Institutionen för systemteknikDepartment of Electrical Engineering

Examensarbete

Method for Acquisition and Reconstruction of

non-Cartesian 3-D fMRI

Examensarbete utfört i Bildbehandlingvid Tekniska högskolan i Linköping

av

Per Thyr

LITH-ISY-EX--08/4058--SE

Linköping 2008

Department of Electrical Engineering Linköpings tekniska högskolaLinköpings universitet Linköpings universitetSE-581 83 Linköping, Sweden 581 83 Linköping

Method for Acquisition and Reconstruction of

non-Cartesian 3-D fMRI

Examensarbete utfört i Bildbehandling

vid Tekniska högskolan i Linköpingav

Per Thyr

LITH-ISY-EX--08/4058--SE

Handledare: Peter Lundberg

Radiofysik, imh, Linköpings Universitet

Olof Dahlqvist Leinhard

Radiofysik, imh, Linköpings Universitet

Maria Magnusson

Bildbehandling, isy, Linköpings Universitet

Examinator: Maria Magnusson

Bildbehandling, isy, Linköpings universitet

Linköping, 7 February, 2008

Avdelning, Institution

Division, Department

Division of Computer Vision LaboratoryDepartment of Electrical EngineeringLinköpings universitetSE-581 83 Linköping, Sweden

Datum

Date

2008-02-07

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Language

� Svenska/Swedish

� Engelska/English

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Report category

� Licentiatavhandling

� Examensarbete

� C-uppsats

� D-uppsats

� Övrig rapport

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URL för elektronisk version

http://www.cvl.isy.liu.se

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-11097

ISBN

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ISRN

LITH-ISY-EX--08/4058--SE

Serietitel och serienummer

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Titel

TitleMetod för insamling och rekonstruktion av icke-kartesisk 3-D fMRI

Method for Acquisition and Reconstruction of non-Cartesian 3-D fMRI

Författare

AuthorPer Thyr

Sammanfattning

Abstract

The PRESTO sequence is a well-known 3-D fMRI imaging sequence. In this se-quence the echo planar imaging technique is merged with the echo-shift technique.This combination results in a very fast image acquisition, which is required forfMRI examinations of neural activation in the human brain. The aim of this workwas to use the basic Cartesian PRESTO sequence as a framework when developinga novel trajectory using a non-Cartesian grid.

Our new pulse sequence, PRESTO CAN, rotates the k-space profiles around theky-axis in a non-Cartesian manner. This results in a high sampling density close tothe centre of the k-space, and at the same time it provides sparser data collectionof the part of the k-space that contains less useful information. This "can- orcylinder-like" pattern is expected to result in a much faster k-space acquisitionwithout loosing important spatial information.

A new reconstruction algorithm was also developed. The purpose was to be ableto construct an image volume from data obtained using the novel PRESTO CANsequence. This reconstruction algorithm was based on the gridding technique,and a Kaiser-Bessel window was also used in order to re-sample the data onto aCartesian grid. This was required to make 3-D Fourier transformation possible.In addition, simulations were also performed in order to verify the function of thereconstruction algorithm. Furthermore, in vitro tests showed that the developmentof the PRESTO CAN sequence and the corresponding reconstruction algorithmwere highly successful.

In the future, the results can relatively easily be extended and generalized forin vivo investigations. In addition, there are numerous exciting possibilities forextending the basic techniques described in this thesis.

Nyckelord

Keywords MRI, PRESTO, PRESTO CAN, rotation, sequence development, image recon-struction, gridding, phase correction

Abstract

The PRESTO sequence is a well-known 3-D fMRI imaging sequence. In this se-quence the echo planar imaging technique is merged with the echo-shift technique.This combination results in a very fast image acquisition, which is required forfMRI examinations of neural activation in the human brain. The aim of this workwas to use the basic Cartesian PRESTO sequence as a framework when developinga novel trajectory using a non-Cartesian grid.

Our new pulse sequence, PRESTO CAN, rotates the k-space profiles around theky-axis in a non-Cartesian manner. This results in a high sampling density close tothe centre of the k-space, and at the same time it provides sparser data collectionof the part of the k-space that contains less useful information. This "can- orcylinder-like" pattern is expected to result in a much faster k-space acquisitionwithout loosing important spatial information.

A new reconstruction algorithm was also developed. The purpose was to be ableto construct an image volume from data obtained using the novel PRESTO CANsequence. This reconstruction algorithm was based on the gridding technique,and a Kaiser-Bessel window was also used in order to re-sample the data onto aCartesian grid. This was required to make 3-D Fourier transformation possible.In addition, simulations were also performed in order to verify the function of thereconstruction algorithm. Furthermore, in vitro tests showed that the developmentof the PRESTO CAN sequence and the corresponding reconstruction algorithmwere highly successful.

In the future, the results can relatively easily be extended and generalized forin vivo investigations. In addition, there are numerous exciting possibilities forextending the basic techniques described in this thesis.

v

Acknowledgments

This work has been a part of collaboration between the Division of Computer Vi-sion Laboratory at the Department of Electrical Engineering, Linköping Univer-sity, the MR-Unit at the Department of Radiation Physics, Linköping Universityand the Center for Medical Image Science and Visualization, Linköping Univer-sity. There are many people connected to these departments that have been veryhelpful and supportive during my work and I would like to send a big thank youto all of you.

An extra thanks to my examiner and supervisors Maria Magnusson, OlofDahlqvist Leinhard and Peter Lundberg for guidance, ideas and suggestions forimprovements. I would also like to wish my co-workers in the MR basement goodluck with there continuous research and thanks for all the interesting discussionsround the coffee table.

vii

Contents

1 Introduction 1

1.1 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Abbreviations 3

3 fMRI and k-space 5

3.1 MRI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.1.1 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.1.2 Gradient-recalled Echo . . . . . . . . . . . . . . . . . . . . . 7

3.2 FID Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Functional MRI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.3.1 BOLD-Contrast . . . . . . . . . . . . . . . . . . . . . . . . 93.4 k-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.4.1 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . 103.4.2 Image Frequencies in k-space . . . . . . . . . . . . . . . . . 113.4.3 Shift Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.5 Geometry Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.5.1 k-space Geometry . . . . . . . . . . . . . . . . . . . . . . . 15

3.6 Artefacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Pulse Sequence 19

4.1 Pulse Sequences in fMRI . . . . . . . . . . . . . . . . . . . . . . . . 194.1.1 Echo-Planar Imaging . . . . . . . . . . . . . . . . . . . . . . 194.1.2 2-D vs. 3-D . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.1.3 Multishot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2 PRESTO Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3 Scanner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.4 Development Environment . . . . . . . . . . . . . . . . . . . . . . . 254.5 PRESTO CAN Sequence . . . . . . . . . . . . . . . . . . . . . . . 25

4.5.1 Gradient Separation . . . . . . . . . . . . . . . . . . . . . . 264.5.2 Gradient Rotation . . . . . . . . . . . . . . . . . . . . . . . 27

4.6 Reference Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

ix

x Contents

5 Reconstruction 31

5.1 Raw Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . 315.1.1 Pre-processing . . . . . . . . . . . . . . . . . . . . . . . . . 315.1.2 PRESTO Reconstruction . . . . . . . . . . . . . . . . . . . 335.1.3 Data Correction . . . . . . . . . . . . . . . . . . . . . . . . 33

5.2 Image Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2.1 Angle Interpolation . . . . . . . . . . . . . . . . . . . . . . 355.2.2 Gridding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.2.3 Further Image Processing . . . . . . . . . . . . . . . . . . . 39

6 Simulation 41

7 Results 45

7.1 PRESTO Reconstruction . . . . . . . . . . . . . . . . . . . . . . . 457.2 PRESTO CAN Reconstruction . . . . . . . . . . . . . . . . . . . . 46

8 Discussion 55

8.1 Further Development . . . . . . . . . . . . . . . . . . . . . . . . . . 56

9 Conclusions 57

Bibliography 59

A User Parameters 61

Chapter 1

Introduction

Functional MRI (fMRI) is a medical imaging method in which magnetic resonanceimaging (MRI) is used to detect areas of the brain that are used for specific tasks,e.g. motor activities, or language generation. This is an extremely useful techniquefor investigating the human brain prior to neurosurgery, or for basic neurologicalresearch as it makes it possible to investigate the brain function in detail.

Short data acquisition times are very important for fMRI as the brain pro-cesses are quite rapid. MRI data is acquired in the k-space and must be Fouriertransformed in order to provide an image. Corrections, due to imperfection inthe magnetic fields and hardware, are also necessary. Furthermore, 2-D fMRI-acquisition techniques are most widely used presently. Data is then acquired ina Cartesian 2D grid in k-space by collecting each 2D slice sequentially. How-ever, acquiring data in such a restricted manner limits the possibilities to improveacquisition speed and also to reduce image artefacts.

In this thesis, a novel method for acquiring 3-D fMRI data of the human brainwas developed and called PRESTO CAN. 3-D fMRI allows very fast acquisitionwithout sacrificing other important aspects such as SNR, or matrix size. The rea-son is that the acquisition trajectory in the k-space then can be optimized moreefficiently, which almost always is equivalent to collecting data in a non-Cartesiangrid. The PRESTO CAN method uses a radial grid to acquire the k-space data.Such an acquisition procedure allows a sparse data collection of the parts of thek-space that contain less information, leading to significant savings in image acqui-sition time. A consequence of such a procedure is that the reconstruction methodto obtain the image become much less trivial and the scanner cannot reconstructthe image by itself. A stand-alone implementation was developed for reconstruc-tion of the k-space data. This reconstruction algorithm needed to handle both there-gridding onto a Cartesian grid and the artefacts occurring due to the rotation.

1

2 Introduction

1.1 Aim

The work presented in this thesis aims to implement the PRESTO CAN se-quence for Philips 1.5T Achieva scanner and the base for a reconstruction al-gorithm capable of handling data exported from the PRESTO CAN sequence.The PRESTO CAN sequence should be based on the PRESTO sequence and forthe re-gridding onto a Cartesian grid, the Kaiser-Bessel window function [9] shouldbe used. Due to time limitation, the reconstruction algorithm should only be ableto handle to most basic examination parameters (excluding for example keyhole,SENSE and multishot).

Chapter 2

Abbreviations

Here follows a compilation of abbreviations used in this thesis.

B0 - Strong static magnetic field.

B1 - Rotating magnetic field. Also know as the RF pulse.

Echo - Reformed version of the FID.

ES - Echo-shift.

EPI - Echo Planar Imaging.

FID - Free Induction Decay.

fMRI - Functional MRI.

