UNIVERSITÁ DIPISA

DOTTORATO DI RICERCA ININGEGNERIA DELL’INFORMAZIONE

### N

### OVEL METHODS FOR ACCELERATED DYNAMIC

### MRI

DOCTORALTHESIS

Author

Giuseppe Valvano

Tutor (s)

Prof. Luigi Landini Eng. Nicola Martini, PhD

Supervisor at ETH - IBT

Prof. Dr. Sebastian Kozerke

Reviewer (s)

Prof. Michela Tosetti Prof. Anna Maria Bianchi

The Coordinator of the PhD Program

Prof. Marco Luise

Pisa, September 2016 XXIX

**Acknowledgements**

### T

HIS DISSERTATION represents the summary of three years of research experi-ence. During these years I’ve been working in collaboration with several in-stitutes: University of Pisa, Fondazione Toscana Gabriele Monasterio and the Institute for Biomedical Engineering (University and ETH Zurich). Given these col-laborations, I met several people, that contributed in this work and in my development. Acknowledgements to these people are not only needed, but also a pleasure for me.Firstly I would like to thank Prof. Luigi Landini for giving me the opportunity to start this Ph.D. project and for encouraging my research. I’m grateful for your support and for trusting me in choosing my research topic and pursuing my objectives. A special thank to my tutor and friend Nicola Martini, that taught me almost everything about MRI. Thank you for your patience and for your help during my experiments. Thank you for the many passionate chats we had about MRI (although sometimes they ruined lunch to other colleagues!). I wish to thank Prof. Dr. Sebastian Kozerke for letting me spend some time with his amazing group in Zurich and for serving in my Ph.D. committee. You have been an excellent mentor for me. Thank you or your support and for your thorough comments about my work. I wish to thank Dr. Dante Chiappino for allowing me to use the MRI scanner in Massa, Maria Filomena Santarelli and Vincenzo Positano for their support during my first experiences in Pisa. I would like to thank also the two referees of this thesis, Prof. Michela Tosetti and Prof. Annamaria Bianchi, for their comments and thorough review of this work.

Besides the people that collaborated the most with me, I would like to thank those with whom I’ve been sharing great part of the daily routine. I would like to thank all the guys I’ve been working with in Massa: Daniele, Alessandro and Gianmarco. Thank you for all the interesting chats and laughs we had. Thank also to all the people I’ve been working with in in Pisa for many chat and nice days spent together during my first year: Giulio, Nicola, Valentina and Maria Sole. Thank to the amazing guys that worked with me during my period in Zurich: Adrian, Claudio and Christian.

Last but not least, I’d like to thank the people that were closer to me during these years: my family, friends and my lovely partner Mina.

**Ringraziamenti**

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UESTA TESI DI DOTTORATOcostituisce la sintesi di tre anni di ricerca. In questianni ho collaborato con diversi istituti: l’Università di Pisa, la Fondazione To-scana Gabriele Monasterio e l’Institute for Biomedical Engineering (Università ed ETH di Zurigo). Per via di queste collaborazioni ho incontrato diverse persone che hanno contribuito al mio lavoro e alla mia maturazione. Dei ringraziamenti a queste persone non sono solo doverosi, ma costituiscono anche un grande piacere per me.

Per iniziare vorrei ringraziare il Prof. Luigi Landini per avermi dato l’opportunità di intraprendere il percorso di Dottorato e per avermi sempre incoraggiato durante questi anni. Sono grato per il suo supporto e per avermi permesso di scegliere questo interes-sante tema di ricerca e raggiungere i miei obbiettivi. Un ringraziamento speciale al mio tutor ed amico Nicola Martini, che mi ha insegnato quasi tutto ciò che conosco riguardo la MRI. Grazie per la tua pazienza e per il tuo aiuto durante i miei esperimenti. Grazie per le molte chiacchierate appassionate sulla MRI (sebbene a volte abbiamo rischiato di annoiare i colleghi durante il pranzo!). Desidero ringraziare il Prof. Dr. Sebastian Ko-zerke per avermi permesso di trascorrere del tempo nel suo fantastico gruppo di ricerca all’ETH di Zurigo, e per aver preso parte alla commissione di Dottorato. Lei è stato un eccellente mentore per me. Grazie per il suo supporto e per i suoi rigorosi commenti sul mio lavoro. Vorrei ringraziare il Dr. Dante Chiappino per avermi permesso di usare lo scanner MRI per i miei esperimenti a Massa, Maria Filomena Santarelli e Vincenzo Positano per il loro supporto durante le mie prime esperienze a Pisa. Ringrazio anche i due revisori di questa tesi, Prof.ssa Michela Tosetti e Prof.ssa Annamaria Bianchi, per i loro commenti e la loro rigorosa revisione di questo lavoro.

