Towards quantitative connectivity analysis: reducing tractography biases

Towards quantitative connectivity analysis: reducing tractography biases

Gabriel Girarda,d,⇤, Kevin Whittingstallb,c, Rachid Deriched, Maxime Descoteauxa,c

aSherbrooke Connectivity Imaging Lab (SCIL), Computer Science Department, Faculty of Science, Universite de Sherbrooke, 2500 BoulevardUniversite, Sherbrooke, QC, Canada J1K 2R1

bDepartment of Diagnostic Radiology, Faculty of Medicine and Health Science, Universite de Sherbrooke, 12e Avenue Nord, Sherbrooke, QC,Canada J1H 5N4

cSherbrooke Molecular Imaging Center, Department of Nuclear Medicine and Radiobiology, Faculty of Medicine and Health Science, Universitede Sherbrooke, 12e Avenue Nord, Sherbrooke, QC, Canada J1H 5N4

dProject Team Athena, INRIA Sophia Antipolis Mediterranee, 2004 Route des Lucioles BP 93, 06902 Sophia Antipolis Cedex, France

Abstract

Tractography is biased by the position, the shape, the size and the length of white matter bundles. A proportion of

connections reconstructed from tractography is thus erroneous and not equally distributed in all white matter bundles.

Hence, quantitative measures of connectivity based on streamlines distribution in the brain such as streamline count

(density), average length and spacial extent or volume are biased by erroneous streamlines produced by tractography

algorithms. In this paper, solutions are proposed to reduce biases in streamlines distribution. We first propose to opti-

mize tractography parameters in terms of connectivity. Then, we propose to relax the tractography stopping criterion

with a novel stopping criterion based on tissue partial volume estimation maps, calculated from a T1-weighted image.

Additionally, we propose a particle filtering method using anatomical information to enforce streamlines connect-

ing gray matter regions and reducing the proportion of streamlines prematurely stopping. We show that optimizing

tractography parameters, stopping and seeding strategies can reduce the biases in position, shape, size and length of

streamlines distribution. These tractography biases are quantitatively reported in both real and synthetic data. This is

the first critical step towards producing tractography results for quantitative structural connectivity analysis.

Keywords: White Matter Tractography, Di↵usion MRI, Anatomical MRI, Connectivity Analysis, Particle Filtering

1. Introduction1

Di↵usion-weighted (DW) magnetic resonance imaging (MRI) tractography is used to reconstruct white matter2

(WM) pathways between brain regions. A growing number of connectomics studies exploit structural properties of3

these pathways or streamlines to make ’connectivity’ comparisons between groups or individuals (Fornito et al., 2013;4

Hagmann et al., 2008; Ng et al., 2013; Sporns, 2010). However, white matter bundles have various position, shape,5

size and length making their reconstruction a challenge for tractography algorithms (Jbabdi and Johansen-Berg, 2011;6

⇤E-mail address: [email protected] (G. Girard)

Preprint submitted to NeuroImage February 4, 2014

Jones, 2010; Jones et al., 2012; Smith et al., 2012). Bundles positioned in partial volume with Cerebrospinal Fluid7

(CSF) are harder to completely reconstruct because streamlines propagation is more likely to be stopped. Narrow8

bundles are harder to reconstruct because more likely to be a↵ected by error in the tracking mask, potentially stop-9

ping the streamline propagation. Curved bundles are also harder to reconstruct because noise can make the tracking10

direction harder to follow in curved regions, especially because discrete steps in the estimated tangent direction are11

taken. Lastly, length of white matter bundles raise two contrary e↵ects that bias their reconstruction: i) seeding from12

the white matter increases the density because there are more streamlines that are initiated in longer bundles than in13

shorter bundles, ii) longer bundles are harder to completely recover because of premature stops, which decreases the14

density of streamlines.15

Therefore, it is clear that streamline reconstruction is biased by the seeding strategy, the stopping and masking cri-16

terion and the tractography parameters themselves. Hence, quantitative measures of connectivity based on streamlines17

distribution in the brain such as streamline count (density), average length and spacial extent or volume are biased18

by erroneous streamlines produced by tractography algorithms (Jbabdi and Johansen-Berg, 2011; Jones, 2010; Jones19

et al., 2012). Yet these e↵ects are rarely addressed and reported in the literature even though they may lead to incorrect20

connectivity measures between areas. It is thus crucial and timely for the DW-MRI community to tackle tractography21

limitations before it can be robustly used in connectomics studies.22

In the majority of cases, tractography is done inside a mask defined by a white matter segmentation of the T1-23

weighted image or fractional anisotropy (FA) thresholded mask (Cote et al., 2013; Hagmann et al., 2007; Li et al.,24

2012b; Smith et al., 2012; Tournier et al., 2012, 2011). Starting from an initial point within the mask, the tractography25

process follows di↵usion orientations in the forward and backward directions until a stopping criterion is reached.26

Typical stopping criteria are when the tracking takes a step outside the tracking mask or when no valid propagation27

direction is available (Toosy et al., 2004; Tournier et al., 2012, 2011). The tracking mask selection and the stopping28

parameters are thus very important as they will determine when the streamline is included in the reconstructed white29

matter pathway. Discrete binary masks derived from thresholded FA or T1-weighted images result in aggressive30

stopping criteria which can have a strong impact on connectivity results. For example, Cote et al. (2013) studied31

streamlines produced by tractography pipelines using the Tractometer (tractometer.org) system analysis on the32

FiberCup dataset (Fillard et al., 2011; Poupon et al., 2010, 2008). Out of all tractography pipelines that found the33

seven out of seven true bundles of the FiberCup (6,360 out of 57,096), between 58% and 97% of streamlines did not34

connect gray matter (GM) regions. Although these observations are based on a phantom mimicking a coronal slice35

(Poupon et al., 2008) of the brain, similar observations are seen in real brain imaging data. For instance, in (Hagmann36

et al., 2007), authors reported that one third to half of streamlines did not reach the WM/GM interface mask and thus,37

2

are excluded from the structural connectivity analysis.38

To overcome the e↵ect of binary masks, one can use tissue partial volume estimation (PVE) maps obtained from39

a structural T1-weighted image (Zhang et al., 2001). The two major di↵erences between tissue masks and tissue PVE40

maps are i) voxels near the boundary between distinct tissues are gray and ii) the subcortical gray matter is mainly41

gray on both WM and GM PVE maps. The discretization of such gray regions creates holes in the white matter mask42

or makes some white matter pathways narrower. This makes streamlines stop prematurely in these regions. This43

problem is especially important when tracking corticospinal fibers or fibers involved in the motor system as shown in44

(Girard et al., 2012). Recently, Smith et al. (2012) proposed a method called Anatomically-Constrained Tractography45

(ACT) taking advantage of the tissue PVE maps. They proposed relaxing the stopping criterion by using WM, GM46

and CSF PVE maps to determine when a streamline stops and if it is included or excluded in the reconstruction.47

Therefore, biological tissue properties are used to better determine the tracking mask and stopping criteria. They48

proposed to threshold interpolated PVE maps to define stopping criterion. However, authors observed that subcortical49

gray matter have low PVE values, leading to streamline going through these regions, connecting other gray matter50

regions or reaching CSF regions excluding the streamline. Authors suggested to cut streamlines going through binary51

segmentation of the subcortical gray matter and include only valid segments. However, it is challenging to choose52

which regions to define in such a binary mask construction.53

Aside from stopping criterion, the seeding strategy can also bias the streamlines distribution and change estima-54

tion of the brain connectivity. The seeding mask defines all the potential voxels where streamlines are initiated. It55

can either be the tracking mask (Centuro et al., 1999; Huang et al., 2004; Tournier et al., 2012, 2011), a WM/GM56

interface mask (Li et al., 2012b; Smith et al., 2012) or a region of interest (Behrens et al., 2007; Huang et al., 2004;57

