Chapter_1.lyx

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\begin_body

\begin_layout Section
Background
\end_layout

\begin_layout Standard

\end_layout

\begin_layout Subsection

\lang british
Introduction 
\end_layout

\begin_layout Standard

\emph on
\lang british
Diffusion MRI
\emph default
 (dMRI) is the principal non-invasive method that provides information about
 the directional structure of neural tracts found in white matter and the
 cortex
\lang english

\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
are you sure about the cortex? what exactly do you mean?
\end_layout

\end_inset


\lang british
.
 dMRI acquires one or more 
\begin_inset Formula $T_{2}$
\end_inset

-weighted reference images, and a collection of diffusion-weighted images
 (see Fig.
\begin_inset space ~
\end_inset


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

\end_inset

) that attenuate the 
\begin_inset Formula $T_{2}$
\end_inset

 signal according to the amount of diffusion along prescribed gradient direction
s 
\begin_inset CommandInset citation
LatexCommand cite
key "DiffMRIBook"

\end_inset

.
 The information is not complete and the tracts cannot be reconstructed
 in full detail 
\begin_inset CommandInset citation
LatexCommand cite
key "denislebihan2006aap"

\end_inset

.
 However, some spatial structures and patterns can be visualised.
 These are usually represented as trajectories 
\begin_inset CommandInset citation
LatexCommand cite
key "WWS+08,MCC+99"

\end_inset

 or connectivity maps 
\begin_inset CommandInset citation
LatexCommand cite
key "Behrens2003NatureNeuroscience"

\end_inset

.
 The unique new area of study that aims to reconstruct the neural tracts
 from diffusion data is called diffusion tractography.
 Other types of tractography are based in staining using for example luxol-fast
 blue 
\begin_inset CommandInset citation
LatexCommand cite
key "margolis5nal"

\end_inset

 but these can only be used with 
\emph on
in vitro
\emph default
 brains and they lack ease of reproducibility.
 For non-human brains as for example in macaque there are even 
\emph on
in vivo
\emph default
 methods for tracing down to single axons 
\begin_inset CommandInset citation
LatexCommand cite
key "Tuch2005"

\end_inset

 however, these are not available or recommended for human studies as they
 are highly invasive.
\end_layout

\begin_layout Standard

\lang british
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
Diffusion was introduced in the mid-1980s (Le Bihan and Breton 1985
\begin_inset CommandInset citation
LatexCommand cite
key "le1985imagerie"

\end_inset

; Merboldt et al.
 1985
\begin_inset CommandInset citation
LatexCommand cite
key "merboldt1985self"

\end_inset

; Taylor and Bushell, 1985
\begin_inset CommandInset citation
LatexCommand cite
key "taylor1985spatial"

\end_inset

)
\end_layout

\end_inset


\end_layout

\begin_layout Subsection

\lang british
Molecular diffusion
\end_layout

\begin_layout Standard

\lang british
Molecular diffusion is a process that occurs incessantly in biological materials
, fluids in particular, 
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout

\lang british
Should we say everywhere rather than fluids?
\end_layout

\end_inset

 and accounts for a number of interesting phenomena; the dMRI signal measures
 the history of random (Brownian) displacements of spin-labelled hydrogen
 protons (spins) resolved in the direction of a magnetic field gradient.
 Though the actual probability displacement function of the protons is unaffecte
d by the presence or variation in the magnetic field, the cumulative phase
 change in the spins reflects the changes in the position-dependent spin
 frequency induced by the field gradient.
 Components of the diffusion motion along the direction of the gradient
 induce such changes.
 The signal change due to cumulative dephasing is greatest when this coincides
 with a direction that allows greater random displacements, e.g.
 because of the orientation of a microstructure within which the proton
 is moving.
 It is this link between the directional dependence of the dMRI signal and
 the orientations of the supposed underlying brain fibres that provides
 the unique insights of diffusion tractography.
 In dMRI we observe that the protons will move more along the directions
 of the axons, and move less perpendicular to that direction.
 
\end_layout

\begin_layout Standard

\lang british
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename last_figures/three_slices_0_20_100.png
	scale 40

\end_inset


\end_layout

\begin_layout Plain Layout

\lang british
\begin_inset Caption

\begin_layout Plain Layout
Three slices from diffusion data sets gathered with zero gradient strength
 on the left, medium gradient strength on the middle and high gradient strength
 on the right.
\end_layout

\end_inset


\end_layout

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

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard

\lang british
Anisotropy is one of the terms that are very common in diffusion terminology.
 Anisotropy means that the mean square displacement of the particles is
 greater along some directions than along others.
 On the other hand, isotropy means that the mean square displacement is
 equal in all directions i.e.
 complete lack of anisotropy.
 It is this level of anisotropy that is the basis of dMRI as a method of
 investigation of the structure of biological materials.
 For a biological interpretation of the signal measured with dMRI see 
\begin_inset CommandInset citation
LatexCommand citep
key "Bea02"

\end_inset

, 
\begin_inset CommandInset citation
LatexCommand cite
key "ZLW+03"

\end_inset

 and 
\begin_inset CommandInset citation
LatexCommand cite
key "DiffMRIBook"

\end_inset

 p.
 105.
 
\end_layout

\begin_layout Subsection

\lang british
Acquisition sequences
\begin_inset CommandInset label
LatexCommand label
name "sub:Acquisition-sequences-in-use"

\end_inset


\end_layout

\begin_layout Standard

\lang british
MRI data are collected by changing certain magnetic fields on and off in
 a prescribed sequence, known as pulse sequence 
\begin_inset CommandInset citation
LatexCommand cite
key "DiffMRIBook,MoriNeuron2006"

\end_inset

.
 The pulse sequence determines the content, quality, contrast and resolution
 of the image.
 MR images primarily reflect the signal from hydrogen nuclei from water
 and fat concentrations.
 The hydrogen nuclei possess a magnetic dipole which is often referred to
 as spin.
 These dipoles can align themselves with an externally applied magnetic
 field.
 The MRI scanner generates a strong, static magnetic field 
\begin_inset Formula $B_{0}$
\end_inset

 which is typically measured in Tesla (
\begin_inset Formula $T$
\end_inset

).
 A second magnetic field is applied for only a brief duration and oscillates
 at radio frequences; known as the RF pulse 
\begin_inset CommandInset citation
LatexCommand cite
key "moriBook"

\end_inset

.
\end_layout

\begin_layout Standard

\lang british
RF pulses are used primarily for excitation and refocusing.
 In the excitation phase spins will rotate away from their preferred orientation
 along 
\begin_inset Formula $B_{0}$
\end_inset

.
 Excited spins precess about 
\begin_inset Formula $B_{0}$
\end_inset

 at a frequency 
\begin_inset Formula $\nu$
\end_inset

 given by the Larmor equation 
\begin_inset Formula $\nu=\gamma B$
\end_inset

 where 
\begin_inset Formula $\gamma$
\end_inset

 is a constant known as the gyromagnetic ratio.
 The precessing part that is perpendicular to the direction of 
\begin_inset Formula $B_{0}$
\end_inset

 decays exponentially with a time constant 
\begin_inset Formula $T_{2}$
\end_inset

 and the spins realign themselves exponentially in the direction of 
\begin_inset Formula $B_{0}$
\end_inset

 with a time constant 
\begin_inset Formula $T_{1}$
\end_inset

.
 
\begin_inset Formula $T_{1}$
\end_inset

 and 
\begin_inset Formula $T_{2}$
\end_inset

 vary with tissue but 
\begin_inset Formula $T_{2}<T_{1}$
\end_inset

 for the same tissue type 
\begin_inset CommandInset citation
LatexCommand cite
key "mcrobbie2006mpp"

\end_inset

.
 The generated magnetic field from the coherently precessing spins induces
 a current in the receiver coils; this current is the signal used to generate
 MR images and corresponds to image brightness.
 The more coherent the phase of the precessing spins the higher the brightness
 in the image pixels.
 However, with time, spins lose their phase coherence.
 Signal loss from both T2 decay and dephasing is called 
\begin_inset Formula $T_{2}^{*}$
\end_inset

 signal loss (
\begin_inset Formula $T_{2}^{*}<T_{2}$
\end_inset

).
 Often, a second RF pulse is applied at some time 
\begin_inset Formula $TE/2$
\end_inset

 after excitation and flips the spins in the plane perpendicular to 
\begin_inset Formula $B_{0}$
\end_inset

.
 If the conditions stay the same all spins will be back in phase at a time
 
\begin_inset Formula $TE$
\end_inset

 after the excitation pulse.
 The moment of spin refocus is called a spin echo and creates the measured
 signal.
 Acquisition sequences which use a refocusing pulse are called spin echo
 pulse sequences; and gradient echo sequences otherwise.
\end_layout

\begin_layout Standard

\lang british
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\noindent
\align center

\lang british
\begin_inset Graphics
	filename figures/stejskal-tanner2.eps
	lyxscale 50
	scale 50

\end_inset


\end_layout

\begin_layout Plain Layout

\lang british
\begin_inset Caption

\begin_layout Plain Layout

\lang british
Pulsed Gradient Spin Echo (PGSE)
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\lang british
\begin_inset CommandInset label
LatexCommand label
name "Flo:pgse"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard

\lang british
The additional magnetic fields generated by an MRI scanner are called magnetic
 field gradients or simply gradients (
\begin_inset Formula $G$
\end_inset