FOV - Field of View.

FT - Fourier transform.

GM - Gradients in the measurement direction.

GP - Gradients in the phase direction.

GS - Gradients in the slice direction.

GX - Gradients in the X direction.

GY - Gradients in the Y direction.

GZ - Gradients in the Z direction.

in vitro - In phantoms or tubes.

in vivo - In the body.

k-space - Another name for the Fourier domain, commonly used in MRI.

3

4 Abbreviations

MR - Magnetic Resonance.

MRI - Magnetic Resonance Imaging.

PARADISE - Philips Acquisition Research ADvanced Integrated Software En-vironment.

PRESTO - PRinciples of Echo-Shifting with a Train of Observations.

PRESTO CAN - The sequence developed in this thesis, see PRESTO.

Profile - A 2-D plane in the k-space. Usually referring to the acquisition plane.

RF pulse - Radio Frequency pulse.

Slice - A 2-D plane in the spatial domain.

TE - Echo Time from excitation.

TR - Repetition Time.

Chapter 3

fMRI and k-space

The physics behind magnetic resonance, MR, is an interesting field to study butis beyond the scope of this report. In this chapter, the theories behind topicsmore close to the aim of the report are treated. Such topics are for examplefunctional MRI, k-space and Fourier transformation. To fully understand thisreport some of the basic in MRI has to be known in advance. Some useful referencesthat cover the basics in MR are Peter Rinck’s Magnetic Resonance in Medicine [11]and Joseph Hornak’s hypertext book The Basics of MRI [4], which also includesanimations.

3.1 MRI

Magnetic resonance imaging, MRI, is an imaging technique using the magnet reso-nance phenomena [11]. By exposing the patient to a combination of three magneticfields in the scanner, the patient emits a signal that can be measured and recon-structed to an image of the anatomy of interest. The three magnetic fields are thestatic field, the gradient field and the radiofrequency field. The strongest of thesemagnetic fields is the static field, B0. Scanners used for body or brain examina-tions usually have a B0 between 1.5-3T. The B0 field is parallel to the patient andthe z-direction of the scanner, see figure 3.7. Three gradient coils in the scannerproduce gradient magnetic fields G =(Gx, Gy, Gz), which are much weaker thanB0. These fields are used to make small changes in the total magnetic field andare thereby able to isolate the area of interest during examination. The gradientfields are also used for positioning during readout of the emitted signal. Above thegradient coils, there are coils that construct a rotating magnetic field, B1, in thexy-plane, see figure 3.1. Because of the rotation frequency and the duration of thisfield, it is usually named radiofrequency pulse, RF-pulse. The coils also receivethe emitted signal from the patient and they are therefore called transmit/receivecoils.

For the theory in this thesis, it is sufficient to use classic mechanics to explainthe principles of MRI instead of using quantum mechanics. The most common

5

6 fMRI and k-space

Figure 3.1. An example how the rotating magnetic field B1, can be constructed. Bymaking the transmitted signals from the two coils phase shifted by π/2, the resulting B1-field will start to rotate around the z-axis. When the B1-field rotates with the Larmorfrequency, the precessing spins will obtain resonance and thereby deviate from the B0-direction.

nuclei used for MRI is the hydrogen nucleus 1H, proton1. Hydrogen nucleus areused because they possess the property of spin and they are common in the humanbody2. Spin is a fundamental property of nature, like electrical charge or mass.Different tissues consist of different amount of hydrogen nucleus, i.e. hydrogendensity. Simplified, these changes in density are measured in MRI. As earlierpointed out, MRI can use other nucleus with spin instead of the hydrogen nucleus,thereby the term spin density is more common. Looking at a small group ofhydrogen nuclei, it can be seen as they have a net spin pointing in some randomdirection. When adding a strong magnetic field, B0, these spins will align withthe strong field. The property of spin will make them precess around the B0-direction, just like a gyro. Depending on the constant gyro-magnetic ratio, γ,(42.575 MHz per Tesla for hydrogen nuclei) of the nuclei and the strength of themagnetic field, the spins will precess with a certain frequency. This frequency iscalled Larmor frequency. By turning on the rotating radiofrequency field, B1, thephase of the precessing spins will become coherent and start to deviate from theB0 axis. The duration of the RF-pulse is proportional to the deviation of the spins.Only RF-pulses with the Larmor frequency will affect the spins, this is called toexcitate the spins3. After the RF-pulse is turned off, the spin will return to theirequilibrium. During this process, energy will be emitted. This energy signal, calledfree induction decay signal, FID, is measured and used to reconstruct the image[11].

1The 1H have one proton but no neutron and are therefore usually called proton [8].2The most common nuclei in MRI is 1H. E.g. 13C, 17O and 31P could also be used [11], [8].3When the frequency of the RF-pulses is the same as the Larmor frequency, the spin obtain

resonance.

3.1 MRI 7

Figure 3.2. After the spins have been excitated, the relaxation starts. The arrowsrepresent the spins. This process can be divided into two parts, the spin-lattice relaxationand the spin-spin relaxation. The spin-lattice relaxation is that the spins return toequilibrium and will, after the time T1, be aligned with B0. On top of this, the spins willstart to dephase in the xy-plane. This is the spin-spin relaxation with the correspondingtime T2. Since T2 is always shorter than T1, the spins will be dephased before they areback in equilibrium.

3.1.1 Relaxation

When the RF-pulse is turned off, the spins will return to their equilibrium [4].It is possible to measure the energy signal while the spins are deviated from B0,because then there is a signal component perpendicular to the B0-direction. Thissignal will decrease in intensity over time. The first reason for this is that thespins return to equilibrium. This process is called spin-lattice relaxation or theT1-relaxation. The time it takes for this relaxation is T1

4. The second processoccurring when the RF-pulse is turned off is that the spins starts to dephase, seefigure 3.2. This signal decrease is called spin-spin relaxation and T2 is the time ittakes until the spins are dephased5. Another time parameter is the T ∗2 . T ∗2 is acombination of T2 and field inhomogeneities in the static magnetic field B0. Therelationships within these three time parameters are T ∗2 < T2 < T1.

3.1.2 Gradient-recalled Echo

Since the spins are dephased before they are in equilibrium, (T ∗2 < T1), it is possibleto recall a reformed version of the FID, an echo. The PRESTO sequence, used inthis project, uses the gradient-recalled echo technique, GRE, to recall echoes [11].By adding a gradient, the dephasing, due to spin-spin relaxation and local fieldinhomogeneities, of the spins will accelerate. By changing the sign of this gradient,the spins will start to rephase. After a while, when the areas of the two gradientsare equal, the spins are back in phase and an echo appears. The time betweenthe RF-pulse and the echo is referred to as TE. It is possible to receive severalechoes until the spins are back to equilibrium. The origin of echoes when using thegradient fields can be seen in figure 3.3. Using GRE sequence, the wanted effects

4T1 is the time it takes until the magnetization has returned to 63% of its orignal value in

the z-direction.5T2 is the time it takes until the magnetization has returned to 37% of its orignal value in

the xy-plane.

8 fMRI and k-space

Figure 3.3. One way to reconstruct the FID is to use a gradient-echo sequence, GRE.After the RF-pulse is turned off, the spins start to dephase. This process is acceleratedby adding a gradient, GM . When switching the sign of this gradient, the spins will startto rephase. When the areas of the two gradients are equal, the spins will be in phaseagain and an echo has been constructed. This echo is a reformed version of the FID. Thetime from the RF-pulse to the echo is called TE.

of local field inhomogeneities are not cancelled out and it is thereby possible touse for fMRI examination6.

3.2 FID Equation

Because of the nature of the MR signal, or FID, the wanted image can be re-constructed using the well-known Fourier Transform, FT. The following equationsexplain why FT is appropriate when analysing the FID. The equations are gen-erally acknowledged in MRI but different authors use different notations. Thenotation used here is from Ljunggren’s article A Simple Graphical Representationof Fourier-Based Imaging Methods [7] in a 2D case.

In MRI, the frequency domain is called k-space, see further section 3.4. Thefollowing equations will show that the measured MR signal can be seen as thek-space corresponding to the imaged anatomy. By changing the magnetic fieldgradients G =(Gx, Gy, Gz), the position k =(kx, ky, kz) in the k-space changes.These gradients origin from the three coils seen in figure 3.7. By varying thesegradients, the total magnetic field can be changed in different parts of the examinedobject. In this way the protons, and their relative relation, can be "controlled" indifferent parts of the object. It can be showed that the following equation is validfor the k-space trajectory,

k(t) = γ

t∫

0

G(t′)dt′. (3.1)

That is, the position k in k-space at time t depends on the gradient G and the

6compared to the spin-echo technique.

3.3 Functional MRI 9

gyro-magnetic ratio constant γ. This is an important equation in this thesis, asone of the aims is to reconfigure the positioning of k-space samples.

Then, the FID signal S(t) measured by the receiver coils can be expressed as

S(t) =

∫ρ(r)eik(t)·rdr. (3.2)

Here ρ(r) corresponds to the spin distribution at the point given by r = (x, y, z) inthe image domain. Extracting ρ out of the received signal gives us the informationneeded to image the anatomy of interest.

This can be compared to the definition of Fourier transform, see further section3.4.1, and it is realized that S(t) is the Fourier transform of the wanted spindensity ρ

ρ̂(k(t)) = S(t) =

∫ρ(r)eik(t)·rdr. (3.3)

Therefore, by placing the received signal value S(t) at its right position in thek-space, k(t), the whole k-space can be determined. Then, the only thing thatremains to reconstruct an image of ρ is an inverse Fourier transform. In articlesrelated to MRI, the Fourier transform is used instead of the inverse Fourier trans-form. Since the only difference between the transformations is that the result ismirrored, the Fourier transform is also used during reconstruction in this thesis.

3.3 Functional MRI

Even though the medical application of the sequence and reconstruction methoddeveloped here is not a part of this thesis an introduction to functional MRI,fMRI, is helpful. Knowing the application of the sequence gives both restrictionsand liberties during the sequence design, e.g. time restrictions and the possibilityto use more efficient sampling patterns.

Functional MRI is an imaging technique mainly used for study the brain func-tion over time. The basic idea of fMRI is to register which part of the brain thatis activated during stimuli. It should be noted that the result from an fMRI ex-amination is the brain activation areas but fMRI itself does not measure brainactivity. Instead, fMRI measures the increased blood flow that is caused by theincreased metabolism in the area of neural activity. This means that fMRI is anindirect method of measuring brain activation [5].