Parker and Alexander, 2005; Toosy et al., 2004; Tournier et al., 2012, 2011). Seeding from a whole white matter58

mask biases the number of reconstructed streamlines in bundles with various lengths because streamlines are more59

likely to be initialized in longer white matter bundles, covering a larger part of the white matter mask (Jones, 2010;60

Smith et al., 2013). For example, if two linear bundles have the same size but one twice the length of the other, the61

number of streamlines in the longer bundle will be approximately doubled. This increase of density is not related62

to the connectivity of the bundle, it is a seeding bias of the tractography. Recently, Smith et al. (2013) proposed a63

method called Spherical-deconvolution informed filtering (SIFT) to reduce local bias in the streamlines density. The64

method filters the tractography results to improve the fit between the streamlines distribution in each voxel and the65

fiber Orientation Distribution Function (ODF) estimated from DW-MRI. SIFT produces streamlines that are more66

representative of measured di↵usion information. In particular, SIFT reduces density bias resulting from the seeding67

strategy. However, streamlines are still a↵ected by the choice of masking and stopping criterion, and tractography68

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parameters used by the tractography algorithm.69

In this work, we show that careful tractography parameters selection and optimal seeding, masking and stopping70

criterion choices can significantly reduce the biases in position, shape, size and length of streamlines distribution.71

Firstly, inspired by the work of Smith et al. (2012) which uses anatomical information from a T1-weighted image for72

tractography, we propose a novel probabilistic stopping criteria based on tissue PVE maps. Secondly, we propose73

a particle filtering method using anatomical information for tractography to enforce streamlines connecting gray74

matter regions. Thirdly, we make recommendations on the most important tractography parameters and optimize the75

parameter selection in terms of global connectivity using the Tractometer (Cote et al., 2013) evaluation strategy from76

synthetic and real data. Our overall contribution is a new tractography framework optimized in terms of quantitative77

connectivity, which reduce tractography biases in position, shape, size and length of white matter bundles.78

2. Method79

2.1. Streamline Tractography80

In this work, we relax the tractography stopping criterion using tissue partial volume estimation (PVE) maps. Since81

our proposed strategies do not represent new tractography algorithms as such, we compared and applied these relax-82

ations to the state-of-the-art fiber ODF deterministic and probabilistic algorithms (Descoteaux et al., 2009; Tournier83

et al., 2012, 2011). In-house implementation of these tractography algorithms is used, which have been validated84

against MRtrix (Tournier et al., 2012) by the Tractometer (Cote et al., 2013). In our implementation, the fiber ODFs85

are projected on a discrete evenly distributed symmetric sphere of 724 vertices (Daducci et al., 2013) and normalized86

(maximum=1). Propagation directions are always a vector of orientation corresponding to one vertex of the sphere and87

of length �s = 0.2 mm (Tournier et al., 2012). A propagation direction is valid if its corresponding value is greater than88

a threshold ⌧ and form an angle smaller than ✓ with the previous propagation direction (Behrens et al., 2007; Tournier89

et al., 2012, 2011). If there is no valid propagation direction, algorithms assume an error in the fiber ODF and contin-90

ues in the previous propagation direction. This is done for a maximum distance of �undeviated. Implementation details91

are given in Appendix A.92

2.2. Continuous Map Criterion - CMC93

In the current study, we propose a novel approach that takes advantage of the complete WM, GM and CSF94

partial volume estimation (PVE) maps to change the way tractography stopping events are triggered. We call our95

novel strategy Continuous Map Criterion (CMC). It uses PVE maps to define the probability of stopping the tracking96

process. This provides smooth boundaries between tissues as ACT (Smith et al., 2012) and additionally encodes a97

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stopping behavior in subcortical gray matter. Streamlines reaching the cortex and going through large regions of98

low GM PVE such as the subcortical gray matter are proportionality likely to be stopped. Using CMC, streamlines99

can propagate close to subcortical gray matter without having to define binary segmentation blocking some of the100

propagation pathways.101

CMC uses an inclusion map (Mapin) and exclusion map (Mapex) to stop the streamline propagation. An example102

of Mapin and Mapex based on GM and CSF PVE maps are shown in Figure 1. We hypothesize that the amount of103

streamlines stopping in a voxel and included should be proportional to Mapin. Similarly, the amount of streamlines104

stopping at a voxel and rejected should be proportional to Mapex. Using CMC, the probability that a streamline105

continues its propagation at position p is given by106

Pcontinuep = (1 � (Mapin

p + Mapexp ))�s/⇢, (1)

with ⇢ the maps voxel size (⇢ = 1 for voxel size of 1x1x1 mm3) and �s the step size. �s/⇢ allows the probability of107

stopping to be stable with respect to the step size �s. Otherwise, since the tracking probability is evaluated at each108

tracking step, using a step size �s < ⇢ will increase the probability of stopping the tractography and decrease the109

probability when �s > ⇢. Alternatively, Pcontinue can be computed and adjusted to the step size following Equation 1110

for each voxels and used directly. If the tracking process stops, the streamline is included (added to the estimated set111

of streamlines) with a probability given by112

Pincludedp = Mapin

p /(Mapinp + Mapex

p ), (2)

otherwise the streamline is excluded (rejected from estimated set of streamlines). Trilinear interpolation is done over113

Mapin and Mapex to get the probability of continuing the propagation (Equation 1) and the probability of including114

the streamline (Equation 2).115

2.3. Particle Filtering Tractography - PFT116

In addition to CMC, we propose using a modular add-on to streamline tractography algorithm, called Particle117

Filtering Tractography (PFT) to reduce to number of premature stopping streamlines. Streamline tractography can be118

modeled as a state system evolving over time using noisy measurements, where states are the tracking position, the119

propagation direction and the tracking status (e.g. ’in the WM’ or ’stopped in the GM’), and are connected over time120

by a Markov chain. The particle filtering algorithm is described in Appendix B.121

PFT is initiated before the premature stopping event and weighs propagation pathways based on the PVE maps122

5

Figure 1: The tracking include map Mapin (first row) is the GM PVE map plus all voxels not part of the brain mask. The exclude map Mapex

(second row) is equal to the CSF PVE map.

to enforce the tracking in the white matter, as illustrated in Figure 2. Propagation pathways are chosen to ensure123

streamlines not to stop in the CSF and reach the gray matter (Bloy et al., 2012; Smith et al., 2012). A backtracking124

approach for probabilistic tractography has been proposed in (Smith et al., 2012) which incrementally truncates and125

re-tracks the streamline when it reaches a premature stop. It shows an increase of the white matter bundle coverage126

and helps the reconstruction of some white matter bundles. However, higher backtracking distances can bias the127

streamlines reconstruction, especially in crossing regions. PFT uses a backtracking idea by simultaneously estimating128

many propagation pathways at a short distance of the premature stopping event.129

The proposed Particle Filtering Tractography (PFT) estimates a likely streamline using Mapin and Mapex (see130

Section 2.2) whenever the tractography reaches a stopping criterion excluding a streamline, as illustrated in Figure 2.131

The key idea is to backtrack �back mm and compute a valid streamline after K = (�back + � f ront)/�s steps, where �back132

and � f ront are respectively the backward and forward distances. If the propagation distance is less than �back, �back is133

set to the propagation distance done so far. The goal is to estimate a likely streamline initialized at �back mm before the134

stopping criterion is reached, and then go � f ront mm further to ensure the local stopping event is solved. That is, the135

streamline stops correctly in an including region or the streamline continues its propagation in the white matter. If the136

streamline stops in an including region, the tracking is done. If the streamline is in the white matter, the tractography137

continues normally until another stopping criterion is reached.138

PFT uses a set {x(i)k ,w

(i)k }Ni=1 of N discrete samples (referred as particles) x(i)

k with an associated weight w(i)k to139

characterize the estimated streamlines distribution. Weights are normalized over all particles to havePN

i=1 w(i)k = 1. A140

6

a) b) c) d) e) f) g)