Oltre alle persone con cui ho collaborato principalmente, vorrei ringraziare coloro con cui ho trascorso gran parte della routine giornaliera. Ringrazio i ragazzi con cui ho lavorato a Massa: Daniele, Alessandro e Gianmarco. Grazie per tutte le chiacchierate interessanti e le molte risate. Ringrazio anche le persone con cui ho lavorato a Pisa, durante il mio primo anno di Dottorato: Giulio, Nicola, Valentina e Maria Sole. I miei ringraziamenti vanno anche ai ragazzi che hanno lavorato con me a Zurigo: Adrian, Claudio e Christian. Infine, vorrei ringraziare le persone che mi sono state più vicino durante questi anni: la mia famiglia, i miei amici e la mia compagna Mina.

**Summary**

### A

CCELERATEDMAGNETICRESONANCEIMAGING(MRI) is among the mostim-portant topics in technological research of the MRI scientific community. Fast acquisitions are in fact needed to improve the image quality and the patient comfort during the MRI exam.

A MRI exam is composed by the acquisition of several images able to represent a given volume of interest with several image contrast. This gives the possibility to characterize the biological tissues by increasing the amount of information available for the diagnosis. The acquisition time of each image depends mainly on its dimensions (in terms of acquisition volume and spatial resolution), and it may be long for some applications. In dynamic MRI, the need for cardiac and respiratory synchronization further increases the acquisition time. For this reason some advanced techniques, like 4D Flow MRI, have not been widely used in the clinical routine.

To reduce the scan time in MRI it is possible to use several techniques as well as their combination. Non-cartesian acquisition trajectories reduce the scan time by sampling more efficiently the k-space. Parallel Imaging and Compressed Sensing (CS), instead, reduce the acquisition time by undersampling the k-space. Artefacts resulting from the undersampled k-space are then removed exploiting the knowledge of coil sensitivity maps (in parallel MRI) or the compressibility of MRI images (in CS-MRI). During the years, CS and related techniques like Matrix Completion, have gained an increasing interest among the scientific community.

A faithful reconstruction using CS, requires incoherent k-space sampling and a non-linear reconstruction that enforces compressibility of the target image. Therefore, in fast MRI, it is important to design incoherent acquisition strategies and reconstruction methods that can improve the reconstruction accuracy.

In this doctoral thesis I describe novel acquisitions strategies and reconstruction methods for accelerated dynamic MRI. The main contributions of this thesis are:

• A novel non-cartesian trajectory based on a stack of variable density spirals is proposed for accelerated 3D CS-MRI and 2D dynamic CS-MRI.

• A novel reconstruction method, based on Low Rank plus Sparse Matrix Comple-tion, is proposed for accelerated 4D Flow MRI.

The thesis is organized as following:

• Part I describes the application of a new non-cartesian trajectory based on variable density spirals to accelerated MRI:

– Chapter 1 introduces the basic principles of MRI and describes the state-of-the-art techniques for accelerated MRI.

– Chapter 2 describes the proposed Variable Density Randomized Stack of Spi-rals trajectory for accelerated 3D CS-MRI.

– Chapter 3 discusses the application of incoherent Variable Density Spirals to dynamic 2D CS-MRI.

• Part II describes the application of acceleration techniques to 4D Flow MRI: – Chapter 4 introduces the principles of 4D Flow MRI, an advanced technique

that provides non-invasive quantitative measurements of the temporal evolu-tion of blood flow in-vivo. A summary of the state-of-the-art acceleraevolu-tion techniques for 4D Flow MRI is also provided at the end of the chapter. – Chapter 5 introduces the proposed reconstruction method for accelerated 4D

Flow MRI. The proposed method was demonstrated to accurately estimate velocity field for 8× accelerated 4D Flow MRI acquisitions.