There are different ways of measuring brain activation but the currently mostused is the BOLD-contrast [5]. Actually, BOLD is so common so when speakingof fMRI it often means BOLD-fMRI.

3.3.1 BOLD-Contrast

When activation in brain neurons occurs, the metabolism needs more oxygen andthe body will increase the blood flow to the active tissue. The increase of bloodflow will in turn increase the amount of oxyhemoglobin carrying new oxygen.Oxyhemoglobin is diamagnetic and will not affect the MR signal whereas the

10 fMRI and k-space

outgoing deoxyhemoglobin, also increasing, is paramagnetic and will cause localinhomogeneities (microscopic susceptibility effects) in the magnetic field. Theselocal changes are easiest to identify when looking at a T ∗2 -weighted image, alsocalled T ∗2 -contrast image. By looking at the difference between image before andduring increase of deoxyhemoglobin, changes in contrast corresponds to activatedareas. Contrast technique using this physiological phenomenon is called blood-oxygenation-level dependent contrast, BOLD-contrast [5].

Except from the T ∗2 -weight, speed is also important for an fMRI sequence.Partly because of the quick changes in brain activation due to short stimuli, (4-5seconds), and partly due to the fact that all k-space data in a profile has to becollected during T ∗2 time7.

3.4 k-space

As mentioned before, the k-space is the same as the frequency domain, FD, ora Fourier transform of the object. The k in k-space used in MR originates fromphysicists notation of spatial frequency, e.g. in the propagation of electromagneticwaves [11]. In MRI, the k-space becomes complex.

The size of k-space (bandwidth) corresponds to the spatial resolution. Higherfrequencies included in the k-space correspond to faster changes in the image.The Field of View (FOV) in a direction in the spatial domain is determined bythe corresponding sampling distance, ∆k in the k-space,

FOV ∝1

∆k.

3.4.1 Fourier Transform

The Fourier transform, FT, is the connection between the spatial domain (imagedomain) and the frequency domain (Fourier domain, k-space). Example of the twodifferent domains can be found in figure 3.5. The FT is a well-known transforma-tion and no deeper explanation will be given in this thesis. For a formal proof andthoroughly description, see for example Bracewalls The Fourier Transform and itsApplications [2]. The continuous Fourier transform for one dimension, 1-D FT, isdefined as

F (ω) =

∞∫

−∞

f(t)e−iωtdt,

where F (ω) is the function in k-space and f(t) is in the spatial domain. Thecorresponding inverse Fourier transformation

f(t) =1

2π

∞∫

−∞

F (ω)eiωtdω,

7or when using interleaved technique, the complete shot has to be done during T ∗2

.

3.4 k-space 11

takes the transformed function back to its original domain. The Fourier transformis easily generalized to higher dimensions. It is separable and therefore a 3-D FTcan be computed by three 1-D FT:s in each direction. This is also the case for thediscrete Fourier transformation, DFT,

F (k) =N−1∑n=0

f(n)e−2πi

Nkn (3.4)

and its inverse

f(n) =1

N

N−1∑k=0

F (k)e2πiNkn, (3.5)

where k and n are discrete points in each domain and N the number of samples.For practical use, the Fast Fourier Transform, FFT, is a fast implementation ofthe DFT and have been used in this work. The FFT and its inverse, IFFT, areincluded in Matlab (The MathWorks Inc, Natick, Massachusetts) as the functionsfft and ifft. There are also the functions fft2, ifft2, fftn and ifftn for 2-D andhigher dimensions. One concern with the FFT is that the data has to be sampledin a Cartesian grid. Therefore, it cannot be used directly during reconstructionfor the new PRESTO CAN sequence, section 4.5, which has a different samplingpattern. To solve this problem some kind of interpolation is needed to get thewanted sampling pattern. The data-driven interpolation using the Kaiser-Besselkernel is described in section 5.2.2.

Fast Fourier Transform Shift

The fft2 function needs to have the zero-frequency component in the upper leftcorner. To place the data in the right order before performing fft, Matlab has thefunction fftshift that should be used. Figure 3.4.1 explains how fftshift rearrangesthe data in k-space. It is presumed that the zero-frequency is located in the middleor just to the lower right of the centre before fftshift. This is also done by thescanner/sequence.

3.4.2 Image Frequencies in k-space

One advantage of the PRESTO CAN sequence is that the centre of k-space willhave higher sampling density. This is an advantage since in fMRI there is notthe same need for high frequency information in an image as in a regular MRIexamination [5]. The centre part of the k-space corresponds to low frequenciesin the reconstructed image, i.e. slow changes and contrast, and the outer partcorresponds to the higher frequencies in the reconstructed image, i.e. details.Figure 3.5 shows the connection between position in k-space and appearance inthe image domain.

3.4.3 Shift Theorem

The shift theorem, or translation theorem, for the discrete Fourier transform isneeded to understand the ghost artefacts, section 3.6, and the phase correction

12 fMRI and k-space

Figure 3.4. The k-space has to be rearranged to fit the fft-function. In Matlab, thisis done by using the fftshift and the inverse function ifftshift. The different colouredsquares corresponds to different frequencies. Normally the k-space of an image has thezero-frequency at the centre, or just below to the right of the centre, and has to be shiftedto the upper left corner before fft.

Spatial Domain k-spacef(x, y) F (kx, ky)

f(x− a, y − b) F (kx, ky)e−2πi

N(akx+bky)

f(x, y)e2πiN

(ax+by) F (kx − a, ky − b)

Table 3.1. The transform pairs corresponding to the shift theorem. Translation in onedomain transforms to added phase in the other domain and vice versa.

later on. The shift theorem states that an added phase in one domain correspondsto translation in the other domain and vice versa. Equations describing this the-orem can be seen in table 3.1. As seen in the table, translations in k-space makethe image complex. However, the magnitude of the image will not be affected. Anexample of how the translation in k-space affects the magnitude and the real partof the image is shown in figure 3.6. Even a small translation in k-space gives anadded phase in the spatial domain. In this example, the translation is done withthe same length in both directions, hence the 45◦ direction of the wavefront. Theshift theorem is used when doing the phase correction, see section 5.1.3.

3.5 Geometry Objects

In the literature, it is easy to mix up the names of the different gradients availablein scanner. Here is an attempt clarify the different names. The three gradientcoils are the x-axis gradient coil, the y-axis gradient coil and the z-axis gradientcoil, figure 3.7. The magnet gradient fields from these coils are Gx, Gy and Gz.Usually the axes of an image slice are not the same as the gradient coil axes andtherefore a combination of the coil axes are needed to describe the image slice.A common way to solve this is to introduce the coordinates M , P and S thatrotates with the slices. It is now possible to construct the sequence using thecorresponding gradient GM , GP and GS and let the scanner calculate the neededcombination of the fixed gradients Gx, Gy and Gz, figure 3.8. In this way, it is

3.5 Geometry Objects 13

Figure 3.5. In the figure, the k-space is to the left and the corresponding reconstructedimage is to the right. The upper row shows a typical image and its corresponding k-space.The middle row shows that the centre part of k-space, the low frequencies correspondsto slow variations and the contrast in the image. In the same way, the last row showsthat the high frequency information in the outer part of k-space corresponds to detailsin the image.

14 fMRI and k-space

Figure 3.6. An example of the effect of the shift theorem. From left to right: mag-nitude image, real part of image and k-space. If there is a translation in k-space, thereconstructed image becomes complex. The real part of the image (imaginary also, notpresented here) shows the added phase. The middle row show that even very smalltranslations give a visible phase in the spatial domain.

3.5 Geometry Objects 15

Figure 3.7. The three different gradient coils are used for adding a magnet gradientto the strong static magnetic field. By varying the gradients, different positions in thek-space can be measured.

easier to understand the pulse sequence and then let the scanner rotate the coilgradients to the wanted image slice position.

The first gradient, which corresponds to the x-direction in the image slice, isused to change the frequency of spins precession during readout and is thereforecalled Gfreq, Gf , Greadout,GRO, RO, Gmeasurement or GM .

In the y-direction of the image slice, the gradient changes the spins mutualphase offset and is for that reason named GPC , PC, Gphase or GP .

The third gradient, orthogonal to the x- and y-direction of the image slice, isnormally used to select the slice of interest in the body, it is called GSlice, GS orjust SS. When speaking of 3-D k-space, some literature uses the name second phasefor GS to describe the extra direction. The names used in this thesis are GM , GPand GS , mainly because it is the same index used in the simulation environment.These gradients correspond to the three directions kx, ky and kz in k-space.

3.5.1 k-space Geometry

This project uses 3-D excitation, but the k-space is measured in 2-D planes, chapter4. These planes are referred to as profiles. The reconstructed image volume isoften presented in 2-D planes called slices. The generally accepted name for thecoordinates in k-space are kx, ky and kz (compared to (u, v, w) in the FourierDomain). These are usually enough to describe the k-space position. However,when working with PRESTO CAN it is useful to introduce the discrete coordinatekφ, a coordinate that describes the current k-space profile. The coordinate axesin k-space can be seen in figure 3.9.

16 fMRI and k-space

Figure 3.8. When planning an examination, the image slices are placed at the orienta-tion most advantageous for the area of interest. To fulfil this, a more complex combinationof gradients are needed which in turn makes the sequence development more complex.A common way to solve this is to introduce the coordinates M , P and S that rotateswith the slices. It is now possible to construct the sequence using the correspondinggradient GM , GP and GS and let the scanner calculate the needed combination of thefixed gradients Gx, Gy and Gz.

Figure 3.9. The normally used coordinate system in k-space is shown to the left.Here the different readout-profiles are placed orthogonal to the kz-axis. When usingPRESTO CAN, right figure, a new coordinate, kφ, is introduce to point out the differentreadout-profiles. After interpolation, the coordinate system to the left can be used again.

3.6 Artefacts 17

3.6 Artefacts

With magnetic resonance imaging technique follows some image artefacts. Someare general for MRI, some are specific for different kind of examinations andsome can be derivable from the used sequence. When using EPI sequences, theN/2 ghosting8 is the most obvious artefact [12]. Ghosting artefacts appear as anoverlayed ghost image on the original image. The ghost image is separated fromthe original image by half the image size in pixels, and thereby called N/2 ghosting.

