Chapter_3.lyx

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\begin_body

\begin_layout Section
Tracking with 
\begin_inset Newline newline
\end_inset

Euler Delta Crossings
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LatexCommand label
name "sec:Euler-Delta-Crossings"

\end_inset


\end_layout

\begin_layout Subsection
Overview
\end_layout

\begin_layout Standard
Tractography methods provide tools to resolve major neuronal fibre bundles
 non-invasively and in-vivo
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "Catani2002NeuroImage"

\end_inset

.
 Since the development of the first tractography algorithms
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\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "MCC+99,CLC+99"

\end_inset

 nearly 
\begin_inset Formula $12$
\end_inset

 years ago a great number of methods have been published.
 Often these algorithms depend strictly on the underlying voxel model or
 acquisition paradigm making it difficult for other researchers to apply
 their own reconstruction methods and evaluate their data sets.
\end_layout

\begin_layout Standard
In this work we designed a purely deterministic method which is fast, accurate
 and all-inclusive.
 Most importantly it can have as input model-based or model-free reconstruction
 algorithms of most known algorithms.
 We call this algorithm EuDX.
 Eu stands for Euler integration, D stands for Delta function which is a
 function that checks for many different stopping criteria and X stands
 for fibre crossings.
 EuDX can deal with any number of crossing fibres as long as the reconstruction
 algorithm supports them.
 The purpose of this algorithm is to be faithful to the reconstruction results
 rather than try to correct or enhance them by introducing regional or global
 considerations which is the topic of other methods reviewed below.
 Therefore, EuDX serves mainly as a robust method for quickly inspecting
 different reconstruction results using streamlines.
 EuDX is noise-friendly i.e.
 if a voxel is too noisy then EuDX will stop tracking on that voxel.
 This property is often useful when validating underlying reconstruction
 models.
 Branching is also supported by a combination of trilinear interpolation
 and propagation along multiple peaks per voxel.
 This method is an extension of the method used by Conturo et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "CLC+99"

\end_inset

 and Yeh et al.
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\end_inset


\begin_inset CommandInset citation
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key "Yeh2010"

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 with the additional support for propagation along multiple fibre directions.
\end_layout

\begin_layout Standard
In sections 
\begin_inset CommandInset ref
LatexCommand ref
reference "sub:Tractography"

\end_inset

, 
\begin_inset CommandInset ref
LatexCommand ref
reference "sub:Known-problems"

\end_inset

 we discussed some of the ideas and the problems behind the most popular
 propagation methods; deterministic and probabilistic.
 The focus of this section is to give a more general overview and introduce
 many more methods.
\end_layout

\begin_layout Standard
Most tractography techniques, as pointed out in Sotiropoulos thesis
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\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "sotiropoulos2010processing"

\end_inset

, can be grouped in three categories: a) local, b) global and c) simulated.
 Local approaches propagate a curve from a starting (seed) point using locally
 greedy criteria, i.e.
 tracking sequentially through orientation estimates in adjacent voxels.
 Global approaches identify the best path between two points of interest,
 according to some optimization criterion, rather than identifying paths
 arising from a single point.
 Simulated approaches comprise of algorithms that simulate the diffusion
 process or solve the diffusion equation to reconstruct white matter tracks.
 A detailed literature review is given below.
\end_layout

\begin_layout Subsubsection
Local
\end_layout

\begin_layout Standard

\series bold
Deterministic 
\series default
tractography was the first to appear.
 Tracks (also known as streamlines) are created as trajectories in the form
 of polylines; orthograde and retrograde along an initial direction at a
 specific point (seed) in the 3D volume.
 In FACT
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\end_inset


\begin_inset CommandInset citation
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key "MCC+99"

\end_inset

 tracks are propagated in unequal steps governed by the entry point of the
 streamline in the voxel (see Fig.
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status open

\begin_layout Plain Layout

~
\end_layout

\end_inset


\begin_inset CommandInset ref
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reference "fig:FACTvsEuDX"

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).
 Euler integration with equal steps was used in Conturo et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "CLC+99"

\end_inset

 and similarly Runge-Kutta integration was used in Basser et al.
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\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "BPP+00"

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.
 Deterministic approaches usually stop propagating when a low anisotropy
 region (usually FA < 
\begin_inset Formula $0.2$
\end_inset

) is found.
 This is useful in order to avoid propagation within the CSF where anatomical
 tracts do not exist or within deep gray matter regions where tracking is
 uncertain.
 They usually also check for large angular changes (e.g.
 larger than 
\begin_inset Formula $90{}^{\circ}$
\end_inset

) between successive steps to avoid unrealistically sharp turns.
 
\end_layout

\begin_layout Standard
Deterministic methods can also be utilized when multiple orientations are
 estimated in a single voxel (crossing fibres).
 These orientations can for example be obtained as the principal eigenvectors
 of multiple Tensors fitted to the data
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
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key "Tuch2002ThesisMIT"

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, or from the local peaks of the diffusion ODF estimated using DSI
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\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "wedeen2005mapping"

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 and QBI
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\end_inset


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key "Tuch2004"

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 or from the orientations from the fibre ODFs
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\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "tournier2004direct"

\end_inset

.
 There are different approaches for propagating across voxels where more
 than one fibre orientation has been identified.
 One approach is, upon entering a voxel, to choose the orientation that
 produces the smallest curvature with the incoming path used in Wedeen et
 al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "WWS+08"

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.
 Another approach follows all orientations that do not exceed a curvature
 threshold, by initiating a new streamline per orientation using in Chao
 et al.
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\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "chao2008multiple"

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 and Descoteaux et al.
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\begin_inset CommandInset citation
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key "Descoteaux2009"

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.
 
\end_layout

\begin_layout Standard
An interesting point is that most methods of this category utilize only
 the fibre orientation estimates.
 Tensor deflection tractography (TEND) proposed by 
\lang british
Lazar et al.
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\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "LWT+03"

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\lang english
 is a FACT variant that uses the whole DTI Tensor rather than just its principal
 eigenvector to determine the direction of curve propagation.
 
\end_layout

\begin_layout Standard
All the methods described up to this point provide binary connectivity informati
on i.e.
 a voxel B can be either connected or not connected to the seed S, depending
 on whether a streamline from S passes through B.
 
\end_layout

\begin_layout Standard
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
Streamline tractography utilizes only the fibre orientation estimates.
 In general, more information is available upon post-processing ofDWimages.
 For example, with DTI a diffusion tensor is computed.
 For perfectly spherical tensors the deflection angle is zero, while for
 oblate tensors, the deflection is towards the ellip- soid plane.
 Tensor deflection tractography is less sensitive to noise than streamline,
 as shown in simulations for straight tracts (Lazar et al., 2003).
 It can also propagate through regions of perpendicular fibre crossings,
 where the principal eigenvector of the underlying oblate Tensor is meaningless.
 However, Tensor deflection underestimates curvature for curved tracts.
 Furthermore, TEND results should be interpreted carefully, as an incoming
 direction that coincides with any of the Tensor eigenvectors will not be
 deflected by that Tensor; even if it is perpendicular to the principal
 eigenvector of a highly prolate Tensor (Lazar et al., 2003).
 All the methods presented in this section are deterministic and provide
 binary connectivity information.
 A voxel B can be either connected or not connected to the seed S, depending
 on whether a streamline from S passes through B.
 To tackle this issue, a group of probabilistic tractography approaches
 have been developed and these are presented in the following section.
\end_layout

\end_inset


\lang british
 
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout

\lang british
Other deterministic approaches can be found here A recent deterministic
 algorithm called Tensorlines
\begin_inset CommandInset citation
LatexCommand cite
key "weinstein1999tad"

\end_inset

 shares the problems of the other deterministic algorithms by failing in
 relatively sharp turns and Meyer's Loop.
 Furthermore, because these methods need a continuous field this necessitates
 the use of interpolation e.g.
 nearest, bicubic or trilinear.
 The amount and type of interpolation is another factor that can change
 dramatically the results of tractography and it is very rarely reported.
\end_layout

\end_inset


\end_layout

\begin_layout Standard

\series bold
Probabilistic
\series default
 tractography was introduced by Parker et al.
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\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "Parker2003"

\end_inset

 and Behrens et al.
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\end_inset


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LatexCommand cite
key "Behrens2003MRM"

\end_inset

.
 Here the standard procedure is to calculate a spatial distribution of tracks
 arising from a single seed rather than a single track.
 In each propagation step of each streamline, a random perturbation of the
 underlying fibre orientation estimate is followed.
 Perturbations are generated using functions that characterize the uncertainty
 in the fibre orientation within each voxel.
 A probabilistic index of connectivity (PICo) is defined between a seed
 and an arbitrary point as 
\begin_inset Formula $M/N$
\end_inset

 ; where 
\begin_inset Formula $N$
\end_inset

 is the number of all the tracks that start from the seed and 
\begin_inset Formula $M$
\end_inset

 is the number of tracks that traverse the seed and the arbitrary point.
\end_layout

\begin_layout Standard
Probabilistic approaches mainly differ in the way that the orientation uncertain
ty is assessed.
 Most commonly a Bayesian framework will be used to calculate the posterior
 probability of the reconstruction model's orientation parameters 
\begin_inset CommandInset citation
LatexCommand cite
key "Behrens2003MRM"

\end_inset

, 
\begin_inset CommandInset citation
LatexCommand cite
key "Behrens2007NeuroImage"

\end_inset

, 
\begin_inset CommandInset citation
LatexCommand cite
key "hosey2005inference"

\end_inset

, 
\begin_inset CommandInset citation
LatexCommand cite
key "friman2006bayesian"

\end_inset

 and 
\begin_inset CommandInset citation
LatexCommand cite
key "zhang2009probabilistic"

\end_inset

.
 In Behrens et al.
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\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "Behrens2003MRM"

\end_inset

, 
\begin_inset CommandInset citation
LatexCommand cite
key "Behrens2007NeuroImage"

\end_inset

 Monte Carlo-Markov chain (MCMC) was used to sample the orientation posterior
 distribution.
 In Friman et al.
 
\begin_inset CommandInset citation
LatexCommand cite
key "friman2006bayesian"

\end_inset

 the posterior was computed numerically after using Dirac priors.
 In Zhange et al.
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\begin_inset CommandInset citation
LatexCommand cite
key "zhang2009probabilistic"

\end_inset

 particle filtering was used for the same purpose.
\end_layout

\begin_layout Standard

\series bold
Bootstrap
\series default
 tractography is another method which characterizes the uncertainty of the
 fibre orientation.
 Pajevic et al.
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LatexCommand cite
key "pajevic2003parametric"

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 and Lazar et al.
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LatexCommand cite
key "lazar2005bootstrap"

\end_inset

 are two of the first to apply this method in dMRI.
 This is a non-parametric approach where a diffusion acquisition is repeated
 many times creating a large set of images for the same subject.
 Some images from this set are drawn in random with replacement.
 This process gives a single bootstrap sample.
 Drawing many samples will give a distribution for the fibre orientation.
 The advantage of bootstrap tractography is that no ad-hoc assumptions are
 made on the noise and it is sensitive to all sources of variability that
 affect the acquired data set.
 The disadvantage is that many repeated acquisitions are required; at least
 
\begin_inset Formula $5$
\end_inset

 for DTI according to O'Gorman et al.
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\end_inset


\begin_inset CommandInset citation
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key "o2006just"

\end_inset

.
 