**Sommario**

### L’

ACCELERAZIONE DELL’ IMAGING ARISONANZAMAGNETICA (MRI) è tra i topic di ricerca tecnologica di maggior interesse della comunità scientifica nel campo della MRI. La riduzione del tempo di scansione è infatti necessaria per migliorare la qualità delle immagini ed il comfort del paziente.Durante un esame MRI vengono acquisite diverse immagini aventi lo scopo di rap-presentare un certo volume di interesse con diversi contrasti tra tessuti. Questa poss-sibiltà permette di caratterizzare i tessuti biologici aumentando la quantità di informa-zioni disponibili a supporto della diagnosi. Il tempo di acquisizione di ogni immagine dipende principalmente dalle sue dimensioni (dimensioni del volume di acquisizione e risoluzione spaziale) e può essere abbastanza lungo per certe applicazioni. Nel ca-so della dynamic MRI, la necessità di acquisire le immagini in sincronia con il battito cardiaco e con la respirazione del paziente, può ulteriormente incrementare il tempo di scansione. Tempi di scansione non compatibili con la routine clinica sono responsabi-li del responsabi-limitato utiresponsabi-lizzo di alcune sequenze di acquisizione avanzate, come la 4D Flow MRI.

La riduzione del tempo di scansione è possible utilizzando diverse tecniche. Per esempio le traiettorie di acquisizione non cartesiane sono in grado di ridurre il tempo di scansione attraversando in maniera più efficiente il k-spazio. Al contrario, il Parallel Imaging ed il Compressed Sensing (CS) sono in grado di ridurre il tempo di scansione effettuando un sottocampionamento del k-spazio. In questo caso, gli artefatti da alia-sing risultanti dal sottocampionamento vengono rimossi sfruttando la conoscenza delle mappe di sensività di campo delle bobine (nel caso del parallel imaging) o la com-pressibilità delle immagini MRI (nel caso del CS-MRI). Nel corso degli anni, il CS e tecniche analoghe, come il Matrix Completion, hanno suscitato un grande interesse nella comunità scientifica.

Per ottenere una buona ricostruzione CS è necessario acquisire i dati nel k-spazio in maniera incoerente e ricostruire l’immagine usando metodi non lineari in grado di sfruttare la compressibilità delle immagini. Per questo motivo è importante progettare strategie di campionamento incoerenti e metodi di ricostruzione efficaci, in modo da migliorare l’accuratezza della ricostruzione.

In questa tesi di dottorato descrivo nuove strategie di acquisizione e metodi di rico-struzione per la dynamic MRI accelerata. I principali contributi di questa tesi sono:

• una nuova traiettoria di acquisizione non cartesiana, basata su stack di spirali a densità variabile, per 3D CS-MRI e 2D dynamic CS-MRI.

• un nuovo metodo di ricostruzione, basato su Low Rank plus Sparse Matrix Com-pletion, per la 4D Flow MRI accelerata.

La tesi è strutturata come segue:

• La parte I descrive l’applicazione di una nuova traiettoria non cartesiana di cam-pionamento al MRI accelerato:

– Il Capitolo 1 introduce i principi fisici della MRI e descrive le tecniche dello stato dell’arte per l’imaging accelerato.

– Il Capitolo 2 descrive la traiettoria proposta (Variable Density Randomized Stack of Spirals) per 3D MRI accelerata.

– Il Capitolo 3 descrive l’applicazione di spirali incoerenti a densità variabile al 2D dynamic MRI.

• La parte II della tesi descrive l’applicazione delle tecniche di accelerazione alla 4D Flow MRI:

– Il Capitolo 4 introduce i principi della 4D Flow MRI, una tecnica avanzata che permette la misura non invasiva dell’evoluzione temporale dei flussi san-guigni in-vivo. Alla fine del capitolo vengono descritte in dettaglio lo stato dell’arte riguardo le tecniche di accelerazione della 4D Flow MRI.

– Il Capitolo 5 descrive il metodo proposto per la 4D Flow MRI accelerata. Viene dimostrata l’accuratezza nella stima dei campi di velocità ottenuti con il metodo proposto per acquisizioni ottenute in un ottavo del tempo di scansione nominale.

**List of Publications**

**International Journals**

1. Giovannetti, G., Pingitore, A., Positano, V., De Marchi, D., Valvano, G., Gibiino, F., Aquaro G.D., Lombardi M., Landini L., Santarelli, M. F. (April, 2014). Im-proving sodium Magnetic Resonance in humans by design of a dedicated 23Na surface coil. Measurement, (Vol. 50, pp. 285-292).