).
 Including the applied gradients the magnetic field in the scanner is given
 by 
\begin_inset Formula $B=B_{0}+G_{x}(t)x+G_{y}(t)y+G_{z}(t)z$
\end_inset

 where 
\begin_inset Formula $x,y$
\end_inset

 and 
\begin_inset Formula $z$
\end_inset

 the three orthogonal directions.
 Gradients have a special role in diffusion weighting as we will discuss
 next.
\end_layout

\begin_layout Standard

\lang british
The best known pulse sequence for generating diffusion-weighted images is
 called Pulsed Gradient Spin Echo method (PGSE), also known as the Stejskal
 and Tanner method 
\begin_inset CommandInset citation
LatexCommand citep
key "Stejskal1965JChemPhys"

\end_inset

.
 This has 
\begin_inset Formula $90^{\circ}$
\end_inset

-
\begin_inset Formula $180^{\circ}$
\end_inset

 spin echo pair of RF pulses with one gradient before the second pulse and
 one equal gradient after the second pulse 
\begin_inset CommandInset citation
LatexCommand cite
key "mcrobbie2006mpp"

\end_inset

 (see Fig.
\begin_inset space ~
\end_inset


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

\end_inset

).
 The refocusing is perfect only when the spins do not move between the two
 pulses.
 The diffusion weighted contrast acts as an inverse 
\begin_inset Formula $T_{2}$
\end_inset

 weighting i.e.
 tissues with mobile water molecules give lower signal than more solid tissues
 with smaller mobility.
 
\end_layout

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

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/Reese-TRSE2.eps
	scale 50

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout

\lang british
Twice-Refocused Spin Echo (TRSE)
\end_layout

\end_inset


\end_layout

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

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard

\lang british
Eddy currents caused by the onset and offset of the gradients are a problem
 with PGSE and most recent systems (including Siemens scanners) use Twice-Refocu
sed Spin Echo (TRSE) sequences 
\begin_inset CommandInset citation
LatexCommand cite
key "Reese2003MRM"

\end_inset

 to reduce these artefacts.
 Every time the magnetic field gradients switch they generate currents that
 produce other smaller magnetic fields which disturb the spins.
 The TRSE sequence is an improvement on the PGSE.
 This improvement is achieved by the use of another refocusing pulse surrounded
 by the inverse mirror of the previous diffusion gradients (see Fig.
\begin_inset space ~
\end_inset


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

\end_inset

).
 By adjusting the timing of the diffusion gradients, eddy currents can be
 nulled or greatly reduced.
 This sequence improves the image quality without loss of scanning efficiency
 i.e.
 TR duration and it is the standard in most modern MRI scanners.
 
\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 figures/STEAM2.eps
	scale 68

\end_inset


\begin_inset Caption

\begin_layout Plain Layout
STimulated Echo Acquisition Mode (STEAM)
\end_layout

\end_inset


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

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
In the experiments described in this thesis we used a recent (
\begin_inset Formula $2010$
\end_inset

) Work In Progress (WIP) protocol from Siemens which uses the STEAM (STimulated
 Echo Acquisition Mode) sequence 
\begin_inset CommandInset citation
LatexCommand cite
key "MAB04"

\end_inset

.
 STEAM, is presented in Fig.
\begin_inset space ~
\end_inset


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

\end_inset

 and works in the following way: Three 
\begin_inset Formula $90^{\circ}$
\end_inset

 pulses are used to produce a stimulated echo.
 The first two pulses are separated by a time delay 
\begin_inset Formula $\tau$
\end_inset

.
 After the same delay 
\begin_inset Formula $\tau$
\end_inset

 following the same pulse, a stimulated echo is produced.
 In order to introduce diffusion weighting into the stimulated echo, two
 identical diffusion gradient lobes are applied, one during the first and
 one during the second 
\begin_inset Formula $\tau$
\end_inset

 interval.
 Because the magnetization of the stimulated echo is stored along the longitudin
al axis between the second and the third RF pulses, it does not experience
 any 
\begin_inset Formula $T_{2}$
\end_inset

 or 
\begin_inset Formula $T_{2}^{*}$
\end_inset

 dephasing during the time interval 
\begin_inset Formula $TM$
\end_inset

.
 
\begin_inset Formula $TM$
\end_inset

, however, does contribute to the diffusion gradient separation time, 
\begin_inset Formula $\Delta$
\end_inset

.
 Thus, a high b-value can be obtained without incurring the 
\begin_inset Formula $TE$
\end_inset

-induced signal loss, as compared to the standard spin echo sequence.
 The signal amplitude of the stimulated echo is, however, less than that
 of the corresponding spin-echo sequence with the same 
\begin_inset Formula $TE$
\end_inset

, because the maximum amplitude of the stimulated echo is one-half of a
 spin echo.
\end_layout

\begin_layout Standard
STEAM was the prefered sequence for most of the experiments used in this
 thesis as it gave higher SNR overall which was an advantage crucial at
 high b-values as the signal in those b-values can be quite low.
\end_layout

\begin_layout Subsection

\lang british
Single gradient signal models
\begin_inset CommandInset label
LatexCommand label
name "sub:Diffusion-Maths"

\end_inset


\end_layout

\begin_layout Standard

\lang british
Under the Brownian motion assumption, the diffusion signal strength is described
 by the following model known as the Stejskal-Tanner
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "Stejskal1965JChemPhys"

\end_inset

 formula 
\end_layout

\begin_layout Standard

\lang british
\begin_inset Formula 
\begin{equation}
S_{b}=S_{0}e^{-bD}\label{eq:mono-exponential}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard

\lang british
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
noindent
\end_layout

\end_inset

 where 
\begin_inset Formula $S_{0}$
\end_inset

 is the measured signal when no gradient direction is applied, 
\begin_inset Formula $D$
\end_inset

 is the diffusion coefficient that we wish to measure and 
\begin_inset Formula $b$
\end_inset

 is the b-value -- the crucial experimental diffusion weighting parameter
 which summarises the amount of diffusion sensitising gradient history.
 
\begin_inset Formula $D$
\end_inset

 is often referred to as the diffusivity value or apparent diffusivity coefficie
nt (ADC).
 The units of 
\begin_inset Formula $D$
\end_inset

 are 
\begin_inset Formula $\mathrm{mm^{2}}$
\end_inset

/sec (for water at 
\begin_inset Formula $37^{o}$
\end_inset


\begin_inset Formula $D\approx3\times10^{-3}$
\end_inset


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

/sec), and of 
\begin_inset Formula $b$
\end_inset

 are sec/
\begin_inset Formula $\mathrm{mm^{2}}$
\end_inset

, typically in the range of 
\begin_inset Formula $0$
\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

--
\end_layout

\end_inset


\begin_inset Formula $5,000$
\end_inset

 sec/
\begin_inset Formula $\mathrm{mm^{2}}$
\end_inset

 though some acquisition paradigms can call for very much larger values
 e.g.
 grea- ter than 
\begin_inset Formula $10,000$
\end_inset

 sec/
\begin_inset Formula $\mathrm{mm^{2}}$
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "Canales-Rodriguez2009"

\end_inset

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

\end_inset

.
 In Fig.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "Fig: signals"

\end_inset

 we see the signal decay for different b-values and specific diffusivities.
 The obvious conclusion here is that signal in areas with high diffusivity,
 as in the corticospinal fluid (CSF) where water persists, will always be
 lower than the signal from areas of lower diffusivity as those found in
 fibrous brain structures, as the corpus callosum.
 Furthermore, there is lowest diffusivity and lowest signal loss when the
 gradient direction is into the wall of a fibre; along the fibre has highest
 diffusivity and highest signal loss.
 Free water has uniform diffusivity and uniform signal loss towards all
 gradient directions.
 Signal decreases exponentially with 
\begin_inset Formula $b$
\end_inset

.
 Finally, we can see that the 
\begin_inset Quotes eld
\end_inset

directional effect
\begin_inset Quotes erd
\end_inset

, i.e.
 the contrast between 'along' and 'transverse' signal is greatest when 
\begin_inset Formula $b\sim1,000$
\end_inset

 .
\end_layout

\begin_layout Standard

\lang british
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\noindent
\align center

\lang british
\begin_inset Graphics
	filename figures/various_signals.png
	scale 80

\end_inset


\begin_inset Caption

\begin_layout Plain Layout

\lang british
Signal as a function of 
\begin_inset Formula $b$
\end_inset

 for various values of 
\begin_inset Formula $D$
\end_inset

.
 The directional effect is the difference between signals from gradients
 along and perpendicular to the fibre.
 
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout

\lang british
What do you mean with directional effect?
\end_layout

\end_inset


\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "Fig: signals"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard

\lang british
The b-value 
\begin_inset Formula $b$
\end_inset

 or 
\emph on
diffusion weighting
\emph default
 is a function of the strength, duration, temporal spacing and timing parameters
 of the specific paradigm.
 This function is derived from the Bloch-Torrey equations 
\begin_inset CommandInset citation
LatexCommand cite
key "Callaghan1991OUP"

\end_inset

.
 In the case of the classical Stejskal-Tanner pulsed gradient spin-echo
 (PGSE) sequence (see section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sub:Acquisition-sequences-in-use"

\end_inset

), at the time of readout
\end_layout

\begin_layout Standard

\lang british
\begin_inset Formula 
\[
b=\gamma^{2}G^{2}\delta^{2}\left(\Delta-\frac{\delta}{3}\right),
\]

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
noindent
\end_layout

\end_inset

 where 
\begin_inset Formula $\gamma$
\end_inset

 is the gyromagnetic ratio, 
\begin_inset Formula $\delta$
\end_inset

 denotes the pulse width, 
\begin_inset Formula $G$
\end_inset

 is the gradient amplitude and 
\begin_inset Formula $\Delta$
\end_inset

 the centre to centre spacing.
 
