Fuzzy Fibers:Uncertainty in dMRI Tractography

Thomas Schultz∗, Anna Vilanova†, Ralph Brecheisen,† and Gordon Kindlmann‡

July 15, 2013

Fiber tracking based on diffusion weighted Magnetic Resonance Imaging (dMRI)allows for noninvasive reconstruction of fiber bundles in the human brain. In thischapter, we discuss sources of error and uncertainty in this technique, and reviewstrategies that afford a more reliable interpretation of the results. This includesmethods for computing and rendering probabilistic tractograms, which estimateprecision in the face of measurement noise and artifacts. However, we also addressaspects that have received less attention so far, such as model selection, partial vo-luming, and the impact of parameters, both in preprocessing and in fiber trackingitself. We conclude by giving impulses for future research.

1 Introduction

Diffusion weighted MRI (dMRI) is a modern variant of Magnetic Resonance Imaging that allowsfor noninvasive, spatially resolved measurement of apparent self-diffusion coefficients. Sincefibrous tissues such as nerve fiber bundles in the human brain constrain water molecules such thatthey diffuse more freely along fibers than orthogonal to them, the apparent diffusivity dependson the direction of measurement, and allows us to infer the main fiber direction.

Based on such data, tractography algorithms reconstruct the trajectories of major nerve fiberbundles. The most classic variant is streamline tractography, in which tracking starts at someseed point and proceeds in small steps along the inferred direction. In its simplest form, thisresults in one space curve per seed point. It has been observed that many of the resulting stream-lines agree with known anatomy [11]. Tractography is also supported by validation studies thathave used software simulations, physical and biological phantoms [25].

Tractography is currently the only technique for noninvasive reconstruction of fiber bundlesin the human brain. This has created much interest among neuroscientists, who are looking forevidence of how connectivity between brain regions varies between different groups of subjects

∗ University of Bonn, Germany. E-Mail: schultz@cs.uni-bonn.de† Technical University Eindhoven, Netherlands. E-Mail: {a.vilanova,r.brecheisen}@tue.nl‡ University of Chicago, USA. E-Mail: glk@uchicago.edu

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[58], as well as neurosurgeons, who would like to know the exact spatial extent of specific fiberbundles in individual patients.

However, drawing reliable inference from dMRI is challenging. Even though a randomizedcontrolled study has shown that using dMRI in cerebral glioma surgery reduces postoperativemotor deficits and increases survival times [66], neurosurgeons have observed that some methodsfor tractography underestimate the true size of bundles [36] and they are still unsatisfied withthe degree of reproducibility that is achieved with current software packages [9].

In order to establish tractography as a reliable and widely accepted technique, it is essentialto gain a full understanding of its inherent sources of error and uncertainty. It is the goal of thischapter to give an introduction to these problems, to present existing approaches that have triedto mitigate or model them, and to outline some areas where more work is still needed.

2 Noise and Artifacts

2.1 Strategies for Probabilistic Tractography

It is the goal of probabilistic tractography to estimate the variability in fiber bundle reconstruc-tions that is due to measurement noise. This is often referred to as precision of the reconstructedbundle trajectory [32]. Due to additional types of error in data acquisition and modeling, whichwill be covered later in this chapter, it is not the same as accuracy (i.e., likelihood of a trueanatomical connection) [34]. Current approaches do not account for factors such as reposition-ing of the head or variations in scanner hardware over time, which further affect repeatability inpractice.

Rather than only inferring the most likely fiber direction, probabilistic approaches derive aprobability distribution of fiber directions from the data. The first generation of probabilistictractography methods has done so by fitting the diffusion tensor model to the data, and using theresult to parameterize a probability distribution in a heuristic manner. This often assumes thatthe fiber distribution is related to a sharpened version of the diffusivity profile [37], sometimesregularized by a deliberate bias towards the direction of the previous tracking step [4, 19]. Pro-grammable graphics hardware accelerates the sampling of such models, and enables immediatevisualization of the results [39]. Parker et al. [44] present two different fiber distribution modelsthat are parameterized by measures of diffusion anisotropy. Subsequent work allows for mul-timodal distributions that capture fiber crossings, and uses the observed variation of principaleigenvectors in synthetic data to calibrate model parameters [43].

In contrast to these techniques, which use the model parameters from a single fit, a secondgeneration of probabilistic tractography methods estimates the posterior distribution of fibermodel parameters, based on the full information from the measurements, which includes fittingresiduals. Behrens et al. [3] do so in an objective Bayesian framework, which aims at making asfew assumptions as possible, by choosing noninformative priors. They have later extended the“ball-and-stick” model that underlies their framework to allow for multiple fiber compartments[2].

