft or right thalamus atlas was picked in the MNI space, whichwas normalized to the T1-weighted image in the native space. TheT1 image along with the thalamus atlas were then co-registeredwith the FA map, so that the ROI of the thalamus was brought tothe native space of the DWI. In addition to the thalami, the left andright Brodman’s areas (BA) 17 were also defined using the WFUPickAtlas tool. Second, the Broca’s and the pre-motor regions were

defined using functional MRI signals obtained from the subjectduring the performance of designated language tasks, as reportedearlier (Newton et al., 2007). Fiber tracking was performed withthe three sets of ROIs defined above, using the GeneTrack with thesame parameter settings as in the synthetic experiments.

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Reproducibility of the GeneTrack algorithm is demonstratedn Fig. 2, which shows 100 fibers tracked between left/right tha-amus and left/right BA 17 in the occipital lobe, and betweenhe left Broca’s area and the pre-motor region in five differentn vivo datasets. The data acquisition, fiber tracking and visu-lization parameters are the same among these datasets. It isvident that, although there are small differences among the fiveubjects, which is likely due to inter-subject anatomical varia-ions, the pathways reconstructed are largely consistent acrosshese subjects, all exhibiting good agreement with known neuro-natomy.

. Discussion and conclusion

In this paper, we proposed a new genetic tracking technique,eneTrack, for DTI based white matter fiber tractography. In con-

rast to most other fiber tracking methods, the proposed algorithmearches for globally optimal fiber pathways between a givenair of ROIs. Using the concept from evolutionary biology that

ncludes selection, recombination and mutation, the GeneTracklgorithm iteratively finds fiber pathways that have optimal trade-

ff between fiber geometric smoothness and consistency with theensor field. Experiments with synthetic and in vivo human DTI dataave demonstrated the feasibility and robustness of this new fiberracking technique, and an improved performance over commonlysed probabilistic fiber tracking.

Methods 184 (2009) 375–379 379

Acknowledgment

This study was supported by science innovation grant of SichuanNO29 (Mingyuan Xie) and USA NIH grant R01NS058639 (Adam W.Anderson).

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