\begin_inset Formula $\gamma$
\end_inset

 is a constant which depends on the nucleus, but we can change the other
 three parameters and in that way control the b-value.
\end_layout

\begin_layout Standard

\lang british
Although the PGSE is useful for expository clarity, in reality as indicated
 in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sub:Acquisition-sequences-in-use"

\end_inset

 more complicated but related sequences such as the twice-refocused spin-echo
 (TRSE) 
\begin_inset CommandInset citation
LatexCommand cite
key "Heid2000ISMRM,Reese2003MRM"

\end_inset

 sequence and subsequent refinements such as TRASE 
\begin_inset CommandInset citation
LatexCommand cite
key "Auerbach2004ISMRM"

\end_inset

 are employed as a means of removing the distortion effects from eddy currents
 resulting from the initial and final ramps of the gradient pulses.
 
\end_layout

\begin_layout Standard

\lang british
An important point is that we can control the size of b-values by changing
 the strength and timings of the gradient pulses, and that depending on
 the b-value, we can expect different amount of signal loss.
 In Fig.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig: directional"

\end_inset

 we see the directional dependence of the simulated signal of a single fibre
 oriented at 0°  for two b-values.
 Note that the signal is lowest in the direction of the fibre both at high
 and low b-values.
\end_layout

\begin_layout Standard

\lang british
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout

\lang british
\begin_inset ERT
status open

\begin_layout Plain Layout

[th!]
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\noindent
\align center

\lang british
\begin_inset Graphics
	filename figures/directional_signal.png
	scale 80

\end_inset


\end_layout

\begin_layout Plain Layout

\lang british
\begin_inset Caption

\begin_layout Plain Layout

\lang british
Directional dependence of signal for two b-values.
 Signal is drawn as a function of direction for a Gaussian diffusion function
 with a horizontal principal direction (
\begin_inset Formula $0$
\end_inset

°) and two values of 
\begin_inset Formula $b$
\end_inset

.
 
\begin_inset Formula $D$
\end_inset

 is set to 
\begin_inset Formula $0.002$
\end_inset

.
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\lang british
\begin_inset CommandInset label
LatexCommand label
name "fig: directional"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard

\lang british
In Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "eq:mono-exponential"

\end_inset

 we assume that the signal can be expressed by a single exponential term
 (mono-exponential).
 In fact, there is evidence that this assumption may break down at higher
 
\begin_inset Formula $b$
\end_inset

-values and more complicated models have been proposed in order to deal
 with this issue.
 One of these, is the multi-exponential model used by Niendorf et al.
 
\begin_inset CommandInset citation
LatexCommand cite
key "Niendorf1996"

\end_inset

, Mulkern et a.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "mulkern1999multi"

\end_inset

 and Ozarslan et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "ozarslan2006resolution"

\end_inset

 which is expressed as 
\begin_inset Formula 
\begin{equation}
S_{b}=S_{0}\sum_{i}^{N}f_{i}e^{-bD_{i}}\label{eq:multi-exponential}
\end{equation}

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
noindent
\end_layout

\end_inset

 where 
\begin_inset Formula $N$
\end_inset

 is the number of exponents or compartments, 
\begin_inset Formula $D_{i}$
\end_inset

 is the 
\begin_inset Formula $i$
\end_inset

-th diffusion coefficient and 
\begin_inset Formula $f_{i}$
\end_inset

 is the volume fraction of the 
\begin_inset Formula $i$
\end_inset

-th compartment.
 
\end_layout

\begin_layout Standard

\lang british
Another model of higher order is found in diffusion kurtosis imaging (DKI)
 proposed by Jensen et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "jensen2005diffusional"

\end_inset


\begin_inset Formula 
\begin{equation}
\ln(S_{b})=\ln(S_{0})-bD+\frac{1}{6}b^{2}D^{2}K+O(b^{3})\label{eq:kurtosis}
\end{equation}

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
noindent
\end_layout

\end_inset

 where 
\begin_inset Formula $K$
\end_inset

 is the apparent diffusion kurtosis coefficient.
 
\end_layout

\begin_layout Standard

\lang british
By putting together these single gradient models (Eq.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "eq:mono-exponential"

\end_inset

,
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "eq:multi-exponential"

\end_inset

,
\begin_inset space ~
\end_inset


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

\end_inset

) for every gradient direction we can create systems of equations where
 we can fit and identify their unknown parameters.
 In this thesis, we will try to avoid fitting.
 We will concentrate on reconstructing the diffusion signal using a non-parametr
ic Fourier-based approach applied to the combined information from many
 gradient directions.
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout

\lang british
I think we need to make a stronger, clearer statement of what restrictions
 we are imposing.
\end_layout

\end_inset


\end_layout

\begin_layout Subsection

\lang british
Q-space reconstruction
\begin_inset CommandInset label
LatexCommand label
name "sub:Q-space-reconstruction"

\end_inset


\end_layout

\begin_layout Standard

\lang british
Bloch and Torrey
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "torrey1956bed"

\end_inset

 established differential equations governing MR diffusion in non-isotropic
 magnetic fields by analogy with Fick's Laws
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "einstein1956itb"

\end_inset

 for spontaneous dispersion along concentration gradients of inhomogeneous
 substances.
 Callaghan
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "Callaghan1991OUP"

\end_inset

 showed how these bulk properties can be derived by statistical methods
 from the collective spin histories of individual protons.
 When a molecule is at position 
\begin_inset Formula $x_{0}$
\end_inset

, we cannot read exactly where it will be after time 
\begin_inset Formula $t$
\end_inset

, we can only model a distribution of possible locations.
 This motion is described by a propagator 
\begin_inset Formula $P(\mathbf{x};\mathbf{x}_{o},t)$
\end_inset

 which defines the probability of being in 
\begin_inset Formula $\mathbf{x}$
\end_inset

 after a time 
\begin_inset Formula $t$
\end_inset

, starting at 
\begin_inset Formula $\mathbf{x}_{0}$
\end_inset

.
\end_layout

\begin_layout Standard

\lang british
Stejskal and Tanner 
\begin_inset CommandInset citation
LatexCommand cite
key "Stejskal1965JChemPhys"

\end_inset

 showed that the spin echo magnitude 
\begin_inset Formula $S(\mathbf{q},t)$
\end_inset

 from a pulsed gradient spin echo (PGSE) experiment (see section
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "sub:Acquisition-sequences-in-use"

\end_inset

) is directly related to the diffusion propagator by the following (inverse)
 Fourier relation
\begin_inset Formula 
\begin{equation}
S(\mathbf{q},t)=S_{0}\int P(\mathbf{r},t)e^{i\mathbf{q}\cdot\mathbf{r}}d\mathbf{r}\label{eq:fourier}
\end{equation}

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
noindent
\end_layout

\end_inset

 where 
\begin_inset Formula $S_{0}$
\end_inset

 is the signal in the absence of the applied magnetic diffusion gradient
 
\begin_inset Formula $\mathbf{g}$
\end_inset

, 
\begin_inset Formula $\mathbf{r}$
\end_inset

 is the relative spin displacement 
\begin_inset Formula $\mathbf{x}-\mathbf{x}_{0}$
\end_inset

 at diffusion time 
\begin_inset Formula $t$
\end_inset

, 
\begin_inset Formula $\mathbf{q}$
\end_inset

 is the spin displacement wave vector.
 
\begin_inset Formula $\mathbf{q}$
\end_inset

 is parallel with the applied magnetic gradient 
\begin_inset Formula $\mathbf{g}$
\end_inset

.
 With the corresponding direct Fourier transform we can reconstruct the
 diffusion propagator 
\begin_inset Formula $P$
\end_inset

 by measuring the signal in a number of different directions and gradient
 magnitudes.
 Q-space imaging (QSI) and Diffusion Spectrum Imaging (DSI) are the best
 known methods which reconstruct the diffusion propagator in that way.
 In Eq.
\begin_inset space ~
\end_inset


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

\end_inset

 the spin density is implicit in 
\begin_inset Formula $S_{0},$
\end_inset

 in a later chapter we will see a more general interpretation of this relationsh
ip (see Eq.
\begin_inset space ~
\end_inset


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

\end_inset

).
 
\end_layout

\begin_layout Standard

\lang british
Q-space is the space defined by the coordinates of the 3D spin displacement
 wave vectors 
\begin_inset Formula $\mathbf{q}$
\end_inset

 as shown in Eq.
\begin_inset space ~
\end_inset


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

\end_inset

.
 The vector 
\begin_inset Formula $\mathbf{q}$
\end_inset

 parametrises the space of diffusion acquisitions.
 It is related to the applied magnetic diffusion gradient 
\begin_inset Formula $\mathbf{g}$
\end_inset

 by the formula 
\begin_inset Formula $\mathbf{q}=(2\pi)^{-1}\gamma\delta\mathbf{g}$
\end_inset

 
\begin_inset CommandInset citation
LatexCommand cite
key "lenglet2009mathematical"

\end_inset

.
 Every single vector 
\begin_inset Formula $\mathbf{q}$
\end_inset

 has the same orientation as the direction of diffusion gradient 
\begin_inset Formula $\mathbf{g}$
\end_inset

 and length proportional to the strength 
\begin_inset Formula $g$
\end_inset

 of the gradient field.
 We have also 
\begin_inset Formula $\mathbf{q}=k\sqrt{b}\mathbf{\hat{g}}$
\end_inset

, where 
\begin_inset Formula $b$
\end_inset

 is the b-value, 
\series bold

\begin_inset Formula $\mathbf{\hat{g}}$
\end_inset


\series default
 is the unit gradient direction, and 
\begin_inset Formula $k$
\end_inset

 is a multiplication constant which is a function of the timing parameters
 of the acquisition scheme.
 