\end_layout

\begin_layout Standard
Model-based residual bootstrap offers an alternative, since it requires
 only a single data acquisition (Chung et al.
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\end_inset


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key "chung2006comparison"

\end_inset

, Berman et al.
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key "berman2008probabilistic"

\end_inset

, Haroon et al.
\begin_inset space ~
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key "haroon2009using"

\end_inset

, Jones et al.
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\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "jones2008tractography"

\end_inset

).
 A single bootstrap sample can then be generated by permuting freely the
 residuals (or just the signs of the residuals using wild boostrap Jones
 et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "jones2008tractography"

\end_inset

, Whitcher et al.
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\end_inset


\begin_inset CommandInset citation
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key "whitcher2008using"

\end_inset

) between all model predicted values.
 The bootstrap technique was first introduced by Efron
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key "efron1979bootstrap"

\end_inset

 in 
\begin_inset Formula $1979$
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.
\end_layout

\begin_layout Standard

\series bold
Other probabilistic
\emph on
 
\series default
\emph default
approaches estimate the orientation uncertainty as an empirically defined
 function.
 For example, in Parker et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "Parker2003"

\end_inset

 this is determined using the value of FA.
 The higher the FA the higher the confidence on the principal eigenvector
 of the Tensor.
 In Parker and Alexander
\begin_inset space ~
\end_inset

 
\begin_inset CommandInset citation
LatexCommand cite
key "parker2003probabilistic"

\end_inset

, Monte-Carlo simulations are used to predict the orientation uncertainty
 for multiple Tensors and later for PAS
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "parker2005probabilistic"

\end_inset

.
 In Descoteaux et al.
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key "descoteaux2009deterministic"

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, the fODF was used for the same purpose.
 Cook et al.
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key "cook2004modelling"

\end_inset

 used a Watson distribution and Seunarine et al.
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key "seunarine2007exploiting"

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 used a Bingham distribution.
 The work of Bjornemo et al.
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\begin_inset CommandInset citation
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key "bjornemoMICCAI02"

\end_inset

 can also be classified in the same category who created a regularized stochasti
c method for probabilistic tractography.
 This method utilizes the principles of a statistical Monte Carlo method
 called Sequential Importance Sampling and Resampling (SISR).
 This technique is similar with particle filters.
 The disadvantage of the method is that it has strong assumptions for the
 Single Tensor as the reconstruction model.
 However, this is often the case with most tracking algorithms.
\end_layout

\begin_layout Subsubsection
Global
\end_layout

\begin_layout Standard
A limitation of probabilistic tractography is that the probabilistic index
 of connectivity decreases with distance from the seed point (see section
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\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "sub:Tractography"

\end_inset

, 
\begin_inset CommandInset ref
LatexCommand ref
reference "sub:Known-problems"

\end_inset

).
 Another limitation is that it is still sensitive to local noise.
 Global approaches try to overcome these limitations by being distance-independe
nt and by increasing resistance against noise in a global fashion.
 These are achieved by finding an optimal path between two voxels, according
 to a global property 
\begin_inset CommandInset citation
LatexCommand cite
key "DiffMRIBook"

\end_inset

.
\end_layout

\begin_layout Standard
Jbabdi et al.
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\begin_inset CommandInset citation
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key "jbabdi2007bayesian"

\end_inset

 developed a 
\series bold
Global Bayesian
\series default
 model to derive the posterior probability of connections.
 The path trajectories represented by splines are compatible with the local
 fibre orientations in regions with low uncertainty estimates.
 In regions with high uncertainty, the global connectivity information constrain
s the local parameter estimation and affects the path sampling.
\end_layout

\begin_layout Standard

\series bold
Front evolution 
\series default
techniques often employ fast marching techniques.
 The front expands from the seed neighbours to the next neighbouring nodes
 with speeds determined by the local fibre orientations.
 As the front propagates, a time of front arrival can be associated with
 each visited voxel.
 Once all image voxels have been traversed by the front, paths of connection
 can be obtained going backwards in the map of front arrival times.
 Starting from an arbitrary voxel, a gradient descent algorithm can find
 the fastest route back to the seed.
 A connectivity index can be associated with each path, representing either
 the weakest link along the path or the agreement between the path tangents
 and the underlying vector orientation field (see Parker et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
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key "parker2002estimating"

\end_inset

, Tournier et al.
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key "tournier2003diffusion"

\end_inset

, Cambell et al.
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key "campbell2005flow"

\end_inset

, Fletcher et al.
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key "fletcher2007volumetric"

\end_inset

 and Gigandet et al.
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key "gigandet2009global"

\end_inset

).
\end_layout

\begin_layout Standard

\series bold
Graph-based 
\series default
tractography utilises weighted networks (graphs).
 This type of tractography was presented by Iturria-Medina et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
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key "IturriaMedina2007NeuroImage"

\end_inset

, Zalesky et al.
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key "zalesky2008dt"

\end_inset

, Lifshits et al.
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key "lifshits2009combinatorial"

\end_inset

, Fillard et al.
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key "Fillard2009a"

\end_inset

 and Sotiropoulos et al.
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\begin_inset CommandInset citation
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key "sotiropoulos2010brain"

\end_inset

.
 The common concept of these approaches is that each image voxel becomes
 a node in the graph where the edges of the graph connect pairs of neighbouring
 voxels or ROIs.
 The edges are assigned weights that can be representative of any type of
 structural information.
 Anatomical paths are then defined as chains with successive elements being
 neighbouring voxels.
 The weights of the edges are used to determine the path strength.
 The strongest path between any image voxel and a seed can then be identified
 using algorithms that search efficiently the image graph.
\end_layout

\begin_layout Standard

\series bold
Energy Minimization
\series default
 methods 
\begin_inset CommandInset citation
LatexCommand cite
key "kreher2008gibbs"

\end_inset

 try to optimize all tracks from the whole brain volume simultaneously.
 Each tract is represented as a chain of cylinders, whose position and orientati
on can change.
 The method tries to find the set of cylinders that best approximate the
 underlying white matter bundles.
 This is achieved by minimizing the overall energy of all cylinders simultaneous
ly, mimicking natural phenomena e.g.
 the polymerization process which is a process of interacting monomer molecules
 together in a chemical reaction to form three-dimensional networks.
 Many standard algorithms e.g.
 gradient descent are usually employed with this framework; however Gibbs
 sampling is the most common in tractography.
 Kreher et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
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key "Kreher2008ISMRM"

\end_inset

, Reisert et al.
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\end_inset


\begin_inset CommandInset citation
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key "reisert2010global"

\end_inset

, Lazar et al.
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\end_inset


\begin_inset CommandInset citation
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key "lazar2002white"

\end_inset

 and Fillard et al.
\begin_inset space ~
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\begin_inset CommandInset citation
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key "Fillard2009"

\end_inset

 showed results using energy minimization.
 Despite the very promising results shown by Reisert et al.
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\begin_inset CommandInset citation
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key "reisert2010global"

\end_inset

, whose team won the Fibre Cup competition
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key "fillard2011quantitative"

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 the very high computation time was an important drawback of this framework.
\end_layout

\begin_layout Standard

\series bold
Microstructure Tracking 
\series default
is an exciting new family of algorithms that combine global tractography
 and direct microstructure estimation using diffusion weighted imaging data.
 Connectivity via tractography, axon diameter distribution and density estimates
 are all combined in order to inform one another given the common assumption
 that microstructural features remain consistent along fibers.
 MicroTrack
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "sherbondy2010microtrack"

\end_inset

 is a recent example of this category.
 Algorithms of this type require their own acquisition schemes, similar
 to those employed in ActiveAx, developed by Alexander et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "alexander2010orientationally"

\end_inset

.
\end_layout

\begin_layout Subsubsection
Simulated
\end_layout

\begin_layout Standard
This family of methods take a very different approach from what we have
 discussed up until this point.
 They simulate the diffusion of water mole- cules within the brain tissue
 or directly solve Fick’s second law of diffusion in the entire brain.
 In this category belong the work of Batchelor et al.
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\end_inset


\begin_inset CommandInset citation
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key "batchelor2001study"

\end_inset

, Kang et al.
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\begin_inset CommandInset citation
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key "kang2005white"

\end_inset

, Hageman et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
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key "hageman2009diffusion"

\end_inset

 and Hagmann et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
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key "hagmann2003dti"

\end_inset

.
\end_layout

\begin_layout Standard
In Batchelor et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "batchelor2001study"

\end_inset

, the diffusion equation is solved using a finite elements approach.
 Successive diffusion simulations over the entire brain, starting from a
 seed, are performed in Kang et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "kang2005white"

\end_inset

.
 Tractography by simulating fluid flow through a 
\begin_inset Quotes eld
\end_inset

pressure
\begin_inset Quotes erd
\end_inset

 Tensor field is performed in Hageman et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "hageman2009diffusion"

\end_inset

.
 The Navier-Stokes equation is solved using a finite elements approach.
 However, solving a partial differential equation increases execution time.
 Furthermore, it is not always easy with these approaches to obtain a connectivi
ty map across the whole brain volume and there is usually a large number
 of parameters to set.
 
\end_layout

\begin_layout Standard
For further understanding of the current map of all the different tractography
 algorithms see the reviews by Fillard et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "fillard2011quantitative"

\end_inset

, Sotiropoulos et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "sotiropoulos2010processing"

\end_inset

 and Jbabdi et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "Jbabdi2007"

\end_inset

.
 In Fillard et al.
 
\begin_inset CommandInset citation
LatexCommand cite
key "fillard2011quantitative"

\end_inset

 
\begin_inset Formula $10$
\end_inset

 tractographies were simulated using a novel hardware phantom.
 Comparing all these different methods which all have different parameters
 and are based on different underlying models is a difficult process and
 a ground truth from anatomical studies, digital or hardware phantoms is
 highly recommended.
\end_layout

\begin_layout Standard
\begin_inset Note Note
status open

\begin_layout Plain Layout
http://www.docstoc.com/docs/58019487/Diffusion-Tensor-MRI-at-3-Tesla
\end_layout

\end_inset


\end_layout

\begin_layout Subsection
The EuDX Algorithm
\end_layout

\begin_layout Standard
We created an algorithm that has many similarities with the classical determinis
tic methods
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "MCC+99,CLC+99,BPP+00"

\end_inset

 and with more recent ones as those described in Descoteaux et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "Descoteaux2009"

\end_inset

 and briefly in Yeh et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "Yeh2010"

\end_inset

.
 Our concentration was to create a more general tractography algorithm which
 can be used with very different families of anistropic functions and work
 well with multiple crossing fibres.
 This algorithm which we call EuDX is applied usually in native space image
 coordinates and it assumes that the voxel dimensions are equal in all three
 dimensions i.e.
 it assumes equal voxel size in all three dimensions.
 If the provided data do not have isotropic voxel size then a reslicing
 preprocessing step to isotropic is required.
 