Bootstrapping estimates the distribution of anisotropy measures [42] or fiber directions [30,53] by repeated model fitting, after resampling data from a limited number of repeated scans.This has been used as the foundation of another line of probabilistic tractography approaches

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[34, 38]. As an alternative to estimating the amount of noise from repeated measurements, wildbootstrapping takes its noise estimates from the residuals that remain when fitting a model to asingle set of measurements [65]. This has been proposed as an alternative to repetition-basedbootstrapping for cases where only a single acquisition is available [31]. Residual bootstrap-ping [12] builds on the same basic idea, but allows for resampling residuals between gradientdirections, by modeling the heteroscedasticity in them. It has not only been combined with thediffusion tensor model, but also with constrained deconvolution, which allows for multiple fibertractography [27].

2.2 Rendering Probabilistic Tractograms

After estimating the reproducibility of white matter fiber tracts by one of the above-describedmethods, we can represent the results in one of two ways: voxel-centric or tract-centric. Thevoxel-centric representation assigns scores to individual voxels, where each voxel stores the per-centage of tracts passing through it. They represent the reproducibility with which a connectionfrom one voxel position to the seeding region is inferred from the data. The resulting 3D vol-ume data sets are sometimes called probability or confidence maps, and are often visualized byvolume rendering techniques and 2D color maps [39].

Tract-centric techniques include the ConTrack algorithm [57], which assigns a score to eachgenerated tract. It reflects confidence in the pathway as a whole, based on its agreement withthe data and assumptions on fiber length and smoothness. Ehricke et al. [15] define a confidencescore that varies along the fiber, and color code it on the streamline. Jones et al. [29, 31] usehyperstreamlines to visualize the variability of fiber tracts obtained using bootstrap or wild-bootstrap methods. They also demonstrate that using standard streamlines to render all fibervariations equally fails to give an impression of which fibers are stable and which are outliers.

Brecheisen et al. [7] propose illustrative confidence intervals where intervals are based ondistances or pathway scores. Illustrative techniques, i.e., silhouette and outlines, are used to vi-sualize these intervals. Interaction and Focus+Context widgets are used to extend the simplifiedillustrative renderings with more detailed information.

Schultz et al. [55] cluster the voxels in which probabilistic tractography terminates, based onthe seed points from which they are reached. They then derive a per-voxel score that indicateshow frequently the voxel was involved in a connection between two given clusters. Fuzzy fiberbundle geometry is defined by isosurfaces of this score, with different isovalues representingdifferent levels of precision.

3 Other Factors

3.1 Impact of Parameters

One source of uncertainty in dMRI tractography that has not received much attention is param-eter sensitivity. Most tractography algorithms depend on user-defined parameters, which resultsin a poor reproducibility of the output results. Some reproducibility studies for concrete appli-cations have been reported [13, 63]. However, there does not exist an automatic solution that

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resolves the problem in a general manner. The stability of the parameter setting is relevant infor-mation for both neuroscientists and neurosurgeons who are trying to assess whether their fibertracking results are stable. Visualization can play an important role to help this assessment.

Brecheisen et al. [6] build a parameter space by sampling combinations of stopping criteria forDTI streamline tractography. Stopping criteria primarily affect fiber length. The investigationof parameter sensitivity is based on generating a streamline set that covers the whole parameterspace of stopping criteria. Afterwards, selective culling is performed to display specific stream-line collections from the parameter space. This is done by selecting parameter combinationsusing 2D widgets such as the feature histogram displayed in Figure 1. An example feature isaverage fiber density per voxel. These views help the user to identify stable parameter settings,thereby improving the ability to compare groups of subjects based on quantitative tract features.

Figure 1: Main viewports of Brecheisen et al. [6] exploration tool. Top-left: 3D visualizationof fiber tract together with anatomical context and axial fractional anisotropy slice.Top-right: color map view used for selecting individual threshold combinations anddefinition of color detail regions. Bottom-right: feature map view showing changesin quantitative tract features as a function of threshold combination at discrete samplepoints of the parameter space. Bottom-left: cumulative histograms of both thresholdand feature values.

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Jiao et al. [28] introduce a toolkit based on three streamline distances that are used to measuredifferences between fiber bundles. The user can vary parameters that affect the results of thefiber tractography and measure the resulting differences based on these distances. This allowsthem to quantify the variation and reproducibility of the fiber bundles due to different sources ofuncertainty and variation in the tractography input parameters.

Although these methods provide a first step to study uncertainty due to parameter settings, itremains a time consuming exploratory task for the user. This is especially true if parameters arecorrelated, and their interrelation needs to be investigated.