Figure 2: PFT algorithm. (a) A streamline prematurely stops in the CSF (white) and (b) a backtracking step is done. (c,d,e) shows the particlesat three iterations of PFT. PFT estimates the distribution of possible streamlines using probabilistic samples and weighs them using anatomicalinformation. Redish particles have low weight and greenish have high weight. (f) A path is drawn from the particles distribution. (g) Thepropagation process then continues using the principal tractography algorithm (deterministic).

particle x(i)k = [p, v, status] has a the tracking position p, a propagation direction v and a status 2 {active, inactive}141

which represents the tracking process propagating (active) or stopped in an including region (inactive). At each142

iteration k, if status = active, the particle position p and propagation direction v are updated following the probabilistic143

tractography algorithm (see Appendix A). Otherwise, if status = inactive, the tracking reached a valid stopping region144

and is stopped (p and v are not updated). The status stays active with a probability of145

Pactivep = (1 � Mapin

p )�s⇢ ,

following the CMC strategy (see Section 2.2). The exclusion map Mapexp , is used to estimate the likelihood of the146

particle x(i)k , which is the likelihood of a streamline propagation at p. The weight w(i)

k , at time k, of a particle at position147

p is calculated following148

w(i)k = w(i)

k�1 · (1 � Mapexp ).

w(i)k is set to 0 if no valid propagation direction is available for a distance �undeviated (see Section 2.1).149

PFT estimates a valid distribution of streamline around the stopping event and iteratively estimates subsequent150

valid streamlines distribution from the previous one. The resulting streamline is drawn from the final valid streamlines151

distribution. As shown in Figure 2 (c-e), this algorithm generates multiple probabilistic streamlines and penalizes152

particles propagating in the excluding region (Mapex). Reddish particles have low weights and greenish have high153

weights. The output of the PFT is either an inactive streamline ending in the including region (Mapin) or an active154

streamline continuing its propagation in the white matter. If at any iteration k the weights w(i)k = 0 8 x(i)

k , the streamline155

is excluded because no valid streamline is found (e.g. Mapexp = 1 8 x(i)

k ). The principal tractography algorithm used156

(deterministic or probabilistic) is done until the propagation reaches a stopping criterion excluding the streamline, as157

7

determined by the CMC (see section 2.2). In this case, PFT is triggered to find an alternative valid pathway.158

2.4. Seeding From the White Matter - Gray Matter Interface159

In this study, we want tractography algorithms to produce a similar density for bundles with the similar size but160

various lengths. To achieve this, we seed from the WM/GM interface as in (Li et al., 2012b; Smith et al., 2012). We161

propose to define the WM/GM interface mask by segmenting all voxel having a GM PVE > 0.1 and a WM PVE >162

0.1. This results in a ribbon of voxels at the boundary between gray matter and white matter (see Figure 3). Most163

of the voxels of the subcortical gray matter are included in the interface since they are partially segmented as white164

matter and gray matter (Figure 3 (c)). An approach based on a dilatation of the gray matter mask could have been used165

to obtain the WM/GM interface such as (Li et al., 2012b; Smith et al., 2012). Further investigation are required to166

quantify the e↵ect of the definition of the WM/GM interface on tractography, but are outside the scope of this paper.167

The seeding mask contains a partial volume of gray matter, which can lead to premature stopping the streamline168

propagation using CMC. To overcome this, CMC (Equation 1) is only triggered once the streamline has reached169

a position p where Mapinp < 0.01 (e.g. in the white matter). Otherwise, propagation stops only when reaching170

Mapinp = 1 (included), Mapex

p = 1 (excluded) or when no valid direction is available for distance �undeviated (excluded).171

This allows streamlines to exit the initial region before stopping the propagation (see Section 2.2).172

When a voxel is identified to initiate a streamline in it, the seed position is randomly chosen within the voxel173

boundary (Tournier et al., 2012). A trilinear interpolation over the spherical harmonic coe�cients of the fiber ODFs174

image is done to obtain the fiber ODF at the seed position. The fiber ODF is then thresholded to a predefined value175

⌧init. The initial propagation direction is drawn from the empirical distribution defined by thresholded fiber ODF. ⌧init176

aims at starting tractography in a good tangent direction to the bundle.177

(a) (b) (c)

Figure 3: The WM/GM interface. All voxels of the interface have a GM PVE > 0.1 and a WM PVE > 0.1.

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2.5. Datasets for the Experiments178

2.5.1. Synthetic Dataset179

The simulated dataset produced for the IEEE International Symposium on Biomedical Imaging (ISBI) 2013 Re-180

construction Challenge (Daducci et al., 2013) is used to evaluate quantitatively the quality of tractography algorithms.181

The synthetic dataset consists of 27 simulated known ground truth white matter bundles, mimicking challenging182

branching, kissing, crossing structures at angles between 30� and 90�, with various curvature, and radii, as seen in183

Figure 4 (a) and Figure 5. The DWI signal is simulated in each voxel based on the Numerical Fiber Generator (Close184

et al., 2009) and some free-water CSF-like partial volume e↵ects. The simulated signal is obtained using a hindered185

and restricted di↵usion model (Assaf and Basser, 2005), and adding Rician noise. In this study, we used 64 uniformly186

distributed gradient directions using a b-value of b = 1000 s/mm2 at signal to noise ratio (SNR) 10, 20 and 30. The187

dataset has a spherical shape with the extremities of the simulated white matter bundles ending on the surface of the188

sphere. The bundles mask is defined as all voxel having a white matter PVE greater 0.1. The simulated gray matter189

consists of the voxels in the three outer layers of the sphere, obtained by three erosion iterations and intersecting the190

bundles mask. The white matter mask is composed of all voxels of the bundles mask and not part of the gray matter191

mask. The simulated CSF is all none gray matter or white matter voxels (see Figure 4 (a)). The simulated WM/GM192

interface is the fourth outer layer of the sphere and part of the white matter mask. Mapin and Mapex are defined from193

the gray matter mask and the CSF mask (see Figure 4 (b, c)).194

2.5.2. Healthy Brain Dataset195

DWI were acquired on a single volunteer along 64 uniformly distributed directions using a b-value of b =196

1000 s/mm2 and a single b = 0 s/mm2 image using the single-shot echo-planar imaging (EPI) sequence on a 1.5197

Tesla SIEMENS Magnetom (128x128 matrix, 2 mm isotropic resolution, TR/TE 11000/98 ms and GRAPPA factor198

2). An anatomical T1-weighted 1 mm isotropic MPRAGE (TR/TE 6.57/2.52 ms) image was also acquired. Di↵usion199

(a) (b) (c)

Figure 4: Synthetic Dataset. (a) Sphere of CSF (black) with WM (white), connecting GM at the extremity of the WM (green), (b) Mapin, (c)Mapex.