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout

\lang british
When creating a grid in q-space we usually normalize 
\begin_inset Formula $\mathbf{q}$
\end_inset

 by dividing by the 
\begin_inset Formula $\mathbf{q}$
\end_inset

 with the smallest non-negative b-value.
\end_layout

\end_inset


\end_layout

\begin_layout Standard

\lang british
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout

\lang british
\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 last_figures/qspace.png
	lyxscale 40
	scale 35

\end_inset


\end_layout

\begin_layout Plain Layout

\lang british
\begin_inset Caption

\begin_layout Plain Layout
One volume is collected for every sampling point in q-space.
 Picture adapted from Hagmann et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "Hagmann2006Radiographics"

\end_inset

.
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\lang british
\begin_inset CommandInset label
LatexCommand label
name "Flo:qspace"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard

\lang british
Every single point in q-space corresponds to a diffusion weighted image
 i.e.
 a 3D brain volume of measured signal for a specific gradient direction
 and strength (see Fig.
\begin_inset space ~
\end_inset


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

\end_inset

).
 If for example we have programmed the scanner to apply 
\begin_inset Formula $60$
\end_inset

 gradient directions then our data should have 
\begin_inset Formula $60$
\end_inset

 diffusion volumes with each volume obtained for a specific gradient.
 A Diffusion Weighted Image (DWI) is the volume acquired from only one direction
 gradient.
 Hence, in the previous example we would gather 
\begin_inset Formula $60$
\end_inset

 DWI volumes corresponding to 
\begin_inset Formula $60$
\end_inset

 locations (
\begin_inset Formula $\mathbf{q}$
\end_inset

-values) in 
\series bold

\begin_inset Formula $\mathbf{q}$
\end_inset


\series default
-space.
 An alternative way to think of 
\begin_inset Formula $\mathbf{q}$
\end_inset

 is in mathematical terms as the combination of parameters which produces
 the inverse Fourier transform relationship between the diffusion signal
 and the probability displacement distribution.
 In these terms (see Callaghan 
\begin_inset CommandInset citation
LatexCommand cite
key "Callaghan1991OUP"

\end_inset

) 
\begin_inset Formula $\mathbf{q}$
\end_inset

 is the reciprocal of the probability displacement vector 
\begin_inset Formula $\mathbf{r}$
\end_inset

, just as in conventional MRI k-space is the reciprocal parametrisation
 of the space of voxel position 
\series bold

\begin_inset Formula $\mathbf{v}$
\end_inset

.
\end_layout

\begin_layout Standard

\lang british
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout

\lang british
Another way to represent dMRI is the following.
 We can combine all the DWIs together creating a new multidimensional image
 where each voxel corresponds to a profile of diffusion signals measured
 in all the chosen gradient directions 
\begin_inset Formula $\mathbf{g}$
\end_inset

.
 This is the only known way to represent all different acquisition methods
 within the same structure and we are using it as the basis for our novel
 tractography algorithm.
 ARE WE KEEPING THIS?
\end_layout

\end_inset


\end_layout

\begin_layout Standard

\lang british
One problem in the diffusion imaging literature is that names for techniques
 often refer both to a particular type of imaging acquisition, and to a
 particular method to reconstruct the directional organisation of the voxel.
 All dMRI acquisition methods acquire data in q-space, and the methods can
 be categorised by their sampling pattern in q-space.
 
\end_layout

\begin_layout Standard

\lang british
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout

\lang british
\begin_inset ERT
status open

\begin_layout Plain Layout

[th!]
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\noindent
\align center

\lang british
\begin_inset Graphics
	filename figures/Acquisitions_New.png
	lyxscale 16
	scale 30

\end_inset


\end_layout

\begin_layout Plain Layout

\lang british
\begin_inset Caption

\begin_layout Plain Layout

\lang british
A: DSI 604 directions, B: HARDI 65 directions, C: HARDI with 2 shells of
 65 directions in each shell, D: QBI 515 directions.
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\lang british
\begin_inset CommandInset label
LatexCommand label
name "Fig:q-space"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard

\lang british
We refer to a method as a 
\emph on
q-space spherical shell
\emph default
 method if it is an acquisition method using a collection of points in q-space
 that can be thought of as lying on a sphere.
 Examples include techniques referred to as HARDI (High Angular Diffusion
 Imaging), Q-ball Imaging and HYDI (Hybrid Diffusion Imaging)
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout

\lang british
References?
\end_layout

\end_inset

.
 A q-space shell method might involve a  single shell (see Fig.
\begin_inset space ~
\end_inset


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

\end_inset

B,D) e.g.
 QBI, or multiple shells (see Fig.
\begin_inset space ~
\end_inset


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

\end_inset

C) e.g.
 EQBI
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout

\lang british
Reference?
\end_layout

\end_inset

.
 In clinical settings, where we can only use a few directions (less than
 60 with minimum 6) it is recommended to use the SDT (Single Diffusion Tensor
 also known as DTI) reconstruction model which is relatively easy to fit
 as it has only a few parameters.
 
\end_layout

\begin_layout Standard

\lang british
A 
\emph on
q-space 3D Cartesian grid
\emph default
 method is an acquisition method more easily thought of as a collection
 of points regularly distributed through a region in q-space.
 The characteristic example is DSI (diffusion spectrum imaging) (see Fig.
\begin_inset space ~
\end_inset


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

\end_inset

A) and QSI (Q-Space Imaging) 
\begin_inset CommandInset citation
LatexCommand cite
key "king1994q,Callaghan1991OUP"

\end_inset

.
 In chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Cartesian-Lattice-Q-space"

\end_inset

 we will concentrate more on q-space Cartesian grid methods and propose
 a new reconstruction method which we call DNI (Diffusion Nabla Imaging)
 which also uses q-space Cartesian grid data sets.
\end_layout

\begin_layout Subsection
Diffusion Tensor
\end_layout

\begin_layout Standard

\lang british
Assuming that the diffusion propagator is given by a 3-dimensional Gaussian
 distribution from Eq.
\begin_inset space ~
\end_inset


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

\end_inset

 we write
\begin_inset Formula 
\begin{equation}
P(\mathbf{r},t)=\frac{1}{\sqrt{4\pi t^{3}\mid\mathbf{D}\mid}}exp(-\frac{\mathbf{r}^{T}\mathbf{D}^{-1}\mathbf{r}}{4t})
\end{equation}

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
noindent
\end_layout

\end_inset

 where 
\begin_inset Formula $\mathbf{D}$
\end_inset

 is known as the diffusion tensor.
 This Tensor is a 
\begin_inset Formula $3x3$
\end_inset

 positive definite symmetric matrix that can be completely described by
 a centred ellipsoid with 3 principal axes and associated eigenvalues 
\begin_inset Formula $\lambda_{1},\lambda_{2},\lambda_{3}$
\end_inset

.
 The trace of the diffusion Tensor has been found valuable for detecting
 and evaluating brain ischemia and stroke 
\begin_inset CommandInset citation
LatexCommand cite
key "moseley1995clinical,warach1995acute"

\end_inset

.
 Frequently, mean diffusivity (MD) is used instead of the trace defined
 as
\begin_inset Formula 
\begin{equation}
MD=\frac{\mathrm{trace}(D)}{3}=\frac{\lambda_{1}+\lambda_{2}+\lambda_{3}}{3}\label{eq:mean_diffusivity}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard

\lang british
Fractional Anisotropy (FA) is the most common scalar metric used in diffusion
 imaging which is used to characterise the presence or absence of a preferred
 direction for diffusion.
 Like the MD it depends only on the eigenvalues.
\begin_inset Formula 
\begin{equation}
FA=\frac{1}{\sqrt{2}}\frac{\sqrt{(\lambda_{1}-\lambda_{2})^{2}+(\lambda_{2}-\lambda_{3})^{2}+(\lambda_{3}-\lambda_{1})^{2}}}{\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}}}\label{eq:FA_classical}
\end{equation}

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
noindent
\end_layout

\end_inset

 If FA is equal to 1 that means very anisotropic (infinitely prolonged ellipsoid
, a `stick') and if FA is equal to 0 that means completely isotropic (sphere).
 FA is used in clinical studies to diagnose diseases like stroke and cancer
 and assess the progress of therapy 
\begin_inset CommandInset citation
LatexCommand cite
key "maddah_phdthesis2008"

\end_inset

.
\end_layout

\begin_layout Standard

\lang british
Whenever we use FA volumes in our analysis we are implicitly assuming that
 the propagator of the spin displacements in every voxel has a 3D Gaussian
 distribution.
 This assumption is used in most of the diffusion related literature where
 DTI or Diffusion Tensor Imaging is the prevailing term.
 Unfortunately, in reality things are much more complicated; inside our
 brain the axons are semi-permeable (restriction), the water molecules interact
 with many different elements in the complex intra fibre fluid, the fibres
 might cross, kiss, divert or bend inside a voxel or between voxels.
 Assuming a Gaussian distribution is therefore, a non-trivial approximation.
 However, FA is still prevalent as it is easy to calculate and it gives
 similar values across different acquisitions.
 