\end_layout

\begin_layout Standard
In order to create tracks we need to provide initially one or more seed
 points 
\begin_inset Formula $S$
\end_inset

.
 These can be chosen randomly or we can specify them explicitly.
 However, these seed points need to be constrained by the volume's dimensions.
 Every seed point 
\begin_inset Formula $\mathbf{p}_{0}$
\end_inset

 becomes the starting point for the track propagation.
 For the integration we solve for 
\begin_inset Formula $\mathbf{p}_{t}=\mathbf{p}_{0}+\int_{0}^{t}\mathbf{v}(\mathbf{p}(\mathbf{s}))d\mathbf{s}$
\end_inset

 and we perform the integration numerically using Euler's method
\begin_inset Formula 
\begin{equation}
\mathbf{p}_{n+1}=\mathbf{p}_{n}+\mathbf{v}(\mathbf{p}_{n})\Delta s\label{eq: euler}
\end{equation}

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
noindent
\end_layout

\end_inset

 where 
\begin_inset Formula $\Delta s$
\end_inset

 is the propagation step size which should be at least smaller than the
 voxel size and 
\begin_inset Formula $\mathbf{v}$
\end_inset

 is the propagation direction.
 Alternatively, Runge-Kutta of 
\begin_inset Formula $2$
\end_inset

nd and 
\begin_inset Formula $4$
\end_inset

th order could be used.
 However, in this document we only experimented with Euler's method.
 
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

[th!]
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename EuDX_propagation.png
	lyxscale 30
	scale 70

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
In every voxel centre (black dot) there an one or more vectors.
 These vectors represent peaks where their length is equal to their anisotropy
 value and the direction is equal to the direction of the peak e.g.
 calculated from a given ODF.
 EuDX can track multiple peaks starting from a single seed point (star)
 if their anisotropy values are higher than a threshold.
 In that way we can track from the same seed towards different directions
 and support tracking in crossing areas as it is shown here.
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "Flo:EuDX_sketch"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
For EuDX's stopping criteria we can use a standard scalar function like
 FA but we can also use vector functions like Quantitative Anisotropy (QA)
 
\begin_inset CommandInset citation
LatexCommand cite
key "Yeh2010"

\end_inset

 or even the full Orientation Density Function (ODF)
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "aganj2010reconstruction"

\end_inset

.
 The only constraint for these functions is to be greater or equal to zero
 everywhere in the volume.
 In most cases, all these functions try to measure in some way the anisotropy
 of diffusion in every single voxel so we decided to use the letter 
\begin_inset Formula $\mathcal{A}$
\end_inset

 for the purpose of representing all these different functions applied on
 the image grid.
 Therefore, when we write 
\begin_inset Formula $\mathcal{A}(\mathbf{u}_{i})=\alpha_{i}$
\end_inset

 this reads for the peak unit direction of 
\begin_inset Formula $\mathbf{u}_{i}$
\end_inset

 the peak value was 
\begin_inset Formula $\alpha_{i}$
\end_inset

.
 For the simple case of FA, 
\begin_inset Formula $\mathbf{u}$
\end_inset

 is equal with the eigenvector corresponding to the highest eigenvalue,
 
\begin_inset Formula $\mathcal{A}(\mathbf{e})=FA$
\end_inset

.
 For QA which can allow for any number of peaks, where usually we constrain
 it to maximum of 
\begin_inset Formula $3$
\end_inset

, we use 
\begin_inset Formula $\mathcal{A}(\mathbf{u}_{i})=QA_{i}$
\end_inset

 where 
\begin_inset Formula $i\in[1,3]$
\end_inset

 denotes the current peak.
 The peak can be characterised by two things: (a) the anisotropy value 
\begin_inset Formula $\alpha_{i}$
\end_inset

 and (b) the unit direction of the peak 
\begin_inset Formula $\mathbf{u}_{i}$
\end_inset

.
 The concept of tracking with the combination of multiple peaks is presented
 in Fig.
\begin_inset ERT
status open

\begin_layout Plain Layout

~
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:EuDX_sketch"

\end_inset

.
 In order to reduce storing space, the vector 
\begin_inset Formula $\mathbf{u}$
\end_inset

 can be replaced by an index to the closest vertex of an evenly distributed
 and dense unit sphere.
 For generality and simplicity we require this indexing process even for
 the case of the Single Tensor where the peak direction is only one.
 Alternatively, we can see this process as a strategy which always maps
 any representation of the voxel on the sphere.
\end_layout

\begin_layout Standard
The EuDX algorithm can be described further in the following way: 
\begin_inset Formula $\mathcal{A}(\mathbf{u}_{i})$
\end_inset

 is estimated at every point of the volume.
 This represents a composite vector field where every point contains the
 peak directions superimposed to the anisotropy values.
 We create an empty list of tracks 
\begin_inset Formula $T=\emptyset$
\end_inset

 and then we generate random or prespecified seed points.
 In more detail: we select a seed point 
\begin_inset Formula $\mathbf{p}_{0}$
\end_inset

 and start propagating.
 We need to remember that the propagation direction can go forward and backward,
 or better said towards one direction (orthograde) and its opposite direction
 (retrograde).
 For the moment we only propagate towards one direction set by
\begin_inset Formula 
\begin{equation}
\mathbf{v}(\mathbf{p}_{0})=\argmax_{\mathbf{u}}\mathcal{A}(\mathbf{u})\label{eq:argmax_anisotropy}
\end{equation}

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
noindent
\end_layout

\end_inset

 but we need to remember to propagate also towards the opposite direction
 
\begin_inset Formula $-\mathbf{v}(\mathbf{p}_{0})$
\end_inset

.
 As 
\begin_inset Formula $\mathcal{A}$
\end_inset

 can have multiple values in each voxel (representing different peaks) we
 need to remember that when we finish with this track followed by direction
 
\begin_inset Formula $\mathbf{v}$
\end_inset

 we will need to propagate towards the direction of second peak, third peak
 etc.
 This is necessary if and only if 
\begin_inset Formula $\mathcal{A}$
\end_inset

 gives information for multiple peaks as it is common with QA (see section
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "sub:Quantitative-Anisotropy"

\end_inset

).
\end_layout

\begin_layout Standard
Apart from the direction, we also need to check when to stop tracking.
 EuDX takes as input a threshold for anisotropy 
\begin_inset Formula $\mathcal{A}_{thr}$
\end_inset

.
 If 
\begin_inset Formula $\mathcal{A}(\mathbf{p}_{n})<\mathcal{A}_{thr}$
\end_inset

 then EuDX stops propagating.
 Otherwise it appends the point to the current track.
 This can be useful for canceling out any seed points which are in the backgroun
d or in very low anisotropy areas where tracking is not recommended e.g.
 in the CSF.
 One important point here is that 
\begin_inset Formula $\mathcal{A}_{thr}$
\end_inset

 depends on the reconstruction method and it will have a different value
 for every different metric QA, FA, GFA etc.
 Therefore, we expect EuDX to give different results with different 
\begin_inset Formula $\mathcal{A}$
\end_inset

 functions.
\end_layout

\begin_layout Standard
In order to generate a smooth tractography it is recommended to use some
 kind of interpolation; this is in contrast with FACT which does not use
 neighbouring information.
 Here we have been using trilinear interpolation which works in the following
 way: the seed devides the neighbouring area (constrained by the centers
 of the neighbouring voxels) in 
\begin_inset Formula $8$
\end_inset

 regions in 3D (
\begin_inset Formula $4$
\end_inset

 in 2D) and the total contribution of the neighbouring points is added according
 to the weights 
\begin_inset Formula $w$
\end_inset

 of the antipodal side.
 The weights express subvolumes in 3D (subareas in 2D).
 The trilinear interpolation give us the weights 
\begin_inset Formula $w$
\end_inset

 that assign the contribution of the directions of the peaks of the neighbouring
 voxels to the seed's next direction.
 It is important to clarify at this point that trilinear interpolation is
 used only for interpolating the peak directions of the propagation and
 not the values of the peaks.
\end_layout

\begin_layout Standard
We describe here how we use the trilinear interpolation weights in order
 to find the next propagation direction.
 (a) We find the nearest direction from the seed's initial direction 
\begin_inset Formula $\mathbf{v}(\mathbf{p_{n}})$
\end_inset

 to every peak direction 
\begin_inset Formula $\mathbf{u}_{i}$
\end_inset

 of every one of the 
\begin_inset Formula $8$
\end_inset

 corners of the neighbourhood of the seed.
 (b) If 
\begin_inset Formula $\textnormal{arccos}(\mathbf{x}_{i},\mathbf{p}_{n})\leqq\theta_{thr}$
\end_inset

 we count the corresponding weight; otherwise we continue to the next weight.
 We simultaneously check for the condition 
\begin_inset Formula $\mathcal{A}(\mathbf{u})<\mathcal{A}_{thr}$
\end_inset

.
 (c) All the adjacent weights will contribute to the new direction according
 to the following formula
\begin_inset Formula 
\begin{equation}
\mathbf{v}^{'}(\mathbf{p}_{n})=\sum_{m}w_{m}\mathbf{v}(\mathbf{p}_{m})\label{eq:trilinear_weights}
\end{equation}

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
noindent
\end_layout

\end_inset

 where 
\begin_inset Formula $\mathbf{v}(\mathbf{p}_{m})=\dfrac{\mathbf{v}^{'}(\mathbf{p}_{m})}{||\mathbf{v}^{'}(\mathbf{p}_{m})||}$
\end_inset

 is normalized.
 d) The next point is calculated with Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "eq: euler"

\end_inset

.
 e) We insert the new point in the track and continue tracking until one
 of the stopping criteria is met.
 The next step will be to repeat the a-e steps for the opposite direction
 of the initial peak direction 
\begin_inset Formula $\mathbf{v}(\mathbf{p}_{o})$
\end_inset

 and for the smaller peaks as described above.
 Finally, we will have to repeat the procedure for the next seed point until
 all seed points are visited.
 When all seeds have been visited we will have in our hands the entire tractogra
phy 
\begin_inset Formula $T$
\end_inset

.
 A formal description of EuDX is given in Alg.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Alg:EuDX_Main"

\end_inset

 and Alg.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Alg:EuDX_Core"

\end_inset

.
\end_layout

\begin_layout Standard
Apart from the anisotropic threshold 
\begin_inset Formula $\mathcal{A}_{thr}$
\end_inset

 and angular threshold 
\begin_inset Formula $\theta_{thr}$
\end_inset

 other anisotropic criteria are also incorporated in EuDX.
 These are: a) The total sum of weights 
\begin_inset Formula $TW$
\end_inset

, which checks there is enough overall neighbouring contribution to continue
 tracking (default value of 
\begin_inset Formula $0.5$
\end_inset

).
 This is very useful in edges or corners where tracking should stop.
 b) It is possible for a track to get trapped in a loop and start looping
 for ever.
 We can check for that using a maximum length threshold or a threshold for
 maximum number of points describing a track 
\begin_inset Formula $MNP$
\end_inset