9

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18

19 20 21 22 23 24 25 26 27

Figure 5: Synthetic WM bundles. Small bundles are blue (1, 2, 3, 5, 8, 9, 16, 19, 21, 25, 27), medium bundles are green (12, 20, 22, 26) and largebundles are red (4, 6, 7, 10, 11, 13, 14, 15, 17, 18, 23, 24).

data were upsampled to 1 mm isotropic resolution using a trilinear interpolation (Dyrby et al., 2011; Girard et al.,200

2012; Smith et al., 2012; Tournier et al., 2012). The T1-weighted image was registered to a 1 mm isotropic DWI201

using FSL/FLIRT (Jenkinson and Smith, 2001). Quality control was done to make sure the registration was done202

robustly by manual inspection. The Fractional Anisotropy (FA) map and color-FA were overlaid on the T1-weighted203

image to make sure optimal alignment between images. The Brain Extraction tool (FSL/BET (Smith, 2002)) and204

FSL/FAST (Zhang et al., 2001) were also used to extract both binary and PVE maps of the WM, GM and CSF. Mapin205

and Mapex are respectively set as the gray matter PVE and CSF PVE maps. Additionally, all voxels not in the brain206

(using the brain mask estimated with FSL/BET) are set to 1 in the Mapin in order to keep streamlines exiting the207

brain mask. Di↵usion tensors, FA, fiber ODFs reconstruction and all tractography algorithms are carefully detailed208

in Appendix A.209

White matter bundles have been manually segmented using the Fibernavigator software (scilus.github.io/210

fibernavigator/) and using FreeSurfer T1-weighted image white matter and gray matter segmentations (Fischl211

et al., 2004). Streamlines are colored by their orientation (the vector connecting their extremities) using the standard212

red-green-blue convention (red: leftright, green: anteriorposterior, blue: inferiorsuperior) (Calamante et al., 2012;213

Pajevic and Pierpaoli, 1999).214

2.6. Quantitative Connectivity Evaluation215

To compare and evaluate estimated streamlines, we use the Tractometer (Cote et al., 2013) connectivity analysis.216

We computed the four metrics of the Tractometer, namely the Valid Connections (VC), the Invalid Connections (IC),217

the No Connections (NC) and the Average Bundle Coverage (ABC). The definition of these connectivity metrics are218

10

listed in Appendix C. Additionally, based on these metrics, we define two other metrics to quantify the uniformity219

and the general accuracy of streamlines: i) The Connection to Seed Ratio (CS R), which is the relation between the220

number of connections estimated and the number of seeds S used by the tractography algorithm, i.e. CS R = VC+ICS .221

If all seeds produce a streamline, CS R = VC+ICVC+IC+NC . ii) The Valid Connections to Connection Ratio (VCCR), which222

is the relation between valid connections and all connections estimated VCCR = VCVC+IC . All metrics are reported in223

percentages (%).224

The valid and invalid connections cannot be directly computed on real data. We thus define a connection as any225

streamline connecting two gray matter regions and having a minimum length �min = 10 mm and a maximum length226

deltamax = 300 mm (Tournier et al., 2012). Using this definition, the connection to seed ratio from real brain data,227

CS Rr, can be estimated. Here, the index r is used to indicate that CS R is computed from real data, as opposed to the228

index s for synthetic data. Since we cannot compute VCCR in the brain, an increase of CS Rr can be due to an increase229

in invalid connections, and hence it must be interpreted carefully.230

2.6.1. What makes a good tractography algorithm?231

An optimal tractography algorithm should produce global connectivity metrics with the following properties : i)232

all seeds should lead to streamlines connecting gray matter regions (low no connections NC, high connection to seed233

ratio CS R), ii) all connected gray matter regions should match the ground truth (high valid connections VC, low234

invalid connections IC, high valid connection to connection ratio VCCR and iii) streamlines should cover all voxels235

within a bundle (high average bundle coverage ABC). Hence, parameters are chosen to increase the valid connections236

to connections ratio (VCCRs) on synthetic data and the connections to seeds ratio (CS Rs and CS Rr) on both real and237

synthetic data. CS Rs and VCCRs are computed seeding from all white matter voxels of the synthetic dataset (8483238

seeds). CS Rr is computed seeding randomly in 10,000 voxels in the white matter mask of the brain dataset.239

3. Results240

3.1. Choosing Optimal Tractography Parameters241

3.1.1. Maximum Deviation Angle ✓det = 45�, ✓prob = 20�242

To evaluate the best maximum deviation angle ✓, other parameters were set to zero (⌧ = 0, ⌧init = 0, �undeviated =243

0 mm) and PFT is not used. Figure 6 (a) shows valid connections to connection ratio VCCRs, and connection to244

seed ratio CS Rs and CS Rr for deterministic and probabilistic tractography varying the maximum deviation angle ✓.245

Higher ✓ decreases the stopping issue in the white matter for both algorithms, by allowing the propagation to curve246

more rapidly. This can lead to an increase in invalid connections IC if the direction associated to more than one white247

11

matter bundle is used to propagate the streamline. This is especially visible on real data from the CS Rr metric (Figure248

6 (a), black curve). Thus, lower ✓ is preferred to reduce this e↵ect. Using deterministic tractography, the metrics on249

synthetic data tend to stabilize using ✓det > 45�. On real data, they stabilize for ✓det > 60�. Based on these results,250

we fix ✓det = 45�, which gives the best results on synthetic data. Probabilistic tractography shows higher values for251

VCCRs using angle in range 15� < ✓ < 30�. CS Rs reaches maximum value using 20� < ✓ < 40� (Figure 6 (a)). We252

set ✓prob = 20� since it provides good results on synthetic and real data.253

3.1.2. Fiber ODF Threshold ⌧ = 0.1 and ⌧init = 0.5254

Figure 6 (b) shows connection to seed ratio CS Rs, valid connection to connection ratio VCCRs and CS Rr varying255

the fiber ODF threshold ⌧. We set ✓ based on Section 3.1.1, keep other parameters to zero (⌧init = 0, �undeviated = 0 mm)256

and PFT is not use. The objective of the ⌧ parameter is to remove noisy directions from the fiber ODF, but keep true257

and small white matter volume fraction contributions in the fiber ODF. We see that ⌧ 2 [0.0 � 0.2] has little e↵ect258

on synthetic data for deterministic tractography. ⌧ > 0.4 tends to reduce CS Rs, removing some of the maxima.259

Increasing ⌧ parameter has a positive e↵ect on VCCRs using probabilistic tractography, but tends to decrease CS Rs260

using ⌧ > 0.5. CS Rr decreases with the increase of ⌧. Removing some of the propagation directions increases the261

stopping issue in the white matter, especially when tracking in low white matter partial volume fraction regions, where262

the fiber ODF has lower values. We thus fix ⌧ = 0.1 since it does not reduce the score of metrics on synthetic data.263

However, CS Rr is reduced by 15%. Figure 7 shows examples of streamline varying ⌧ seeding 1,000 streamlines using264

probabilistic tractography from a single voxel of the synthetic dataset. Thresholding the fiber ODF helps reducing265

erroneous streamlines. Then, we tested the e↵ect of varying ⌧init, using ⌧ = 0.1. It has little e↵ect on CS Rs, VCCRs,266

and CS Rr when ⌧init � 0.5 (see Figure 6 (c)). Lower value of ⌧init reduces CS Rs, especially using low SNR. We fix267

⌧init = 0.5. It provides better initial propagation direction that is tangent to the white matter bundle.268

3.1.3. Undeviated Propagation Distance �undeviated = 1 mm269

Figure 6 (d) shows how varying the value of the undeviated propagation distance �undeviated a↵ects tractography270

results. We set ✓ based on Section 3.1.1, ⌧ and ⌧init based on Section 3.1.2, and PFT is not use. Increasing �undeviated271

decreases the number of streamlines stopping in the white matter by allowing the tracking to propagate through regions272

where propagation direction are missing. �undeviated has little e↵ect on synthetic data, meaning that the propagation273

stops rarely in the white matter. This is not the case on real data, where �undeviated increases connection to seed ratio274

CS Rr, especially using deterministic tractography. Even if increasing the value of �undeviated tends to increase CS Rr,275

we see that bigger value of �undeviated produces more erroneous streamlines exiting the white matter bundles. This276

parameter has similar e↵ects as increasing the step size when no valid direction are available and thus, produces277