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout

\lang british
The CHARMED model of Assaf and Basser seems to characterise hindered and
 restricted diffusion in a useful way.
 You might use it to improve this paragraph?
\end_layout

\end_inset


\end_layout

\begin_layout Subsection

\lang british
Orientation Distribution
\begin_inset CommandInset label
LatexCommand label
name "sub:Orientation-Distribution"

\end_inset


\end_layout

\begin_layout Standard

\lang british
Since one of the primary interests of dMRI is the way that the signal depends
 on the direction of underlying fibre orientations; it is the orientation
 information of the diffusion propagator (see Eq.
\begin_inset space ~
\end_inset


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

\end_inset

) that is principally of interest.
\end_layout

\begin_layout Standard

\lang british
One possible approach would be to replace the diffusion probability density
 function with an isosurface, which is a surface that passes through all
 points of equal probability density value.
 For instance, an isosurface of a 3D Gaussian distribution is an ellipsoid.
 A more commonly used technique that is less sensitive to noise involves
 the computation of the Orientation Distribution Function (ODF) from the
 displacement distribution 
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "Hagmann2006Radiographics,Tuch2004,wedeen2005mapping"

\end_inset

.
 An ODF may be considered a spherical polar plot whose radius in a given
 direction is proportional to the integral of the diffusion probability
 density function in that direction.
 For ease of visualization, we colour-code the surface according to the
 diffusion direction (
\begin_inset Formula $(x,y,z)=(r,g,b)$
\end_inset

, where 
\begin_inset Formula $r=\text{red}$
\end_inset

, 
\begin_inset Formula $b=\text{blue}$
\end_inset

, and 
\begin_inset Formula $g=\text{green}$
\end_inset

).
 An orientation distribution function or isosurface can be plotted for each
 individual MR imaging voxel (see Fig.
\begin_inset space ~
\end_inset


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

\end_inset

).
\end_layout

\begin_layout Standard

\lang british
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout

\lang british
\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 last_figures/pdf_iso_odf.png
	lyxscale 30
	scale 30

\end_inset


\end_layout

\begin_layout Plain Layout

\lang british
\begin_inset Caption

\begin_layout Plain Layout
Top: The reconstruction of the 3D displacement probability distribution
 also known as the diffusion propagator or ensemble average propagator (EAP)
 or diffusion spectrum produced from the inverse Fourier transform of the
 diffusion signal.
 Two approaches that may be used to simplify the visual representation of
 the EAP are shown.
 Left: the replacement of the displacement distribution with an isosurface.
 Right: the computation of the commonly used Orientation Distribution Function
 (ODF).
 This displacement distribution simulates the crossing of two fibres.
 In general, the ODF is used essentially to identify the primary directions
 of the underlying fibres.
 Picture adapted from Hagmann et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "Hagmann2006Radiographics"

\end_inset

.
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\align center
\begin_inset CommandInset label
LatexCommand label
name "Flo:ODF"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard

\lang british
The ODF expresses the probability of a spin displacing into a differential
 solid angle about a possible fibre direction 
\begin_inset Formula $\hat{\mathbf{u}}$
\end_inset

.
 This is used in order to model and visualise the directional information
 in diffusion propagator and in simple words it just projects the diffusion
 function on to the sphere by integrating over the radial coordinate of
 the diffusion function.
 The ODF representation symbolised below with 
\begin_inset Formula $\psi$
\end_inset

 sacrifices all the radial information but retains the relevant directional
 information:
\end_layout

\begin_layout Standard

\lang british
\begin_inset Formula 
\begin{equation}
\psi(\hat{\mathbf{u}})=\int_{0}^{\infty}P(r\hat{\mathbf{u}})r^{2}dr
\end{equation}

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
noindent
\end_layout

\end_inset

 where 
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\lang english

\begin_inset Formula $\hat{\mathbf{u}}$
\end_inset


\family default
\series default
\shape default
\size default
\emph default
\bar default
\noun default
\color inherit
\lang british
 is a unit normal vector, and 
\begin_inset Formula $r$
\end_inset

 is the radial coordinate in the diffusion space.
 By construction 
\begin_inset Formula $\psi(\mathbf{\hat{u})}$
\end_inset

 is a probability distribution over 
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\lang english

\begin_inset Formula $\hat{\mathbf{u}}$
\end_inset


\family default
\series default
\shape default
\size default
\emph default
\bar default
\noun default
\color inherit
\lang british
.
 
\end_layout

\begin_layout Standard

\lang british
The ODF is a function on the sphere.
 The sphere is usually represented by a discrete spherical grid with evenly
 distributed points.
 It is a common procedure to identify the direction of the underlying fibres
 from the points where the maximum values are found (see Fig.
\begin_inset space ~
\end_inset


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

\end_inset

).
 This procedure is also known as 
\emph on
peak finding
\emph default
.
\end_layout

\begin_layout Subsection

\lang british
Tractography
\begin_inset CommandInset label
LatexCommand label
name "sub:Tractography"

\end_inset


\end_layout

\begin_layout Standard

\lang british
Tractography algorithms integrate local estimates of diffusion direction
 information.

\lang english
 Once we know the orientation of fibres at every voxel, we can join these
 directions up to reconstruct complete tracks and hence approximate anatomical
 tracts.
 In its simplest form, this consists of starting at a seed location and
 following the preferred direction until we reach a new voxel.
 We can then change to this voxel's referred direction and carry on until
 an entire track is propagated (see Fig.
\begin_inset space ~
\end_inset


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

\end_inset

).
 
\end_layout

\begin_layout Standard

\lang british
The two best known families of algorithms for track propagation (also known
 as track integration, tracking or tractography) are 
\emph on
deterministic
\emph default

\begin_inset space ~
\end_inset


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

\end_inset

 and 
\emph on
probabilistic
\emph default

\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "Behrens2007NeuroImage"

\end_inset

.
 A track propagation algorithm belongs to the probabilistic domain if the
 fibre model that is being used incorporates uncertainty i.e.
 errors in estimating the orientation of the fibre at every voxel.
 In the case it does not assume any uncertainty along the path of the track
 then it belongs to the deterministic domain.
 In chapter
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Euler-Delta-Crossings"

\end_inset

 we will give a short overview of many other track propagation methods including
 global tractography
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "Kreher2008ISMRM"

\end_inset

.
 
\begin_inset Note Note
status open

\begin_layout Plain Layout

\lang british
Tractography algorithms are based in local estimates of diffusion direction.
 
\end_layout

\end_inset


\end_layout

\begin_layout Standard

\lang british
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename last_figures/tensor_trajectory.png
	lyxscale 50
	scale 40

\end_inset


\end_layout

\begin_layout Plain Layout

\lang british
\begin_inset Caption

\begin_layout Plain Layout
The white line shows the track obtained by connecting up a set of voxels
 based on the direction of the axis of the maximum Tensor eigenvalue and
 is an example of deterministic tractography.
 The color is a complementary way of coding the Tensor ellipsoid direction
 where red denotes left-right, green denotes back-front and blue denotes
 up-down.
 
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\align center

\lang british
\begin_inset CommandInset label
LatexCommand label
name "Flo:tensortrack"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard

\lang british
One of the simplest and earliest deterministic methods is called Fibre Assignmen
t by Continuous Tracking (FACT) 
\begin_inset CommandInset citation
LatexCommand cite
key "MCC+99"

\end_inset

 (see Fig.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:FACTvsEuDX"

\end_inset

).
 The FACT algorithm starts through the input of an arbitrary point in the
 volume and then propagates in both directions i.e.
 forward and backward.
 Perhaps the most interesting part with these tracking methods is the way
 which they decide when to stop tracking.
 FACT uses a single threshold variable
\end_layout

\begin_layout Standard

\lang british
\begin_inset Formula 
\[
R={\displaystyle \sum_{i}^{s}\sum_{j}^{s}|\mathbf{e}_{i}\cdot\mathbf{e}_{j}|/s(s-1)}
\]

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
noindent
\end_layout

\end_inset

 where 
\begin_inset Formula $s$
\end_inset

 is the number of neighbouring voxels and
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
 
\begin_inset Formula $\mathbf{e}$
\end_inset

 is the eigenvector corresponding to the highest eigenvalue in each voxel.

\family default
\series default
\shape default
\size default
\emph default
\bar default
\noun default
\color inherit
 The simplest case for defining a neighbourhood of a voxel is to use all
 
\begin_inset Formula $26$
\end_inset

 other adjacent (touching) voxels.
 Now, let's think of how 
\begin_inset Formula $R$
\end_inset

 will behave in different neighbours.
 When adjacent fibres are aligned strongly 
\begin_inset Formula $R$
\end_inset

 will be near to 1 as the absolute value of each dot product will come closer
 to 1 as the normalised vectors become more co-linear.
 On the other side, 
\begin_inset Formula $R$
\end_inset

 will be smaller in regions without consistency in fibre direction.
\begin_inset Note Note
status open

\begin_layout Plain Layout

\lang british
, where 
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none

\begin_inset Formula $R=0$
\end_inset

 in perfect isotropic conditions IAN: don't believe this is true - when
 isotropic - I think it should be 2/
\begin_inset Formula $\pi$
\end_inset

 which equals about 0.637.
\end_layout

\end_inset


\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
 In voxels with 
\begin_inset Formula $R$
\end_inset

 less than a prespecified threshold e.g.
 0.8 the tracks will stop being propagated and FACT will terminate.