 (default maximum value 
\begin_inset Formula $1000$
\end_inset

 points).
 c) Finally, we need to check that we are always inside the image volume.
 The 
\begin_inset Formula $3$
\end_inset

 dimensions of the volume are hold in variable 
\begin_inset Formula $V$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float algorithm
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

[th!]
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
textbf{Input} $
\backslash
mathcal{A}$, $
\backslash
mathcal{I}$, $S$, $U$, $
\backslash
Delta s$, $V$ 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2.5em} $
\backslash
mathcal{A}_{thr}$,$
\backslash
theta_{thr}$, $TW$, $MNP$ 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
textbf{Output} $T$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash

\backslash

\end_layout

\begin_layout Plain Layout

$T 
\backslash
leftarrow 
\backslash
emptyset$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
textbf{For} seed in $S$ 
\backslash
textbf{Do} 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} 
\backslash
textbf{For} peak in ($
\backslash
mathcal{A}$, $
\backslash
mathcal{I}$) 
\backslash
textbf{Do} 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{4em} track $
\backslash
leftarrow$  EUDX
\backslash
_Core(seed,peak) 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{4em} 
\backslash
textbf{append}($T$,track) 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} 
\backslash
textbf{EndFor} 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
textbf{EndFor} 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
caption{EuDX -- All tracks}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "Alg:EuDX_Main"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
In Alg.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Alg:EuDX_Main"

\end_inset

 we see how EuDX creates an entire tractography with tracks grown from seeds
 
\begin_inset Formula $S$
\end_inset

.
 The core algorithm which is the same for every seed and every peak is given
 in Alg.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Alg:EuDX_Core"

\end_inset

.
 In Alg.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Alg:EuDX_Main"

\end_inset

 the input parameters are: 
\begin_inset Formula $\mathcal{A}$
\end_inset

 the 4D volume holding the peak values, 
\begin_inset Formula $\mathcal{I}$
\end_inset

 the 4D volume with the indexed directions of each peak in relation to unit
 sphere 
\begin_inset Formula $U$
\end_inset

.
 
\begin_inset Formula $U$
\end_inset

 is an array of size 
\begin_inset Formula $N\times3$
\end_inset

 where 
\begin_inset Formula $N$
\end_inset

 is the number of vertices in the sphere.
 
\begin_inset Formula $S$
\end_inset

 is an 
\begin_inset Formula $M\times3$
\end_inset

 array with the precomputed 
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\lang british
seeds.
 
\family default
\series default
\shape default
\size default
\emph default
\bar default
\noun default
\color inherit
\lang english

\begin_inset Formula $\Delta s$
\end_inset


\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\lang british
 is the propagation step size.
 We also input different stopping thresholds
\family default
\series default
\shape default
\size default
\emph default
\bar default
\noun default
\color inherit
\lang english
: 
\begin_inset Formula $\mathcal{A}_{thr}$
\end_inset

 defines the lowest possible peak value that allows tracking to continue.
 
\begin_inset Formula $\theta_{thr}$
\end_inset

 is the maximum allowed angle between the current propagation direction
 and the next direction.
 
\begin_inset Formula $TW$
\end_inset

 checks the overall contribution of the neighbourhood for the next propagation
 direction and 
\begin_inset Formula $MNP$
\end_inset

 checks that a track does not pass from the same point more than a number
 of times.
 
\begin_inset Formula $MNP$
\end_inset

 checks this condition by counting the number of current points appended
 in a track as the track grows.
 Usually, only parameters 
\begin_inset Formula $\mathcal{A}$
\end_inset

, 
\begin_inset Formula $\mathcal{I}$
\end_inset

 ,
\begin_inset Formula $S$
\end_inset

 and 
\begin_inset Formula $V$
\end_inset

 need to be updated with different datasets; the rest of the parameters
 can have default values which can be used across all experiments.
\end_layout

\begin_layout Standard
\begin_inset Float algorithm
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
textbf{Input} seed, peak, $
\backslash
Delta S$, $V$ 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
textbf{Output} track  
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash

\backslash

\end_layout

\begin_layout Plain Layout

$track 
\backslash
leftarrow 
\backslash
emptyset$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

delta, i
\backslash
_direction $
\backslash
leftarrow$ Initial
\backslash
_Direction(seed) 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
#propagate orthograde 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

direction $
\backslash
leftarrow$ i
\backslash
_direction 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
textbf{While} delta is True 
\backslash
textbf{Do}
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} delta, n
\backslash
_direction $
\backslash
leftarrow$ New
\backslash
_Direction(direction)
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} 
\backslash
textbf{If} delta is False 
\backslash
textbf{Do} 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{4em} 
\backslash
textbf{Break} 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} 
\backslash
textbf{EndIf} 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} point $
\backslash
leftarrow$ point + $
\backslash
Delta s$$
\backslash
times$n
\backslash
_direction 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} 
\backslash
textbf{append}(track,point)
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} direction $
\backslash
leftarrow$ n
\backslash
_direction 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
textbf{EndWhile} 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

delta $
\backslash
leftarrow$ True 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
#propagate retrograde 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

direction $
\backslash
leftarrow$ - i
\backslash
_direction 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
textbf{While} delta is True 
\backslash
textbf{Do} 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} 
\backslash
#Same as above 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
caption{EuDX
\backslash
_Core}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "Alg:EuDX_Core"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
The core part of EuDX is given in Alg.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "Alg:EuDX_Core"

\end_inset

.
 This is called for every seed and every peak.
 The algorithm starts by calling the function Initial_Direction which finds
 the closest peak direction to follow from the nearby voxels.
 If the values of these peaks are lower than 
\begin_inset Formula $\mathcal{A}_{thr}$
\end_inset

 then it returns False otherwise it returns True and the vector with the
 initial direction.
 The variable delta is used as check point that no stopping criteria are
 met.
 As long as delta is True we can continue tracking using Euler integration
 (see Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "eq: euler"

\end_inset

).
 The function New_Direction calculates the new direction of the propagation
 by accumulating the weights created by trilinear interpolation as shown
 in Eq.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:trilinear_weights"

\end_inset

 and finds the nearest direction for the eight neighbouring voxels.
 It also checks for the total weight threshold 
\begin_inset Formula $TW$
\end_inset

 and if we are inside the volume's boundaries given by 
\begin_inset Formula $V$
\end_inset

.
 After this is done we update the points as shown in Eq.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq: euler"

\end_inset

 and repeat the same procedure until one of the stopping criteria is met
 by checking variable delta.
 EuDX stems its D letter from that delta function.
 After tracking towards the orthograde direction has stopped and delta is
 now False, we repeat the same procedure as before for the retrograde direction
 in order to create the complete track (see Alg.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Alg:EuDX_Core"

\end_inset

).
 
\end_layout

\begin_layout Standard
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
Initial_Direction find the closest direction to follow if all A_thr are
 low then it returns False (Delta)
\end_layout

\begin_layout Plain Layout
\begin_inset Float algorithm
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
textbf{Input} seed, peak, $A$ 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
textbf{Output} True/False, direction  
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

point=Nearest
\backslash
_Interpolation(seed)
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
textbf{If} $A$[point]< aniso
\backslash
_thr 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} return False 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
textbf{Else}
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} return True, direction
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
caption{EuDX Initial Direction}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "Alg:EuDX Initial Direction"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
Calculates the weights for trilinear interpolation and finds nearest direction
 / check total_weight threshold (max total weight is 1- that a peak has
 enough weight so we are close at a peak and that all other peaks of the
 neighbourhood contribute.
\end_layout

\begin_layout Plain Layout
\begin_inset Float algorithm
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
textbf{Input} 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
textbf{Output} 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

weights=Trilinear
\backslash
_Interpolation(seed)
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
textbf{If} out of volue  
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} return False 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

total
\backslash
_weight=0 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
textbf{Else}
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} return True, direction
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
textbf{For} 8 corners
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} Delta, direction = Nearest
\backslash
_Direction
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} 
\backslash
textbf{If} Delta 
\backslash
textbf{is} False
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{4em} 
\backslash
textbf{Continue}
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} total
\backslash
_weight += W[corner]
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} new
\backslash
_direction += W[corner]$
\backslash
times$direction
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
textbf{If} total
\backslash
_weight < total
\backslash
_weight
\backslash
_thr
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{4em} return False
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
textbf{Else}
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{4em} return True, normalized(new
\backslash
_direction)
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
caption{EuDX New Direction}
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "Alg:EuDX New Direction"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
Just compares and finds the current direction with the peak direction and
 gives back the neirest or False.
\end_layout

\begin_layout Plain Layout
\begin_inset Float algorithm
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
textbf{Input} seed, peak, $A$ 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
textbf{Output} True/False, direction  
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout


\backslash
textbf{If} maximum
\backslash
_peak < aniso
\backslash
_thr 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} return False 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
textbf{For} all peaks 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} 
\backslash
textbf{If} peak < $A$
\backslash
_thr 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{4em} 
\backslash
textbf{Break}
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} angle = angle(peak direction, current direction)
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
textbf{If} maximum
\backslash
_angle > angle
\backslash
_thr 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hspace*{2em} return False
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
textbf{Else} return True, direction (closest peak - angular)
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
caption{EuDX Nearest Direction}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "Alg:EuDX Nearest Direction"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Results with software phantoms
\begin_inset CommandInset label
LatexCommand label
name "sub:Results-with-digital"

\end_inset


\end_layout

\begin_layout Standard
In order to validate the performance of EuDX we created a 3D software phantom
 with the method described in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sub:Digital-Phantoms"

\end_inset

 using eigenvalues 
\begin_inset Formula $\lambda_{\parallel}=1.7\times10^{-3}$
\end_inset

 
\lang british

\begin_inset Formula $m^{2}/sec$
\end_inset


\lang english
 and 
\begin_inset Formula $\lambda_{\perp}=0.1\times10^{-3}m^{2}/sec$
\end_inset

 which are in the range typically found in the human brain 
\begin_inset CommandInset citation
LatexCommand cite
key "canalesrodriguez2009mdq"

\end_inset

.
 The software phantom consists of two parts: a) a diagonal orbit and b)
 an elliptical orbit with axes ratio 
\begin_inset Formula $\nicefrac{\lambda_{2}}{\lambda_{1}}=0.6$
\end_inset

.
 Both parts were added together to create a crossing configuration and partial
 volume effects are assumed negligible.
 The average thickness of both parts was 
\begin_inset Formula $5$
\end_inset

 voxels.
 The elliptical orbit did not complete a full ellipse in order to avoid
 creating looping tracks which have no anatomical relevance.
 We created two experiments using the same data set of size 
\begin_inset Formula $64\times64\times64\times102$
\end_inset

 where the 
\begin_inset Formula $3$
\end_inset

 first dimensions give the size of the volume and the last dimension is
 the number of diffusion weighted volumes (
\begin_inset Formula $101$
\end_inset