Several factors give raise to ghost artefacts. B0 inhomogeneities, gradient im-perfections, eddy-currents and inaccurate timing in sampling are some of them.The main reason for ghosting to occur in EPI is that the above mentioned factorsaffect the k-space rows in the phase encoding direction different. This is due tothe alternating readout directions. Hence, the ghosting will appear in the phaseencoding direction after reconstruction. Due to the above mentioned imperfec-tions, both translation errors and phase errors occur in k-space data. These errorscannot be completely compensated for because the influence of the imperfectionschanges over time. However, some major improvement can be done by using theinformation gathered during the reference scan, section 4.6. In figure 3.10, it canbe seen that the ghosting effect origins from both translation and phase errors. Inthis simulation, FT has been made separate on even and odd rows and the finalimage has then been constructed by adding these two resulting images. Abovein the figure, the original image is reconstructed without adding any k-space im-perfections. Here it can be seen that ghosting occur in both of the reconstructedimages but these effects are cancellated when adding them together. This is notthe case when every even row are translated or multiplied with a phase. The phaseshift theorem, mentioned in section 3.4.3, gives that these changes will make phaseand translation modification in the image domain. Since only every other line isaffected by these modifications, the cancellation during the addition of the twoimages will not be perfect and thereby ghosting will appear. The two lower rowsin the figure show how these modifications of k-space data cause ghost artefacts.The same results will be received if the two separated k-spaces are added beforeFT. The division of k-space in the simulation is just for explanation. Duringreconstruction, the prominent ghost due to translation is corrected.

8also known as Nyquist ghosting.

18 fMRI and k-space

Figure 3.10. The upper row shows that if k-space is divided into two separate k-spaces,by taking every second line, the FT of these k-spaces will result in two images withghosts. When adding these two images together, the result is a ghost free image dueto cancellation. The same result would be received if FT was performed direct on theoriginal k-space. The two lower rows show how modification, on every other line, bytranslation or phase multiplication, will result in ghost artefacts because the cancellationdoes not become complete. During reconstruction, the prominent ghost due to translationis corrected.

Chapter 4

Pulse Sequence

In MR, a pulse sequence is a timing diagram of RF pulses and gradients that areused to image the body. A pulse sequence controls the scanner when to transmitand when to receive. There is an abundance of sequences used in different kindsof examinations and for different needs of contrast. This chapter contains thesequences used as base for this project, the development environment and howthe PRESTO CAN sequence is developed. During sequence development, thegradients in a pulse sequence are divided into objects, depending on the purposeof the gradients. After an optimization of the objects, they become lobes in thetiming diagram. In generally speaking, gradients, objects and lobes are the samething.

4.1 Pulse Sequences in fMRI

As pointed out earlier in section 3.3.1, the most important demands on an fMRIpulse sequence is that it is T ∗2 -weighted and the speed of the acquisition. In 1977, agroup of scientists, with Sir Peter Mansfield1 in charge, developed a pulse sequencethat complied with these demands. The method was called echo-planar imaging,EPI. In the beginning, the capacity of the scanner gradients was a limitation.This is, however, not the case today and EPI has become one of the most commontechniques used with fMRI [5].

4.1.1 Echo-Planar Imaging

There have only been small changes from the original EPI sequence (1977) to theone described here. This sequence is easy to implement and very useful in fMRI.Certainly, there are other sequences as radial, spiral, propeller etc. [1]. However,basic EPI is simple and already used in the PRESTO sequence, see section 4.2.

We start with a small 2-D example on how the gradients affect the positioningin k-space and we keep in mind equation (3.1). The left side in figure 4.1 is a

1Nobel Prize in Physiology or Medicine 2003 together with Paul Lauterbur

19

20 Pulse Sequence

Figure 4.1. A pulse sequence is shown to the left and the corresponding k-space trajec-tory is shown to the right. This simple 2-D example shows how the smaller gradients inGP , called blips, moves the position in k-space one step in the ky direction. The largergradients in GM make moves in the kx direction. Different size and sign moves differentdistance and direction (along the kx-axis).

sequence diagram. Only the phase encoding gradients, GP , and the measurementgradients, GM , are shown (RF and GS are left out). To the right side in the figure,the resulting trajectory is shown. Integration of the first gradient moves us outto the far right in the kx direction. Using lower amplitude or a shorter durationwould not take us as far. The short gradient in GP is called a blip (2) and is usedto jump to the next row in the ky direction. Gradient number (3) has double size(area) but the opposite sign of gradient number (1) and will thereby make a moveto the other side of the k-space. The last blip (4) makes another short move inthe ky direction. Different combinations of gradients would of course result in adifferent trajectory.

Moving on to a more advanced sequence diagram, we look at the 3-D EPI,figure 4.2 and table 4.1. The principle is the same but now we begin in the lowerleft corner of k-space. This is done by using the prephasing lobes py and obj0.Alternation in sign of the m lobes combined with blips in GP will make the EPItrajectory in one profile. The objects obj1 and pyr are used to go back to the centreof each profile. In this sequence diagram, we have added the GS diagram and theRF time diagram. Because this is a 3-D EPI, GS is not only used during excitation,as in 2-D EPI, but also to move to the desired position in the kz direction. Insteadof excitating a slice, we now excitate a volume, a slab. This is done with s_ex andthe RF-pulse. The sequence diagram is repeated with different amplitude of pz tochange the kz-position between slice readouts. To make the sequence balanced, adephasing lobe pzr, with opposite sign of pz, return the trajectory to kz = 0.

In theory, the measurement of the k-space is now complete and a 3-D FFT willreturn the wanted image volume. However, in practice there first has to be somecorrection due to e.g. field homogeneity. This is called the reconstruction phaseand is described in chapter 5.

4.1.2 2-D vs. 3-D

In 2-D MRI techniques, only a slice of the body is excitated and reconstructed. Byobtaining several slices next to each other, a volume containing several 2-D k-spaces

4.1 Pulse Sequences in fMRI 21

Figure 4.2. The outline for a 3-D EPI sequence. GS , GP and GM denote the gradientsin slice-, phase and measurement direction. Explanation of each gradient can be foundin table 4.1. Note that the amplitude of pz and pzr changes from repetition to repetitionto move in kz direction between profiles. The trajectory after repeating this 3-D EPIsequence can be seen to the right in the figure.

Gradient objects in EPI sequenceGS Gradients in the slice direction.GP Gradients in the phase direction.GM Gradients in the measurement, or readout direction.s_ex Slice (slab in 3D) selection gradient. Used during excitation.pz Prephasing lobe. Move to start position in kz direction.pzr Return to centre of kz-axis.blip Move to next line in ky direction.py Prephasing lobe. Move to start position in ky direction.pyr Return to centre of ky-axis.m Readout in kx direction.obj0 Prephasing lobe. Move to start position in kx direction.obj1 Return to centre of kx-axis.

Table 4.1. Explanation of the gradient objects used for the 3-D EPI sequence in figure4.2.

22 Pulse Sequence

is received. In 3-D MRI, a volume is excitated and the resulting k-space becomesa continuous volume [1]. There are different advantages and disadvantages withthe two volume techniques depending on the examination. Usually, 3-D MRI takeslonger time and not used in fMRI. One advantage with 3-D MRI is that becausethe complete volume is excitated for every profile, the whole volume contributesto the measured signal and thereby a better SNR can be achieved. This is to becompared to the 2-D MRI where only the excitated slice contributes to the signal.Another advantage is that there are more possibilities for processing the imagedata when the k-space is continuous. There are some artefacts specific for 2-DMRI and some for 3-D MRI that makes it hard to do a general statement of whichtechnique is the best.

The PRESTO sequence, used as a base for this project, is a 3-D technique.The, in this project developed, PRESTO CAN sequence uses the possibilities inthe 3-D k-space and the fact that during an fMRI examination changes in lowerfrequencies are more interesting and thereby can shorten the acquisition time.

4.1.3 Multishot

Using a single RF-pulse during the acquisition of one profile can cause suscep-tibility artifacts and geometrical distortions [12]. Another concern is the timelimitation because of the fast transversal relaxation, T ∗2 . To be able to make afull acquisition in one excitation, the size of the k-space has to be reduced whichwill cause reduced spatial resolution. To solve these problems, multiple excitationsduring one profile acquisition, multishot, are introduced. There are different tech-niques how to construct the trajectory for each excitation. The trajectory used inPRESTO and PRESTO CAN is called interleaved EPI and is a good technique forreducing ghosting artefacts. When using interleaved EPI, the spacing between therows in the trajectory is increased. For each new excitation, a similar trajectory isperformed, however shifted in the ky direction. In this way, an interleaved patternis received, figure 4.3, and the profile becomes completely sampled. The numberof acquisitions required to fill the profile is called the EPI-factor. Completing theprofile in only one acquisition corresponds to an EPI-factor of one, singleshot. Ifthe EPI factor is two, the ky-blips are double in size to perform the right trajec-tory. Using multishot increases the total examination time, so there is a balancebetween image quality and acquisition time.

4.2 PRESTO Sequence

The PRESTO sequence is not as common as the EPI sequence in MR, but it isoften used in fMRI because of its ability to make the repetition time, TR, shorterthan the echo time, TE. The advantage of this is that the acquisition time getsshorter while the image still is strongly T ∗2 -weighted, i.e. the demands for a goodfMRI imaging according to section 3.3.1. PRinciples of Echo-Shifting with a Trainof Observations, PRESTO, sequence is based on extra gradients that shifts theecho into the next TR-period and in this way manages to get TE>TR [6].

4.2 PRESTO Sequence 23

Figure 4.3. Using interleaved EPI, a multishot technique, the spacing between the rowsin the trajectory is increased and multiple excitations, shots, with shifted trajectoriesare performed to fill the profile. A higher image quality can be achieved but with thedisadvantage of longer examination time. Interleaved EPI can be used in both PRESTOand PRESTO CAN. In this thesis, the PRESTO CAN is only evaluated using singleshot.

The PRESTO sequence can be divided into two parts, echo-shift, ES, and thetrain of observation. The echo-shift is the part that shifts the echo to the nextTR-period. By changing the added gradients, it could shift the echo even furtherbut in this study we only move it to the next TR-period. The two new gradientsneeded for the echo-shifting is placed just after the RF-pulse and just before thenext RF-pulse [1]. The first gradient is called dephasing lobe and is used to fullydephase the spins that just have been excitated. Because of this there will be noecho during this TR-period, see figure 4.4. The second gradient, rephasing lobe,is double in size compared to the dephasing lobe and thereby the spins will stillbe out of phase, but in the "opposite direction". When the spins are out of phase,they will not be affected by the upcoming RF-pulse. The second dephase lobe willnow make the spins in phase, just as they where directly after the first RF-pulse,and an echo can be produced. This can be comprehended by summing up thethree gradients that the spins have been affected by and see that they now havecancelled each other. The second dephase lobe will also make the spins, that areexcitated by the second RF-pulse, delayed to the next TR-period.