\family default
\series default
\shape default
\size default
\emph default
\bar default
\noun default
\color inherit
 An important problem with FACT is that it fails to track in areas where
 there are crossings.
 In this case, it only tracks one of the major pathways of the crossing
 area .
\begin_inset Note Note
status open

\begin_layout Plain Layout
TALK ABOUT THE F-E PICTURE - SAY AS WELL THAT FACT PROPAGATES IN NON-FIXED
 STEPS AND EULER INTERGRATION IN FIXED STEPS
\end_layout

\end_inset


\end_layout

\begin_layout Standard

\lang british
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Graphics
	filename last_figures/FACT_EUDX.png
	lyxscale 50
	scale 70
	rotateOrigin center

\end_inset


\end_layout

\begin_layout Plain Layout

\lang british
\begin_inset Caption

\begin_layout Plain Layout

\lang british
Left: FACT propagates with different steps indicated by the entering point
 on each voxel.
 Right: Other deterministic methods use constant steps.
 In this case interpolation of the neighbouring directions is necessary.
 For example in EuDX trilinear interpolation is used.
 Picture adapted from Stamatopoulos
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "sotiropoulos2010processing"

\end_inset

.
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\lang british
\begin_inset CommandInset label
LatexCommand label
name "fig:FACTvsEuDX"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard

\lang british
Another popular deterministic approach commonly used in the the field of
 fluid dynamics for flow simulation was applied to the field of dMRI by
 
\begin_inset CommandInset citation
LatexCommand cite
key "CLC+99"

\end_inset

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

\end_inset

.
 The authors made the assumption of a continuous vector field where the
 track is propagated by the solution of a system of differential equations
 subject to an initial condition, the position of the seed point.
 Here the authors propose that a track can be represented by a 3D curve
 
\begin_inset Formula $\mathbf{r}$
\end_inset

 parametrised by the arc length 
\begin_inset Formula $s$
\end_inset

 of the track.
 This is provided by the iterative solution of the differential equation
\begin_inset Formula 
\begin{equation}
d\mathbf{r}(s)/ds=\mathbf{e}(\bm{r}(s))
\end{equation}

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
noindent
\end_layout

\end_inset

 where 
\series bold

\begin_inset Formula $\bm{e}$
\end_inset


\series default
 is the primary direction of the Tensor or a different model.
 The solution of this system of Ordinary Differential Equations (ODE) in
 its simplest case is given by iterative methods like Euler integration
 bound to an initial condition 
\begin_inset Formula $\bm{r}(0)=\bm{r}_{0}$
\end_inset

 where 
\begin_inset Formula $\bm{r}_{0}$
\end_inset

 is the seed point
\end_layout

\begin_layout Standard

\lang british
\begin_inset Formula 
\begin{equation}
\bm{r}(s_{1})\sim\bm{r}(s_{0})+\alpha\mathbf{e}_{1}(\bm{r}(s_{0}))
\end{equation}

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
noindent
\end_layout

\end_inset

 where 
\begin_inset Formula $0<\alpha\leq1$
\end_inset

 defines the integration step length.
 In Euler integration we are using the first two terms of the Taylor expansion.
 Conturo et al.
\begin_inset space ~
\end_inset


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

\end_inset

 used Euler integration and Basser et al.
\begin_inset space ~
\end_inset


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

\end_inset

 used a higher order approximation called 
\begin_inset Formula $4^{th}$
\end_inset

-order Runge-Kutta scheme.
 
\end_layout

\begin_layout Standard

\lang british
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout

\lang british
\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 last_figures/colourtracks4.png
	lyxscale 50
	scale 35

\end_inset


\end_layout

\begin_layout Plain Layout

\lang british
\begin_inset Caption

\begin_layout Plain Layout
Deterministic whole brain tractography based on EuDX, generated using 
\begin_inset Formula $\texttt{DIPY}$
\end_inset

 and visualised using 
\begin_inset Formula $\texttt{FOS}$
\end_inset

.
 The colour encodes the orientation of the mid-segment of every track using
 a colourmap based on Boy's real projective plane immersion 
\begin_inset CommandInset citation
LatexCommand cite
key "Demiralp-2009-CLF-url"

\end_inset

.
 
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\lang british
\begin_inset CommandInset label
LatexCommand label
name "Flo:Dipytracks"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard

\lang british
Wedeen et al.
\begin_inset space ~
\end_inset


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

\end_inset

 showed that one could derive the local orientation field of vectors 
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none

\begin_inset Formula $\mathbf{e}$
\end_inset

 from the local maxima of the ODF calculated in each voxel.
 In that way one could visualise crossing distributions and depict crossing
 fibres.

\family default
\series default
\shape default
\size default
\emph default
\bar default
\noun default
\color inherit
 The authors suggest that diffusion MRI with sufficient signal to noise
 ratio (SNR) could make tractography a mathematically well-posed problem.
 However, much longer scanning time is needed in order to reach the necessary
 resolution and this is often impractical.
 Our novel tractography algorithm (EuDX) belongs in the deterministic domain
 and it is presented in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Euler-Delta-Crossings"

\end_inset

 (see Fig.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:FACTvsEuDX"

\end_inset

, 
\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:Dipytracks"

\end_inset

).
\end_layout

\begin_layout Standard

\lang british
In summary, the deterministic algorithms propagate tracks by making a series
 of discrete locally optimum decisions.
 These are fast, simple and easy to interpret.
 Usually, we depict them using tracks (also known as streamlines or polylines).
 The main disadvantage of deterministic algorithms is that they are vulnerable
 to local noise.
 
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout

\lang british
R-K2 is less vulnerable to noise because it looks a few steps ahead but
 it can also generate more non-existing tracks that we will actually have
 no use of.
\end_layout

\end_inset


\end_layout

\begin_layout Standard

\lang british
Probabilistic tractography is meant to deal with this problem of noise and
 propagate tracks even in regions where the tracking is unclear.
 This is made possible by assuming that uncertainty exists concerning the
 orientation of the fibre at each point of the track.
\end_layout

\begin_layout Standard

\lang british
\begin_inset Float figure
wide false
sideways false
status collapsed

\begin_layout Plain Layout

\lang british
\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 last_figures/tractogram_human_connectome.png
	lyxscale 30
	scale 40

\end_inset


\end_layout

\begin_layout Plain Layout

\lang british
\begin_inset Caption

\begin_layout Plain Layout

\lang british
The 3D distribution of voxels connected to the seed voxel is called a tractogram.
 
\begin_inset Note Note
status open

\begin_layout Plain Layout

\lang british
Maxime page 213
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\lang british
\begin_inset CommandInset label
LatexCommand label
name "Flo:tractogram"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Imagine a particle in a seed voxel moving in a random fashion with a constant
 speed within the brain white matter.
 The transition probability to a neighbouring point depends on the local
 orientation distribution or underlying model.
 This yields high transitional probabilities along the main fibre directions.
 Hence, the particle will move in parallel to the fibre direction with a
 higher probability than in a perpendicular direction.
 In this probabilistic method, we start a large number of particles from
 the same seed point, let the particles move randomly according to the local
 ODF and count the number of times a voxel is reached by the path of a particle
 (connectivity values).
 The random walk is stopped when the particle leaves the white matter volume.
 Probabilistic tractography is commonly depicted as a tractogram (see Fig.
\begin_inset space ~
\end_inset


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

\end_inset

).
 In a tractogram, we count and store the number of tracks that go through
 each voxel and visualise the entire volume of the stored values.
 
\end_layout

\begin_layout Standard

\lang british
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout

\lang british
\begin_inset ERT
status open

\begin_layout Plain Layout

[th!]
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\noindent
\align center

\lang british
\begin_inset Graphics
	filename figures/probtrack2.png
	lyxscale 25
	scale 45

\end_inset


\end_layout

\begin_layout Plain Layout

\lang british
\begin_inset Caption

\begin_layout Plain Layout

\lang british
A simplified example showing in (i) and (ii) the same data set.
 (i) The yellow line shows the result of deterministic tractography which
 is given by a single trajectory and in (ii) is given by connectivity matrix
 depicting in red the probability of different pathways throughout the hole
 slice.
 For the ease of understanding, only 3 possible pathways are depicted.
 Finally, in (iii) an example is given where it is shown that probabilistic
 tractography weights more closer connections.
 However, it can track further deep than deterministic tractography.
 
\begin_inset Note Note
status open

\begin_layout Plain Layout

\lang british
Ian: Should the word 'tract' be changed to 'track' in this figure legend?
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\lang british
\begin_inset CommandInset label
LatexCommand label
name "fig:deterministic_probabilistic"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard

\lang british
We will try to illustrate the difference between the two approaches (determinist
ic and probabilistic) using a simplified 2D example shown in Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:deterministic_probabilistic"

\end_inset

.
 Generalising afterwards in 3D is straightforward.
 