12

Determ

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(a) ✓ (b) ⌧ (c) ⌧init

Determ

in

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Pro

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(d) �undeviated (e) � f ront (f) �back

Figure 6: Valid connection to connection ratio (VCCRs), and connection to ceed ratio (CS Rs) obtained on the synthetic dataset and CS Rr obtainedon the brain dataset. (a) The maximum deviation angle ✓ (�), (b) the fiber ODF threshold ⌧, (c) the initial fiber ODF threshold ⌧init , (d) the maximumundeviated propagation distance �undeviated (mm), (e) the PFT forward tracking distance � f ront (mm), (f) the PFT backward tracking distance �back(mm). The vertical dashed line indicates the chosen value for each parameters.

similar behavior as using a bigger step size. For this reason, we set �undeviated to a maximum distance of 1 mm, half the278

size of the di↵usion space voxel size. This increases connection to seed ratio CS Rr (see Figure 6 (d), black curve) and279

produces qualitatively good results on real data, allowing streamlines to propagate through voxels where propagation280

directions are missing.281

13

(a) seed position (b) ⌧ = 0.0 (c) ⌧ = 0.1 (d) ⌧ = 0.2

Figure 7: A thousand streamlines were initiate at the seed voxel (a). (b,c,d) show resulting streamlines varying the fiber ODF threshold ⌧ parameteron probabilistic tractography. Increasing ⌧ reduce the number of erroneous streamlines.

3.1.4. Forward and Backward Distances �back = 2 mm, � f ront = 1 mm282

Figure 6 (e) quantifies the e↵ect of �back and � f ront distances using PFT on connection to seed ratio CS Rs, valid283

connection to connection ratio VCCRs, and CS Rr, using best parameters from Sections 3.1.1, 3.1.2 and 3.1.3. First,284

we set �back = 0 and vary � f ront, as seen in Figure 6 (e). We see that an increase of the connection to seed ratios with285

� f ront � 0.2 and a small decrease of VCCRs, especially for deterministic tractography. Based on these observations,286

we set � f ront = 1 mm to spatially stay close to the stopping issue, and then, parameter �back is varied. Figure 6 (f) shows287

an increase of the connection to seed ratios until �back � 2 mm, where it tends to stabilize. �back = 2 mm, � f ront = 1 mm288

provide good results and are relatively small.289

3.1.5. Number of Particles N = 25290

The number of particles N should be large enough to produce a good approximation of the distribution and small291

enough to keep to computation requirement low (Arulampalam et al., 2002; Doucet et al., 2001). In our experiments,292

we observe (results not shown) that all metrics are stable using N � 15. In this work, results are obtained using293

N = 25.294

SNR=10 SNR=20 SNR=30

Algorithm ABC

s

CS

Rs

VC

CR

s

ABC

s

CS

Rs

VC

CR

s

ABC

s

CS

Rs

VC

CR

s

Deterministic In-house 18.3 48.5 58.3 24.1 53.5 66.1 25.8 54.7 68.9In-housePFT 51.1 86.6 49.5 53.2 90.2 57.6 54.7 91.1 61.3

Probabilistic In-house 31.6 32.6 38.5 38.8 38.7 48.3 42.2 39.4 53.5In-housePFT 59.3 85.7 36.0 63.4 89.4 47.0 63.7 91.1 49.7

Table 1: Comparaison of deterministic and probabilistic tractography algorithms on synthetic data with SNR 10,20 and 30. In-house: trackingwithin a binary mask, In-housePFT : in-house tracking using CMC and PFT. All metrics are reported in %.

14

3.2. Connectivity Analysis on Synthetic Data295

Table 1 shows the average bundle coverage ABCs, the connection to seed ratio CS Rs and the valid connection296

to connection ratio VCCRs for our in-house probabilistic and deterministic tractography algorithms used with and297

without the Particle Filtering Tractography (PFT) (in-housePFT ). Results are shown on synthetic data at SNR 10,298

20 and 30. PFT increases the connection to seed ratio CS Rs by 37.1% on average using deterministic tractography299

and by 51.8% on average using probabilistic tractography. Out of the 6 experiments shown in Table 1, in-housePFT300

algorithms have on average 89.0% of connection to seed ratio CS Rs, against 44.6% for in-house algorithms. This301

means that on average, the tracking connects gray matter regions from a seed position twice often using particle302

filtering approach. However, in-housePFT shows a decrease in valid connection to connection ratio VCCRs by 8.3%303

on average using deterministic tractography and 2.5% on average using probabilistic tractography. VCCRs is always304

lower for probabilistic algorithms than for deterministic algorithms (see Table 1). Thus, the decrease of VCCRs is305

expected for in-housePFT deterministic tractography since a probabilistic algorithm is used with PFT. The average306

bundle coverage ABCs is always higher for in-housePFT , which suggests that that streamlines recovered by PFT307

propagate in regions previously not covered by streamlines produced by the default algorithms.308

Next, in Table 2, we study the e↵ect of Particle Filtering Tractography (PFT) on connection to seed ratio CS Rs309

on individual bundles previously shown in Figure 5, seeding from its WM/GM interface using S NR = 20 dataset.310

We grouped bundles by their size, and computed the average (µ) and the standard deviation (�) of CS Rs and VCCRs.311

As shown in Table 1, Table 2 shows a decreases in valid connections to connection ratio VCCRs using in-housePFT312

(deterministic: 7.1%, probabilistic: 3.6%) but the standard deviation � of VCCRs is reduced by 7.9% for deterministic313

tractography and 8.6% for probabilistic tractography. This decrease is higher for small white matter bundles. This314

Bundles AlgorithmsDeterministic Probabilistic

In-house In-housePFT In-house In-housePFT

Size(mm2) C

SR

s

VC

CR

s

CS

Rs

VC

CR

s

CS

Rs

VC

CR

s

CS

Rs

VC

CR

s

Small (11) µ 8.5 29.4 56.8 88.3 49.0 18.3 41.4 86.5 38.6� 1.2 19.7 35.6 5.8 23.2 18.7 34.7 4.7 22.9

Medium (4) µ 17.5 54.9 56.4 93.7 51.1 40.2 44.4 93.6 40.8� 2.1 12.9 6.0 2.9 5.4 15.7 16.5 2.7 7.1

Large (10) µ 33.3 60.3 68.3 95.2 61.1 45.8 56.2 94.5 51.7� 11.0 17.2 20.6 2.0 17.2 17.1 21.8 2.2 17.2

All (25) µ 19.8 45.8 61.3 91.9 54.2 32.8 47.8 90.8 44.2� 13.4 23.1 27.7 5.3 19.8 21.9 28.5 5.2 19.9

Table 2: Comparaison between in-house algorithms and in-house algorithms using PFT on synthetic bundles reconstruction. Bundles 14 and 18have been omitted because they share a common ending region, making CS Rs, VCCRs not practicable over these bundles independently. In-house:tracking within a binary mask, In-housePFT : in-house tracking using CMC and PFT. All metrics are reported in %.