) plus 
\begin_inset Formula $1$
\end_inset

 unweighted b0 volume (see Fig.
\begin_inset ERT
status open

\begin_layout Plain Layout

~
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:crossing_eudx_sims"

\end_inset

(i) lower right corner).
 We generated b-vectors and b-values by using a keyhole Cartesian sampling
 grid 
\begin_inset CommandInset citation
LatexCommand cite
key "Tuch2002ThesisMIT"

\end_inset

 (see Fig.
\begin_inset ERT
status open

\begin_layout Plain Layout

~
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Fig:q-space"

\end_inset

); therefore, DSI, GQI, EIT or DTI were all suitable for the signal reconstructi
on into ODFs or Tensors respectively for these type of data.
 In contrast, QBI is not suitable because it assumes a spherical grid in
 q-space (see section
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "sub:Q-space-reconstruction"

\end_inset

).
 For both experiments we used a high SNR of 
\begin_inset Formula $100$
\end_inset

 as the main goal was to validate the algorithm on good conditions.
 We will discuss later the validation process with human data sets where
 SNR is naturally lower.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename last_figures/crossing_eudx.png
	lyxscale 30
	scale 45

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
(i) Tractrography generated by EuDX with DSI as the reconstruction method
 on a software phantom containing two intersecting bundles.
 A slice from the b0 volume of the phantom is also shown at the lower right
 corner.
 (ii) and (iii) In higher resolution we can see that tracks traveling from
 the two different bundles cross unimpeded in the intersection area of the
 two simulated bundles.
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "Flo:crossing_eudx_sims"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
In the first experiment (see Fig.
\begin_inset ERT
status open

\begin_layout Plain Layout

~
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:crossing_eudx_sims"

\end_inset

) we generated 
\begin_inset Formula $200,000$
\end_inset

 uniformly sampled random seeds in the entire 3D volume of the phantom.
 We used DSI reconstruction with standard parameters: q-space grid size
 
\begin_inset Formula $16$
\end_inset

, hanning filter width 
\begin_inset Formula $32$
\end_inset

, and radial integration range from 
\begin_inset Formula $2.1$
\end_inset

 to 
\begin_inset Formula $6$
\end_inset

 at steps of 
\begin_inset Formula $0.2$
\end_inset

.
 As we presented earlier, EuDX expects as input the peaks and the directions
 of the peaks.
 For every voxel we used DSI to create the corresponding ODF (sampled on
 an evenly distributed sphere of 
\begin_inset Formula $642$
\end_inset

 vertices and 
\begin_inset Formula $1,280$
\end_inset

 faces) and from the ODF we used the peak finding function introduced in
 section
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "sub:Peak-Finding"

\end_inset

.
 We further removed the peaks which had values less that 
\begin_inset Formula $70\%$
\end_inset

 of the highest peak and normalized the rest so that the maximum peak equals
 to 
\begin_inset Formula $1$
\end_inset

.
 From now on we will call this output function from the ODF; PK (PeaK anisotropy
).
 PK is different from QA because we do not remove the isotropic component
 neither we normalize with the overall maximum ODF value as it is common
 practice with QA (see section
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "sub:Quantitative-Anisotropy"

\end_inset

).
 We also calculated FA and zeroed the peaks with FA values lower than 
\begin_inset Formula $0.2$
\end_inset

.
 In that way we ensured that there will be no tracking in the background
 area of the phantom.
\end_layout

\begin_layout Standard
For EuDX we used parameters 
\begin_inset Formula $\mathcal{A}_{thr}=0.2$
\end_inset

, 
\begin_inset ERT
status open

\begin_layout Plain Layout

$
\backslash
Delta s=0.5$
\end_layout

\end_inset

, 
\begin_inset Formula $\theta_{thr}=60^{\circ}$
\end_inset

 and 
\begin_inset Formula $TW=0.5$
\end_inset

.
 The results are shown in Fig.
\begin_inset ERT
status open

\begin_layout Plain Layout

~
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:crossing_eudx_sims"

\end_inset

.
 The tracks are colour-coded with their orientation; this is defined as
 the unit vector connecting the first with the last point of each track.
 Because we generated random seeds that went everywhere in the phantom some
 of them had to fall in the crossing area.
 In accordance with what we discussed in the previous section we know that
 if a seed falls in a crossing region, EuDX will propagate towards both
 directions of the crossing if and only if multiple peaks are supported
 from the underlying reconstruction method.
 We can confirm this by observing the result at Fig.
\begin_inset ERT
status open

\begin_layout Plain Layout

~
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:crossing_eudx_sims"

\end_inset

.
 We can see that the crossing area is well represented and that the propagation
 was successful.
 We can also observe that not all tracks travel the entire distance from
 end to end.
 Otherwise we would expect to see all tracks of the entire elliptic part
 with one colour and the tracks of the straight orbit all with a different
 colour because they have different orientation.
 It seems that this observation had something to do with the discrete nature
 of the phantom especially near the bundles' boundary areas and of course
 the functionality of EuDX.
 For this purpose, we created a different experiment to evaluate this finding.
 
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename last_figures/eudx_mask_tracks_sims.png
	lyxscale 30
	scale 60

\end_inset


\begin_inset Caption

\begin_layout Plain Layout
On the left panel we see the b0 volume of the software phantom with 
\begin_inset Formula $4$
\end_inset

 ROIs; A, B, C and D presented with different colours.
 
\begin_inset Formula $2,000$
\end_inset

 seeds were generated inside each ROI and we then measured the amount of
 tracks which reached the other regions.
 On the right panel we see that when the DTI (Single Tensor) was used for
 the reconstruction; the tracking started from A reached only C.
 However, when EITL was used the tracks reached both areas B and C as it
 was expected.
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "Flo:eudx_fromABCD"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
In the second experiment we generated 
\begin_inset Formula $2,000$
\end_inset

 seeds inside specific regions of the bundles (see Fig.
\begin_inset ERT
status open

\begin_layout Plain Layout

~
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:eudx_fromABCD"

\end_inset

).
 More precisely, the end-point areas of the phantom denoted with A, B, C
 and D.
 The goal was to count the percentage of tracks which reached any of the
 other end-point areas.
 We tried this with 
\begin_inset Formula $4$
\end_inset

 different reconstruction methods: Single Tensor, EITL, DSI and GQI.
 The Single Tensor was fitted using weighted least squares.
 For EITL we used for radial sampling from 
\begin_inset Formula $0$
\end_inset

 to 
\begin_inset Formula $5$
\end_inset

 with steps of 
\begin_inset Formula $0.4$
\end_inset

 and Gaussian weighting of 
\begin_inset Formula $0.05$
\end_inset

.
 GQI had sampling length of 
\begin_inset Formula $1.2$
\end_inset

 and DSI had the same parameters as in the previous experiment.
 The orientations of the peaks for all the methods were found in a reconstructio
n sphere of 
\begin_inset Formula $642$
\end_inset

 vertices.
 The same peaking function was also used with the ODFs of EITL, GQI and
 DSI.
 For these 
\begin_inset Formula $3$
\end_inset

 methods the peaking procedure was the same as described in the previous
 experiment.
 The single peak of the DTI reconstruction was found by the eigenvector
 corresponding to the maximum eigenvalue.
 EuDX expects the orientation input of the peaks as indices on a unit sphere.
 Therefore, for the Single Tensor case; for the eigenvector which corresponds
 to the maximum eigenvalue we found the vertex in the reconstruction sphere
 with the minimum angular distance.
 An alternative way would be to calculate the ODF directly from the Tensor
 in an analytical way.
 Then we could continue as we do with the other methods i.e.
 find the peaks in the ODF with our standard peak finding method.
 We did not follow this approach because theoretically the orientation of
 the peak of the Tensor ODF is identical with the direction of the eigenvector
 which corresponds to the maximum eigenvalue and the reconstruction sphere
 is densely sampled.
 
\end_layout

\begin_layout Standard
The results of this study are summarized in Tab.
\begin_inset ERT
status open

\begin_layout Plain Layout

~
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Tab:EuDX_AtoB"

\end_inset

.
 A first general observation is that independently from which end-point
 ROIs the tracks start there will always be a percentage of tracks which
 will stop before reaching the other end-points.
 We can clearly see this phenomenon by observing the last column of each
 sub-table.
 The last column that we symbolize with 
\begin_inset Formula $\emptyset$
\end_inset

 holds the percentage of tracks which did not reach any of the other end-points
 e.g.
 tracks which started from seeds in region A never reached regions B, C
 or D.
 We can observe in these columns that the best case was 
\begin_inset Formula $3.7\%$
\end_inset

 loss from reconstructions using EITL and the worst case was 
\begin_inset Formula $43.5\%$
\end_inset

 loss using DTI.
\end_layout

\begin_layout Standard
To simplify comparisons within Tab.
\begin_inset ERT
status open

\begin_layout Plain Layout

~
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Tab:EuDX_AtoB"

\end_inset

 we can use a maximum standard error of difference (MaxSED) between any
 two corresponding entries in any pair of rows.
 MaxSED
\begin_inset Formula $=100z_{\alpha/2}\sqrt{2p(1-p)/n}$
\end_inset

, where 
\begin_inset Formula $z_{\alpha/2}=\textrm{\alpha/2}$
\end_inset


\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
th percentile of standard normal distribution,
\family default
\series default
\shape default
\size default
\emph default
\bar default
\noun default
\color inherit
 
\begin_inset Formula $n=2,000$
\end_inset

 is the number of seeds in each row (ROI), and 
\begin_inset Formula $p=0.5$
\end_inset

 gives the worst case (largest) standard error
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
.
 
\begin_inset Formula $p$
\end_inset

 is the probability that tracks from seeds in the row ROI will terminate
 in the column ROI.
 If we take 
\begin_inset Formula $\alpha=0.0001$
\end_inset

, 
\begin_inset Formula $z_{\alpha/2}=3.29$
\end_inset

, then MaxSED
\begin_inset Formula $=5.2\%$
\end_inset

.
 
\family default
\series default
\shape default
\size default
\emph default
\bar default
\noun default
\color inherit
The same value can be used as a conservative MaxSED in comparisons involving
 Tab.
\begin_inset ERT
status open

\begin_layout Plain Layout

~
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:PMT_table"

\end_inset

 for which the effective 
\begin_inset Formula $n$
\end_inset

 is greater than 
\begin_inset Formula $2,000$
\end_inset

.
 Differences greater than MaxSED are highly significant (
\begin_inset Formula $p<0.0001$
\end_inset

).
 
