dti report

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Visualization of Diffusion Tensor Imaging

Masters Thesis

Guus Berenschot

May 2003

Eindhoven University of Technology

Department of BioMedical Engineering

Biomedical Imaging and Informatics

Supervisors:

dr. A. Vilanova i Bartroli

prof. dr. ir. B.M. ter Haar Romeny

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II

Abstract

Diffusion Tensor Imaging (DTI) is a fairly new Magnetic Resonance Imaging technique.It shows the diffusion (i.e. random motion) of water molecules in tissue. The apparentdiffusion coefficient (ADC) is a measure for the amount of diffusion in tissue. Whitematter in brain tissue and muscles are structured tissues, meaning that there is clear

orientation in these tissues due to nerve fiber bundles and muscle fibers. In structuredtissue the ADC is direction dependent, it is larger in the direction along structures thanin the directions perpendicular to it. DTI measures the ADC in six directions, fromthe ADC in these six directions is a symmetric diffusion tensor matrix is derived. Afterdiagonalization of the diffusion tensor matrix three eigenvectors and three correspond-ing eigenvalues are obtained. The eigenvector with the largest eigenvalue correspondsto the main diffusion direction. The other two eigenvectors correspond to directionsperpendicular to this direction.

The Maxima Medical Center (MMC) uses DTI to study the development of theneonatal brain. The Magnetic Resonance Laboratory (MRL) uses it to study functional

properties of muscles.A visualization tool is developed to visualize the DTI data. It is concentrated on

data obtained by the MMC an MRL. This tool contains Multi Planar Reconstruc-tion (MPR) planes, glyphs and fiber tracking. On the MPR, several textures can bedisplayed that provide information about the diffusion tensor. The MPR planes arealso used to give context to the other visualizations, the glyphs and fiber tracking.Glyphs are icons that represent the local diffusion tensor. Two types of glyphs can beused: cuboids and ellipsoids. Both types of glyphs depict the three eigenvectors andcorresponding eigenvalues of the diffusion tensor. Fiber tracking simplifies the diffu-sion tensor tensorfield to the vectorfield of the main eigenvector. If we consider this

vectorfield as a velocity field and drop a free particle on it, this particle will follow atrajectory due to the velocity field. The found trajectory can be seen as a fiber thatrepresents a bundle of nerve fibers in the brain or muscle fibers. Fiber tracking showsglobal information about orientation in tissue.

Fiber tracking gives problems in regions where the first two eigenvectors of thediffusion tensor are about the same size. This happens for example in regions wherefibers are crossing. The main eigenvector is not reliable in these regions. A possiblesolution to this problem is presented.

Results obtained with the DTI tool show promising results for further investigationof DTI. Specially the fiber tracking together with the MPR planes is considered, by the

MMC and MRL, as a very useful visualizing tool for DTI.

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III

Samenvatting

Diffusie Tensor Imaging (DTI) is een betrekkelijk nieuwe Magnetic Resonance Imagingtechniek. Het laat de diffusie (d.w.z. willekeurige beweging) van water moleculen inweefsel zien. De apparent diffusion coefficient (ADC) is een meting voor de hoeveel-heid diffusie in weefsel. De witte stof in hersenen en spieren zijn gestructureerd, ditbetekent dat er orientatie in dit weefsel zit ten gevolge van zenuw bundels en spierfibers. In gestructureerd weefsel is de ADC richting afhankelijk, het is groter in de

richting langs de structuren dan in de richting loodrecht op deze richting. DTI meet deADC in zes richtingen, van de ADC in deze zes richtingen wordt een symmetrische dif-fusie tensor matrix afgeleid. Door diagonalisatie van de diffusie tensor matrix wordendrie eigenvectoren met drie corresponderende eigenwaarden verkregen. De eigenvec-tor met de grootste eigenwaarde correspondeert met de belangrijkste diffusie richting.De andere twee eigenvectoren corresponderen met de richtingen loodrecht op de dezerichting.

Het Maxima Medisch Centrum (MMC) gebruikt DTI om de ontwikkeling van hetneonatale brein te onderzoeken. Het Magnetic Resonance Laboratory (MRL) gebruiktDTI om de functionele eigenschappen van spieren te bestuderen.

Om DTI data te visualiseren is een visualisatie tool ontwikkeld. Het is gericht opdata verkregen van het MMC en MRL. De tool bevat Multi Planar Reconstruction(MPR) vlakken, glyphs en fiber tracking. Op de MPR vlakken kunnen verschillendetextures worden afgebeeld die informatie verschaffen over de diffusie tensor. De MPRvlakken zijn ook gebruikt om context te geven aan de andere twee visualisaties, deglyphs en de fiber tracking. Glyphs zijn iconen die de locale diffusie tensor represen-teren. Twee soorten glyphs kunnen worden gebruikt: cuboids en ellipsoids. Beidengeven de drie eigenvectoren met de drie corresponderende eigenwaarden van de diffusietensor weer. Fiber tracking vereenvoudigt het diffusie tensor tensorveld tot het vec-torveld van de belangrijkste eigenvector. Als we dit vectorveld als een snelheidsveld

beschouwen en er een vrij deeltje op laten vallen zal dit deeltje een traject afleggen tengevolge van het snelheidsveld. Dit traject kan worden gezien als een bundel van zenuwfibers in de hersenen of spier fibers. Fiber tracking laat globale informatie zien van deorientatie in weefsel.

Fiber tracking geeft problemen in gebieden waar de eerste twee eigenvectorenongeveer dezelfde waarde hebben. Dit gebeurt bijvoorbeeld in regios waar fiber bun-dels elkaar kruisen. De belangrijkste eigenvector is niet betrouwbaar in deze regios.Een mogelijke oplossing voor dit probleem is gepresenteerd in dit verslag.

De resultaten verkregen met de DTI tool zijn veelbelovend. Vooral de fiber trackingsamen met de MPR vlakken worden als een zeer nuttige visualisatie methode ervaren

door het MMC en het MRL.

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Contents

Abstract II

Samenvatting III

Abbreviations and Symbols VI

1 Introduction 1

2 Diffusion Tensor Imaging 3

2.1 Diffusion in Tissue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.2 Restricted Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.3 Anisotropic Diffusion . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Aquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 The diffusion tensor and its properties . . . . . . . . . . . . . . . . . . 7

2.4 Interpretation Diffusion Tensor . . . . . . . . . . . . . . . . . . . . . . 7

2.4.1 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . 7

2.4.2 Anisotropy Indices . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Tensor Visualization 11

3.1 2D Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1.1 Colorcoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.1.2 Glyphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1.3 Color-coded vectors . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 3D Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.1 Volume Rendering . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.2 Glyphing in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

IV

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CONTENTS V

3.2.3 Fiber tracking or Streamlines . . . . . . . . . . . . . . . . . . . 19

3.2.4 Hyperstreamlines . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Surface Building 26

4.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2.1 Surface Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2.2 Data structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 DTI Tool 34

5.1 Multi Planar Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 34

5.2 Glyphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.3 Fiber tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6 Results 42

6.1 Optical nerve of a cow . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6.2 Healthy Volunteer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6.3 Patient with Tumor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.4 Neonates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.4.1 Neonate without abnormalities in the brain . . . . . . . . . . . 48

6.4.2 Neonate with small hemorrhages . . . . . . . . . . . . . . . . . 48

6.4.3 Neonate with asphyxia during delivery . . . . . . . . . . . . . . 50

6.4.4 Neonate with a large hemorrhage . . . . . . . . . . . . . . . . . 50

6.5 Rat Brain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6.6 Mouse Muscle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.7 Surface Building in a healthy volunteer . . . . . . . . . . . . . . . . . . 53

7 Conclusion and Future Work 57

7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

A Created Classes 60

Bibliography 63

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CONTENTS VI

Abbreviations and Symbols

Abbreviations

BM/i2 BioMedical Imaging and InformaticsCC Corpus CallosumDTI Diffusion Tensor ImagingDWI Diffusion Weighted ImagingFLTK Fast Light ToolKitLUT Look-Up-TableMMC Maxima Medical Center, VeldhovenMPR Multi Planar ReconstructionMRI Magnetic Resonace ImagingMRL Magnetic Resonance LaboratoryNMR Nuclear Magnetic ResonancePOI Point Of InterestROI Region Of InterestTE echo timeVTK Visualization ToolKit

Symbols

integration stepsize duration gradient pulse gyromagnetic ratio phase effect tortuosityi i

th eigenvalue

ei ith eigenvector

ei,j edge of a surface between vertices i and j, vertex j is connected tovertex i

Ei,j edge of a surface between vertices i and j, vertex j is traced fromvertex i

Radius1 largest radius of the cross section of hyperstreamlines andhyperstreamprismas

Radius2 smallest radius of the cross section of hyperstreamlines andhyperstreamprismas

TDi track direction i, for surface buildingvin incoming vectorvout outgoing vectorvprop propagation vector