15

means that, on average, the valid connection to connection ratio VCCRs is less biased by various bundle sizes and315

shapes. Most importantly, in-housePFT shows a clear increase of connection to seed ratio CS Rs using both proba-316

bilistic and deterministic tractography (see Table 2). It reflects more alternative connections found due to the stopping317

criterion relaxation of in-housePFT . CS Rs shows increases from 45.8% ± 23.1% to 91.9% ± 5.3% for deterministic318

tractography and from 32.8% ± 21.9% to 90.8% ± 5.2% for probabilistic tractography. This means that streamlines319

are more uniformly distributed amongst white matter bundles having various shapes and sizes. This is observed by a320

higher increase of CS Rs on small white matter bundle and a smaller standard deviation � of CS Rs among individual321

bundle reconstruction.322

3.3. Connectivity Analysis on Real Data323

Table 3 shows the distribution of included and excluded streamlines on real data. We quantify the increase of324

connection to seed ratio CS Rr and compare the streamlines density for bundles of various lengths. The streamlines325

are obtained seeding from the WM/GM interface. The streamlines included are those ending in the gray matter and326

having a length �min � 10 mm. The distribution of included and excluded streamlines in the brain (see Table 3)327

shows that for deterministic tractography, 50.0% of the seeds produced included streamlines and 21.2% of the seeds328

produced excluded streamlines either ending in the CSF or in the white matter (62.9% and 13.6% for probabilistic329

tractography respectively). Using PFT, the streamlines previously included (not using PFT) are exactly the same,330

but additionally 19.0% of the excluded streamlines are recovered by the particle filtering approach (indicated in the331

ExtraPFT column), using deterministic tractography and 10.7% using probabilistic tractography. These additional332

streamlines do not share the same length distribution as the previously included streamlines. This can be observed by333

the higher average length of the recovered streamlines (50.6mm and 54.6mm) than average length of the other included334

streamlines (32.5mm and 37.2mm), using deterministic and probabilistic algorithms respectively.335

Finally, Table 4 shows the streamlines count and their average length for several real data bundles using deter-336

Algorithm Included Streamlines Excluded StreamlinesExtraPFT CSF WM �min

Det

erm

inis

tic In-house 50.0% 2.6% 18.6% 28.8%32.5mm 42.8mm 29.7mm 7.2mm

In-housePFT50.0% 19.0% 0.1% 3.4% 27.6%32.5mm 50.6mm 57.2mm 34.5mm 6.7mm

Prob

abili

stic

In-house 62.9% 3.4% 10.2% 23.5%37.2mm 48.9mm 34.5mm 8.3mm

In-housePFT62.9% 10.7% 0.1% 3.2% 23.2%37.2mm 54.6mm 56.7mm 38.9mm 7.3mm

Table 3: Streamlines distribution seeding from WM/GM interface. Included streamlines end in the GM. ExtraPFT shows the pourcentage ofstreamlines included using PFT. Excluded streamlines either stop in the CSF or in the WM, or end in the GM but have a length smaller than�min = 10 mm.

16

Algorithm BundlesAll CST CC SLF ILF UF U1 U2

Det

erm

inis

tic

MRtrix (WM) 100,000 584 7,959 711 772 4 105 10942.5 mm 112.4 mm 86.6 mm 114.9 mm 84.5 mm 52.0 mm 24.9 mm 42.6 mm

In-house (WM/GM) 100,000 159 3,139 217 377 101 619 96132.4 mm 125.0 mm 96.2 mm 119.0 mm 92.4 mm 64.2 mm 18.8 mm 33.2

In-housePFT (WM/GM) 100,000 289 4,078 312 450 82 639 1,19437.3 mm 130.7 mm 96.4 mm 120.9 mm 93.7 mm 62.4 mm 31.4 mm 37.2

ExtraPFT (WM/GM) 100,000 603 6,982 526 621 57 700 1,64850.7 mm 136.6 mm 97.3 mm 125.3 mm 94.6 mm 68.9 mm 36.4 mm 43.8 mm

Prob

abili

stic

MRtrix (WM) 100,000 525 8,385 658 361 1 87 7541.8 mm 109.0 mm 71.4 mm 114.2 mm 98.3 mm 41.8 mm 26.6 mm 44.7 mm

In-house (WM/GM) 100,000 152 2,851 149 401 85 659 1,32636.9 mm 131.7 mm 98.7 mm 125.9 mm 101.3 mm 64.2 mm 33.2 mm 41.4 mm

In-housePFT (WM/GM) 100,000 242 3,742 156 495 69 714 1,23136.7 mm 136.6 mm 97.4 mm 127.7 mm 107.4 mm 68.3 mm 34.0 mm 41.9 mm

ExtraPFT (WM/GM) 100,000 756 8,373 204 684 55 884 1,18754.9 mm 142.5 mm 96.3 mm 131.8 mm 81.9 mm 71.8 mm 38.0 mm 47.4 mm

Table 4: Comparaison between MRtrix, in-house and in-housePFT algorithms on brain bundles. ExtraPFT are streamlines additionnaly includedusing PFT. The table shows the streamlines count among bundles and algorithms. The average length of the streamlines of each bundle is shown.From left to right: All streamlines, the corticospinal tract (CST), the Corpus Callosum (CC), the Superior Longitudinal Fasciculus (SLF), theInferior Longitudinal Fasciculus (ILF), the Uncinate Fasciculus (UF), the association fibers between the precentral gyrus and postcentral gyrus(U1) and the association fibers between the superior frontal gyrus and middle frontal gyrus (U2).

ministic and probabilistic tractography. Each experiment reports 100,000 included streamlines using the in-house337

and the in-housePFT algorithms, seeding from the WM/GM interface. For comparison, 100,000 streamlines with de-338

fault MRtrix parameters are reported. We also randomly select 100,000 streamlines included using the particle filter339

(ExtraPFT ). In-housesPFT streamlines can be seen as a fraction of in-house streamlines plus a fraction of ExtraPFT340

streamlines (previously excluded streamlines). We observe in Table 4 that shorter bundles (Uncinate Fasciculus (UF),341

and short association fibers U1 and U2 ) have a higher streamlines count using in-house and in-housePFT algorithms342

than using MRtrix (e.g. U1: 619, 639 and 105 streamlines using in-house, in-housePFT and MRtrix deterministic343

tractography respectively). Longer bundles (the corticospinal tract (CST), the corpus callosum (CC), the superior lon-344

gitudinal fasciculus (SLF) and the inferior longitudinal fasciculus (ILF)) are overrepresented seeding from the white345

matter (e.g. CST: 159, 289 and 584 streamlines using in-house, in-housePFT and MRtrix deterministic tractography346

respectively). However, the streamline count is generally higher for in-housePFT in long white matter bundles (e.g.347

CC: 3,139 and 4,078 streamlines using in-house and in-housePFT deterministic tractography respectively). This can348

be observed in Figures 8, 9, 10, and 11. Longer white matter bundles are well reconstructed using MRtrix, seeding349

from the white matter mask, because there are more seeds that are initiated in these bundles than in shorter bundles.350

However, seeding from the WM/GM interface with in-housePFT algorithms provides a less bias reconstruction of351

bundles with respect to their length. Finally, MRtrix reconstructed the UF with the lowest streamlines density. It is352

likely caused by both the seeding strategy and the use of a binary white matter tracking mask.353

17

Determ

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(a) MRtrix (b) In-house (c) In-housePFT (d) ExtraPFT

Figure 8: Deterministic and probabilistic tractography of the Corticospinal Tracts (CST) in sagittal and coronal views. (a) MRtrix, (b) in-housealgorithms with CMC, (c) in-house algorithms with CMC and PFT, (d) additonally included streamlines using PFT. In-housesPFT streamlines canbe seen as a fraction of in-house streamlines plus a fraction of ExtraPFT .

Determ

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Pro

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(a) MRtrix (b) In-house (c) In-housePFT (d) ExtraPFT

Figure 9: Deterministic and probabilistic tractography of the association fibers of between the superior frontal and the middle frontal gyrus (U1)in sagittal view. (a) MRtrix, (b) in-house algorithms with CMC, (c) in-house algorithms with CMC and PFT, (d) additonally included streamlinesusing PFT. In-housesPFT streamlines can be seen as a fraction of in-house streamlines plus a fraction of ExtraPFT .