By making the size of the ES gradient corresponding to the size of the k-space,artefacts occurring due to ES can be avoided. In the lower part of the figure, asuccession of k-space positions is drawn. This illustrates how the position in thek-space jumps a multiple of the k-space size when the ES gradients are added toGP and GM . The succession also shows that the echoes always arise in the same

24 Pulse Sequence

Figure 4.4. Using echo-shifting, new gradients GES are added to the sequence. Thenew gradients are placed just after the RF-pulse, dephasing lobe, and just before thenext RF-pulse, rephasing lobe. This combination of gradients shifts the echo to the nextTR-period and TE>TR. The lower part of the figure shows the corresponding positionin 2-D k-space, when adding GES gradients to GP and GM . To avoid artefacts, the sizeof ES gradient has to correspond to a multiple of the k-space size. Here it can be seenthat the echo appears at the same multiple of k-space as it originates. The first figure isthe starting point, then a new figure is drawn after each gradient.

k-space position as they originate. In 3-D PRESTO these echo-shift gradients areadded to all gradient direction (GS , GP and GM ).

The second part of PRESTO is an EPI sequence, as in figure 4.2, where thechanging of sign during readout will refocus the spin for each row of k-space. Therefocus will cause multiple echoes during one excitation. There is one echo for eachrow and the timing of the sequence makes the maximum of each echo in the middleof the rows. The echoes during one excitation will change in amplitude betweenthe rows. The timing of the sequence will place the maximum echo in the centreof k-space. When combining the two parts, it is important that the EPI sequenceis balanced, i.e. the sum of the gradients during a TR-period have to be equal tozero. Otherwise, the ES will start to drift and the jumps in the k-space will notbe a multiple of the k-space size. A complete 3-D PRESTO sequence can be seenin figure 4.5. Here, the gradients from EPI and ES occurring at the same timeare merged together, which makes the sequence look a bit confusing. The reasonfor this is to make it easier to optimize the different gradients before the sequenceis sent to the scanner. To be able to develop the PRESTO CAN sequence, thedifferent gradients have to be extracted, see further section 4.5. The PRESTOsequence will be used both as a reference and as a starting point for developingPRESTO CAN.

4.3 Scanner 25

Figure 4.5. A PRESTO sequence showing that the echo occurs in the next TR-period.To make the time optimization, the gradients have been merged together and thereby itis more difficult to see the EPI part of PRESTO. Here, only five rows are measured, foreach profile of k-space between the RF-pulses, compared to ten rows in figure 4.2.

4.3 Scanner

The MR scanner, used during this project is a 1.5T Achieva scanner (PhilipsMedical System, Best, The Netherlands) [14], figure (4.6), placed at the Centerfor Medical Image Science and Visualization, CMIV [3].

4.4 Development Environment

CMIV has a full research agreement with Philips Medical Systems and therebyPhilips provides a simulation and development environment called Philips Acqui-sition Research ADvanced Integrated Software Environment, PARADISE. PAR-ADISE contains all the features that the real scanner controller has, except fromthe possibility to do the actual scan (e.g. do not return any data). PARADISEalso contains some helpful programs for programming and evaluation of pulse se-quences. Due to restrictions in research agreement no profound explanation ofPhilips sequences are possible in this thesis.

4.5 PRESTO CAN Sequence

This section presents how the EPI part of PRESTO sequence is extracted and howthe PRESTO CAN sequence is constructed. The traditional way to collect the k-space volume is to measure profile by profile in the kz-direction and construct aCartesian grid. This is the case for regular PRESTO. If we instead rotate theprofiles, we receive a non-Cartesian grid more looking like a can, see figure 3.9.This is the difference between the PRESTO sequence and the PRESTO CAN

26 Pulse Sequence

Figure 4.6. The MR scanner used during this thesis is a Philips 1.5T Achieva placedat CMIV. Picture by courtesy of CMIV.

sequence. The advantage of PRESTO CAN compared to PRESTO is that fewerprofiles are needed to receive a large FOV, keep the high sampling density for lowfrequencies in k-space (slower changes in the image) and still have an acceptablespatial resolution (high frequencies in k-space). Fewer profiles result in fasterk-space acquisition that corresponds to higher time resolution.

4.5.1 Gradient Separation

As mentioned before, the ES gradients and the EPI gradients are merged togetherwhen it is possible. Therefore, before rotation can be performed, the EPI gradientshave to be separated from the ES gradients. Since the rotation is around the ky-axis, only the EPI gradients in GM and GS will be rotated. However, because areference scan is needed, section 4.6, the EPI gradients in GP direction are isolatedas well. Because a reference scan is used in the PRESTO sequence as well, theseare already isolated. Since the k-space profiles are rotated instead of piled up inthe kz direction as in PRESTO, the pz and pzr gradients are set to zero before theyare merged together with the ES gradients. When separating the EPI gradientand the ES gradient in the measurement direction, the area of the EPI gradientsare identified first. Then the scanner makes the time optimization through aniteration scheme. In this way, the duration and timing of the gradients becomecorrect without any further adjustments. By knowing the area and duration ofthe EPI gradients, obj0 and obj1 can be calculated. So far, the earlier merged

4.5 PRESTO CAN Sequence 27

Figure 4.7. The PRESTO sequence is a combination between EPI technique and echo-shift technique. The first step in developing the PRESTO CAN sequence is to isolatethese to two parts. This separation can be seen in the figure. A table of the gradients iscompiled in table 4.2. The gradients are divided into three groups, base, xbase and fin.This is useful because the scanner can not only handle gradients, it can also handle agroup of gradients. Note that we here have chosen to give the ES gradient in M directionthe opposite sign compared to figure 4.4 and 4.5.

gradients and the sum of the new gradients are the same. This is used to confirmthat the separation is performed correct. The separation is illustrated in figure 4.7and table 4.2.

4.5.2 Gradient Rotation

Since the PRESTO CAN sequence is a further development of the PRESTO se-quence, the user interface for parameter adjustment is the same. Only the enablerotation parameter and the subparameter rotation factor have been added. If theenable rotation is set to no, the PRESTO CAN sequence will be the same as thePRESTO sequence. In this state of development, the number of slices is deter-mined by the number of sampling points in the measurement direction. The valueof the rotation factor changes the number of profiles. So far in this project, therotation factor is set to one, the predetermined value.

After the EPI gradients have been isolated, it is simple to perform the rota-tion by using the built-in rotation function in the scanner. Since the profiles arecentered round the rotation axis, it is enough to do a 180◦ rotation to cover thek-space. The profiles are evenly distributed. The rotation function is actuallyan orientation matrix that is updated with the new rotation angle between everyprofile. Except from the rotation angle, the orientation matrix needs to know

28 Pulse Sequence

Gradient objects in the PRESTO sequenceEPI gradientss_ex Slice (slab in 3D) selection gradient. Used during excitation.pz Prephasing lobe. Move to start position in kz direction.pzr Return to centre of kz-axis.blip Move to next line in ky direction.py Prephasing lobe. Move to start position in ky direction.pyr Return to centre of ky-axis.m Readout in kx direction.obj0 Prephasing lobe. Move to start position in kx direction.obj1 Return to centre of kx-axis.

Echo-shift gradientsr0 Dephasing lobe in kz direction.d Rephasing lobe in kz direction.pd0 Dephasing lobe in ky direction.pd1 Rephasing lobe in ky direction.mc0 Dephasing lobe in kx direction.md Rephasing lobe in kx direction.

Table 4.2. The gradients in a PRESTO sequence can be divided into EPI gradients andES gradients. The corresponding timing diagram can be seen in figure 4.7.

the rotation axis and which gradients or groups of gradient that are rotated. InPRESTO CAN, obj0, obj1 and xbase (see figure 4.7) are rotated. This will notaffect the gradients in the GM direction, since GM rotates as well, but it will af-fect the gradients in the fixed XYZ coordinate system. As pointed out earlier, theprofiles are not displaced in the kz direction and the GP gradients pz and pzr aretherefore set to zero. However, to make the rotation, some gradients are needed inthe GZ direction and these are added by the rotation function. Since the readoutnot only takes place in the kx direction, a readout gradient sequence is added aswell. An excerpt from the PRESTO CAN sequence showing the EPI gradients forfive consecutive profiles can be seen in figure 4.8. To make the figure more clear,the ES gradients have been excluded. Have in mind that the ES gradient will staythe same throughout the rotation.

Since the rotation is done after time optimization, the changing amplitude ofthe EPI gradients is not compensated for. This becomes a problem after a 90◦ ro-tation when the amplitude for obj0 and obj1 has changed in sign. Because of this,the maximum gradient strength for the scanner could be exceeded. A compensa-tion for this has been implemented by adding extra gradient before optimizationand then remove it. This static compensation prolong the gradient duration evenwhen it is not necessary, so a further development is needed.

4.6 Reference Scan 29

Figure 4.8. Five different k-space profile seen in the kx-kz plane. When rotating, thereadout-gradients still are in the GM direction because the MPS-axes rotates with theprofiles. However, looking at the gradients in the fixed XYZ-system, it can be seenthat a readout-sequence has been added in the GZ direction. The gradients amplitudedependency on rotation angle is visible in the XYZ-system. As seen in the figure, thesign of the GX gradients changes after 90◦ rotation. This is not the case for the GZgradient because it is enough to perform an 180◦ rotation to fill the k-space.

4.6 Reference Scan

The main purpose of a reference scan, in an EPI sequence, is to see how thedata during readout differs depending on the direction in which the data has beencollected. Then, the difference between these rows are used to correct the k-spacecollected during the actual examination. In this way, the direction of the readoutwill not be a factor during the reconstruction [12]. If these imperfections are notcorrected before reconstruction there will be severe artefacts, see section 3.6.

Since the received signal is strongest for ky = 0, this row is collected. Thisis done by turning off the py gradients, the pyr gradients and the blips duringthe reference scan. Now the (ky = 0)-row will be collected several times but fromdifferent directions, i.e. the phase encoded axis is replaced by a time axis forthe (ky = 0)-row. Since the profiles in a PRESTO sequence are parallel to eachother, it is enough to calculate calibrations for the middle profile, kz = 0, andthen re-use it for all other profiles. However, in the PRESTO CAN sequence theprofiles are rotated and calibration has to be done for each profile. This demandsfurther development of the PRESTO CAN sequence. An easy solution to thisproblem is to run a complete PRESTO CAN sequence before, or after, the actualexamination with the above mentioned gradients turned off2. Since the referencescan is an isolated occurrence during a fMRI examination, the increased time forthe complete reference scan can be accepted.