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CONTENTS VII

D Diffusion tensorI Identity matrix

A Attenuation MR signal caused by diffusionADC Apparent Diffusion CoefficientCa anisotropy measure for deviation from spherical caseCl anisotropy measure for linear diffusionCp anisotropy measure for planar diffusionCs anisotropy measure for spherical diffusionD Diffusion coefficientFA Fractional AnisotropyRA Relative AnisotropyRMS Root Mean Square displacement

S Signal intensityT absolute temperature

b b-value, represents the sensitivity of a measurementf friction coefficientg gradient pulsehyperscale user-defined factor to scale the diameter of the hyperstreamlines and

hyperstreamprismas

k Bolzmann constantmax2 maximum second eigenvalue with a FA higher then 0.3scalefactor scale factor to scale the diameter of the hyperstreamlines and

hyperstreamprismastdiff diffusion timewpunct user-defined parameter in tensorline technique

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Chapter 1

Introduction

The Biomedical Imaging and Informatics (i.e. BM/i2) department of Biomedical Engi-neering started last year with a new project about the visualization of Diffusion TensorImaging. Diffusion Tensor Imaging (DTI) is a fairly new MR imaging technique thatprovides information about the amount of diffusion (i.e. motion of water molecules)and preferred diffusion directions. It provides a diffusion tensor, i.e. 3x3 matrix, foreach voxel in the data. DTI is an extension to Diffusion Weighted Imaging (DWI),which only provides information about the amount of diffusion. This project is donein cooperation with the Maxima Medical Center (MMC) and the Magnetic ResonanceLaboratory (MRL).

The MMC is one of the ten hospitals with a neonatal intensive care unit. This unitis specialized in treating critically ill newborns. Some of these newborns, have braindamage caused by a lack of oxygen (i.e. cerebral hypoxic ischemic brain damage), dueto problems with the bloodflow. DWI proved already to be a useful technique to detectthis. Diffusion Tensor Imaging, might also provide information about structure andthe development of the neonatal brain.

The MRL does research on the functional properties of muscles. It is proven thatthe spatial structure of muscle strongly determines the functional properties it. Forexample, the distribution of stresses and strains largely depends on the orientation ofmuscle fibers. Also the length of the fibers and the amount of fibers are parameters todetermine functional properties of a muscle. Diffusion Tensor Imaging showed already

that it provides information about the orientation of fibers in a muscle.

Visualizing the DTI data of the MMC and MRL is complex. Most visualizationtechniques display just one scalar value and the diffusion tensor, is a 3x3 matrix. Thefact that the most interesting structures are 3-dimensional (3D), makes the visualizationeven more complicated. The goal of this study is to find a useful visualization methodto visualize the DTI information of the MMC and MRL. First a literature study ofthe visualization techniques that are already available for DTI is done. Thereafter themost useful techniques for the MMC and MRL are implemented. A problem of one ofthe selected techniques is considered into detail and a possible solution to this problemis presented.

1

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CHAPTER 1. INTRODUCTION 2

Chapter 2 describes the basic principles of Diffusion Tensor Imaging. In chapter

3 the results of the literature study of the visualization methods of DTI is described.

Chapter 4 describes a concrete problem of a selected techniques and a possible solution.Chapter 5 describes a visualization tool that has been developed for visualizing the DTI

data. Thereafter, in chapter 6 results of visualizing data of the MMC and MRL with

this tool are presented. Finally, in chapter 7, conclusions and recommendations for

future work will be presented.

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Chapter 2

Diffusion Tensor Imaging

Diffusion tensor imaging is a non-invasive technique to measure diffusion of watermolecules in vivo. This chapter describes the physical basis of diffusion in tissue andhow this can be used [1] [2]. It also describes how a diffusion tensor can be measuredwith a diffusion sensitized MR sequence and how this tensor can be understood.

2.1 Diffusion in Tissue

2.1.1 Diffusion

Diffusion is the random translational, or Brownian, motion of molecules or ions that isdriven by internal thermal energy. In systems with a concentration gradient of diffusingmolecules, diffusion leads to a netto displacement of the diffusing molecules, i.e. flux.In isotropic solutions (i.e. solutions without a concentration gradient) the probabilityof displacement of molecules is equal in all directions, and the mean molecular dis-placement, and the flux, is zero. The mobility of the molecules can be characterizedby a physical constant, the diffusion coefficient, i.e. D. In case of a pure liquid this isalso called self-diffusion coefficient. D is related to the root mean square displacement(RMS), which is the root of the mean displacement of a molecule over a given time.The Einstein relation which describes this relation in 1-D is:

RMS=

2Dtdiff (2.1)

with tdiff the time over which the diffusion is measured (diffusion time). The diffusioncoefficient is closely related to the size of molecules, and temperature (T) according tothe Stokes-Einstein equation:

D = (kT)/f (2.2)

in which k the Bolzmann constant, T the absolute temperature and f the frictioncoefficient, which depends on the size of the particle and the viscosity of the fluid.

3

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CHAPTER 2. DIFFUSION TENSOR IMAGING 4

2.1.2 Restricted Diffusion

So far, only unrestricted isotropic diffusion has been described. We assume thatmolecules can diffuse in every direction an infinite distance, so the distance thatmolecules diffuse (i.e. diffusion distance) increases in proportion to the square rootof time. When a container limits the diffusion, the molecules are reflected when theyreach a boundary. In this situation, the diffusion distance increases linearly with thesquare root of time only for short diffusion times, and reaches a plateau at longertimes. In biological tissues, physical barriers often restrict diffusion. These barrierscan be membranes and organelles, which are generally partially permeable. This leadsto an intermediate situation, where the diffusion distance increases in proportion tothe square root of time (for long diffusion times), but the diffusion distances are muchsmaller than for free diffusion. The diffusion coefficient measured by nuclear magnetic

resonance (NMR) is also called apparent diffusion coefficient (ADC). This coefficienttakes into account that it is not a measure of the intrinsic diffusion coefficient (D),but a coefficient that depends on the interactions of the diffusing molecule with thecellular structures over a given diffusion time. For the measurements described lateron we can assume that the diffusion time is long enough to get a constant ADC. TheADC can be described as follows:

ADC= D/2 (2.3)

In this equation D is the intrinsic diffusion coefficient and is the tortuosity, which isa measure of the hindrance imposed by physical barriers.

2.1.3 Anisotropic Diffusion

The tortuosity of tissue can be different for each direction in space. In a pure liquidwhen there is no hindrance to diffusion (so =1) or in a sample with barriers that arenot coherently oriented ( >1), the tortuosity is the same in all directions and diffusionis isotropic. In a sample with highly oriented barriers, the tortuosity is different foreach direction in space and is leading to anisotropic diffusion. So in samples withoriented structures, the tortuosity perpendicular to the structure will be larger thanthe tortuosity along the structure and the ADC values will respectively be smaller and

larger. For this reason the ADC values in certain directions are directly related to thegeometry of these structures (see figure 2.1).

Tissues with a highly oriented structure which show anisotropic diffusion, are forexample the brain and muscles. For muscles the macroscopic and the microscopicstructure are responsible for the anisotropy in diffusion (see figure 2.2 a). In braintissue, bundles of myelinated axons are responsible for the anisotropy. Axons are apart of nerve cells (neurons) that are located in the brain (see figure 2.2 b). The cellbodies are located in the grey matter, which is mainly located in the outer part of thebrain. The axons are located in the white matter. They can be 5-10.000 times as longas the cell body is wide. The axons are surrounded by a myelin sheath which is buildup by layers of membranes. This layer causes a higher restriction and thus a higher

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Figure 2.1: Diffusion in isotropic sample (left) vs. diffusion in an anisotropic sample (right). In theisotropic case the diffusion is similar in all directions, in the anisotropic case the diffusion is larger inone direction than in the other

anisotropy. After the age of 26 weeks (after conception) myelination starts and willcontinue up to 2 years of age. This is also the reason that the anisotropy in the brainof newborns is lower than in adult brain tissue.

a) b)

Figure 2.2: a) The structure of a muscle: a muscle is built from fibre bundles and fibres that containof myofibrils. All are oriented in a parallel direction. b) Schematic representation of a neuron. In thebrain, bundles of axons are responsible for anisotropy

2.2 Aquisition

Diffusion Tensor MRI is based on traditional Magnetic Resonance Imaging (MRI).The reader is referred to books explaining MR basics for detailed information, e.g.Vlaardingerbroek et al. [3]. The method used for the diffusion measurements is the

pulsed-field gradient (PFG) method. This method was first introduced by Steljskal

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Figure 2.3: Diffusion sensitized spin-echo sequence(A) The 90 excitation and 180 refocusing pulsesform a spin-echo at the echo-time TE.(B) The pulse sequence of (A) can be sensitized for diffusionby applying a pair of pulsed-field gradients, with amplitude g, surrounding the 180 pulse. Thedefinition of the gradient pulse duration, , and the time separation between the gradient pulses, ,is indicated. [5]

and Tanner [4]. They incorporated a pair of diffusion-sensitizing linear magnetic-fieldgradients into a Hahn spin-echo sequence (figure 2.3). The purpose of the gradientpulses is to magnetically label the transverse magnetization of spins within a moleculeas a function of spatial position. The gradient causes the Larmor frequency of thenuclear spins to become spatially dependent with respect to the gradient direction.The phase effect of a gradient pulse g of duration on a spin at position r is givenby:

(r) = gr (2.4)

in which is a the gyromagnetic ratio.