\end_layout

\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
small
\backslash
addtolength{
\backslash
tabcolsep}{-5pt}
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="3" columns="2">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="6">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
DTI
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
A
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
B
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
C
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
D
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\emptyset$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
A
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

--
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
76.4%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
23.6%
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
B
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

--
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
59.8%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
40.2%
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
C
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
79.9%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

--
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
20.1%
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
D
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
56.5%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

--
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
43.5%
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
hspace{1.5em}
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="6">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
EITL
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
A
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
B
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
C
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
D
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\emptyset$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
A
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

--
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
63.4%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
8.7%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
27.9%
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
B
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
65.6%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

--
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
5.6%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
28.8%
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
C
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
14.5%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

--
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
76.8%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
8.7%
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
D
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.5%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
95.8%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

--
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3.7%
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
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\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="6">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
DSI
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
A
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
B
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
C
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
D
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\emptyset$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
A
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

--
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
65.3%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
9.6%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.0%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
25.1%
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
B
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
72.6%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

--
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.0%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
5.5%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
21.9%
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
C
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
14%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.0%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

--
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
79.9%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6.1%
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
D
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.0%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
10.6%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
84.8%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

--
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4.6%
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
hspace{1.5em}
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="6">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
GQI
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
A
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
B
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
C
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
D
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\emptyset$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
A
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

--
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
57.8%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
8.7%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.0%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
33.5%
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
B
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
67.2%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

--
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.0%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
10.7%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
22.1%
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
C
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
37.5%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.0%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

--
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
55.5%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
7%
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
D
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.0%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
22.3%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
72.3%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

--
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
5.4%
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Every sub-table shows the percentage of tracks which started from areas
 A, B, C or D (rows) and reached the other areas A, B, C, D or 
\begin_inset Formula $\emptyset$
\end_inset

 (columns) using EuDX with input from different reconstruction methods DTI,
 EITL, DSI, GQI.
 The column 
\begin_inset Formula $\emptyset$
\end_inset

 symbolizes the number of tracks which did not reach any of the A, B, C,
 D areas.
 For example, by looking only the first row of each sub-table we easily
 observe that the crossing area was well represented by EITL, DSI and GQI
 but not from DTI as it was expected because the Single Tensor cannot resolve
 crossings.
 We can also observe by comparing all the rows that in the more curve branch
 AB of the phantom fewer tracks reached their target than in the diagonal
 branch CD.
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "Tab:EuDX_AtoB"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
When DTI is compared with any of EITL, DSI and GQI, each of the connections
 
\begin_inset Formula $A\rightarrow\emptyset$
\end_inset

, 
\begin_inset Formula $B\rightarrow\emptyset$
\end_inset

, 
\begin_inset Formula $C\rightarrow\emptyset$
\end_inset

, 
\begin_inset Formula $D\rightarrow\emptyset$
\end_inset

 is significantly more frequent with DTI.
 Similarly 
\begin_inset Formula $A\rightarrow B$
\end_inset

, and 
\begin_inset Formula $B\rightarrow A$
\end_inset

 are less frequent with DTI, as are 
\begin_inset Formula $C\rightarrow D$
\end_inset

, and 
\begin_inset Formula $D\rightarrow C$
\end_inset

.
 By contrast 
\begin_inset Formula $A\rightarrow C$
\end_inset

, and 
\begin_inset Formula $C\rightarrow A$
\end_inset

 are more frequent with DTI, as are 
\begin_inset Formula $B\rightarrow D$
\end_inset

, and 
\begin_inset Formula $D\rightarrow B$
\end_inset

.
\end_layout

\begin_layout Standard
Between EITL and DSI we note that for tracks starting in 
\begin_inset Formula $B$
\end_inset

, 
\begin_inset Formula $B\rightarrow\emptyset$
\end_inset

 more frequently with EITL whereas 
\begin_inset Formula $B\rightarrow A$
\end_inset

 less frequently.
 For tracks starting in 
\begin_inset Formula $D$
\end_inset

, 
\begin_inset Formula $D\rightarrow C$
\end_inset

 more frequently with EITL than DSI , and 
\begin_inset Formula $D\rightarrow B$
\end_inset

 less frequently with EITL then DSI.
\end_layout

\begin_layout Standard
Between EITL and GQI, for tracks starting in 
\begin_inset Formula $A,$
\end_inset

 
\begin_inset Formula $A\rightarrow\emptyset$
\end_inset

 less often with EITL than with GQI, whereas 
\begin_inset Formula $A\rightarrow B$
\end_inset

 and 
\begin_inset Formula $A\rightarrow C$
\end_inset

 occur more often.
 However 
\begin_inset Formula $B\rightarrow\emptyset$
\end_inset

 more often for EITL than GQI.
 By contrast 
\begin_inset Formula $C\rightarrow D$
\end_inset

 occurs more frequently, and 
\begin_inset Formula $C\rightarrow A$
\end_inset

 less frequently.
 Also 
\begin_inset Formula $D\rightarrow B$
\end_inset

 occurs less.
\end_layout

\begin_layout Standard
Finally, comparing DSI with GQI, 
\begin_inset Formula $A\rightarrow\emptyset$
\end_inset

 less often with DSI whereas 
\begin_inset Formula $A\rightarrow B$
\end_inset

 and 
\begin_inset Formula $B\rightarrow A$
\end_inset

 more often.
 Similarly 
\begin_inset Formula $C\rightarrow D$
\end_inset

 and 
\begin_inset Formula $D\rightarrow C$
\end_inset

 more often.
 Correspondingly 
\begin_inset Formula $D\rightarrow B$
\end_inset

 happens more often with GQI.
\end_layout

\begin_layout Standard
So, why are there always a number of tracks which will not reach the other
 end-points? This happens because most of the area in high-dimensional volumes
 (3D volumes) is concentrated close to the boundaries of the volumes; therefore
 many random seeds will fall close to the boundaries.
 In addition, the boundaries are not smooth (see left panel of Fig.
\begin_inset ERT
status open

\begin_layout Plain Layout

~
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:eudx_fromABCD"

\end_inset

) and for this reason the tracks which will start close to the boundaries
 will most likely not travel long distances and because the number of seeds
 which will fall close to the boundaries is not negligible; the number of
 tracks which will not travel far is not negligible as well.
 This is something to have in mind when generating tractographies.
 A naive solution would be to increase the resolution of the volume by interpola
tion.
 This approach has also its disadvantages; the diffusion data sets are already
 very large and increasing the resolution would result in a further increase.
 Interpolation on the other hand is usually associated with smoothing which
 will reduce the peak resolution in relation to high b-values where the
 signal is generally low.
 For this purpose we decided to not preprocess the phantom any further.
\end_layout

\begin_layout Standard
Another general observation is that in the curved branch less tracks reached
 their target than in the diagonal branch.
 This is to be expected because the discretization is more asymmetric in
 the curved branch (see Fig.
\begin_inset ERT
status open

\begin_layout Plain Layout

~
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:eudx_fromABCD"

\end_inset

-left panel) therefore, more tracks close to the boundaries will stop propagatin
g because they reach background voxels where anisotropy is much lower.
 This is in agreement with EuDX which is designed to stop in low anisotropic
 regions as defined by parameter 
\begin_inset Formula $\mathcal{A}_{thr}$
\end_inset

.
\end_layout

\begin_layout Standard
In the right panel of Fig.
\begin_inset ERT
status open

\begin_layout Plain Layout

~
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:eudx_fromABCD"

\end_inset

 we see that the tracks which started from ROI A, reached only ROI C when
 we used the peaks from the Single Tensor reconstruction as input to EuDX.
 However, when we used EITL the tracks which started from A, reached both
 B and C.
 The tracks did not reach ROI D because of the angular threshold 
\begin_inset Formula $\theta_{thr}$
\end_inset

.
 In contrast, tracks which traveled from the right part of the bundle did
 reach ROI C.
 This happened because the trilinear interpolation close to the intersection
 area gave more weight to the directions of the neighbouring voxels and
 the diverting angle was less than 
\begin_inset Formula $\theta_{thr}$
\end_inset

.
 This was another confirmation about the behaviour of EuDX in crossing areas.
 This identified that even if the seeds have not fallen in the intersection
 region of the two bundles, the tracks which will reach the crossing region
 will follow the branch to which they are closer to in terms of spatial
 and angular distances.
 This is possible because of the usage of trilinear interpolation.
 
\end_layout

\begin_layout Standard
Independently from the deterministic tractography described previously we
 repeated the same experiments with probabilistic tracks generated from
 the Camino Diffusion MRI Toolkit 
\begin_inset Foot
status open

\begin_layout Plain Layout
\begin_inset Formula $\texttt{cmic.cs.ucl.ac.uk/camino}$
\end_inset


\end_layout

\end_inset

.
 We will call these PMT tracks where PMT stands for Probabilistic MultiTensor
 Tractography.
 In order to create the PMT tracks we fitted a Two Tensor model in the data
 with both Tensors enforced to be prolate.
 For the fitting method we used the standard non-linear fit provided in
 Camino with positivity constraints.
 For this purpose we used the command 
\begin_inset Formula $\texttt{modelfit}$
\end_inset

 with options 
\begin_inset Formula $\texttt{-model}$
\end_inset

 
\begin_inset Formula $\texttt{cylcyl}$
\end_inset

 
\begin_inset Formula $\texttt{nldt\_pos}$
\end_inset

.
 The probabilistic tractography was generated using Pico PDFs sampled from
 a Watson distribution (
\begin_inset Formula $\texttt{picopdfs}$
\end_inset

 
\begin_inset Formula $\texttt{-pdf}$
\end_inset

 
\begin_inset Formula $\texttt{-watson}$
\end_inset

).
 In order to generate the streamlines we used the Camino command 
\emph on

\begin_inset Formula $\texttt{track}$
\end_inset


\emph default
 which took as input the PDFs from before and a seed file with the mask.
 
\end_layout

\begin_layout Standard
As previously, we performed two experiments; one where the seeds can be
 anywhere in the simulated bundles and a second where the seeds were constrained
 in the pre-specified ROIs A, B, C or D (see Fig.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:eudx_fromABCD"

\end_inset

).
 The results of the first experiment can be seen in Fig.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:PMT_phantom"

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename eudx_last/PMT.png
	lyxscale 20
	scale 50

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Probabilistic tractography performed on the software phantom shown in Fig.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:crossing_eudx_sims"

\end_inset

.
 We use Pico and a Two Tensor model (PMT) to generate the tracks.
 (i) 
\begin_inset Formula $1$
\end_inset

 iteration of the Pico (showing 
\begin_inset Formula $321$
\end_inset

 tracks).
 (ii) A detail of the left bundle.
 Some of the tracks are diverting on a zig-zag fashion although the phantom
 at these voxels has only one main direction.
 (iii) Focus on the crossing region of the phantom.
 Many of the PMT tracks are successfully propagating towards both pathways.
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "Flo:PMT_phantom"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
For illustration purposes we see in Fig.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:PMT_phantom"

\end_inset

(i) the result with 
\begin_inset Formula $321$
\end_inset

 tracks of 
\begin_inset Formula $1$
\end_inset

 iteration of the Camino 
\begin_inset Formula $\texttt{track}$
\end_inset

 command.
 In Fig.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:PMT_phantom"

\end_inset

(ii) we see a detail of the left bundle.
 It is obvious that some of the tracks are diverting vigorously on a zig-zag
 fashion although the phantom at these voxels has only one main direction.
 In Fig.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:PMT_phantom"

\end_inset

(iii) we zoom in the crossing region of the phantom where we observe that
 many of the PMT tracks are successfully propagating towards both pathways
 through the crossing region.
\end_layout