2. Giovannetti, G., Valvano, G., Virgili, G., Giannoni, M., Flori, A., Frijia, F., De Marchi, D., Hartwig, V., Landini, L., Aquaro,G. D., and Pingitore, A., (November, 2015) Design and simulation of a dual-tuned 1H/23Na birdcage coil for MRS studies in human calf, Applied Magnic Resonance. (Vol. 46, Issue 11, pp. 1221-1238).

3. Santarelli, M.F., Positano, V., Martini, N., Valvano, G., Landini, L. (January, 2016). Technological innovations in Magnetic Resonance for early detection of cardiovascular diseases, Current Pharmaceutical Design. (Vol. 22, Number 1, pp. 77-89).

4. Valvano, G., Martini, N., Landini, L. and Santarelli, M. F. (July, 2016), Variable density randomized stack of spirals (VDR-SoS) for compressive sensing MRI. Magnetic Resonance in Medicine. (Vol. 76, Issue 1, pp.59-69).

5. Valvano, G., Martini, N., Huber, A., Santelli, C., Binter, C., Chiappino, D., Lan-dini, L., and Kozerke, S. (October ,2016), Accelerating 4D Flow MRI by Exploit-ing Low-rank Matrix Structure and Hadamard Sparsity. Magnetic Resonance in Medicine. (doi: 10.1002/mrm.26508)

**International Conferences/Workshops with Peer Review**

1. Valvano, G., Martini, N., Chiappino, D., Landini, L., Santarelli, M.F. (May, 2015) Random Delayed Spirals for Compressive Sensing Cine MRI. Proceedings 23rd meeting International Society Magnetic Resonance in Medicine(pp. 3633).

2. Valvano, G., Martini, N., Chiappino, D., Santarelli, M.F., Landini, L. (August, 2015) A novel 3D Cartesian random sampling strategy for Compressive Sensing Magnetic Resonance Imaging. Proceedings IEEE Engineering in Medicine and Biology Society, Milan 2015. (pp. 7502).

**List of Abbreviations**

Symbols

2D Two Dimensional.

3D Three Dimensional.

A

ADMM Alternating Direction Method of Multipli-ers.

C

CS Compressed Sensing.

F

FCSA Fast Composed Splitting Algorithm.

FISTA Fast Iterative Soft Thresholding Algorithm. FISTA LS Fast Iterative Soft Thresholding Algorithm

for L + S reconstruction.

FOV Field of View.

I

ISTA Iterative Soft Thresholding Algorithm. ISTA LS Iterative Soft Thresholding Algorithm for

L + S reconstruction. M

MRI Magnetic Resonance Imaging.

mSSIM mean Structure Similarity Index. N

NMR Nuclear Magnetic Resonance.

P

PC Phase Contrast.

PCA Principal Components Analysis.

PI Parallel Imaging.

PSF Point Spread Function.

R

R-SoS Randomized Stack of Spiral.

RF Radio Frequency.

S

SENSE SENSitivity Encoding.

SNR Signal to Noise Ratio.

SoS Stack of Spirals.

SoVDS Stack of Variable Density Spirals.

SVD Singular Values Decomposition.

T

TPSF Transform Point Spread Function.

tR-SoS temporally Randomized Stack of Spirals. V

VDR-SoS Variable Density Randomized Stack of Spi-rals.

**Contents**

**Acknowledgements**I

**Summary**V

**List of Publications**IX

**List of Abbreviations**XI I Accelerated MRI 1

**1 Accelerated Magnetic Resonance Imaging** 3

1.1 Principles of Magnetic Resonance Imaging . . . 4

1.1.1 Magnetic Resonance Physics . . . 4

1.1.2 Magnetic Resonance Imaging Hardware . . . 5

1.1.3 Signal equation and image acquisition . . . 5

1.2 Accelerated MRI . . . 8

1.2.1 Non-Cartesian MRI . . . 8

1.2.2 Parallel Imaging . . . 9

1.2.3 Compressed Sensing MRI . . . 10

**2 Variable Density Randomized Stack of Spirals (VDR-SoS) for Compressed **
**Sens-ing MRI** 15
2.1 Introduction . . . 16