Note that only motion in the direction of the gradient causes a change in phase ofthe spin. If we consider a spin that was at position r0 during the first gradient pulseand at position r1 during the second, then the change in phase of this individual nuclearspin, , by moving form r0 to r1 is given by:

(r1 r0) = g(r1 r0) (2.5)

When there is no diffusion, the phase difference is zero since the two identical gradientpulses exert and equal the phase effect. The phase effect is cancelled out in that casebecause of the fact that the 180 pulse in the spin-echo sequence (figure 2.3) reversesthe sign of the phase angle. This causes the static spins to be all in phase, which gives

a maximum echo signal. Diffusing spins will not be in phase, which is leading to asignal attenuation. The amount of attenuation A caused by diffusion is determined bythe ADC and a so-called b-factor (b), which describes the strength of the diffusionsensitivity.

A = exp(bADC) (2.6)

The b-factor for the gradients displayed in figure 2.3 is:

b = 2g22( /3) (2.7)

In which is the gyromagnetic ratio, the duration of the gradient pulse, and thetime between the onset of the two gradient pulses. Of course, the intensity of the signal

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is also affected by T2 relaxation. To know the amount of this effect, a measurementthat is not diffusion weighted (S(0)) is required. The measured signal S(b) is:

S(b) = AS(0) (2.8)

From equation 2.12 and 2.8 the ADC can be calculated:

ADC = 1

bln(

S(b)

S(0)) (2.9)

2.3 The diffusion tensor and its properties

In case of anisotropic diffusion, diffusion cannot be characterized by a single scalarcoefficient. A tensor representation(D) is then required:

D =

Dxx Dxy DxzDyx Dyy DyzDzx Dzy Dzz

(2.10)

The diagonal elements Dxx,Dyy and Dzz represent the ADC along the x,y and z axesin the laboratory frame (i.e., the frame in which the gradients are applied). The off-diagonal elements represent the correlation between the diffusion in perpendicular di-rections. The tensor D is symmetric (i.e. Dij = Dji, with i, j = x , y , z) and positive. Inthe isotropic case and the case that the main diffusion directions coincide with the labo-ratory frame, there is no correlation between the perpendicular directions (i.e. Dij = 0,with i = j). The attenuation (A) can be described by:

A = exp(

i=x,y,z

j=x,y,z

bijDij) (2.11)

or, because D is symmetric:

A = exp(bxxDxx byyDyy bzzDzz 2bxyDxy 2bxzDxz 2byzDyz) (2.12)

To determine the diffusion tensor, D, one should collect diffusion weighted imagesalong at least 6 directions and an image that is not diffusion weighted (i.e. b = 0).This minimal set of images may be repeated for averaging, this to improve the signal-to-noise ratio (SNR). From equation 2.9 the ADC can be calculated for each direction. The

tensor can be calculated from these ADC values and a rotation matrix that dependendson the direction of the applied gradients. The gradients that are used depend on thescanner and the used sequence. More details about which gradients are commonly usedin the MMC are described in Janssen [6].

2.4 Interpretation Diffusion Tensor

2.4.1 Eigenvalues and Eigenvectors

In order to determine the main diffusion directions in every voxel one needs to diago-

nalize the diffusion tensor. This diagonalization provides three eigenvectors ( e1, e2, e3)

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with three corresponding eigenvalues (1, 2, 3) and can be seen as the three maindiffusion directions. The eigenvector corresponding to the largest eigenvalue will point

in the direction of the largest diffusion coefficient, the vector with the smallest eigen-value points in the direction with the smallest diffusion coefficient. Mathematically,the diagonalization of the diffusion tensor can be described as follows:

Dei = iei ei = 0 (2.13)The solutions (i) to equation 2.13 are the eigenvalues ofD. The vectors ei associatedwith each eigenvalue are the eigenvectors ofD. Since the null vector is omitted 2.13 canbe re-written as (D iI)ei = 0, where I represents the identity matrix. This impliesthat the matrix D iI is singular and its determinant is zero, which corresponds tothe eigenvalues being the solutions to the secular equation:

|D

iI

|= 0 (2.14)

For each eigenvalue i the corresponding eigenvector ei can be found by solving:

(D iI)ei = 0 (2.15)The eigenvector-eigenvalue pairs together contain all information available in the orig-inal tensor. In general the eigenvalues may be real or complex. However, for a realsymmetric tensor, like the diffusion tensor, the eigenvalues are always real and theeigenvectors will be orthogonal to each other. Since the diffusion coefficient is a posi-tive quantity, the eigenvalues will always be positive.

2.4.2 Anisotropy Indices

There are several scalar indices that indicate the amount of diffusion anisotropy ofthe diffusion tensor. The indices described here are based on the eigenvalues of thetensor and are rotationally and scale invariant. Some indices only distinguish betweenisotropic and anisotropic, other indices also divide the anisotropy in two cases: planar(pancake shape) and linear (cigar shape) anisotropy. The next section describes indicesthat are found often in literature. A more complete overview of anisotropy indices canbe found in Janssen [6].

Fractional and Relative Anisotropy

Two anisotropy indices that are often found in literature are the fractional and relativeanisotropy(FA respectively RA). These indices are defined as follows [7]:

FA =

(1 2)2 + (2 3)2 + (1 3)2

2(21 + 22 +

23)

(2.16)

RA =

(1 2)2 + (2 3)2 + (1 3)2

2(1 + 2 + 3)(2.17)

The range of these indices is between zero and one, with zero the isotropic and one the

anisotropic case.

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Linear and Planar Anisotropy Indices

Westin et al. [8] divided diffusion anisotropy into three basic cases depending on therank of the tensor. Linear case, Cl: diffusion is mainly in the direction correspondingto the largest eigenvalue (see figure 2.4 a). Planar case, Cp: diffusion is restricted toa plane spanned by the two eigenvectors corresponding to the two largest eigenvalues(see figure 2.4 b). Spherical case Cs: isotropic diffusion (see figure 2.4 c)

a) b) c)

Figure 2.4: a) linear case, b) planar case, c) spherical case

In Westin et al. [8] the indices are defined as follows:

Cl =1 2

1 + 2 + 3(2.18)

Cp =2 3

1 + 2 + 3(2.19)

Cs = 33

1 + 2 + 3(2.20)

Ca = 1 Cs = Cl + Cp =1 + 2 231 + 2 + 3

(2.21)

A very similar definition can be found in Westin et al. [9]. It can be shown that thethree indices fall in the range [0,1], and that they sum to unity:

Cs + Cp + Cl = 1 (2.22)

To depict the space of possible anisotropies, barycentric coordinates are used (see figure

2.5). For every point in the triangle, there is a corresponding ellipsoid for which theanisotropy indices correspond to the points barycentric coordinates. In the figure,the three ellipsoids accompanying the corners of the triangle are representative of theellipsoids which correspond to those corners. At each vertex of the triangle, one of theanisotropy indices is one, while the the others are both zero. Along the sides of thetriangle, one of the anisotropy indices is zero, and the other two indices sum to one.

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Figure 2.5: Barycentric coordinate map

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Chapter 3

Tensor Visualization

Anisotropic diffusion is characterized by a second order symmetric tensor. This cor-responds to 6 different values per sampled position. In MR scanners, the samples areusually acquired in a 2D rectilinear grid or a stack of 2D images which in 3D corre-sponds to a 3D cartesian grid (i.e. volume). The sampled point and its surroundingarea in 2D are called pixel and in 3D voxel. Visualizing these 6 scalar values separatelyin a 2D slice (see figure 3.1) or by 3D volume rendering techniques makes it difficult toextract any meaningful information. Therefore other methods need to be used whichhelp the interpretation of the data.

Figure 3.1: The 6 different components of the diffusion tensor D displayed separately. The position inthe image corresponds to the position in the tensor. Note that the off diagonal images are symmetric. [6]

3.1 2D Methods

2D methods are understood as methods that take just one slice into account for visu-alization. These methods are often suitable for displaying local information in certain

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CHAPTER 3. TENSOR VISUALIZATION 12

a) b) c)

Figure 3.2: a) Reference isotropic ADC image. b) Color coding of the linear anisotropy (Cl), hue

LUT shown on the left. c) Color coding of Cp, hue LUT shown on the left.

areas of the data. It is almost impossible to understand global structures, since theseare basically 3D structures.

3.1.1 Colorcoding

From the tensor several scalar values, e.g. anisotropy indices, can be derived. Thesevalues have a more intuitive meaning than the six separated tensor components. Thesevalues can be shown using colorcoding. The colorcodings described in this section are

applied to 2D planes. Note that these colorcoding methods can also be used to colorglyphs or fibers that will be described in the next sections.

Colorcoding anisotropy indices

Anisotropy indices can be used to visualize the tensor information. These indicesprovide only information about how anisotropic the tensor is. The six values for thedifferent directions that are measured by DTI are reduced to one scalar value.