\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
footnotesize
\backslash
addtolength{
\backslash
tabcolsep}{-5pt}
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="1" columns="2">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="6">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
PMT100
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
A
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
B
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
C
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
D
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\emptyset$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
A
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

--
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\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
22%
\end_layout

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</cell>
<cell alignment="center" valignment="top" usebox="none">
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\begin_layout Plain Layout
45.2%
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</cell>
<cell alignment="center" valignment="top" usebox="none">
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0.1%
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<cell alignment="center" valignment="top" usebox="none">
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32.6%
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</cell>
</row>
<row>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
B
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
40.8%
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

--
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0.6%
\end_layout

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<cell alignment="center" valignment="top" usebox="none">
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11.1%
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<cell alignment="center" valignment="top" usebox="none">
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47.5%
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\end_inset
</cell>
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<row>
<cell alignment="center" valignment="top" usebox="none">
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\begin_layout Plain Layout
C
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
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28.9%
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<cell alignment="center" valignment="top" usebox="none">
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0.4%
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</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

--
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
51%
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19.7%
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</cell>
</row>
<row>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
D
\end_layout

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</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
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0.1%
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</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
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\begin_layout Plain Layout
3.8%
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</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
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85.5%
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</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

--
\end_layout

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\end_layout

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</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

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10.6%
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</row>
</lyxtabular>

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
hspace{1.5em}
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="6">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
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<row>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
PMT5000
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
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\begin_layout Plain Layout
A
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
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\end_layout

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\end_layout

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D
\end_layout

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</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
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\begin_inset Formula $\emptyset$
\end_inset


\end_layout

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</cell>
</row>
<row>
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A
\end_layout

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</cell>
<cell alignment="center" valignment="top" usebox="none">
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status open

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--
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\end_layout

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</cell>
<cell alignment="center" valignment="top" usebox="none">
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22.6%
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<cell alignment="center" valignment="top" usebox="none">
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44.%
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B
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status open

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</cell>
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0.7%
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9.3%
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49.1%
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<row>
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C
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28.2%
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0.3%
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</cell>
<cell alignment="center" valignment="top" usebox="none">
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\begin_inset ERT
status open

\begin_layout Plain Layout

--
\end_layout

\end_inset


\end_layout

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</cell>
<cell alignment="center" valignment="top" usebox="none">
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50.2%
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21.3%
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</cell>
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<row>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
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\begin_layout Plain Layout
D
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</cell>
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0.3%
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</cell>
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4.2%
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</cell>
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85.2%
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<cell alignment="center" valignment="top" bottomline="true" usebox="none">
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10.3%
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</row>
</lyxtabular>

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\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
In comparison with Tab.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "Tab:EuDX_AtoB"

\end_inset

 we observe that PMT performed worse that EuDX with DSI, GQI or EIT.
 This becomes evident by observing the 
\begin_inset Formula $\emptyset$
\end_inset

 columns of the corresponding tables where EuDX had a lower percentage of
 tracks which did not reach any of the other end-points.
 The difference between PMT100 and PMT5000 is in the number of iterations
 which was 
\begin_inset Formula $100$
\end_inset

 and 
\begin_inset Formula $5,000$
\end_inset

 respectively.
 Increasing the number of iterations did not increase considerably the performan
ce of PMT.
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "Flo:PMT_table"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
The results from the second experiment where the seeding takes place only
 in specific regions of the phantom are summarized in Tab.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:PMT_table"

\end_inset

.
 The distinction between PMT100 and PMT5000 is in the number of iterations
 run from Camino 
\begin_inset Formula $\texttt{track}$
\end_inset

 command.
 The default value recommended is 
\begin_inset Formula $5,000$
\end_inset

 however for this experiment we do not see a significant difference between
 
\begin_inset Formula $100$
\end_inset

 and 
\begin_inset Formula $5,000$
\end_inset

 iterations.
 In comparison with Tab.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Tab:EuDX_AtoB"

\end_inset

 we observe that PMT performed worse that EuDX with any of DSI, GQI or EIT.
 This becomes obvious when comparing the 
\begin_inset Formula $\emptyset$
\end_inset

 columns of the corresponding tables where we see that EuDX had a significantly
 lower percentage of tracks that did not reach any of the other end-points.
 Further detailed comparison of EITL with PMT shows that the two-way connections
 
\begin_inset Formula $A\rightarrow B$
\end_inset

 & 
\begin_inset Formula $B\rightarrow A$
\end_inset

, and 
\begin_inset Formula $C\rightarrow D$
\end_inset

 & 
\begin_inset Formula $D\rightarrow C$
\end_inset

 are stronger with EITL.
 By contrast 
\begin_inset Formula $A\rightarrow C$
\end_inset

 & 
\begin_inset Formula $C\rightarrow A$
\end_inset

 are stronger with PMT.
\end_layout

\begin_layout Standard
This is an important outcome as probabilistic approaches are usually supposed
 to perform better than deterministic 
\begin_inset CommandInset citation
LatexCommand cite
key "Behrens2007NeuroImage"

\end_inset

, 
\begin_inset CommandInset citation
LatexCommand cite
key "Parker2003"

\end_inset

.
 In this study we have not investigated if PMT performs poorly because of
 the Two Tensor fit or because of the probabilistic orientation sampling.
 However, we plan to investigate this further in the future.
\end_layout

\begin_layout Subsection
Results with humans
\end_layout

\begin_layout Standard
In this section we investigate how EuDX performed with real data sets.
 There are mainly two ways to validate tractographies with real data sets
 of healthy humans.
 The first is to directly ask for feedback from specialist neuroanatomists
 who have good understanding of the underlying white matter anatomy and
 the second is to compare against the results of another published tractography
 method which has already gone through anatomical validation from experts.
 We investigated both approaches.
 In order to do that we generated tractographies (see Fig.
\begin_inset ERT
status open

\begin_layout Plain Layout

~
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:arcuate_and_whole_brain"

\end_inset

B) and asked expert neuroanatomists Professor Luigi Cataneo and Dr.
 Nivedita Agarwal from the Center for Brain/Mind Science (CiMeC) in Trento,
 Italy to validate the quality of the data in relationship to their knowledge
 of anatomy.
 We then asked them to manually label bundles of their scientific interest
 in different subjects.
 We show here two of the many known bundles that they segmented.
 In Fig.
\begin_inset ERT
status open

\begin_layout Plain Layout

~
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:arcuate_and_whole_brain"

\end_inset

A we see Arcuate Fasciculus (AC) which is described in the literature as
 belonging in the language pathways 
\begin_inset CommandInset citation
LatexCommand cite
key "Frey2008"

\end_inset

, 
\begin_inset CommandInset citation
LatexCommand cite
key "rilling2008evolution"

\end_inset

, 
\begin_inset CommandInset citation
LatexCommand cite
key "glasser2008dti"

\end_inset

 and 
\begin_inset CommandInset citation
LatexCommand cite
key "makris2005segmentation"

\end_inset

.
 In Fig.
\begin_inset ERT
status open

\begin_layout Plain Layout

~
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:cst_cst_bcc"

\end_inset

A we see the right Corticospinal bundle
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "verstynen2011vivo"

\end_inset

 in another healthy subject.
 
\end_layout

\begin_layout Standard
In order to help the medical practitioners to perform the manual labeling
 we developed an interactive visualization application (available in 
\begin_inset ERT
status open

\begin_layout Plain Layout

$
\backslash
texttt{fos.me}$
\end_layout

\end_inset

).
 This is based on a novel tractography clustering method which is the topic
 of the next chapter.
 This application has the capability to create an accessible representation
 of the initial tractography into bundles of interest (BOIs).
 In this concept, a bundle is a collection of tracks with similar spatial
 and shape characteristics.
 After the BOIs are created, the medical practitioners can select one or
 more BOIs interactively and hide the parts of the tractography which are
 not interesting to their investigations.
 This application is further explained in Chapter
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "sub:Future-work"

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename eudx_last/subj5_double.png
	lyxscale 30
	scale 60

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
A: right arcuate fasciculus generated by EuDX and segmented by expert neuroanato
mists.
 The tracks are in MNI coordinates and we visualize simultaneously the T1
 slice (X=
\begin_inset Formula $29$
\end_inset

) 
\begin_inset Note Note
status open

\begin_layout Plain Layout
subject 5
\end_layout

\end_inset

 for the same subject.
 B: the sagittal view of the whole brain tractography of the same subject
 is shown.
 For visualization purposes we are depicting only tracks of length from
 
\begin_inset Formula $120$
\end_inset


\begin_inset space ~
\end_inset

mm to 
\begin_inset Formula $150$
\end_inset


\begin_inset space ~
\end_inset

mm.
 
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "Flo:arcuate_and_whole_brain"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
The tractographies shown in Fig.
\begin_inset ERT
status open

\begin_layout Plain Layout

~
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:arcuate_and_whole_brain"

\end_inset

 and at the left panel of Fig.
\begin_inset ERT
status open

\begin_layout Plain Layout

~
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:cst_cst_bcc"

\end_inset

 were calculated with EuDX parameters 
\begin_inset Formula $\mathcal{A}_{thr}=0.02$
\end_inset

, 
\begin_inset ERT
status open

\begin_layout Plain Layout

$
\backslash
Delta s=0.5$
\end_layout

\end_inset

, 
\begin_inset Formula $\theta_{thr}=60^{\circ}$
\end_inset

, 
\begin_inset Formula $TW=0.5$
\end_inset

 and with input from QA of GQI reconstruction with sampling length 
\begin_inset Formula $\lambda=1.2$
\end_inset

.
 The data sets were generated at a 3T scanner (TIM Trio, Siemens) in Medical
 Research Council, Cognition and Brain Sciences Unit, Cambridge, UK.
 We used Siemens advanced diffusion work-in-progress sequence, and STEAM
 
\begin_inset CommandInset citation
LatexCommand cite
key "merboldt1992diffusion,MAB04"

\end_inset

 as the diffusion preparation method.
 The field of view was 
\begin_inset Formula $240\times240\, mm^{2}$
\end_inset

, matrix size 
\begin_inset Formula $96\times96$
\end_inset

, and slice thickness 
\begin_inset Formula $2.5\, mm$
\end_inset

 (no gap).
 
\begin_inset Formula $55$
\end_inset

 slices were acquired to achieve full brain coverage, and the voxel resolution
 was 
\begin_inset Formula $2.5\times2.5\times2.5\, mm{}^{3}$
\end_inset

.
 A 
\begin_inset Formula $102$
\end_inset

-point half grid acquisition with a maximum b-value of 
\begin_inset Formula $4,000\, s/mm^{2}$
\end_inset

 was used.
 The total acquisition time was only 
\begin_inset Formula $14\, min\,21\, s$
\end_inset

 with TR=
\begin_inset Formula $8,200\, ms$
\end_inset

 and TE=
\begin_inset Formula $69\, ms$
\end_inset

.
 