2This can be done by entering the development mode in the scanner.

Chapter 5

Reconstruction

Due to the sampling pattern in the PRESTO CAN sequence, the scanner cannot,as usually, reconstruct the image volume by itself. This is instead performedon a stand-alone computer. To make the reconstruction possible, an in-housereconstruction algorithm, using Matlab, was implemented. This reconstruction isdivided into two parts, the raw data processing and the image processing, figure5.1. In the raw data processing, imperfections due to hardware or magnetic fieldinhomogeneities etc. are corrected. The actual reconstruction of the image isconducted during the image processing.

For the evaluation of PRESTO CAN reconstruction, only singleshot has beenused, i.e. one excitation per profile. Even if this will result in unnecessary artefacts,it is used to simplify the evaluation.

5.1 Raw Data Processing

The data that is exported from the scanner is called raw data. The raw dataconsist of two files, one including the actual complex k-space data and the otherincluding some information about the data. The second file is used when readingthe data file into Matlab. In this way, the data is placed in the right order and thebelonging information is connected to it. The information includes for examplek-space position, image size, image position, dynamics1, readout direction and soon.

5.1.1 Pre-processing

Even though the exported data is called raw data, some pre-processing is alreadyperformed by the scanner. It is not obvious which corrections are done. However,the flipping of rows due to the different readout directions in EPI is one of them.

1different images taken of the same area of interest. Can be used to show changes.

31

32 Reconstruction

Figure 5.1. Diagram over the reconstruction process. Before the scanner export thek-space data, some pre-processing of the data occurs. Imperfections due to hardware ormagnetic field inhomogeneities etc. are corrected in the raw data process. During imageprocessing, the actual image volume is reconstructed. Before the image is displayedfor examination, some image enhancement can be made. This is not included in thereconstruction algorithm developed in this thesis.

5.1 Raw Data Processing 33

Figure 5.2. This reconstruction scheme was implemented for the PRESTO sequence.The phase inversion corrects for the T1 contrast enhancement, section 5.1.3. Since nore-gridding is needed, the FT in ky and kz follows directly after the corrections. Thecutting in the X direction of the image volume, due to oversampling in the k-space, has tobe performed before translation correction to fit the internal reference scan data receivedduring export of raw-data.

5.1.2 PRESTO Reconstruction

A PRESTO sequence image volume, reconstructed by the scanner, has been usedas a reference volume to evaluate the PRESTO CAN sequence and the reconstruc-tion algorithm. One step in the development of the PRESTO CAN reconstructionalgorithm was to implement an algorithm for reconstruction of raw data from thePRESTO sequence. This was implemented successfully and the algorithm, figure5.2, became the base for the PRESTO CAN algorithm. The main difference be-tween this reconstruction and the scanner reconstruction is that the scanner makessome image enhancements before displaying them. Since this was not the aim ofthis thesis, no further effort has been done to improve this part of the algorithm.

5.1.3 Data Correction

Usually the scanner performs corrections to avoid artefacts before reconstructionof the image. This is, however, not done when raw data is exported from thescanner. The raw data includes some basic correction data which is sufficientfor correcting the k-space from the PRESTO sequence. However, because of therotation in PRESTO CAN the correction data has to be calculated from a separatereference scan.

T1 Contrast Enhancement

When looking at the data or the reference data, every second profile has the inversephase, as if the phase of the excitation pulse alternates. This is presumably done bythe scanner to minimize some artefacts by making destructive interference betweenremaining magnetization after excitation. These changes of is not a problem fora normal PRESTO sequence since such a phase shift corresponds to a translationin the z direction (according to the shift theorem, section 3.4.3) and can easilybe corrected for after reconstruction. For PRESTO CAN however, with rotationbetween profiles, the phase shift must be corrected for before reconstruction. Sincethis data is produced by singleshots, the solution to this problem is to multiplyevery second profile with e−iπ, i.e. −1. In figure 5.3 (a,b), it is visible how thephase inversion expresses itself and how the discontinuities disappear after thecorrection. This is a correction between the profiles.

34 Reconstruction

Translation Correction

When reading every second row in k-space in the opposite direction, as in EPI,unwanted translations between the rows will occur. Sub-pixel correction of thesetranslations in the kx direction, can be performed by correcting phase shifts inthe image domain. This is due to the shift theorem, section 3.4.3. First, both thedata and the reference data is Fourier transformed in the kx direction. Then thecorrection data is calculated from the reference data by subtracting, in pairs, thephase of rows with opposite direction. Then, rows collected in the opposite direc-tion will be adjusted to remove the unwanted phase shifts. Finally, an inverse FTis performed to return to k-space for further corrections. The result of translationcorrection can be seen in figure 5.4 (a,b). Here the correction is performed on thereference data itself to evaluate the correction.

Phase Correction due to Time

During examination, there will be a phase drift over time, beginning after eachexcitation. This drift is more obvious in this data because singleshot has beenused. Since all the rows in the reference scan represent ky = 0, they should allhave the same phase. The drift can be seen in figure 5.4 (a). By using the middlerow as reference row, the needed correction data can be calculated by subtractingthe phase of the other rows. Then the correction is performed by multiplyingthe k-space, row by row, with the corresponding correction data. Since only thephase of k-space is adjusted, the absolute values in k-space will not be affected.The result after translation and phase correction of the reference scan can be seenin figure 5.4 (b) and (c). The translation correction and phase correction areperformed within the profiles.

The middle row is chosen as reference row, because this row corresponds to,in time, the ky = 0 of the original data. This has been proved to be a bettersolution than using the first row as reference row, which perhaps would be a moreinstinctive choice. Later on in this project, multishot will be used and the phasecorrection will probably not be needed.

During the evaluation of the reconstructed image volume, section 7.2, the con-clusion is drawn that this correction actually deteriorate the result and is thereforetemporarily excluded from the reconstruction algorithm until further investigationis performed.

Phase Correction due to Rotation

When looking at the PRESTO CAN raw data, there seems to be a phase alter-ation between profiles, depending on rotation angle. Since the profiles are evenlydistributed, the first profile and the last profile should almost have the same phase,only mirrored. This is not the case, instead there is a discontinuity at the tran-sition between the last and the first profile, as seen in figure 5.3. Also seen inthe figure, is that the phase at the centre of kx drift between profiles. Since thispoint coincides with the rotation axis, it should be the same for all profiles. Thedifference between this point at the first profile and the other profiles is used as

5.2 Image Processing 35

Figure 5.3. Phase correction due to rotation. Due to an artefact reduction techniqueused by the scanner, there is a 180◦ phase alternation between the profiles that causediscontinuities (a). After this has been corrected for (b), there still is a discontinuitybetween the last and the first profile. There is also a phase drift at the rotation point,kx = 0. By adjusting the profile to avoid this phase drift, the discontinuity disappearsas well (c). At the bottom of the figure, the first and the last profile has been put nextto each other to show that the phase now is continuous.

correction data. This is corrected for by multiplying every profile with the cor-responding correction data. After this correction the discontinuity disappear aswell, see figure.

5.2 Image Processing

During image processing, the k-space is re-gridded onto Cartesian coordinates, soa 3-D FT can be used to reconstruct the image. Some filtering and weighting isalso performed. Finally, the image has to be resized because of oversampling ofthe k-space performed by the scanner.

5.2.1 Angle Interpolation

An interpolation function in kφ direction is implemented to make it possible touse even fewer profiles and still keep good image quality. During evaluation ofthe PRESTO CAN sequence, the interpolation factor is set to 1, i.e. no angularinterpolation occurs. The results of this function have not been investigated inthis thesis.

Assuming that the sampling distance in the phase direction will result in aperfect reconstruction, the number of profiles needed can be calculated. With

36 Reconstruction

Figure 5.4. The images to the left show the magnitude from a reference scan profile. Tothe right is the corresponding phase. The top images (a) are before corrections. Even ifthe rows all represent ky = 0, the intensity of the signal decrease over time. This decreasewill be reduced if multishot is used. The middle images (b) show how the translationcorrection eliminates the zigzag pattern in the original profile. Even if the magnitudeof the profile seems to be correct, the phase drift has to be corrected. As seen in thebottom images (c), the phase correction only affects the phase image and the rows of thereference scan are now aligned.

5.2 Image Processing 37

Figure 5.5. Data-driven interpolation is used in the gridding-technique. In data-driveninterpolation, every data sample add its contribution to adjacent points in the new grid.The figure shows how one data points from a rotated profile is distributed to the nearbypoints in the Cartesian grid.

the demand of equal or shorter sampling distance to produce "perfect" reconstruc-tion in the radial direction, it can be shown, with some approximations, that thenumber of profiles becomes

Nprofile ≥π

2Nphase

where the Nphase is the number of samplings points in phase direction, i.e. k-spacesize in ky direction. A rule of thumb gives the demand for "good" reconstruction,

Nprofile ≥ Nphase.

5.2.2 Gridding

To be able to use FFT to reconstruct the image, the k-space data has to be re-sampled onto a Cartesian grid. There are several ways to do this re-sampling. Afast implementation would be to use simple bilinear interpolation. This approachis rejected by Schomberg et al. [13], since it can produce extra artefacts. Insteadthey recommend using "gridding" described in O’Sullivans article from 1985 [9].This technique is based on data-driven interpolation using a window function, W ,as a convolution kernel. In data-driven interpolation [10], every data sample addsits contribution to adjacent points in the new grid, figure 5.5. The size of thecontribution distance is a predefined value. The gridding method is a trade-offbetween speed and image quality.

The rectangular function is the ideal apodization function, which correspondsto the sinc-function as a convolution kernel. However, the sinc-function is prac-tically impossible because of its infinitive size. Therefore, another trade-off hasto be done and O’Sullivan [9] proposed the Kaiser-Bessel window W (kx), figure5.6. Schomberg et al. [13] also recommend the Kaiser-Bessel window after their

38 Reconstruction

Figure 5.6. Normalized 1-D Kaiser-Bessel window function W (kx) and the correspond-ing inverse Fourier transform w(x) both in normal and log-scale. A triangle-function isalso plotted (dashed) in both domains. The sides of the reconstructed image correspondsto w(0.5). The major advantage with the Kaiser-Bessel window, compared to e.g. thetriangle function, is the a? lmost non-existent sidelobes which results in minimal aliasingin the reconstructed image volume.

comparison with other windows. The major advantage with the Kaiser-Bessel win-dow, compared to e.g. the triangle function, is the almost non-existent sidelobeswhich results in minimal aliasing in the reconstructed image volume. Convolution(or interpolation) with W (kx) · W (kz) in k-space corresponds to multiplicationwith w(x) · w(z) in the image and therefore the need for low sidelobes to avoidaliasing (w(0.5) corresponds to the sides of the image, figure 5.6 (b) and (c)). Thesloping shape of the headlobe can be perfectly compensated for by inverse windowweighting.