18

Determ

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(a) MRtrix (b) In-house (c) In-housePFT (d) ExtraPFT

Figure 10: Deterministic and probabilistic tractography of the Corpus Callosum (CC) in coronal and axial views. (a) MRtrix, (b) in-housealgorithms with CMC, (c) in-house algorithms with CMC and PFT, (d) additonally included streamlines using PFT. In-housesPFT streamlines canbe seen as a fraction of in-house streamlines plus a fraction of ExtraPFT .

Determ

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Pro

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(a) MRtrix (b) In-house (c) In-housePFT (d) ExtraPFT

Figure 11: Deterministic and probabilistic tractography of the association fibers between the precentral and the postcentral gyrus (U1) in coronalview. (a) MRtrix, (b) in-house algorithms with CMC, (c) in-house algorithms with CMC and PFT, (d) additonally included streamlines using PFT.In-housesPFT streamlines can be seen as a fraction of in-house streamlines plus a fraction of ExtraPFT .

19

4. Discussion354

Optimal parameters. We used the Tractometer strategy (Cote et al., 2013) to investigate the influence of tractography355

parameters on synthetic and real datasets. Best tractography parameters were chosen in terms of two new global356

connectivity metrics: 1) Connection to seed ratio CS R and 2) Valid connection to connection ratio VCCR. The357

proposed metrics provide useful information to optimize tractography algorithms. Hence, using the deterministic and358

probabilistic tractography algorithms described in 2.1, we recommend ✓det 2 [45, 60]�, ✓prob 2 [20, 30]�, ⌧ 2 [0.1, 0.2],359

⌧init 2 [0.2, 0.5] and �undeviated set to a maximum of half the acquisition voxel size. Taken together, setting these360

tractography parameters accordingly is the first step towards reducing position, shape, size and length biases.361

Deterministic versus probabilistic tractography. We observed from Tables 1 and 2 that deterministic tractography362

always shows better performance in terms of valid connection to connection ratio VCCRs and similar or better perfor-363

mance in terms of connection to seed ratio CS Rs. Probabilistic tractography shows an average bundle coverage ABCs364

always higher with both in-house and in-housePFT algorithms. Deterministic tractography tends to reduce the propor-365

tion of invalid connection IC in comparison to probabilistic tractography, but decreases the average bundle coverage366

ABC. Thus, the tractography algorithm (deterministic or probabilistic) must be choose to be the most suitable for the367

streamline analysis (less IC or more ABC).368

New seeding, stopping and masking strategies. Through our novel Continuous Map Criterion (CMC) and Particle369

Filtering Tractography (PFT), we have showed injecting anatomical prior information into tractography seeding,370

stopping and masking reduces biases in the distribution of streamlines. CMC determines where are the valid and371

invalid stopping regions. PFT gives a more uniform distribution of streamlines, finding alternative valid pathways for372

streamlines stopping in invalid regions, such as in the white matter or in the CSF. It uses the partial volume estimation373

(PVE) maps to reduce the biases in long and curved bundles. It increases the average bundle coverage ABC and the374

connection to seed ratio CS R. Qualitatively, PFT provides a better coverage of knows white matter pathways of the375

brain and helps reducing bias in the streamlines distribution. We showed that this relaxation of the stopping criterion376

enhances the density of complex streamline bundles (e.g. high curvature or tight white matter pathways). This is in-377

line with recent works in the literature that anatomical information and filtering can help reduce tractography biases378

(Bloy et al., 2012; Li et al., 2012a; Smith et al., 2012, 2013).379

Reducing the position, shape and size biases. White matter bundles have various position, shape and size, making380

their reconstruction a challenge for tractography algorithms. Bundles positioned in partial volume of CSF are harder to381

completely reconstruct because streamlines propagation is more likely to be stopped. Narrow bundles are more likely382

20

to be a↵ected by error in the tracking mask that could stop the streamline propagation, making their reconstruction383

harder. Because tractography follow tangent directions of bundles, curved bundles are harder to reconstruct. Noise384

can make the tracking direction harder to follow in curved region, especially because discrete steps in the estimated385

tangent direction are taken. CMC reduces biases in position and size by making smooth boundary between distinct386

tissues. PFT reduces biases in position, shape and size by finding alternative pathways when errors in the propagation387

lead to premature stops.388

Reducing the length bias. There are two contrary e↵ects that bias the streamlines density due to white matter bundles389

length: i) seeding from the white matter increases the density because there are more streamlines that are initiated in390

longer bundles than in shorter bundles, ii) longer bundles are harder to completely recover because of premature stops391

(in WM or CSF), which decreases the streamlines density. Seeding from WM/GM interface reduces i) by initiating392

the propagation at extremities of white bundles. Thus, bundles of similar size, but various lengths, have a similar393

number of seeds initiated in them. The premature stop bias of ii) is reduced using CMC and PFT in the same fashion394

as the position, size and shape biases.395

In the end, in-housePFT generates streamlines connecting gray matter regions together with more than 95% of396

success rate for streamlines reaching a length of 10 mm, for both deterministic and probabilistic tractography. PFT397

improves streamlines reconstruction distribution and can be triggered in conjunction with any streamline tractography398

algorithm. Our results suggest that that streamlines recovered by PFT propagate in regions previously not covered by399

streamlines. PFT reduces the proportion of prematurely stopping streamlines and can have a positive e↵ect on brain400

connectivity studies.401

It is worth pointing out that tractography algorithms based on graph models or energy minimization method,402

often referred to as global tractography algorithms, can also encode anatomical information and enforce connections403

between gray matter regions. For instance, the graph-based tractography algorithm proposed in (Iturria-Medina et al.,404

2008) penalized pathways going through CSF PVE when searching for a shortest path. Global tractography techniques405

have shown promising results in recent years (Mangin et al., 2013) and their development is of interest. However, in406

most cases, anatomical information is not used in global tractography algorithms. Reconstructed pathways are thus407

not guaranteed to connect gray matter regions, prematurely stopping in the white matter. Incomplete pathways bias the408

structural connectivity analysis. Moreover, interpretation of connectivity based on global tractography is challenging,409

making ’classical’ streamline tractography often used in connectomics studies (Fornito et al., 2013).410

Finally, Particle Filtering Tractography (PFT) does not address the issue of invalid connections IC. Many included411

streamlines result from noise and errors in the propagation directions and manage to connect gray matter regions. They412

do not represent anatomical connections as such. In this sense, one of the next big challenge is to reduce the invalid413

21

connections and perform better brain structural connectivity estimation. We believe PFT, CMC and the proposed414

tractography parameters are important steps towards tackling this challenge.415

5. Conclusion416

We have shown that optimizing tractography parameters, stopping and seeding strategies can reduce the biases in417

position, shape, size and length of streamlines distribution. These tractography biases are quantitatively reported in418

both real and synthetic data. These findings are critical for future quantitative structural connectivity analysis. We have419

therefore proposed a novel framework for tractography. Information from the T1-weighted image must be included420

in tractography and can no longer be ignored. This represents a paradigm shift in tractography and strengthen the421

message that tractography cannot be a DW-MRI-only technique, as also proposed by Smith et al. (2012). Other prior422

information could be included from brain atlases, white matter bundles probability maps, blood vessels (Vigneau-423

Roy et al., 2013) map or functional connectivity maps. Our novel tractography framework is flexible to these future424

add-ons and is therefore promising for new developments in quantitative connectomics.425

6. Acknowledgment426

The authors wish to thank Emmanuel Caruyer, Ph.D. for the development and sharing of the synthetic data used427

in this study, and the Tractometer team (tractometer.org) for the tractography evaluation system.428