\end_layout

\begin_layout Standard

\lang british
Let's imagine that we have a 2D slice where in each pixel we have calculated
 a vector showing the primary direction for that specific pixel.
 This vector 
\begin_inset Formula $\mathbf{e}$
\end_inset

 could have been calculated from the Tensor as the principal eigenvector
 or as a principal direction of a different model e.g.
 the maximum point of the ODF.
 Let's now think that we want to find the best track from seed A to seed
 Z.
 When using deterministic tractography we are using only the local direction
 information in every pixel therefore in a direction field as this of Fig.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:deterministic_probabilistic"

\end_inset

(i) we will have to use only one track and this is depicted in the diagonal
 pathway with yellow colour.
 However, there are other possible tracks as well in this diagram e.g rather
 than taking the diagonal we could go first up from A and then right.
 
\end_layout

\begin_layout Standard

\lang british
Probabilistic tractography aims to identify all the possible tracks by assigning
 to each one of them a weight.
 All the weights of all the tracks together sum to 1.
 This is possible by generating samples from a probability distribution
 for every pixel.
 In this simple example shown in Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:deterministic_probabilistic"

\end_inset

(ii) the orientation of the blue vectors is represented by a single parameter,
 angle 
\begin_inset Formula $\omega$
\end_inset

.
 
\begin_inset Formula $\omega$
\end_inset

 here is a random variable that takes values from a Probability Density
 Function (PDF).
 We have many possible directions to move next but with different probabilities.
 The weights of all directions again sum to 1.
 After this explanation we can identify in Fig.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:deterministic_probabilistic"

\end_inset

(ii) that the most likely track is again the diagonal (with deep red) but
 there are other possible tracks (with lighter red) that are less likely.
 In the same diagram we show with 3 combined red arrows some of the many
 directions that are possible in each point.
 However, some are more certain than others.
\end_layout

\begin_layout Standard

\lang british
Let's try now to understand how a track is valued as more probable than
 others.
 In Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:deterministic_probabilistic"

\end_inset

(iii) we have drawn a very simple image with only two pixel rows and we
 are assuming that the probability of moving along the primary direction
 (shown with blue arrow) is 0.9 and there is only a secondary direction given
 by 0.1 (1-0.9) i.e.
 for ease of understanding, we assume only 2 possible directions.
 We can see that there is a discontinuity in position K.
 In that point, an Euler based deterministic approach (without interpolation)
 has to stop at K (yellow line).
 The probabilistic method will continue tracking and it will generate two
 tracks that both reach the target.
 The probability of each track is calculated by multiplying the probability
 of a specific direction of each point.
 Therefore, the shorter and dark red track will have 
\begin_inset Formula $p_{s}=0.9^{7}\times0.1$
\end_inset

 and the longer and lighter red track will have 
\begin_inset Formula $p_{l}=0.9^{10}$
\end_inset

.
 It is obvious that 
\begin_inset Formula $p_{s}>p_{l}$
\end_inset

 and that the darker red track is more likely to exist according to this
 method.
\end_layout

\begin_layout Standard

\lang british
This method of multiplying the probabilities at each voxel along the paths
 has been proposed by many 
\begin_inset CommandInset citation
LatexCommand cite
key "FW05"

\end_inset

,
\begin_inset CommandInset citation
LatexCommand cite
key "anwander2007cbp"

\end_inset

, 
\begin_inset CommandInset citation
LatexCommand cite
key "bjornemoMICCAI02"

\end_inset

, 
\begin_inset CommandInset citation
LatexCommand cite
key "HTJ+03"

\end_inset

,
\begin_inset CommandInset citation
LatexCommand cite
key "Hosey2005MagResMed"

\end_inset

,
\begin_inset CommandInset citation
LatexCommand cite
key "PPC+05"

\end_inset

 and used in many software packages as well FSL FDT 
\begin_inset CommandInset citation
LatexCommand cite
key "Behrens2003MRM,behrens2005rca,Behrens2003NatureNeuroscience"

\end_inset

 and others.
 However, there are some problems which are discussed next.
\end_layout

\begin_layout Subsection

\lang british
Known problems
\begin_inset CommandInset label
LatexCommand label
name "sub:Known-problems"

\end_inset


\end_layout

\begin_layout Standard

\lang british
Although the probabilistic methods are able to identify many known tracts,
 they miss several large tracts such as the visual pathways LGN-MT and callosal
 MT 
\begin_inset CommandInset citation
LatexCommand cite
key "Sherbondy2008JVision"

\end_inset

.
 The visual pathways are useful test cases for algorithmic development and
 testing because they diverge and bend significantly.
 Probabilistic methods have the advantages that they expand the track search
 space beyond deterministic algorithms and that they can easily expand with
 complex models supporting crossings (usually not more than 2 crossings).
 However, they do not compute an accurate probability of brain connections.
 The phraseology 
\begin_inset Quotes eld
\end_inset

connection probabilities
\begin_inset Quotes erd
\end_inset

 or 
\begin_inset Quotes eld
\end_inset

estimation of global connectivity
\begin_inset Quotes erd
\end_inset

 or 
\begin_inset Quotes eld
\end_inset

the probability of the existence of a connection through the data field,
 between any two distant points
\begin_inset Quotes erd
\end_inset

 found in 
\begin_inset CommandInset citation
LatexCommand cite
key "Behrens2003MRM"

\end_inset

 can be very misleading because someone might believe that they represent
 the actual connectivity profile of the subject.
 For example, we are certain (with probability 1) that LGN (lateral geniculate
 nucleus) is connected to V1 and V2 (primary visual cortex) in any healthy
 brain however the estimated connection probability in FDT is much less
 than 1.
 In addition, current probabilistic algorithms fail to identify pathways
 even when they are known to exist or in a few cases they generate pathways
 even when they do not exist 
\begin_inset CommandInset citation
LatexCommand cite
key "Behrens2007NeuroImage,jbabdi2007bfg,sherbondy2006mma"

\end_inset

.
 For example, no connections between left MT+ and the posterior portion
 of Corpus Callosum were found in PiCo 
\begin_inset CommandInset citation
LatexCommand cite
key "PHW03"

\end_inset

 or FDT although it is well established that they do exist.
\end_layout

\begin_layout Standard

\lang british
In 2008, Sherbondy et al.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "Sherbondy2008JVision"

\end_inset

 in order to try to deal with the problems explained in the previous paragraph
 introduced an algorithm that separates the pathway sampling and scoring
 steps.
 In that way the scoring does not depend any more on whether we are tracking
 from seed A to seed B or from B to A; therefore it assumes symmetry when
 the other probabilistic methods do not.
 At the same time Sherbondy's method assumes independence between different
 tracts i.e.
 pathway 
\begin_inset Formula $A\rightarrow B$
\end_inset

 is independent from 
\begin_inset Formula $A\rightarrow C$
\end_inset

 or 
\begin_inset Formula $K\rightarrow L$
\end_inset

.
 This does not happen in most other methods where a pathway depend on other
 pathways starting from the same seed.
 Sherbondy's method was designed to estimate connections that are known
 to exist.
 The disadvantage of this method is that it needs a lot of user interaction
 to add the known tracks and the user needs to be a specialist in white
 matter anatomy otherwise the results might be biased.
 Other tools like FSL's FDT uses waypoint masks to reduce the effect of
 this problem.
 But these too need to be defined by the user.
 
\end_layout

\begin_layout Standard

\lang british
\begin_inset Float table
wide false
sideways false
status open

\begin_layout Plain Layout
\align center

\lang british
\begin_inset Tabular
<lyxtabular version="3" rows="6" columns="3">
<features tabularvalignment="middle">
<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

\end_layout

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

\begin_layout Plain Layout

\size small
\lang british
Deterministic
\end_layout

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

\begin_layout Plain Layout

\size small
\lang british
Probabilistic
\end_layout

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

\begin_layout Plain Layout

\size small
\lang british
Voxel Noise Resistance
\end_layout

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

\begin_layout Plain Layout
Less
\end_layout

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

\begin_layout Plain Layout

\size small
\lang british
More
\end_layout

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

\begin_layout Plain Layout

\size small
\lang british
Non-existing Tracts
\end_layout

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

\begin_layout Plain Layout

\size small
\lang british
Yes
\end_layout

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

\begin_layout Plain Layout

\size small
\lang british
Yes
\end_layout

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

\begin_layout Plain Layout

\size small
\lang british
Execution Time
\end_layout

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

\begin_layout Plain Layout

\size small
\lang british
Fast
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</cell>
<cell alignment="center" valignment="top" usebox="none">
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\begin_layout Plain Layout

\size small
\lang british
Slow
\end_layout

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

\begin_layout Plain Layout

\size small
\lang british
Memory Size
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\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
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\begin_layout Plain Layout

\size small
\lang british
Less
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\end_inset
</cell>
<cell alignment="center" valignment="top" usebox="none">
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\begin_layout Plain Layout

\size small
\lang british
More
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</cell>
</row>
<row>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\size small
\lang british
Biased on Tract Length
\end_layout

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</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
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\begin_layout Plain Layout
Yes
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</cell>
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
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\size small
\lang british
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</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Plain Layout

\lang british
\begin_inset Caption

\begin_layout Plain Layout

\lang british
Known problems with deterministic and probabilistic tractography 
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\lang british
\begin_inset CommandInset label
LatexCommand label
name "tbl:detvsprob"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard

\lang british
A small summary of the comparative strengths of deterministic and probabilistic
 tractography is given in the qualitative Tab.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "tbl:detvsprob"

\end_inset

 where we can see that although deterministic tractography will most likely
 stop more frequently at a noisy voxels it is much faster to calculate than
 probabilistic tractography.
 They both can generate non-existing tracts because of propagation errors
 or errors in the reconstruction step.
 For noise related problems e.g.
 motion and eddy correction and possible solutions see 
\begin_inset CommandInset citation
LatexCommand cite
key "Hosey2005MagResMed,BPP+00,jones1999osm,HM96,ZHK+06,andersson2002mbm,WirestamMRM2006"

\end_inset

, and for methodology and ideas comparing across subjects see 
\begin_inset CommandInset citation
LatexCommand cite
key "KanaanPsych2006,moriBook,maddah_phdthesis2008,deriche1990dcm,DavisTMI02,ODonnell_MICCAI09,DoughertyPNAS2005,DauguetNeuroImage2007"

\end_inset

.
 We showed in the previous section that probabilistic tractography usually
 gives higher weight to shorter pathways.
 In the deterministic tractography an opposite weighting is required as
 longer pathways will have higher representation in the datasets and so
 it is more likely to have more seeds along a long track rather than along
 a short track.
 Therefore, a normalization by length both for probabilistic and deterministic
 tractography is highly recommended.
 