The basic gridding method can be divided into three steps, see figure 5.7. Firstthe convolution F =W ∗F̂ is performed to resample the data in F̂ onto a Cartesiangrid. Then the inverse FT2 is performed to receive fw. Finally the wanted imagef is obtained as f = fw/w. In PRESTO CAN, the gridding is only performedin the kx-kz plane since the ky direction already is in Cartesian grid. The 2-Dconvolution kernel can be constructed by multiplying two 1-D kernels. Some extrasteps are added for further improvement and the final algorithm becomes,

1. The sampled data is divided with the local sampling density D(kx, kz),

FD(kx, kz) =F̂ (kx, kz)

D(kx, kz).

2. Re-sampling onto Cartesian grid using the Kaiser-Bessel window functionW (kx, kz),

F (kx, kz) =W (kx, kz) ∗ FD(kx, kz) =∑FD(kx, kz)W (kx − a, kz − b).

3. A normalization function is calculated,

K(kx, kz) =∑ W (kx − a, kz − b)

D(a, b).

2As mentioned in section 3.2, the FT is used instead

5.2 Image Processing 39

4. As an option, normalization can be performed,

FK(kx, kz) =F (kx, kz)

K(kx, kz). (5.1)

5. Inverse Fourier Transform3 (in the implementation, the complete k-space isFourier transformed by a 3-D FFT),

fw(x, z) = F−12 [FK(kx, kz)] .

6. To compensate for the non-ideal window, division withw(x, z) = F−1

2 [W (kx, kz)] gives

f(x, z) =fw(x, z)

w(x, z).

The local density function is dependent on the radial distance r from origin, thesampling distance ∆r and the number of profiles Nφ. The function is valid whenthe rotated profiles are even distributed over 360◦. It can be shown that thesampling density becomes

D = r∆r2π

Nφ, r 6= 0

D =π( ∆r

2 )2

Nφ, r = 0.

The implemented gridding function re-grids the real data and imaginary dataseparate. This is done plane by plane in ky direction. The re-gridding function isimplemented as C-code in Matlab to increase speed, still this part of the recon-struction takes the longest time.

5.2.3 Further Image Processing

Before the 3-D FT is performed, an LP-filter can be applied on the k-space. Thisreduces the ringing artefact (Gibbs phenomenon) that can occur because of thesharp edges of k-space. The filter is smoothened by a cosine-weight to furtherreduce the artefact. Because of oversampling of k-space, the reconstructed imagehas to be cut to the right FOV. Information about this is included in the raw-datainformation file.

3As mentioned in section 3.2, the FT is used instead.

40 Reconstruction

Figure 5.7. The basic gridding method can be divided into three steps. First theconvolution F = W ∗ F̂ is performed to resample the data in F̂ onto a Cartesian grid.Then the Fourier transform is performed to receive fw. Finally the wanted image fis obtained as f = fw/w. Some extra steps are added for further improvement of thealgorithm.

Chapter 6

Simulation

By using tables of Fourier transforms, for example in Bracewell’s book The FourierTransform and its Applications [2], k-spaces used for simulation can be con-structed. The phantom used in these simulations is similar to the well-knownShepp-Logan phantom. This phantom consists of an elliptic disc (intensity value1.02) surrounded by an elliptic ring (intensity value 2.00). To make the phantommore complex, six objects with different size and intensity values are added (inten-sity value 1.75, 1.75, 1.75, 1.75, 0.5, 0.5). Since the following simulations are usedfor evaluation of the gridding technique and the Kaiser-Bessel window, no arte-facts where added to these k-spaces. Even though the k-space received from thePRESTO CAN sequence is a 3-D volume, the gridding is performed in 2-D (kx-kzplanes), section 5.2.2. Therefore are the simulations executed on 2-D k-spaces.

Figure 6.1. To the right in the figure is a Shepp-Logan like phantom used for thesesimulations. This image is reconstructed from a Cartesian gridded k-space. The leftimage is reconstructed from a radial grid comparable to the PRESTO CAN. The artefactin the corner arises during compensation for the non-ideal window. This is because ofdivision with values close to zero.

Figure 6.1 shows a comparison between the reconstructed phantom from aradial sampling pattern (left), and a Cartesian sampling pattern (right). Bothhave 128 samples in the kx direction and to make the size of k-spaces similar,

41

42 Simulation

Figure 6.2. By changing the number of rotated rays, Nφ, the image quality decreases.Here Nφ = 128, 64, 32 and 16 was used. Using less number of rays than stated in section5.2.1 (ky = 128) will result in worse image quality. However, the images can still beuseful, e.g. for planning the examination.

the number of rotated rays, Nφ in the radial grid is equal to the number of kysampling points in the Cartesian grid (128 points). Both reconstructions includelow-pass filtering before FT. The artefact appearing in the corners of the left imagearise during the compensation of the Fourier transformed window w, section 5.2.2.The values in the corner of this window are close to zero and thereby work asan amplifier at division. An improved gridding algorithm using double samplingdensity has been developed to avoid this artefact, however this has not yet beenevaluated nor implemented in the reconstruction algorithm.

Even though a more complex phantom is necessary to estimate the least numberof Nφ, this simulation shows the effect that different Nφ have on the resultingimage, figure 6.2. Using less number of rays than the stated in section 5.2.1(ky = 128) will result in worse image quality. However, the images can still beuseful, e.g. for planning the examination.

In section 5.2.2, normalization was added as an option to the gridding al-gorithm. Simulations, figure 6.3, establish that this normalization is preferable.Except from the changed intensity values, the artefacts becomes more intensewithout normalization. This is more prominent for lower number of Nφ.

A simulation of 3-D k-space can be seen in figure 6.4. Once again, a phantomsimilar to the Shepp-Logan phantom was used. The size of k-space is kx = 64,

43

Figure 6.3. When not using the optional normalization, the artefacts become moreprominent, especially for lower values on Nφ. Here Nφ = 128 and 32. Note the changedvalues of the colour scale compared to earlier images in this chapter.

ky = 64 and Nφ = 64. The objects intensity is the same as for the 2-D phantom.By decreasing Nφ with a factor 4, artefacts become more dominant. Still, thebasic shapes of the objects are visible.

Figure 6.4. Reconstruction of a simulated 3-D k-space phantom. The size of the k-spaces (a) are kx = 64, ky = 64 and Nφ = 64 and (b) are kx = 64, ky = 64. Nφ = 16.A decrease of Nφ with a factor 4 results in more artefacts. Still, the basic shapes of theobjects are visible.

Chapter 7

Results

This chapter displays the results from the evaluation of the novel PRESTO CANsequence. An evaluation of the PRESTO reconstruction algorithm, used as baseand reference for the PRESTO CAN reconstruction, is also included.

To reduce the amount of raw-data during reconstruction, the resolution of theimages is deliberately poor. To reduce the complexity of the k-space data structure,singleshot was used. Unfortunately, this simplification introduces artefacts thatare not corrected for during the reconstruction. New scanner software release andtime limitation, prevented further tests during this work.

7.1 PRESTO Reconstruction

For evaluation of the algorithm implemented to reconstruct raw-data from thePRESTO sequence, the scanner-reconstructed image volume was used. The ex-ported image volume from the scanner has the Philips export format, PAR/REC.This image volume is not the final result used for clinical investigation, but goodenough for this evaluation. The scanner reconstruction includes image enhance-ment and intensity scaling, which make the comparison more difficult. To makethe comparison more fair, linear interpolation to increase the resolution was per-formed on our reconstructed image volume. Because of the differences in imageenhancement, the PAR/REC-files have not been imported into Matlab for a deepercomparison. Only visual comparisons of the images are performed for evaluation.

First an in vitro examination was performed, figure 7.1. The phantom usedwas a spherical container filled with 2.7l liquid (including 0.10% sodium azide and0.10% Magnavest). The refill hole is visible in both the scanner reconstruction(b) and our reconstruction (a). To receive a more detailed image, an in vivoexamination was performed. The brain of a healthy volunteer was imaged, figure7.2. Image (b) and (d) are two image slices from the scanner reconstruction.Here the scanner’s image enhancement is more obvious, when comparing withthe reconstructed images (a) and (c). Differences in intensity scale are prominentbetween the images, especially when looking at the black "background".

45

46 Results

Figure 7.1. A spherical phantom was scanned to evaluate the PRESTO reconstructionalgorithm. No image enhancement, except interpolation to increase the resolution, hasbeen performed on our reconstructed image (a) compared to the scanner reconstructedimage (b).

7.2 PRESTO CAN Reconstruction

Two new in vitro examinations (PRESTO and PRESTO CAN), with the samephantom as above, was performed for evaluation of the PRESTO CAN sequenceand the corresponding reconstruction algorithm. Exported raw-data from thePRESTO examination was reconstructed and used as a reference. Both the re-construction of the PRESTO data and the PRESTO CAN data were performedin Matlab. The same user parameters were used for both examinations, see ap-pendix A. Since PRESTO CAN rotates the profiles, the FOV differs in the Zdirection compared to PRESTO. Because of the use of an external reference scanin the PRESTO CAN reconstruction, the order of the reconstruction steps differsbetween the two algorithms. This results in different k-space size when comparingthem.

In figure 7.3 a sequence of images shows how the different corrections, section5.1.3, affect an image slice of the resulting PRESTO CAN volume. Image (a)corresponds to no corrections. The major reason for the nonsense in this imageis the alternating sign of the phase between every second profile, mentioned insection 5.1.3. After correcting for T1 contrast enhancement, the shape of thephantom can be seen in (b). Here the N/2 ghosting artefact is visible. Thisartefact is removed after translation correction (c). As claimed in section 5.1.3,phase correction improves the image quality. Here however, the correction appearsto deteriorate the image (d). This is confirmed after the phase correction due torotation, where the resulting image without phase correction (f) becomes smoothercompared to the mottled image (e). The following images in this chapter excludephase correction.

Correction of phase errors is an important part of the reconstruction and there-fore a comparison between the phase of k-space for PRESTO and

7.2 PRESTO CAN Reconstruction 47

Figure 7.2. To receive a more detailed image volume, a brain was scanned. Thereare differences in image quality between our reconstructed images (a) and (c) and thescanner reconstructed images (b) and (d). However, our implementation is only expectedto reconstruct the image properly, not to perform image enhancement.