Scalar values can be colorcoded with a look-up-table (i.e. LUT). A look-up-tableholds an array of colors. Associated with the table is a minimum and maximum scalar

value into which the scalar values are mapped. Scalar values greater than the maximumrange are clamped to the maximum color, scalar values less than the minimum range areclamped to the minimum color value. In figure 3.2 colorcoding of the linear anisotropy(Cl, see equation 2.18) and planar anisotropy (Cp, see equation 2.19) on one slice ofthe brain with a hue LUT is shown. The colors of a hue LUT and the values to whichthe colors are mapped are also shown in the figure.

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Figure 3.3: Color coding scheme for color coded vectors. A-P is anterior-posterior (front-back), R-Lis left-right and F-H is the feet-head direction [13]

Colorcoding main diffusion direction

A method that is commonly used in medical context is the colorcoding of the maindiffusion direction [10] [11]. In this case the tensor is simplified to the main eigenvectorand the FA. Of course, also other anisotropy indices can be used instead of the FA.

A color is assigned to each voxel using the main eigenvector of the diffusion ten-sor. This vector is normalized and at each voxel the absolute value of the x, y and zcomponents are used as the red, green, and blue channels. Therefore a red pixel in theimage corresponds to a vector oriented in left-right direction, green in anterior-posterior(front-back) direction and blue in feet-head direction. For instance, if the x component

of a vector is much larger compared to the y and z components the color assigned willbe close to pure red. Otherwise, the color will be a mixture of red, green and bluedepending on the magnitudes of the vector components, i.e. on the direction of thevector. Figure 3.3 shows this color coding scheme with the appropriate color paintedonto a sphere.

In regions with low anisotropy, the main diffusion direction is meaningless. Toavoid representing directions in regions of low anisotropy, the pixel intensity can bescaled with, e.g. the Fractional Anisotropy (FA). In regions with high anisotropy theintensity of the color will be very bright, while in the less important isotropic regionsthe color will appear dark. Figure 3.4 b shows an example of a slice in which the maineigenvector is colorcoded. Figure 3.4 c shows an example of a colorcoded main diffusiondirection weighted with the FA. Figure 3.4 a is an isotropic ADC image and is addedas a reference. By scaling the intensity with the FA, it is directly clear what the mostanisotropic regions in the image are. A disadvantage of this method is that it is notintuitive what the meaning of the color is. Also one color can represent more thanone direction because of the symmetry of the sphere, see for example the two markedregions in figure 3.4 c. Several other colormaps are described in Pajevic et al. [12].

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a) b) c)

Figure 3.4: a) Isotropic ADC image b)Direction main eigenvector colorcoded. c) Main eigenvector

colorcoded and weighted with local FA. Two regions are marked here with about the same color,however the main diffusion direction is different.

3.1.2 Glyphing

To show the information of the diffusion tensor a glyph can be used. A glyph is anicon that represents the data at a specific location. This method can be very useful fordisplaying detailed information in a small local area. When looking to bigger areas thismethod will be less useful because the information will be too detailed to see anythingin it. However in 2D glyphs can give the impression that regions are connected in a 2Dslice. In fact these connections are 3D, and cannot be displayed in a 2D slice.

Ellipsoids

The diffusion tensor matrix is symmetric and has positive eigenvalues see chapter 2.3.These properties make an ellipsoid a natural geometric representation of the diffusiontensor. The direction of axes of the ellipsoid are determined by the eigenvectors. Thelength of the axis are determined by the size of the eigenvalues. The use of ellipsoidsfor DTI is shown by e.g. Pierpaoli et al. [14] and Basser et al. [15]. Laidlaw [16]normalized the size of the ellipsoid to obtain a more continuous visual appearance.

Advantages of this technique is that it displays most of the information available

from the tensor, i.e. the eigenvectors and eigenvalues. A problem of using an ellipsoidas glyph is that it is often difficult to see the orientation and shape of it. It is, forexample, difficult to distinguish between an face-on flat ellipsoid and a sphere. Anexample of the normalized ellipsoids in the brain is giving in figure 3.5.

Boxes

Worth et al. [13] presented a method that is very similar to the ellipsoids described inthe previous section. Instead of using spheres as basic representation the authors usedcuboids which are scaled according the eigenvalues and eigenvectors. The advantage

of this method compared with the ellipsoids is that it is easier to see the direction of

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a) b)

Figure 3.5: a) Overview of the position in brain where ellipsoids are placed. b) Ellipsoids zoomed,colorcoding according to Fractional Anisotropy (see section 3.1.1), background is an isotropic ADCimage.

the second and third eigenvector of the boxes. Figure 3.6 compares the ellipsoids withthe boxes. Both are oriented in the same way and have the same eigenvectors andeigenvalues.

a) b)

Figure 3.6: a) Cuboid b) Ellipsoid. The ellipsoid and cuboid represent the same tensor in the sameorientation. It can be seen that the orientation of the cuboids is much clearer than that of the ellipsoids.

Spheres, planes, lines

Westin et al. [9] suggest a composite shape with a linear, planar, and spherical com-ponent. The size of these components are scaled according to Cl, Cp and Cs, seeequations 2.18, 2.19, 2.20. The orientation of the linear component is in direction ofthe main eigenvector. The normal of the plane is oriented in direction of the thirdeigenvector. Figure 3.7 a shows this display technique compared to the traditional

ellipsoid shape. Figure 3.7 b shows this technique applied to an artificial dataset of a

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crossing fiber. Advantage of Westins method that its is clear to see what the orientationof the local tensor is and also what the dominating shape is.

a) b)

Figure 3.7: a) Comparison of an ellipsoid and a composite shape depicting the same tensor with

eigenvalues 1=1, 2=0.7 and 3=0.4, b) Composite shape applied to an artificial dataset of a crossingfiber [9]

3.1.3 Color-coded vectors

The method that Peled et al. [17] suggest is a combination of glyphing and colorcoding.It simplifies the tensor to the main eigenvector (e1) and an anisotropy index, the relativeanisotropy (RA) in this case. They display the vector RA e1. The vector is displayedon the background of an anatomical image. The blue headless arrows, which can be seenas glyphs, represent the in-plane components of the vector. The out-plane componentsare shown in colors ranging from green to yellow to red, with red indicating the highestvalue for this component, this can be seen as colorcoding (see figure 3.8). A problem ofthis method is that the length of the vector is determined by the RA and the in-planecomponent, which makes it difficult to interpret.

Figure 3.8: Color-coded vectors: The blue headless arrows represent the in-plane components of thevector. The out-plane components are shown in colors ranging from green to yellow to red, with redindicating the highest value for this component [17]

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3.2 3D Methods

Until now we just described 2D methods, we were just looking at one slice of the data.However, often we are interested in understanding the 3D structures, which is difficultto get from the 2D methods. It is difficult to extend the 2D-methods to 3D directly.The glyphing methods described in the previous section can be also applied in 3D.However, visual cluttering becomes a serious problem, another problem is that justlocal structure is shown and not global structure, such as fibers.

This chapter describes a volume rendering method, which is a visualization methodoften used for 3D visualization of scalar volume data. Thereafter, it describes theglyphing in 3D. Finally, the method that is by far the most popular, fibertracking isdescribed and an extension to this, the hyperstreamlines.

3.2.1 Volume Rendering

Kindlmann et al. [18] [19] introduced a method for visualizing diffusion tensor fields bydirect volume rendering. The philosophy behind this method is to display only someof the information, but display that information densely within a volume. Kindlmannet al. present methods for assigning color, opacity and shading to each location inthe dataset based on the properties of the diffusion tensor information. Although lowopacity is assigned to unimportant data, visual cluttering is still a problem. The com-pression of 3D dataset into a 2D image makes it difficult for a viewer to extract precisepositional information from the final image. This makes it difficult to pick out the pathof a certain fibrous structure from its neighborhood. Also, a lack of interactivity limitsa users understanding of the image. An example of a volume rendering of the brain isdepicted in figure 3.9. The color and opacity are mapped with a barycentric map (seealso figure 2.5), these maps are also shown in figure 3.9. For more details about thismethod, the reader is referred to Kindlmann et al. [18] [19].

3.2.2 Glyphing in 3D

There are two limitations of glyphing methods when applied to a 3D dataset. First, byvisualizing every sample point in a 3D dataset, only the outermost layer of the dataset

can be displayed on the screen. The internal data points will be blocked. Second,continuity inherent in biological tissues will not be properly represented in the finalimage. For example, the neural fibers in the brain that connect different anatomicalregions would be difficult to locate within an array of ellipsoids. To get an idea of theseproblems an example of the ellipsoids applied to 3D can be seen in figure 3.10

Westin et al. [9] showed a 3D application of his method (i.e. spheres/planes/lines,see section 3.1.2) on an artificial dataset of three crossing fibers. First they decide whichglyphs in the data to display. Than they threshold the anisotropy indices and displaysdepending on which index is high only a sphere, a plane or a line, see figure 3.11.