3.2 Model Uncertainty and Selection

Methods for tractography that seek to recover more than a single fiber direction in a given areahave to make a judgement about how fiber directions can be meaningfully recovered from thedMRI data. The combination of measurement noise, partial voluming, and the practical con-straints on how many diffusion weighted images may be acquired create uncertainty in thenumber of fibers present. Qualitatively different than the angular uncertainty in a single fiberdirection, the traditional focus of probabilistic tractography, this uncertainty can be a viewed asa kind of model selection uncertainty, which is described further in Section 4.1.

Uncertainty in fiber number has been handled by different tests that either statistically sampleor deterministically choose a level of model complexity (with an associated fiber number) froma nested set of models. Behrens et al. [2] use Automatic Relevance Determination (ARD) toprobabilistically decide the number of “sticks” (fibers) in their ball-and-multiple-stick model.Within their probabilistic tractography, this achieves Bayesian Model Averaging [22] of thefiber orientation. For deterministic tractography, Qazi et al. [49] use a threshold on the (single,second-order) tensor planarity index cp [64] to determine whether to fit to the diffusion weightesimages a constrained two-tensor model [45] that permits tracing two crossing fibers. Schultz etal. compare different strategies for deciding the appropriate number of fiber compartments, basedon the diminishing approximation error [54], thresholding compartment fraction coefficientsof a multi-fiber model [56], or by learning the number of fiber compartments using simulateddata and support vector regression [52], which represents uncertainty in the form of continuousestimates of fiber number.

Much of the work on determining the number of per-voxel fiber components has been de-scribed outside of any particular tractography method, but may nonetheless inform tractographicanalysis. Alexander et al. [1] use an F-Test to find an appropriate order of Spherical Harmonic(SH) representation of the ADC profile. Jeurissen et al. [26] decide the number of fibers bycounting significant maxima in the fiber orientation distribution after applying the SH deconvo-lution (constrained by positivity) of Tournier et al. [60]. The SH deconvolution of Tournier etal. [60] in some sense involves model selection, because the deconvolution kernel is modeled bythe SH coefficients of the voxels with the highest FA, presumably representing a single fiber.

Aside from the question of counting fibers, other work has examined more broadly the ques-tion of which models of the diffusion weighted signal profile are statistically supported. Bret-thorst et al. [8] compute Bayesian evidence (see Section 4.1) to quantify the fitness of variousmodels of the diffusion weighted signal, producing maps of model complexity in a fixed ba-boon brain, and of evidence-weighted averages of per-model anisotropy. Freidlin et al. [17]

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choose between the full diffusion tensor and simpler constrained tensor models according to theBayesian information criterion (BIC) or sequential application of the F-Test and either the t-Testor another F-Test.

3.3 Partial Voluming

Tractography works best in voxels that contain homogeneously oriented tissue. Unfortunately,many regions of the brain exhibit more complex structures, where fibers cross, diverge, or dif-ferently oriented fibers pass through the same voxel [1]. This problem is reduced at highermagnetic field strength, which affords increased spatial resolution. However, even at the limitof what is technically possible today [20], a gap of several orders of magnitude remains to thescale of individual axons.

Super-resolution techniques combine multiple images to increase effective resolution. Mostsuch techniques use input images that are slightly shifted with respect to each other and initialsuccess has been reported with transferring this idea to MRI [46]. However, due to the fact thatMR images are typically acquired in Fourier space, spatial shifts do not correspond to a change inthe physical measurement, so it is unclear by which mechanism repeated measurements shouldachieve more than an improved signal-to-noise ratio [50, 47]. It is less controversial to com-pute images that are super-resolved in slice-select direction [18, 51] or to estimate fiber modelparameters at increased resolution via smoothness constraints [41].

Track density imaging [10] uses tractography to create super-resolved images from diffusionMRI. After randomly seeding a large number of fibers, the local streamline density is visualized.It is computed by counting the number of lines that run through each element of a voxel gridwhose resolution can be much higher than during MR acquisition. Visually, the results resemblethose of line integral convolution, which had been applied to dMRI early on [23, 67].