\end_layout

\begin_layout Subsection
Segmentation
\begin_inset CommandInset label
LatexCommand label
name "sub:Segmentation"

\end_inset


\end_layout

\begin_layout Standard
The white matter contains pathways known as fibre tracts that connect functional
 areas of the brain.
 A diagram of commonly found fibre tracts is sketched in Fig.
\begin_inset space ~
\end_inset


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

\end_inset

 and real fibre tracts are shown in Fig.
\begin_inset space ~
\end_inset


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

\end_inset

.
 The white matter contains three types of fibre tracts: commissural, association,
 and projection 
\begin_inset CommandInset citation
LatexCommand cite
key "o2006cerebral"

\end_inset

.
 
\end_layout

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\begin_inset Float figure
wide false
sideways false
status open

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\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 last_figures/Gray751.png
	scale 50
	rotateOrigin center

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Diagram showing principal systems of association fibres in the cerebrum.
 The white matter fibre tracts are large bundles of axons that interconnect
 the gray matter processing areas both within and across hemispheres.
 The association fibres connect fibres from the same hemisphere.
 Picture from Gray's Anatomy #751 
\begin_inset CommandInset citation
LatexCommand cite
key "gray_anatomy"

\end_inset

.
\end_layout

\end_inset


\end_layout

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

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Commissural tracts connect related regions of the two cerebral hemispheres.
 Association fibres connect regions in the same hemisphere.
 Association fibres come in various sizes: the smallest fibres are completely
 within the cortex, the medium ones are called u-fibres and connect one
 gyrus to the next, and the longest association bundles connect different
 lobes.
 Finally, projection fibres connect the cortex and subcortical structures
 such as the thalamus, basal ganglia, and spinal cord.
 
\end_layout

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wide false
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[th!]
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\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename last_figures/association_pathways2.png
	scale 50

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Fibre pathways are so densely packed in the real brain that a segmentation
 algorithm of some kind looks like a possible solution for the neurosurgery
 planning of the future and further understanding of the brain connectivity.
 Picture from virtual hospital 
\begin_inset CommandInset citation
LatexCommand cite
key "williams1999human"

\end_inset

.
 
\end_layout

\end_inset


\end_layout

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

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
A major issue with white matter is that it is particularly dense (see Fig.
 
\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:real_brain"

\end_inset

) i.e.
 the boundaries between different tracts are very difficult to distinguish,
 many bundles cross or touch other bundles (see Fig.
\begin_inset space ~
\end_inset


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

\end_inset

) and most bundles diverge into smaller and smaller tracts as they reach
 and enter gray matter areas.
 
\end_layout

\begin_layout Standard
Previously it was not possible to automatically create white matter models
 similar to these anatomical atlas diagrams of fibre tracts in vivo.
 However, methods are now available to estimate white matter fibre tracts
 using diffusion MRI.
 In this thesis we present a method for segmentation of the trajectories
 estimated from diffusion MRI by automatically grouping them into anatomical
 regions or to be more accurate regions of similar proximity and shape character
istics.
 This is the main topic of chapter 
\lang british

\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Highly-Efficient-Tractography"

\end_inset


\lang english
 where we propose QuickBundles a highly efficient algorithm for tractography
 segmentation.
\end_layout

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wide false
sideways false
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[th!]
\end_layout

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\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename last_figures/bundles_invert2.png
	lyxscale 20
	scale 40

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
An example of a tractography segmentation based on labeling provided by
 a neuroanatomist.
 An important aim of this thesis is to automatically find the labels or
 simplify the work of an expert by clustering tractographies.
 CST: Corticospinal Tract, FX: Fornix, CG: Cingulum Bundle, SLF: Superior
 Longitudinal Fasciculus, ARC: Arcuate Fasciculus, IOF: Inferior Occipitofrontal
 Fasciculus, CC-FM: Corpus Callosum Forceps Major, UNC: Uncinate Fasciculus.
 
\end_layout

\end_inset


\end_layout

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

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
There are three main goals which should be satisfied by an automatic tractography
 segmentation algorithm
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand cite
key "o2006cerebral"

\end_inset

: automatic grouping of like trajectories into regions, region correspondence
 across subjects, and anatomical labeling of regions.
 Our ability to perform automatic, subject-specific definition of the white
 matter fibre tracts has applications in neuroanatomical visualization, neurosurg
ical planning, and neuroscientific studies of white matter integrity, structure,
 and variability.
 An example of a segmentation is shown in Fig.
\begin_inset space ~
\end_inset


\lang british

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

\end_inset

.
 The data set and labels used in this figure are from Pittsburgh Brain Competiti
on
\begin_inset Foot
status open

\begin_layout Plain Layout

\lang british
\begin_inset Formula $\texttt{braincompetition.org}$
\end_inset


\end_layout

\end_inset

.
\end_layout

\begin_layout Subsection

\lang british
Foreword
\end_layout

\begin_layout Standard

\lang british
In Chapter
\begin_inset space ~
\end_inset


\lang english

\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Cartesian-Lattice-Q-space"

\end_inset

 we concentrate on the problem of establishing multiple fibre directions
 in single voxels.
 This is a very important problem as it is considered that at least 30-90%
 of the brain contains crossings.
 We focus on comparing methods which acquire data in a Cartesian grid in
 q-space (see section
\begin_inset space ~
\end_inset


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

\end_inset

) and we contribute with a new method which we call the Equatorial Inversion
 Transform (EIT).
 In addition, we show results with a new version of Generalized Q-sampling
 Imaging (GQI2) which was not yet used with simulated or real data sets.
 Both methods have impressive accuracy on low angle crossings.
\end_layout

\begin_layout Standard
In Chapter
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Euler-Delta-Crossings"

\end_inset

 we concentrate on the problem of integrating the directional information
 from voxel to voxel in order to create streamlines (tra- cks).
 The streamlines approximate anatomical tracts as we discussed in section
\begin_inset space ~
\end_inset


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

\end_inset

.
 We contribute with a new deterministic algorithm and show how it can produce
 different results depending on the underlying reconstruction model and
 compare it against a probabilistic method (PICo).
 We show that our algorithm named EuDX overperforms in simulation tests
 in approximating tract integrity along long distances and gives more uniform
 results with real data sets against PICo.
\end_layout

\begin_layout Standard
In Chapter
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Highly-Efficient-Tractography"

\end_inset

 we concentrate on the problem of simplifying large numbers of streamlines
 which are common in current tractography analysis procedures.
 We contribute with an average linear time clustering algorithm (QuickBundles)
 which groups the streamlines by accounting a distance metric.
 This metric combines spatial and shape characteristics of the streamlines.
 QuickBundles provides representative streamlines which can be used for
 interaction, registration, queries and many other applications.
 This algorithm opens the way towards accurate segmentation of tractographies.
 Such an advancement is of great importance as described in section
\begin_inset space ~
\end_inset


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

\end_inset

.
 
\end_layout

\begin_layout Standard
In Chapter
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "sec:ThesisConclusion"

\end_inset

 we conclude the thesis and give some ideas of the work that we plan to
 do in the near future.
 
\end_layout

\begin_layout Standard
In the next chapters we will use the following definitions.
 Define a track (or streamline) as a polyline 
\begin_inset Formula $s=\{\mathbf{x_{1}},...,\mathbf{x_{n}}\}$
\end_inset

, where 
\begin_inset Formula $\mathbf{x\in}\mathbf{\mathbb{R}}^{3}$
\end_inset

.
 Then the entire tractography is defined as 
\begin_inset Formula $T=\{s_{1},...,s_{m}\}\sim\mathfrak{T}$
\end_inset

, where usually the number of tracks is 
\begin_inset Formula $|T|\simeq2\times10^{5}-2\times10^{6}$
\end_inset

.
 For an anatomical (physical) bundle (tract) e.g.
 Arcuate Fasciculus we use 
\begin_inset Formula $\upsilon$
\end_inset

 and for a fibre bundle we use 
\begin_inset Formula $u$
\end_inset

 where 
\begin_inset Formula $u\subset T$
\end_inset

, which approximates 
\begin_inset Formula $\upsilon$
\end_inset

.
 We think it is very important to always have in mind that we can only approxima
te a real tract.
 
\end_layout

\end_body
\end_document