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Figure 3.9: Volume rendering of the brain, with color and opacity determined by the barycentricmaps shown. [19]

Figure 3.10: Ellipsoid technique applied to 3D. [20]

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Figure 3.11: Left) Original tensor data displayed with sphere plane line technique. Right) Tensor

data after thresholding the anisotropy indices. [9]

3.2.3 Fiber tracking or Streamlines

Fiber tracking is a fairly new technique. It gives a better understanding of the fiberstructure in for example the brain. Water in white matter diffuses more along brainfibers than perpendicular to it. Therefore we can assume that the main eigenvectorof the diffusion tensor gives us a good estimate of the fiber orientation. The diffusiontensor is simplified to the vectorfield of the main eigenvector.

There are several techniques to visualize vectorfields. Streamlines were presented

by [] . Later on this technique was adapted for the DTI community to visualize tensorsand called fiber tracking. The first successful fiber tracking results were presented inXue et al. [21], Morgan et al. [22] and Conturo et al. [23].

If we consider the vectorfield of the main eigenvector as a velocity field, fiber trackingcan be seen as follows. We drop a particle on the velocity field and follow stepwise thetrace of it. The trace is the curve tangent to the vectorfield. The found trajectory canbe seen as a fiber. The starting point of the particle is the seed point. The particlestops following the tract when a stopping criterium is reached.

This section describes the principles of the tracking itself. It describes how seedpoints and stopping criteria for the tracking are defined. Finally it describes some of

the problems of fiber tracking possible solutions to this and it shortly explains about adifferent class of fiber tracking techniques.

Tracking

The trajectory is followed in a continuous vectorfield. Since the data that is measuredby MR-scanners is discrete, it has to be interpolated. This interpolation can be donein the main eigenvector vectorfield. However, a more robust method is to interpolatein the tensordata and to calculate the eigenvectors from this interpolated data [23].In literature often first order trilinear interpolation is used. Trilinear interpolation is

linear interpolation of points within a box (3D) given values at the vertices of the box.

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The values of the vertices of the box are the values of the nearest voxels. For example,Pajevic [24] suggests a more sophisticated way to approximate a continuous vectorfield.

If we drop a particle in the continuous vectorfield, the displacement of that particle(dx), over a certain amount of time, (dt), can be described by:

dx = e1dt (3.1)

If a particle is followed over a larger time, the trajectory, x(t) can be described by thefollowing integral:

x(t) =t

e1(t)dt (3.2)

A solution to this equation can be found using numerical integration techniques. Thesimplest form of numerical integration is the first order Eulers method:

x(i + 1) = x(i) + e1(i) (3.3)

where x(i + 1) the position is at time i + 1. is the stepsize of the integration. Thisstepsize influences the accuracy of the integration. The accuracy of the Euler method isin order ofO(2). This method is used in early approaches of fiber tracking. Currentlymost fiber tracking techniques use the more accurate second order Runge Kutta:

x(i + 1) = x(i) +

2(e1(i) + e1(i + 1)) (3.4)

in which e1(i + 1) is calculated using the Euler method, see equation 3.3. The error

of this method isO

(3

). To find a trajectory, a seed pointx

(0) must be defined. Theother points of the fiber, i.e. x(i + 1) with i ranging from 0 to the number of points onthe trajectory, can be calculated with equation 3.4.

Until here we did not mention the sign of the main eigenvector, e1, which is inde-terminate, i.e., it can be positive or negative. The tract should be followed in bothdirections, starting from the seed point (i.e. +e1(0) and e1(0)) to find a completetrajectory. Once the direction of following the particle is determined, e1 should bechosen to point along the trajectory consistently. This can be done by taking the dotproduct between the eigenvector at the last position an at the current position. If theresult is positive the sign must be preserved; if the result is negative the sign must beswapped.

Seed points

In literature there are mainly three ways used to define seed points:

Manual definition of a region of interest (ROI) or point of interest (POI) fromwhich the fibers start (e.g. Basser et al. [25]).

Initiate tracking in all voxels whereafter only the tracks that enter a predefinedROI are selected [23], [26]. Methods that start tracking in all voxels are called

exhaustive search approaches.

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Figure 3.12: The exhaustive search approach will have less problems finding branching paths. Forexplanation figure, see text. [27]

Start tracking in 2 ROIs, only tracks that cross both ROIs are selected [26].

The exhaustive search approaches are computationally very expensive, but will haveless problems with finding branching fiber paths, since fiber tracking is initiated in theentire volume. This is explained by the example in figure 3.12. Suppose figure 3.12 A)represents the shape of a white matter tract of interest with an anatomical landmarkindicated by a white circle. If tracking is initiated from the landmark, there are fourpossibilities for the results, each representing one branch of the tract, see figure 3.12B). This is because a propagation result from one pixel can delineate only one line.Conversely, the line propagation can be initiated from all pixels and all propagationresults that penetrate the anatomical landmark are searched, which leads to morecomprehensive delineation of the tract of interest, see figure 3.12 C).

Stopping criteria

When fiber tracking is applied in brain tissue, it should stop following fibers in regionsof low anisotropy, where the main eigenvector is not clearly defined. This can be doneby thresholding, e.g., by using the Cl, FA or RA (see equations 2.18, 2.16, 2.17). Ofcourse also other anisotropy indices can be used for this. Problem with the FA andRA are that they can also be high in regions with planar anisotropy. When there isplanar anisotropy the first two eigenvectors are about the same size. In this case the

main eigenvector is not reliable any more. Therefore the author thinks that its betterto use Cl which gives a value for the reliability of the main eigenvector.

Another criterium to stop tracing fibers is to define a maximum allowable anglebetween to steps. For anatomical reasons we can assume that a fiber is more or lesssmooth. When the angle between two steps is bigger than a certain threshold thisis caused by noise or a partial volume effect, such as crossing fibers, which will bedescribed later on. For the maximum allowable angle, e.g., Tench et al. [28] suggests10 per 1-mm step and Basser et al. [25] a radius of smaller than 2 voxels.

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Smoothing techniques

Since the data obtained by DTI is often very noisy, all kind of noise removal techniquesare presented in literature. The methods vary from Gaussian blurring to more sophisti-cated methods that maintain main structures in the data. These smoothing operationscan be performed in the raw data, in the main eigenvector vectorfield as for exampleis done in Tench et al. [29] and Vemuri et al. [30], but also during fiber tracking alongthe fiber as e.g. in Zhukov et al. [31].

Regions with high planar anisotropy

As pointed out before, there is no clear eigenvector defined in regions with planaranisotropy (equation 2.19). This planar anisotropy can be due to crossing or kissingfibers, or a branching or merging fiber bundle, see figure 3.13. In regions with crossingor kissing fibers it is not clear which direction to follow.

Figure 3.13: Examples of regions where planar anisotropy exists; Left) a kissing fiber. Middle) acrossing fiber. Right) a braching/merging fiber. Regions that are marked grey are planar anisotropic.

Weinstein et al. [32] presented a solution to this problem, the tensorline technique.They try to find the most likely direction of propagation of the fiber. The propagationvector ( vprop) is a combination of three vectors, e1 (local main eigenvector), vin andvout. The propagation vector is calculated according:

vprop = Cle1 + (1 Cl)((1 wpunct) vin + wpunct vout) (3.5)

in this wpunct is a user controlled parameter that will be described later on. vin isthe direction of the previous propagation direction and vout is the incoming directiontransformed by the tensor matrix (D) according:

vout = D vin (3.6)

When the local Cl is high the propagation vector will be dominated by e1. If the localtensor shape is spherical or planar with the incoming vector parallel to the plane thepropagation vector is a combination of vin and vout ( vin vout). In case that the local

tensor shape is planar and the incoming vector is not parallel to the planar shape, the

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propagation vector will be a combination of vin and vout, with the amount of vin andvout controlled by the user defined parameter wpunct. Problem with this technique is

that just one fiber is followed, so a branching fiberpath will never be found. In regionswhere crossing or kissing of fibers occurs, it is more likely to find a crossing fiber thana kissing fiber.

Zhang et al. [33] assume that the planar shape of diffusion anisotropy is caused bya sheet-like structure of fibers. His approach, called Streamsurfaces, connects planarisotropic regions and forms planes in these positions. The results could locate and visu-alize white matter architectures with the linear and planar diffusion shapes separately,see figure 3.14.

The solutions presented before are all based on conventional DTI data. New scan-ning techniques scan in more than 100 directions evenly spaced in 3D. With these

so-called high angular scans it is possible to find more than one dominant diffusiondirection in a voxel. This technique is specially useful in cases when there is a mixtureof more than one fiber tract within a voxel [34] [27].

Figure 3.14: Planar regions depicted with green planes. In linear isotropic regions fiber tracking isperformed, fibers displayed red. The ventricles (blue) and the inside skull of the head are added as areference [33].

Energy Minimization Techniques

The approaches described till now are stepwise following the main local eigenvector.There are other techniques that try to find the energetically most favorable path be-

tween two predetermined pixels. The energy of a path is determined by the anisotropy,

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the alignment with the vectorfield and stiffness. Problem with these approaches is thatone has to have prior knowledge about the starting and ending point of a fiber. Refer-

ences and a more detailed description can be found in [27].