Appendix A. Streamline Tractography429

Appendix A.1. Local Reconstruction Technique430

Di↵usion Tensor estimation and corresponding Fractional Anisotropy (FA) map generation were done using MR-431

trix (Tournier et al., 2012). From this, the single fiber response function was estimated from all FA values above a432

threshold of 0.7, within the WM binary mask. This single fiber response was used as input to spherical deconvolution433

(Descoteaux et al., 2009; Tournier et al., 2007) to compute the fiber ODFs, with spherical harmonic order 8, at every434

voxel. In this work, we used the e�cient implementation publicly available in MRtrix (Tournier et al., 2012).435

Appendix A.2. Implementation Details436

In this study, we used deterministic and probabilistic streamline tractography algorithms. In our implementation,437

fiber ODFs are projected on a discrete evenly distributed symmetric sphere of 724 vertices (Daducci et al., 2013) and438

normalized (maximum=1). Propagation directions are always a vector of orientation corresponding to one vertex of439

22

the sphere and of length �s = 0.2 mm. ’Overshoot’ errors have been observed using a large �s in curved structure,440

and small �s increases the computation requirement and is prone to noise in di↵usion direction (Tournier et al., 2012).441

�s = 0.2 mm provides good results and it is consistent with observations made in (Tournier et al., 2012). The single442

di↵erence between probabilistic and deterministic algorithms is the way the propagation direction vi+1 is chosen.443

Given a position pi, a propagation direction vi, the maximum deviation angle ✓, and the fiber ODF threshold ⌧, the444

discrete set of potential propagation directions can be estimated: all discrete directions on the sphere with an associate445

value greater than ⌧ and within the aperture cone define by ✓ and vi. The maximum deviation angle ✓ between two446

consecutive steps (or a minimum radius of curvature R = �s2·sin(✓/2) (Behrens et al., 2007; Tournier et al., 2012)), limits447

the high angle variations of streamlines and addresses the smoothness assumption of WM fibers. The fiber ODF448

threshold ⌧ removes some of the noise directions of the fiber ODF. Given the discrete set of potential propagation449

directions, vi+1 is :450

• Deterministic: The chosen propagation direction vi+1 is the closest aligned maximum of the fiber ODF with the451

previous propagation direction. Maxima of the fiber ODF are defined as any values greater than all its neighbors452

in a cone of an angle of ⇡16 (⇡ 11�). Using the 724 vertices sphere, this provides good results ensuring a direction453

is the maximum in a neighborhood of 6 to 9 vertices. Other methods exist to extract maxima of the fiber ODF454

such as those proposed in (Bloy and Verma, 2008; Descoteaux et al., 2009; Ghosh et al., 2013; Tournier et al.,455

2012). Further investigation on the maxima extraction method on brain connectivity study are of interest but456

outside the scope of this paper.457

• Probabilistic: The chosen propagation direction vi+1 is drawn from the empirical distribution defined by the458

fiber ODF values of the potential propagation directions (Behrens et al., 2007; Parker and Alexander, 2005). The459

higher the value associated with a direction (vertex) is, the higher the probability of propagating the streamline460

in this direction is.461

The new tracking position is pi+1 = pi + �s · vi+1. If the discrete set of potential propagation directions is empty,462

vi+1 = vi. The tractography algorithm assumes an error in the fiber ODF and continues in the previous propagation463

direction. This is done for a maximum distance of �undeviated. This can be seen as allowing the step size �s to increase464

up to the size of �undeviated if there is no propagation direction locally available. MRtrix and most other algorithms stop465

the propagation if no valid direction is available.466

From an initial seeding position, the streamline propagates by making discrete steps of size �s in the initial467

propagation direction until a stopping criteria is reached. Then, the same is done in the opposite initial direction,468

creating the streamline. The initial propagation direction is obtained following the seeding strategy (see Section 2.4.469

23

The next propagation directions are obtained following the tractography algorithm.470

Once the tractography is done, streamlines with length within the interval [�min mm, �max mm] are included in the471

estimated set of streamlines and excluded otherwise. The minimum length criterion (�min) ensure that connections472

are between a minimal distanced gray matter regions. The maximum length (�max) criterion eliminates spurious473

streamlines that loop around or have impossible trajectories.474

Appendix B. Particle Filtering475

The particle filter model has been widely used for localization (Arulampalam et al., 2002; Doucet et al., 2001) us-476

ing sensor measurements to estimate position. Recently, it has also been used for white matter tractography (Pontabry477

and Rousseau, 2011; Savadjiev et al., 2010; Zhang et al., 2009). Particle filtering methods aim to estimate a sequence478

of target state variables X0:t = {Xk, k = 0, ..., t} from a sequence of observation variables Y0:t = {Yk, k = 0, ..., t}.479

The goal is to sequentially estimate the posterior distribution p(Xk |Y0:k). X0:t is a first order Markov process such480

that Xk |Xk�1 ⇠ p(Xk |Xk�1) with a known initial distribution p(X0) and Y0:k are conditionally independent if X0:k are481

known. The posterior distribution p(Xk |Y0:k) is represented by a set of random samples with associated weights and482

compute estimates of the target distribution based on the samples and weights (Arulampalam et al., 2002; Doucet483

et al., 2001). {x(i)k ,w

(i)k }Ni=1 denotes the set of N discrete random samples that characterize the posterior distribution,484

where {x(i)k , i = 1, ...,N} is the set of random samples, {w(i)

k , i = 1, ...,N} their associated weights. The weight of a485

sample x(i)k at time k corresponds to its weight at time k � 1 times the likelihood of the observation y(i)

k . Weights are486

then normalized over all particles to havePN

i=1 w(i)k = 1. Such a discrete model su↵ers of degeneracy since the variance487

of the weights increases over time, leading to a situation where all samples except one have a weight close to zero.488

To overcome this problem a resampling method is apply when a significant degeneracy is observed. The degeneracy489

problem can be observed when the number of e↵ective samples Ne f f falls below some threshold NT (Arulampalam490

et al., 2002). Ne f f is obtained following491

Ne f f = 1/NX

i=1

(w(i)k )2.

The resampling eliminates samples with low weights and concentrates on samples that have high weights. The resam-492

pling generates N new samples with equal weights from the current discrete estimation of p(Xk |Y0:k) (Arulampalam493

et al., 2002). In this study, the resampling is done when Ne f f < NT = N/10 (Arulampalam et al., 2002).494

24

Appendix C. Global Connectivity Metrics Defined in the Tractometer495

The Tractometer is a novel tractography evaluation system based on new global connectivity measures detailed in496

(Cote et al., 2013). Here, we recall them for completeness.497

• Valid Connections (VC): streamlines connecting expected regions of interest (ROIs) and not exiting the expected498

bundle mask (Cote et al., 2013) (see Figure C.1 (a)).499

• Invalid Connections (IC): streamlines connecting unexpected ROIs or streamlines connecting expected ROIs500

but exiting the expected bundle mask. These streamlines are spatially coherent, have managed to connect ROIs,501

but do not agree with the ground truth (Cote et al., 2013) (see Figure C.1 (b)).502

• No Connections (NC): streamlines that do not connect two ROIs. These streamlines either stop prematurely503

due for example to angular constraints or due to hitting the boundaries of the tracking mask (Cote et al., 2013)504

(see Figure C.1 (c)).505

• Average Bundle Coverage (ABC): Average of the number of voxels crossed by streamlines divided by the506

total number of voxels in the bundle (Cote et al., 2013). This is the average proportion of bundles covered by507

streamlines.508

(a) seed position (b) VC (c) IC (d) NC

Figure C.1: A thousand streamlines were initiate at the seed voxel (a). (b,c,d) show examples of Valid Connections (VC), Invalid Connections(IC) and No Connections (NC) on the synthetic dataset (Cote et al., 2013).

25

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