4 Perspectives

4.1 Evidence for Model Selection

Many of the methods for finding per-voxel fiber count (or more generally the per-voxel signalmodel) described in Section 3.2 share two notable properties which may be reconsidered andrelaxed in future research. First, they deterministically calculate the single best model, withhard transitions between the regions best explained by one model versus another [1, 17, 49, 54,56, 26]. Yet we know that partial voluming (Sect. 3.3) creates smooth transitions between dif-ferent neuroanatomic tissue regions. Though computational expensive, Markov Chain MonteCarlo (MCMC) sampling of both model parameter space and the set of models enables av-eraging information from more than one model [2, 8]. Second, most methods work within aparticular hierarchical set of linearly ordered models (SH of different orders [1], ball and mul-tiple sticks [2], sum of higher-order rank-1 terms [56]). One can easily imagine configurations,however, that confound such a linear ordering: an equal mix of two fibers crossing and isotropicdiffusion (perhaps due to edema), or a mix of one strong fiber and two weaker equal-strengthfibers. Furthermore, there is rarely objective comparison or reconciliation between disjoint setsof models.

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An informative perspective on these situations may be gained by directly visualizing, eitheron data slices or by some form of volume rendering, the fitness of a large palette of possiblemodels. In a Bayesian setting, the model fitness can be quantified by the marginal likelihood ofthe data x given the model Mk, or the model evidence, computed by integrating over the modelparameter space θθθ k [35].

P(x|Mk)︸ ︷︷ ︸evidence

=∫

P(x|θθθ k,Mk)︸ ︷︷ ︸likelihood

P(θθθ k|Mk)︸ ︷︷ ︸prior

dθθθ . (1)

Bretthorst et al. [8] have pioneered the calculation and visualization of model evidence for dMRI,but many possible directions are left unexplored, including the application to counting fibers, andto models that account for intra-voxel fanning or bending [40].

4.2 Reproducibility, Seeding, and Preprocessing

The reproducibility of tractography depends on many factors. The manual placement of seedpoints is an obvious concern. Detailed written instructions improve reproducibility between op-erators [13], especially across sites [63]. Combining multiple seed regions with logical operatorsmakes the results more reproducible [24, 21] and seeding protocols for up to 11 major fiber bun-dles have been developed this way [63]. Warping individual brains to a standard template hasalso been reported to increase reproducibility [21, 62]. Selecting streamlines from a whole braintractography via three-dimensional regions of interest [5] or semi-automated clustering [61] isan alternative way to reproducibly extract fiber bundles.

When the same person places the seeds on a repeated scan, the resulting variability is generallyhigher than when different observers follow a written protocol to place seeds in the same data[13]. Within the same session, measurement noise is the main limiting factor [14]. Betweensessions, differences in exact head positioning and other subtle factors increase the variabilitynoticably [62].

Reproducibility suffers even more when repeating the measurement on a different scanner[48]. Even a pair of nominally identical machines has produced a statistically significant bias inFractional Anisotropy [62]. Improving calibration between sessions or scanners via software-based post-processing appears possible [62], but has not been widely explored so far.

More time consuming measurement protocols generally afford better reproducibility. Eventhough Heiervang et al. [21] report that the improvement when using 60 rather than 12 gradi-ent directions was not statistically significant, Tensouti et al. [59] report a clear improvementbetween 6 and 15 directions, which continues – at a reduced rate – when going to 32 direc-tions. Farrel et al. [16] use 30 directions and demonstrate a clear improvement when averagingrepeated measurements.

Finally, reproducibility depends on the tractography algorithm [59], its exact implementation[9], as well as the methods used for pre-processing the data [33, 62] and their parameter settings.Given that the reproducibility of tractography will be crucial for its wider acceptance in scienceand medicine, more work is needed that specifically targets these problems.

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5 Conclusion

Reproducibility of dMRI tractography is a fundamental problem that limits the acceptance ofthis technique in clinical practice and neuroscience research. Although some effort has beenmade to include uncertainty information in the tractography results, several open issues remainthat need further investigation.

Probabilistic tractography is established, but visualization research has concentrated on de-terministic streamline-based techniques, and few techniques have been developed to visualizethe information obtained by probabilistic methods. There are several sources of uncertainty inthe tractography visualization pipeline. However, only a few of them have been explored, and ifat all studied, they are often considered independently with no connection to each other. Tech-niques that investigate the impact of parameters on the fiber tracking results and that aim toreduce the impact of user bias through parameter selection have been investigated only recently.Model selection and data preprocessing have hardly been studied with respect to their effects ontractography results.

Techniques that allow the combined analysis of uncertainty from different sources in the sameframework, and that facilitate an understanding of their influence on the final tractography resultare still missing. One challenge faced by visualization systems that aim to aid understanding ofthese uncertainties is to display this additional information efficiently and effectively, withoutcausing visual clutter.

Ultimately, uncertainty visualization should contribute to making fiber tracking a more reli-able tool for neuroscience research, and to conveying the information needed for the decisionmaking process in clinical practice.

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