3.2.4 Hyperstreamlines

Hyperstreamlines [35] are an extension to the fiber tracking techniques described in theprevious section. Fiber tracking reduces the tensor information to the main diffusiondirection. The hyperstreamlines also show the second and third eigenvector. The cross-

sectional shape along the fibers is an ellipse. The two main axes ( Radius1 and Radius2,see figure 3.15) of this ellipse are determined by the second and third eigenvector andvalue and a constant scalefactor.

Radius1 = e22scalefactor (3.7)

Radius2 = e33scalefactor (3.8)

The scalefactor is used to get control of the size of the eigenvalues, which can vary alot depending on the datatype.

Problems with the hyperstreamlines can be visual cluttering when displaying largeamount of fibers. In regions with low anisotropy the second and third eigenvaluesare relatively large, resulting in a large cross-section of the fiber. This is not veryintuitive, since the regions that are the least interesting get bigger. An example ofhyperstreamlines can be seen in figure 3.16.

Figure 3.15: Cross section hyperstreamline.

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Figure 3.16: Hyperstreamlines colorcoded according the FA. In areas with low FA the tubes are

thick, in areas with high FA the tubes are thin.

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Chapter 4

Surface Building

In chapter 3.2.3, we mentioned the problem with fiber tracking in regions with planaranisotropy. We stated that the fiber tracking should stop in these regions because themain eigenvector is not reliable. However, these planar regions can be due to crossingfibers or branching fiber bundles. In these cases, it is principally not right to stopthe tracking. The methods suggested in literature to solve this problem have severaldisadvantages. The tensorline technique (see section 3.2.3) has the disadvantage that just one fiber is followed, in case of a kissing fiber this is probably not the correctfiber, also a branching fiber bundle will never be found. High angular scans are notavailable for us at the moment, so this is not further investigated. Zhang et al. [20]builds streamsurfaces in regions with high planar anisotropy. They do not try to find

crossing or branching fiber bundles with it. This chapter describes a new technique tosolve these problems.

4.1 Methods

We suggest a new approach that only tries to display the information that is there inthe data, leaving to the user the interpretation of it. This means that fiber tracking isdone in regions with high linear anisotropy. If a region with high planar anisotropy isentered by the fiber tracking, then a 2D tracking is started. The 2D tracking follows

the local plane, defined by the first two eigenvectors. If more positions with a highplanar anisotropy are found a surface is displayed between these positions. If one ofthe positions that is found has a high linear anisotropy, then fiber tracking is initiatedagain from this position.

This method will be able to find branching fiber paths, since fiber tracking is ini-tiated again if points with high linear anisotropy are found. In regions where fibersare crossing or kissing, all fibers from all sides will be displayed. In the middle wherethe crossing or kissing is, a plane will be displayed. Advantage of this method is thatassumptions are not made. Just the structures that are there in the data are displayed,it is up to the user to interpret this data.

26

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CHAPTER 4. SURFACE BUILDING 27

4.2 Algorithm

This section describes the algorithm to build the surfaces. It describes how the surfacesare defined, the data structures, and the implementation. The algorithm is partiallybased on the algorithm of Zhang et al. [20].

4.2.1 Surface Definition

First the surface has to be defined. The surface is build in a hexagonal grid. Thismeans that a point, i.e. vertex, in a complete planar region has six points (i.e. vertices)surrounding it, which forms six triangles.

The neighboring vertices of a vertex can be found by tracking from the vertex in six

equally spaced directions, in the local plane of the vertex. Figure 4.1 gives an exampleof a surface that can be found with the algorithm that will be described. The verticesare indicated with numbers. The number corresponds to the order in which the verticesare found. The tracked edges are indicated with Ei,j, with i the number of the vertexfrom which the edge is tracked and j the number of vertex that is at the other side ofthe edge.

Tracking is performed only in directions that were not tracked before (i.e. trackedfrom a different vertex). This means that after finding the first planar point the trackingis performed in all six directions. From the second point the tracking is done in threeor less directions. In the example of figure 4.1 the tracking from point two is done in

three directions,E2

,7

,E2

,8

and,E2

,9

. If edges are not tracked, but just connected, theyare denoted with ei,j, with i and j the vertices that are connected.

The first six direction in which tracking is performed (i.e. TD1), with a stepsize ,are defined by the first two eigenvectors (i.e. e1 and e2) in vertex 1 according:

TD1(n) = (e1 cos(n

3) + e2 sin(

n

3)) (4.1)

with n ranging from 0 to 5. In figure 4.1 this is shown by the edges E1,j, with j rangingfrom 2 till 6. Note that only 5 edges and 5 vertices are found in this case, this meansthat at one vertex no planar anisotropy was found. If at this point linear anisotropy isfound this vertex is defined as a new seed point for the fiber tracking.

The definition of the next tracking directions is less straightforward. The nextexample shows how the vectors are defined for the vertex 2 in figure 4.1. First weproject the edge E1,2, which is the vector pointing the direction from vertex 1 to vertex

2, on the plane in vertex 2, E2:

E2 = (e1 E1,2)e1 + (e2 E1,2)e2 (4.2)

e1 and e2 are the first two eigenvectors in vertex 2. After normalizing this vector, andmultiplying it with the integration stepsize, , the edge E2,8 is found:

E2,8 =E2

| E2 |

(4.3)

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Figure 4.1: Example of a surface, the vertices are numbered. The edges are indicated with a Ei,j for

tracked edges and ei,j for connected edges.

To find the other vectors in the plane defined at vertex 2, a vector perpendicular toE2,

8 and the third eigenvector has to be defined. Since eigenvectors of a tensor, do nothave the sign defined, we have to make sure that the third eigenvector at vertex 2 hasthe same sign as the third eigenvector in vertex 1. This is done by checking the innerproduct of these two vectors, if the sign is negative the sign of the third eigenvector inpoint 2 (e3) is swapped. The vector perpendicular to E2,8 (i.e. E2,8) is found by:

E2,8 = E2,8 e3 (4.4)

When substituting E2,8 and E2,8 for e1 and e2 respectively in equation 4.1 thetrackdirections for the second point, TD2, can be found:

TD2(n) = E2,8 cos(n

3 ) + E2,8 sin(

n

3 ) (4.5)

with n ranging from 0 to 5. Since three directions are already tracked, we are onlyinterested in the cases with: n = 0 (edge E2,8), n = 1 (edge E2,9) and n=5 (edge E2,7).

4.2.2 Data structure

Three structures are defined to define the geometry:

An array of the surface vertices: SurfaceVertex.

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An array defining the relation between a vertex and its neighbors: VertexNeigh-bors.

An array with triangles that build the final surface: triangles.

The array SurfaceVertex contains the vertices of the surface with the coordinates ofthis vertex (X,Y,Z). In VertexNeighbors is the relation of a vertex with its neighbor-ing vertices is described. It contains all neighboring vertices (i.e. indices of the arraySurfaceVertex) of a vertex, or a -1 if a neighboring vertex is not planar anisotropic.Figure 4.2 a) shows the connections between vertex 1 of figure 4.1 and its neighbor-ing vertices. The vertex marked with -1 is a vertex that can be isotropic or linearanisotropic. If it is linear anisotropic it will serve as a new seed point for the fibertracking. The VertexNeighbors of vertex 1 in figure 4.2 is stored as indicated in fig-

ure 4.2 b). The vertices are ordered in counter clockwise order.

a) b)

Figure 4.2: a) Connection of vertex 1 of figure with its neighbors. The vertex marked with -1 is notplanar anisotropic. b) VertexNeighbors describes the connections of a vertex with its neighbors.

The triangles that are rendered are stored in the array triangles. Each row consists of3 pointers to the indices of the SurfaceVertex array which form a triangle. The completedata structures for the figure 4.1 are shown in figure 4.3. Note that all numbers pointto the indices of the SurfaceVertex.

Inserting points in the data structures

To explain how the vertices are inserted in the arrays (i.e. VertexNeighbors, SurfaceV-ertex and triangles) the first few steps of the surface in figure 4.1 will be explained.

If the first planar anisotropic vertex (i.e. vertex 1) is found the coordinates fromthis vertex (X1, Y1, Z1) are inserted at position one in the SurfaceVertex array. Thefirst row of VertexNeighbors is initialized with zeros, see figure 4.4 a). Six directionsare tracked from vertex 1. Vertex 1 will be called currentVertex since this is the vertexfrom which the neighboring vertices will be generated.

The first vertex that will be searched is vertex 2, a new vertex that is inserted

will be called newVertex from now on. The coordinates of this newVertex point are

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Figure 4.3: The data structures for the example in figure 4.1.

filled in the array SurfaceVertex. A new row is also iniatialized in VertexNeighbors.Since the newVertex is connected to the currentVertex, the currentVertex will be addedin corresponding position of the newVertex row. The position of newVertex will beupdated in the currentVertex row, see figure 4.4 b).