2.2 Trajectory Design . . . 17

2.2.1 Base 2D Trajectory: Variable Density Spiral . . . 17

2.2.2 Randomized Stack of Spirals: Rotations and Random Delays . . 17

2.2.3 VDR-SoS . . . 18

2.3 Methods . . . 20

2.3.1 Compressed Sensing reconstructions . . . 20

2.3.2 Comparison with SoS trajectory . . . 20

2.4 Results . . . 24

2.4.1 Comparison with SoS trajectory . . . 24

2.4.2 Comparison with 3D spirals . . . 27

2.5 Discussion . . . 28

2.6 Conclusions . . . 30

**3 Random delayed spirals in Dynamic MRI** 31
3.1 Random delayed spirals in Dynamic MRI . . . 32

3.1.1 Coherence estimation . . . 32

3.1.2 Coherence reduction . . . 32

3.2 Retrospective undersampling . . . 34

3.3 Results . . . 35

3.4 Discussion and conclusions . . . 35

II Accelerated 4D Flow MRI 37
**4 4D Flow MRI** 39
4.1 Principles of 4D Flow MRI . . . 39

4.1.1 Phase Contrast MRI . . . 39

4.1.2 4D Flow MRI . . . 41

4.2 Accelerated 4D Flow MRI . . . 43

4.2.1 Parallel Imaging and Compressed Sensing . . . 43

4.2.2 Spiral trajectories . . . 44

4.2.3 Temporal acceleration techniques . . . 44

**5 Accelerating 4D Flow MRI by Exploiting Low-Rank Matrix Structure and Hadamard**
**Sparsity** 47
5.1 Introduction . . . 48

5.2 Theory . . . 48

5.2.1 MRI reconstruction using Matrix Completion . . . 48

5.2.2 L + S reconstruction for 4D Flow MRI . . . 50

5.3 Methods . . . 53
5.3.1 Retrospective undersampling . . . 53
5.3.2 Prospective measurements . . . 54
5.4 Results . . . 56
5.4.1 Retrospective undersampling . . . 56
5.4.2 Prospective measurements . . . 57
5.5 Discussion . . . 60
5.6 Conclusions . . . 65
III Conclusions 66
**Appendices** 70

**A Fast Iterative Soft Thresholding Algorithm for L + S reconstruction** 72

### Part I

### CHAPTER

## 1

**Accelerated Magnetic Resonance Imaging**

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AGNETICResonance Imaging (MRI) is a powerful tool for medical diagnosis.With this technique it is possible to acquire images with a large variety of contrasts, allowing to characterize tissues based on their magnetic properties. One of the main limitations of MRI is its long scan time, which can limit some advanced techniques. For example in some applications, like functional MRI, a trade off between spatial and temporal resolution must be chosen. Other applications, like 4D Flow MRI, have had a limited impact in the clinical routine, since their inherently long scan time has hindered their wide-spreading.

Over the years there has been a great interest, among the scientific community, in developing new sampling strategies and reconstruction algorithms that allow to reduce the acquisition time. The aim of this chapter is to introduce the basic principles of MRI and some state-of-the-art techniques for accelerated MRI. The chapter is organized as follows:

• The basic physical principles of MRI are described in Section 1.1.1. • Section 1.1.2 briefly describes the hardware of a MRI scanner.

• Section 1.1.3 introduces the concepts of k-space and the classical procedure used to create an image in MRI.

• A brief summary of non-cartesian acquisition trajectories for accelerated MRI is provided in Section 1.2.1.

• The use of Parallel Imaging acceleration techniques is briefly described in Section 1.2.2.

**1.1**

**Principles of Magnetic Resonance Imaging**

Despite its quantum nature, Nuclear Magnetic Resonance (NMR) can be well described at the macroscopic scale by classical theory [1]. This description is based on some quantum-mechanics properties, and is well suited for the purposes of image recon-struction. For this reason in the following sections I will refer to the classical theory. 1.1.1 Magnetic Resonance Physics

The NMR properties of the matter depend upon the spin value of its particles. Nuclei like1H,13C and23Na, possess a net spin value that give them NMR properties. MRI is usually performed on1H nucleus, that is the most abundant nucleus in the human body. A set of spins, in a static magnetic field B0, aligns in the direction to the magnetic

field axis, exhibiting a net magnetization. The magnetization caused by this polariza-tion process can be manipulated to produce the image in MRI.