48 Results

Figure 7.3. An image slice after: (a) no corrections, (b) T1 contrast enhancement,(c) translation correction, (d) phase correction, (e) rotation correction. Even if thephase correction improves the phase image, according to section 5.1.3, here it appears todeteriorate the image. Image (f) is the result if phase correction is not used.

7.2 PRESTO CAN Reconstruction 49

PRESTO CAN is performed, figure 7.4. This comparison is performed after thecorrections. Without these corrections, the images would look completely different.The PRESTO CAN k-space is re-gridded to make the comparison possible and thecentre profiles are displayed for each direction. Since the phase offset changes fromexamination to examination, a correction is performed before displaying the phaseimages.

The images in figure 7.5 show the magnitude of the k-space for PRESTO CAN(a,c,e) and PRESTO (b,d,f) after correction. These profiles are the same for thephase images, except from displaying the absolute value instead of the phase.

The final images are slices from the reconstructed image volume from PRESTO,figure 7.6, and PRESTO CAN, figure 7.7. Three image slices for each of the viewingdirection X, Y and Z are displayed. The slices are chosen to be comparable betweenthe two sequences. Note the rather good agreement between the correspondingimage slices.

50 Results

Figure 7.4. Comparison between the phase of the k-spaces (PRESTO CAN (a,c,e) andPRESTO (b,d,f)). The different sampling techniques and reconstruction makes the sizesof the k-space differ. The profiles have been chosen to be comparable. The PRESTO CANk-space has been re-gridded according to the reconstruction algorithm.

7.2 PRESTO CAN Reconstruction 51

Figure 7.5. Comparison between the magnitude of the k-spaces (PRESTO CAN (a,c,e)and PRESTO (b,d,f)) after correction. The different sampling techniques and recon-struction make the sizes of the k-space to differ. The profiles have been chosen to becomparable. The PRESTO CAN k-space has been re-gridded according to the recon-struction algorithm. These profiles are the same as the phase images in figure 7.4.

52 Results

Figure 7.6. Reconstructed images from the PRESTO sequence used for evaluation ofthe reconstructed images from the PRESTO CAN sequence, figure 7.7.

7.2 PRESTO CAN Reconstruction 53

Figure 7.7. Reconstructed images from the PRESTO CAN sequence. The image sliceshas been chosen to be comparable to the images from the PRESTO sequence, figure 7.6.

Chapter 8

Discussion

The test result from the comparison between scanner reconstruction and our re-construction of a PRESTO sequence, figure 7.1, shows that no extra artefacts canbe seen in our reconstructed image slice (a) compared to a scanner image slice(b). Neither any distortion nor disturbing changes in intensity that cannot bederivable from image enhancement is to be found. Looking at the more compleximages in figure 7.2, the scanner image enhancement is more obvious. Still, detailsand changes of structure can be seen in all of the images and no severe artefactscan be found. There are differences, however, and our reconstructed images do nothave the same quality as the scanner images. Nevertheless, the result establishesthat our reconstruction algorithm works and can be used as a base for developingthe PRESTO CAN reconstruction algorithm and as reference.

The implemented correction improves the image quality and removes someartefacts, figure 7.3, without distorting the image or making too large adjustmenton the raw-data. Before these corrections can be confirmed as completely correct,further examinations with better scanner adjustments have to be performed. Thisis to eliminate as many artefacts as possible before reconstruction. Use of multishotwill decrease the phase drift since multi-excitation is used. Using multishot willtherefore presumably clarify if the phase correction is needed or if it has to beadjusted to perform better.

Different sizes of the two k-spaces makes the comparison of phase, figure 7.4,and magnitude, figure 7.5, more difficult. However, basic shapes and positions canbe distinguished during comparison. This implies that the developed sequenceworks.

In the final figures the slices through the reconstructed volume of the sphericalphantom is shown from all directions. The same good results as for PRESTOimages are not received. In figure 7.7, images (a-f) appears to be better in theY direction than in the X and Z direction. Images (g,h,i) are all worse thanthe corresponding images in figure 7.6. These directions coincide with the di-rections collected during the rotation and could imply that collection of data ismore sensitive to artefacts during rotation that regular sampling pattern and morecorrections have to be implemented.

55

56 Discussion

Even though the spherical phantom is clearly visible in the reconstructed vol-ume, the reconstruction are far from being acceptable and more development ofthe basic reconstruction has to be done before functionality that improves theacquisition speed can be added. Since the re-gridded PRESTO CAN k-space haslot of resemblance with the PRESTO k-space the conclusion is made that mostwork effort has to be on the corrections in the reconstruction algorithm and notthe sequence itself.

It is too early to predict the fully potential of PRESTO CAN but simula-tions and theoretical discussions implies that PRESTO CAN has the possibility tobecome a useful sequence in fMRI examinations. The minor changes needed to de-velop the PRESTO sequence into the PRESTO CAN sequence is one of the majoradvantages for PRESTO CAN. This gives a stable ground for PRESTO CAN andpossibility to use functionality as keyhole, SENSE etc. without any considerablefurther sequence development. Therefore, we recommend further investigation anddevelopment of the PRESTO CAN sequence.

8.1 Further Development

There are many ideas how to continue to develop the PRESTO CAN sequenceand the reconstruction. In this section, the developing steps closest in time arelisted. In my opinion, these should be ticked of before further improvements aredone. The list below is sorted after priority, beginning with the most important.

• Implement the PRESTO CAN sequence in the new scanner software release.The present patch is not compatible with the new release and therefore nonew data can be collected until this is done.

• Implement multishot handling in the reconstruction algorithm and collectnew data using multishot. This decreases artefacts due to long readout timeand will make further phase correction development possible.

• Add artefacts to the simulation to explore the artefacts specific to rotation.

• Make memory handling more efficient in the reconstruction algorithm. Datasets that are more realistic will increase the size of the exported data.

• Use more detailed phantom to verify the orientation on the reconstructedimages. This will also give a hint of the needed number of profiles to getacceptable image quality.

When the basic reconstruction works, functionality as keyhole, SENSE, interpo-lation of data in time-dimension etc. will improve the acquisition speed. A finalidea is to import the data, after correction and re-gridding, into the scanner andlet the scanner complete the reconstruction, i.e. image enhancement.

Chapter 9

Conclusions

We have implemented a novel scanning trajectory primarily developed for fMRIexaminations. The new sequence, PRESTO CAN, has been applied on a Philips1.5T Achieva scanner. A corresponding reconstruction algorithm has been de-veloped and implemented in Matlab. Our reconstructed image volume has beencompared to an image volume reconstructed from the PRESTO sequence.

The re-gridding from a radial grid onto a Cartesian grid has been successfullyperformed with a Kaiser-Bessel window. This function has been verified by bothsimulations and in vitro examination. During the development of the reconstruc-tion algorithm, most effort has been on correction of the k-space data.

The comparisons of the k-space and of the reconstructed image volume showthat we are almost there. So far, the result from the PRESTO reconstruction isbetter then the result from the PRESTO CAN reconstruction. Nevertheless, theresult shows that we are on track and we recommend further development.

57

Bibliography

[1] Matt A. Bernstein, Kevin F. King, and Xiaohong Joe Zhou. Handbook ofMRI Pulse Sequences. Elsevier Inc., 2004. ISBN 0-12-092861-2.

[2] Ronald N. Bracewell. The Fourier Transform and its Applications. McGraw-Hill, second edition, 1986. ISBN 0-07-Y66454-4.

[3] Center for Medical Image Science and Visualization.URL: www.cmiv.liu.se.

[4] Joseph P. Hornak. The Basics of MRI. A hypertext book on magnetic reso-nance imaging, 1996.URL: www.cis.rit.edu/htbooks/mri/.

[5] Scott A. Huetell, Allen W. Song, and Gregory McCarthy. Functional MagneticResonance Imaging. Sinauer Associates, Inc, 2004. ISBN 0-87893-288-7.

[6] Guoying Liu, Geoffrey Sobering, Jeff Duyn, and Chrit T. W. Moonen. Afunctional mri technique combining principles of echo-shifting with a train ofobservations (presto). Journal of Magnetic Resonance in Medicine, 30:764–768, 1998.

[7] Stig Ljunggren. A simple graphical representation of fourier-based imagingmethods. Journal of Magnetic Resonance, 54:338–343, 1983.

[8] Donald G. Mitchell and Mark S. Cohen. MRI Principles. Elsevier Inc., secondedition, 2004. ISBN 0-7216-0024-7.

[9] J. D. O’Sullivan. A fast sinc function gridding algorithm for fourier inversionin computer tomography. IEEE Transaction on Medical Imaging, MI-4:200–207, 1985.

[10] John M. Pauly. Non-Cartesian Reconstruction. Lecture notes, 2007.URL: http://www.stanford.edu/class/ee369c/notes/non_cart_rec_07.pdf.

[11] Peter A. Rinck. Magnetic Resonance in Medicine. ABW WissenschaftsverlagGmbH, fifth edition, 2003. ISBN 3-936072-12-4.

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60 Bibliography

[13] Hermann Schomberg and Jan Timmer. The gridding method for image recon-struction by fourier transformation. IEEE Transaction on Medical Imaging,14:596–607, 1995.

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Appendix A

User Parameters

The parameter settings used during evaluation of the PRESTO CAN sequenceduring this thesis. The order of parameters in the list below is based on thesimulators Scan List user interface. When working with the scanner, the userinterface look a little bit different, but all the parameters are to be found. Therecan also be some differences between product versions.

1. Select Anatomy → Head_CNS → Functional → fMRI_PRESTO2. Set FOV AP and FOV RL to 2403. Set Voxel size AP, RL and FH to 3.0 mm4. Set number of slices to 325. The matrix size should be 806. Set TR to shortest7. Set TE to short8. Set EPI shot mode to single-shot9. Set fMRI echo stabilization to no10. Set number of dynamic scans to quite a few when testing11. Set Enable radial volume to yes12. Set the rotation factor to 1.

At scanner/examination:13. Chose to export the data before starting the actual scan.14. Turn off scanner reconstruction.15. Chose Quadrature coil (Optional)16. Set Contrast Enhancement to T1 (Optional)

61

Upphovsrätt

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Upphovsmannens ideella rätt innefattar rätt att bli nämnd som upphovsmani den omfattning som god sed kräver vid användning av dokumentet på ovan be-skrivna sätt samt skydd mot att dokumentet ändras eller presenteras i sådan formeller i sådant sammanhang som är kränkande för upphovsmannens litterära ellerkonstnärliga anseende eller egenart.

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c© Per Thyr