The next found vertex, vertex 3, is treated the same as the second vertex. Howeverthis vertex is not only connected to the currentVertex (i.e. vertex 1), but also to thevertex standing on the left of its position in the row of the currentVertex (i.e. vertex2). This vertex will be called leftVertex. These vertices also have to be filled in thecorrect positions in the newVertex row (i.e., vertex 3 row), see figure 4.4 c). It alsohas to be checked whether the vertex on the right is an already existing vertex (i.e., avalue different from zero). This vertex will be called rightVertex. If we do this for allpositions of the currentVertex, the structures will look like figure 4.4 d). Note that if avertex is not planar anisotropic then the array is filled with -1 and for this vertex thereis not a new row initialized in SurfaceVertex and VertexNeighbors.

After treating the first vertex, vertex 2 will become the currentVertex and the firstzeros in the row of vertex 2 will be filled. The filling of a vertex in VertexNeighborscan be summarized by the following function in pseudocode:

Function InsertVertex (newVertex, currentVertex)if newVertex is the first Vertex and planar then

initialize the first row of VertexNeighbors with zeroselse if newVertex is planar then

Insert newVertex at position of the first zero in the currentVertex row.Insert a new row in VertexNeighbors, initialize this to zero and fill the first positionwith currentVertex.if leftVertex is not zero then

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a)

b)

c)

d)

e)

Figure 4.4: a) One found surfaceVertex and its corresponding data structures. b) Data structuresafter two vertices are found c) Data structures after three points are found. d) Data structures afterthe neighboring vertices of vertex 1 are generated. e) The structure triangles is created and filled afterthe neighboring vertices of vertex 1 are generated.

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insert leftVertex at the first empty position in the newVertex rowinsert newVertex at the last empty position in the leftVertex row

end ifif rightVertex is not zero theninsert rightVertex at the last empty position in the newVertex rowinsert newVertex at first position in the rightVertex row

end if

else if Vertex is linear anisotropic or isotropic thenInsert -1 at position of the first zero in the currentVertex row.if leftVertex is not zero then

insert -1 at last empty position in the leftVertex rowend if

if rightVertex is not zero then

insert -1 at first position in the rightVertex rowend if

end if

After a currentVertex in VertexNeighbors is treated, i.e. all positions are unequal tozero, it is checked whether new triangles are formed. Figure 4.4 e) shows the trianglesthat are formed after the first point is treated and how the data structure triangleslooks. The array triangles if formed according the following function in pseudocode:

Function CreateTriangles(currentVertex)for All vertices of row currentVertex do

if This vertex and the vertex right from it exist and have a higher number than

the currentVertex thenThis is a triangle, add a new row to triangles and fill this with the currentVertexand the two vertices next to each other

end if

end for

4.2.3 Implementation

Until here we described small parts of the algorithm with functions. One more functionis defined here. It describes how the tracking directions of the Surface Building are

treated:Function TreatTrackingDirections(tracking directions, currentVertex)for each tracking direction do

track in this direction to find newVertexinsertVertex(newVertex, currentVertex)if newVertex is linear anisotropic then

Define this vertex as a new seed point for fiber trackingend if

end for

The complete fiber tracking algorithm with surface generation can be described in

by the following pseudocode:

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Start fiber tracking described in chapter 5.3.if Cl at a point is lower than the threshold and Cp is higher than threshold and this

is the first plane that is built in this regionthen

Define 6 tracking directions defined by equation 4.1TreatTrackingDirections(tracking directions, currentVertex)CreateTriangles(currentVertex)while There are point in VertexNeighbors equal to zero do

for next point in VertexNeighbors dodefine new tracking directions according equation 4.5TreatTrackingDirections(tracking directions, currentVertex)CreateTriangles(currentVertex)

end for

end while

end if

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Chapter 5

DTI Tool

The Maxima Medical Center (MMC) is doing research on DTI of the neonatal brainand the Magnetic Resonance Laboratory (MRL) works on DTI of rat muscle. Tovisualize the data of them a tool has been created. The goal was to create a tool thatallows the user to visualize the data locally as well as visualizing the data globally.Context for the orientation within the data is necessary.

Multi Planar Reconstruction (MPR) planes (chapter 5.1) to visualize local tensorinformation and as an anatomical reference for other visualizations are used. In order tovisualize local tensor information, glyphing is used (chapter 5.2) and to visualize moreglobal tensor information a fiber tracking algorithm is implemented (chapter 5.3). This

chapter describes the motivation behind these choices and the exact tools that areavailable in the program.

The implementation has been done in C++, using The Visualization ToolKit(VTK) which is an open source, software system for 3D computer graphics, imageprocessing and visualization. VTK consists of a C++ library and supports visualizationand imaging algorithms. Standard interactions as zooming, panning and rotating areavailable already. An interface is created with The Fast Light ToolKit (FLTK),which is also open source. A short description of the new generated classes is givenin appendix A. The DTI tool contains a manual written in html in the help of theprogram.

5.1 Multi Planar Reconstruction

The Multi Planar Reconstruction (MPR) constists of three orthogonal cutting planesthat are oriented relative to the brain as can be seen in figure 5.1. For doctors this canbe a very informative way of visualizing the data, since they are used to looking to 2Dslices that are oriented in this way. The planes can be moved through the volume andcan be made visible or invisible. Several textures can be applied to the MPR:

An MR reference image, e.g. an isotropic ADC image or T2 weighted image.

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CHAPTER 5. DTI TOOL 35

Figure 5.1: Orientation of the 3 orthogonal cutting planes (saggital, coronal and axial)

Colorcoded anisotropy indices: FA, RA, Cp and Cl, described in chapter 3.1.1

Colorcoding main diffusion direction, and colorcoding main diffusion directions

weighted with Fractional Anisotropy, described in chapter 3.1.1

The MR reference image is mapped with a black-white look-up-table. The width of and

the center of this look-up-table can be modified. The anisotropy indices are colorcoded

with a hue look-up-table which also can be modified. The part of the planes that is

outside the data can be made transparent to avoid image cluttering. Figure 5.2 shows

two screenshots of the program with an MPR. This MPR planes are also used to givecontext to other visualizations like glyphing or fiber tracking.

a) b)

Figure 5.2: a) MPR with isotropic ADC image displayed on it without transparant background.b)MPR with color coded main diffusion direcion with a transparent background

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5.2 Glyphing

Glyphs can be displayed on the MPR planes to get detailed local information aboutthe shape of the tensor. There are two options for the shape of the glyphs that canbe displayed. The first option is the ellipsoids (chapter 3.1.2) because this is the mostcommon representation of a tensor. The boxes, also called cuboids, explained in chap-ter 3.1.2 are the other option. The boxes are chosen because of their clear directionalinformation also of the second and third eigenvector. The size of the glyphs is normal-ized and they are colorcoded according the Fractional Anisotropy. By normalizing theglyphs, the information about the amount of diffusion is lost. This colorcoding andthe used look-up-table is the same as in the MPR. Figure 5.3 shows a screenshot ofglyphing on a twisted optical nerve of a cow. Note that the rendering of the boxes isfaster than the rendering of the ellipsoids, because fewer polygons have to be rendered

to approximate the shape of cuboid than an ellipsoid.

Figure 5.3: Local tensor information by glyphing: the glyphs (boxes) are shown on a twisted opticalnerv of a cow. Background image is a T2 weighted MR image.

5.3 Fiber tracking

Fiber tracking has proven to be a very useful way to show global structures in the data(see chapter 3.2.3). It provides global 3D information about the orientation of nervfibers (in brain data) or muscle fibers.

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Integration

The tracking is done with a second order Runge Kutta integration, through the maineigenvector vectorfield. The main eigenvector vectorfield is made continuous by trilinearinterpolation of the diffusion tensor. Literature showed that this gives good results, formore details see chapter 3.2.3. The integration stepsize (, see equation 3.2), whichinfluences the accuracy of the tracking, is user-defined. Reducing this stepsize furtherthan 0.1 times the cross-diameter of a voxel will hardly not improve the accuracy ofthe fiber tracking anymore.

Seed Point placement

Seed points are the points from which the fiber tracking is initiated. Seed points canbe defined in four ways:

Single fiber: Fiber tracking is initiated from one seed point. Just one fiber isdisplayed, the fiber is moved and recalculated if the seed point is moved.

Painting fibers: Fiber tracking is initiated from one seed point. When a new seedpoint is defined the old fiber will be maintained and a new fiber will be calculatedand shown.

Start Region: Seed points are defined by defining a region. Within this regionseed points are created uniformly in a regular grid with a user-defined distance.

Through Region: Seed points are defined in the center of each voxel in the data.Only the fibers that cross a user-defined region are kept and displayed. Of coursethis approach takes much longer, but it will have less problems with findingbranching fiberpaths.

Stopping Criteria

Stopping the fiber tracking can be done by setting a threshold for a minimumanisotropy index. Indices that can be used are the Fractional Anisotropy (FA), Rel-ative Anisotropy (RA) and Linear Anisotropy (Cl). Also a maximum allowable anglecan be defined. This angle is measured over one integrationstep (i.e. ), the units ofthis angle are [degrees/].