\end_layout

\begin_layout Standard
The tractographies were generated in diffusion native space and then linearly
 registered in MNI space.
 For that reason FA volumes were generated from the same data sets using
 Tensor fitting with weighted least squares after skull stripping with FSL
 
\begin_inset Formula $\texttt{bet}$
\end_inset

.
 These FA volumes were again in diffusion native space, therefore we used
 FSL 
\begin_inset Formula $\texttt{flirt}$
\end_inset

 to align them in MNI space.
 For this purpose a standard FA atlas 
\begin_inset Formula $\texttt{FMRIB58}$
\end_inset

 from the FSL toolbox was used as the reference image.
 We then applied the affine transformation matrix from the previous step
 to the initial tractography to align it to MNI space.
 
\end_layout

\begin_layout Standard
In order to help the neuroanatomists to guide themselves with the segmentation
 of bundles, we linearly registered structural MRIs (T1 MPRAGE) images from
 the same subjects to the standard template 
\begin_inset Formula $\texttt{MNI152}$
\end_inset

.
 Therefore, the tractography and the T1 image were registered in the same
 space.
 This was very useful for the neuroanatomists because they could find known
 bundles from the regions they connect in the cortex which are better visible
 in the T1 image rather than in the FA image.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename eudx_last/Subj1_cst_cc_cross.png
	lyxscale 20
	scale 70

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
A: EuDX tracks from the CST segmented by our expert neuroanatomists.
 The tracks are linearly registered in MNI standard space and visible is
 also the T1 slice (Y=
\begin_inset Formula $5$
\end_inset

) from the same subject.
 B: The intersection of the BCC with the right CST.
 This is a confirmation that EuDX can propagate successfully in crossing
 areas of human brain data.
 The T1 slice (Y=
\begin_inset Formula $-1$
\end_inset

) from the same subject is also visible.
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "Flo:cst_cst_bcc"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
In section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sub:Results-with-digital"

\end_inset

 we discussed with simulation experiments that EuDX was able to propagate
 correctly in crossing areas (see Fig.
\begin_inset ERT
status open

\begin_layout Plain Layout

~
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:crossing_eudx_sims"

\end_inset

) by generating tracks towards both directions of the crossings.
 We wanted to confirm if tracking in crossing areas was also robust with
 human data sets where the noise artefacts are less predictable.
 This was indeed confirmed by looking at the intersection of two well known
 bundles: the Body of Corpus Callosum (BCC) and the Corticospinal Tract
 (CST).
 In Fig.
\begin_inset ERT
status open

\begin_layout Plain Layout

~
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:cst_cst_bcc"

\end_inset

B we can see that a part from the BCC shown with red is intersecting the
 CST bundle shown with blue without being diverted from the CST.
 If EuDX was not able to propagate in crossing areas then the tracks from
 BCC would stop in the intersection area or divert towards the direction
 of the CST.
 
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename eudx_last/12_subjects_cingulum.png
	lyxscale 20
	scale 40

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
The left and right Cingulum (CG) bundles of 
\begin_inset Formula $12$
\end_inset

 healthy subjects are presented here.
 Left and right CGs were selected in MNI space from entire EuDX tractographies
 of 1 million seeds and GQI as the reconstruction input.
 We also show the T1 image slice Z=
\begin_inset Formula $10$
\end_inset

 for every subject except from last subject whose T1 was not available.
 For that subject we use the standard MNI template.
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "Flo:cingulum_12_subjects"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
In the next experiment we visually compared specific bundles across different
 healthy subjects.
 We concentrated at a pair of bundles not so often studied in literature:
 the Cingulum bundles; across 
\begin_inset Formula $12$
\end_inset

 healthy subjects (
\begin_inset Formula $20-40$
\end_inset

 years old).
 The Cingulum (CG) is an association fibre tract that runs within the Cingulate
 Gyrus along its entire length.
 It collects axons from the Cingulate Gyrus that travel immediately dorsal
 to the Corpus Callosum (CC) and along the ventral face of the Hippocampus,
 forming a large C-shape trajectory.
 It carries afferent connections from the Cingulate Gyrus to the entorhinal
 cortex.
 Because of its narrow tubular shape, it is often difficult to reveal its
 entire length by a single data set 
\begin_inset CommandInset citation
LatexCommand cite
key "moriBook"

\end_inset

.
 CG has the characteristic that both left and right CGs travel in parallel
 and are very close to each other.
 Often many tractography algorithms do not represent the Cingulum bundle
 very well because it is very close to the CC.
 However, using EuDX we see that both left and right CG bundles were consistent
 across all 
\begin_inset Formula $12$
\end_inset

 subjects (see Fig.
\begin_inset ERT
status open

\begin_layout Plain Layout

~
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:cingulum_12_subjects"

\end_inset

).
 The cingulum was selected using the interactive tool referenced previously
 from the entire tractography.
 The reconstruction parameters and EuDX parameters were the same as before
 with the exception of the number of initial seeds which was 
\begin_inset Formula $1$
\end_inset

 million.
 Furthermore, the tractographies of all subjects were registered together
 with the structural images in MNI space.
 The only exception was with the last subject 
\begin_inset Formula $12$
\end_inset

 whose T1 image was not available and it was visualized together with the
 MNI standard template (
\begin_inset Formula $\texttt{MNI152}$
\end_inset

) (see bottom-right corner of Fig.
\begin_inset ERT
status open

\begin_layout Plain Layout

~
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:cingulum_12_subjects"

\end_inset

).
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout

[th!]
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename eudx_last/all_eudx_and_pico.eps
	lyxscale 50
	scale 60

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
\begin_inset Formula $5$
\end_inset

 tractographies from the same data sets of the same subject.
 The 
\begin_inset Formula $3$
\end_inset

 top and the bottom left are created using our proposed deterministic approach
 (EuDX).
 The bottom right is created using Probabilistic tractography (PMT) .
 
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "Flo:EuDX_vs_PMT"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
In Fig.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:EuDX_vs_PMT"

\end_inset

 we show results generated from a single subject.
 The data had size 
\begin_inset Formula $96\times96\times55\times102$
\end_inset

 and were acquired as described previously in this section.
 We first removed the scalp using 
\begin_inset Formula $\texttt{bet}$
\end_inset

 and secondly created an FA image for this data.
 We masked out all voxels with FA
\begin_inset Formula $<0.2$
\end_inset

 in order to make sure that seeding will take place in areas of high anisotropy.
 We then generated EuDX tracks using DTI, GQI, DSI and EITL reconstructions
 respectively.
 We also used the same seeds to generate probabilistic PMT tracks.
 In Fig.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:EuDX_vs_PMT"

\end_inset

 we can see that EuDX based tracks are more uniform than the PMT tracks.
 We can also observe that the DTI tracks do not represent well the SLF regions
 of the bundles especially on the left hemisphere.
 Finally, we can observe the GQI, DSI and EIT tracks look the most similar
 which was expected as they have comparable angular accuracy (see Fig.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:2_20"

\end_inset

).
 The EuDX parameters used here were 
\begin_inset Formula $\mathcal{A}_{thr}=0.2$
\end_inset

, 
\begin_inset ERT
status open

\begin_layout Plain Layout

$
\backslash
Delta s=0.5$
\end_layout

\end_inset

, 
\begin_inset Formula $\theta_{thr}=60^{\circ}$
\end_inset

, 
\begin_inset Formula $TW=0.5$
\end_inset

 with input from PK of GQI, DSI and EITL correspondingly.
 The Single Tensor was fitted using weighted least squares.
 For the DSI reconstruction we used standard parameters: q-space grid size
 
\begin_inset Formula $16$
\end_inset

, hanning filter width 
\begin_inset Formula $32$
\end_inset

, and radial integration range from 
\begin_inset Formula $2.1$
\end_inset

 to 
\begin_inset Formula $6$
\end_inset

 at steps of 
\begin_inset Formula $0.2$
\end_inset

.
 For GQI we used sampling length 
\begin_inset Formula $1.2$
\end_inset

.
 For EITL we used for radial sampling from 
\begin_inset Formula $0$
\end_inset

 to 
\begin_inset Formula $5$
\end_inset

 with steps of 
\begin_inset Formula $0.4$
\end_inset

 and Gaussian weighting of 
\begin_inset Formula $0.05$
\end_inset

.
 The running time of EuDX was on average 
\begin_inset Formula $11$
\end_inset

 seconds for generating on average 
\begin_inset Formula $70,000$
\end_inset

 tracks.
\end_layout

\begin_layout Subsection
Conclusion
\end_layout

\begin_layout Standard
We showed that EuDX is a fast deterministic tractography method which can
 be used to propagate in crossing and non-crossing areas in simulations
 and human subjects.
 With the help of expert neuroanatomists we confirmed that EuDX can be used
 to find bundles which are known from anatomy like AC, CST and we investigated
 how CG looked between different subjects.
\end_layout

\begin_layout Standard
Currently the neuroanatomists are continuing inspecting and segmenting more
 and more bundles and we hope that in the near future we will have a large
 collection of segmented bundles.
 These will be used to create better tractography algorithms by prior anatomical
 knowledge or exploit supervised learning techniques in order to find similar
 bundles in non-labeled subjects.
\end_layout

\begin_layout Standard
The purpose of EuDX is to be faithful to the reconstruction results rather
 than try to correct or enhance them by introducing regional or global considera
tions which is the topic of other methods which were reviewed earlier.
 Therefore, EuDX serves mainly as a robust method for quickly inspecting
 different reconstruction results using deterministic tractography.
 However, we observed that EuDX performed better than standard probabilistic
 approaches in the simulations.
 This observation needs further exploration and contradicts what is commonly
 assumed i.e.
 that probabilistic tractography will perform better.
 
\end_layout

\begin_layout Standard
We think that most algorithms concentrated on inferring the distributions
 of the major directions in each voxel.
 Nevertheless, not much attention was payed to the actual track density
 and integrity of the bundle.
 We believe this is an important issue especially for clinical applications.
 
\end_layout

\begin_layout Standard
EuDX will stop tracking on voxels with low anisotropy and will not take
 into consideration other voxels as it would happen with other probabilistic
 techniques.
 This property is often useful when validating underlying reconstruction
 models.
 Furthermore we showed that EuDX can take as input many different anisotropic
 functions: FA and QA or PK which have no restrictions for the underlying
 model neither to the number of peaks per voxel.
\end_layout

\begin_layout Standard
The source code for EuDX can be found in module 
\begin_inset Formula $\texttt{dipy.tracking.eudx}$
\end_inset

 which is freely available at 
\begin_inset Formula $\texttt{dipy.org}$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
Reference abbreviations: 
\end_layout

\begin_layout Plain Layout
search for: bibtex style abbreviate
\end_layout

\begin_layout Plain Layout
ieeetr (selected)
\end_layout

\begin_layout Plain Layout
acm
\end_layout

\end_inset


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\end_inset


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\end_inset


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\begin_layout Standard

\lang british
\begin_inset Note Note
status open

\begin_layout Plain Layout

\lang british
\begin_inset CommandInset bibtex
LatexCommand bibtex
bibfiles "diffusion"
options "ieeetr"

\end_inset


\end_layout

\end_inset


\end_layout

\end_body
\end_document