Displaying the fibers

The fibers can be displayed as streamlines, shaded streamtubes, shaded hyperstream-lines and shaded hyperstreamprismas.

Streamlines display the found trajectory with a polyline. The value of the parameterfiberStep determines the accuracy of the approximation of the found trajectory. Fig-

ure 5.4 a) shows a found trajectory with the integrationstep, , indicated. Figure 5.4

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b) shows the approximation of this trajectory with a streamline with the fiberstepindicated. The fiberstep is twice as big as the , this will cause a faster rendering

than a fiberstep as big as the . Of course, if the fiberstep is smaller than the , theaccuracy wont improve anymore. If the rendering is fast enough, it is recommendedto set the fiberstep on the same value as the .

Streamtubes display the found trajectory with a tube surrounding it. The radiusof the tube is controlled by a user-defined parameter. The tube consists of small lineartube parts, the fiberstep determines the length of the tube parts.

The trajectory can also be displayed as a hyperstreamline. Hyperstreamlines areexplained in chapter 3.2.4. The radii of the hyperstreamlines are scaled according thesecond and third eigenvalue (see equation 3.7 and 3.8). Since the size of the eigenvaluesdepends on the data and can be very large or very small, a scalefactor must be defined

different for each dataset. In order to get a default scalefactor automatically, themaximum second eigenvalue with a FA higher than 0.3 is measured, i.e. max2. Thescalefactor is defined as follows:

scalefactor =hyperScale

max2(5.1)

hyperScale is a parameter that is user-defined, default this value is one. It can be usedif one wants to display the radius of the hyperstreamlines bigger or smaller. Also thehyperstreamlines are built from tube parts, with a length defined by the parameterfiberstep.

The hyperstreamprimas are introduced here in order to show the torsion of a fiber.The scaling is the same as with the hyperstreamlines. The difference is that instead ofan ellipsoid a rectangle is used for the diameter (see figure 5.5). Hyperstreamprismasare build from tube parts with four sides.

a) b)

Figure 5.4: a) A trajectory, with the integrationstep (i.e. ) indicated. b) The trajectory of a) isindicated with dashed lines. The solid line shows the approximation of a) with a polyline with thestepsize fiberstep indicated. The fiberstep is twice as big as the , this will cause a faster renderingthan a fiberstep as big as the .

The fibers can be colored according the FA, RA, Cl and principal diffusion direction.

Figure 5.6 compares the shapes described before, colorcoded with the FA.

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a) b)

Figure 5.5: Cross section a) hyperstreamline and b) hyperstreamprisma with Radius1 and Radius2indicated.

a) b) c) d)

Figure 5.6: Comparison between the possible shape of the fibers: a) hyperstreamline, b) hyperstream-prisma, c) streamtube, d) streamline. Note that the torsion is visible in the hyperstreamprisma whileit cannot be seen at the hyperstreamline. The streamlines are not very useful for displaying only onefiber, but are less costly to render and therefore more useful when groups of fibers are rendered.

The hyperstreamlines, hyperstreamprismas and streamtubes are built from tubeparts as pointed out before. An example of a tube part of a Streamtube is shown infigure 5.7. The tube parts are divided in triangles which are used for graphics hardwareacceleration. The number of triangles that are rendered influences the rendering speed.

The length of the tube parts is determined by the fiberstep. Of course, if the fiberstepis increased less tube parts per trajectory will have to be rendered and the renderingwill be faster. The number of triangles that are rendered also depends on the numberof sides of the tube parts. Figure 5.7 a) shows an example of a tube part with 6 sides, itapproximates the part of the trajectory that is depicted with the bold line. Figure 5.7b) shows an example of a tube part with 4 sides, also with the part of the trajectorydepicted with a bold line. When the number of sides is reduced the rendering will befaster, however the shape of the fiber will be less round. For the hyperstreamprismathe number of sides is always four. In general 6 sides for the hyperstreamlines andstreamtubes is enough to get a good approximation of a round shape. Note that the

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accuracy of the tracking itself is not influenced by changing these parameters, it is justthe approximation of the roundness of the tubes.

a) b)

Figure 5.7: Tube parts a) A tube part with 6 sides. The tube part depicts the part of the trajectoryindicated with the bold line. b) A tube part with 4 sides. a) is a more accurate approximation of acylinder shape, while b) is faster to render, since it has to render less triangles

Minimum fiber length can be used to display only fibers that are longer than thatspecific length. This can be useful since often short fibers are less interesting. It alsoreduces the amount of visual cluttering.

Histogram

Meaningful quantitative information is difficult to extract from the fibers. The fiberlength largely depends on the defined stopping criteria and the amount of noise in thedata. This makes it difficult to compare quantitative results between different datasets.The number of fibers mainly depends on the number of seed points that are defined.However, it might be that statistical information about the fiber length is useful in somecases. This still has to be evaluated. A histogram of the fiber length is added to do thisevaluation. It contains information about the mean fiber length, average fiber length,standard deviation. An example of fiber tracking in the brain of a healthy volunteer isshown in figure 5.8 a). The histogram of the fiber lengths is shown in figure 5.8 b).

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a)

b)

Figure 5.8: a) Fiber tracking in a healthy volunteer. The minimum fiber length is 10 mm. b)Histogram of the fibers in a).

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Chapter 6

Results

Diffusion tensor imaging is used for several applications. This chapter shows examplesof the use of the DTI tool for visualizing DTI data. The first section shows results inthe optical nerve of a cow. This is used for a visual validation of the obtained results.Thereafter examples of DTI in the brain are shown. A healthy volunteer is used tovalidate whether fiber tracking in the brain shows the main fiber bundles. Examples ofthe use of the DTI tool in a patient with a tumor and neonates show the use of it forclinical applications. The MRL uses DTI on the brain of rats and on mouse muscle,results of this are also shown. In the last section some results of the presented SurfaceBuilding (see section 4) method are shown.

In the examples theFA

index is often used as a stopping criterium for stoppingthe fiber tracking. We suggested that it would be better to use the Cl index, sincethis index indicates the reliability of the main eigenvector. However literature and ourresearch partners (i.e. MMC and MRL) often use the FA as a stopping criterium. To beable to compare results, the FA is often used as a stopping criterium for fiber tracking.

Note that just screenshots of the program are shown, to get a real good impressionof the 3D structure of the data, one should use the DTI tool, which allows the user tointeract with the data. The computation of the results presented in this chapter andthe interactions with the visualizations are done in real time.

6.1 Optical nerve of a cow

The optical nerve of a cow has a high anisotropy (FA 0.5) and a very clear, straightorientation. These properties make the optical nerve an object that can serve as a visualvalidation for the DTI tool. The optical nerve is twisted before scanning. Figure 6.1 a)shows a slice with a T2 weighted MR image displayed on it, figure 6.1 b) the colorcodedFA and figure 6.1 c) shows the main diffusion direction weighted with the FA. Figure 6.2shows the result of fiber tracking in the optical nerve. The fibers are colorcoded withgreen meaning the direction perpendicular to the plane displayed in figure 6.2 a), bluethe direction perpendicular to the plane displayed in figure 6.2 b), and red, the direction

perpendicular to the other two described directions.

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CHAPTER 6. RESULTS 43

From the results of the optical nerve of the cow can be concluded that fiber tracking

visually shows the expected orientation in oriented tissue.

a) b) c)

Figure 6.1: Slice of twisted optical nerv of a cow. Texture displayed on a) isotropic ADC, b) FractionalAnisotropy (FA), c) Main eigenvector weighted with FA

a) b)

Figure 6.2: Fiber tracking in the twisted optical nerve of a cow. The fibers are colorcoded with greenmeaning the direction perpendicular to the plane displayed in a), blue the direction perpendicularto the plane displayed in figure b), and red, the direction perpendicular to the other two describeddirections.

6.2 Healthy Volunteer

This section describes visualization of the DTI data of a healthy volunteer. It points

out some interesting fiber tracts that can be found in it. This shows that the fiber

tracking method is able to track the main fiber bundles of the brain.

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a) b)

c) d)

Figure 6.3: The Corpus Callosum. a) Seed region, b), c) and d) The fiber tracking results viewedfrom different angles and with different slices on the background.

An interesting brain structure is the Corpus Callosum (i.e. CC), it connects the leftand right brain half. The CC has a relative high anisotropy (FA 0.4, Cl 0.25 inthis example) which makes it easy to find and visualize. Seed points are placed in theCC using the region indicated in figure 6.3 a). The red structure in this figure is partof the Corpus Callosum. Figure 6.3 b), c) and d) show the result of the fibertrackingfrom different view points and with different slices displayed for context. As stoppingcriterium 0.1 is chosen for the linear anisotropy (Cl) and the minimum fiber length isset to 20 [mm].

Figure 6.4 a) shows the optical tract. The optical tract is the connection of eye

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nerves with the rest of the brain. The eye nerves start are at the left part of the image,the anterior side, both tracts splits and go to both parts of the t

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