The polarization equilibrium can be changed if a proper electromagnetic field is applied to the set of spins. Due to quantum phenomena it is necessary to play an electromagnetic wave with a proper frequency (the Larmor frequency):

fL=

γ 2πB0

where γ is the gyromagnetic ratio, a NMR propriety of the element that we are trying
to excite (in the case of1H, _{2π}γ ≈ 42.576M Hz/T ). The typical frequency values of the
electromagnetic pulse applied to excite these spin is the range of the radio frequencies
(RF), depending on the B0 field strength and on the nucleus under examination. In the

most common case of1_{H imaging, on a 3T MRI scanner f}

L≈ 127.7M Hz. Assuming

that the B0 field is parallel to the physical z axis, a RF pulse (whose frequency is equal

to fL) can tilt the net magnetization in the xy plane. The angle between the z axis and

the net magnetization is called flip angle α. The RF pulse causes the appearance of a transversal magnetization Mxy and a longitudinal magnetization Mz:

Mz = M0cos(α)

Mxy = M0sin(α)

After the end of the RF pulse the nuclei release the acquired energy, returning to their initial equilibrium state. Interactions between spins cause a dephasing of the magne-tization, resulting in a decay of the net transverse magnetization Mxy. The decay is

exponential, with a time constant equal to the spin-spin relaxation time T2:

Mxy(t) = Mxy(0)e − t

T2

The longitudinal magnetization Mz returns to its equilibrium position with a time

constant called spin-lattice relaxation time T1:

Mz(t) = Mz(0)(1 − e − t

T1_{)}

The temporal evolution of the net magnetization is well described by the Bloch equa-tions[1]: dM dt = M × γB − Mxx + Mˆ yyˆ T2 −(Mz− M0)ˆz T1

where M = [Mx, My, Mz]T and B = [Bx, By, Bz]T are respectively the

magneti-zation vector and the magnetic field vector in the physical xyz axes. The values T1and

T2 depend on the physical environment where the spins are present (i.e. the biological

tissues).

The magnetization vector precesses around the z axis at the frequency fL, creating

a time varying magnetic flux. A receiver RF coil, tuned at the frequency fL, is able to

measure a changing in the voltage caused by the time varying magnetic flux (due to the Faraday’s law). This signal is measured in MRI scanners and used to produce images. 1.1.2 Magnetic Resonance Imaging Hardware

The hardware of MRI scanners is able to generate and manipulate the magnetization vector in order to create an image. Different components are present:

• The source of the static magnetic field B0.

The static field B0 is necessary to polarize the spins. A magnet is used to generate

a homogeneous field, whose strength ranges from 0.3 to 7T. Depending on the field strength it is possible to use a permanent magnet, an electromagnet or a superconducting magnet.

• The RF transmit coil.

A RF coil, tuned to the Larmor frequency of the nucleus of interest, is used to excite the spins with a transverse electromagnetic field B1.

• The gradient coils.

A set of three gradient coils is embedded in the scanner. Their aim is to generate spatially and temporally varying longitudinal magnetic fields, that are used to en-code spatial informations in the phase of the acquired signal (the image generation will be described in section 1.1.3). The magnetic field, in presence of the gradient coils, varies linearly with the position: B(t, x, y, z) = |B0| + Gx(t)x + Gy(t)y +

Gz(t)z.

• The RF receiver coils.

A set of receiver coils is used to receive the signal produced by the precession of the magnetization.

1.1.3 Signal equation and image acquisition

Image acquisition is performed exciting only a certain region of interest. Assuming a two dimensional (2D) axial acquisition, only a slice in the xy plane should be excited. For this reason the Gz gradient is activated during the excitation pulse, changing the

Larmor frequency of the spins:

fL(z) =

γ

2π(|B0| + Gzz)

In this way it is possible to excite only a subset of spins by changing the bandwidth of the RF excitation pulse. The frequency content of the excitation defines the shape of the excited slice along the z axis. A spatial encoding is necessary also in the xy plane. For this reason, after the end of the excitation, the imaging sequence is modified to include

RF Gz Gy Gx echo ky kx k-Space MRI pulse sequence

Figure 1.1: Example of gradient echo sequence and corresponding k-space filling. A trapezoidal Gz waveform is played out at the same time of the excitation pulse to select a slice along thez axis. Then a trapezoidalGywaveform is played out in order to change thekyvalue. Finally a readout gradient Gxis played out to create an echo that is acquired, filling thekyrow of the k-space matrix.

Gx and Gy gradient waveforms. In presence of activation of the gradients, the spins

accrue a phase that depends on their position r = [x, y]T: φ(t, r) = γ Z t 0 (Gx(τ )x + Gy(τ )y)dτ = γ Z t 0 G(τ ) · rdτ

where the time reference t = 0 is positioned at the center of the RF excitation pulse and G(τ ) = [Gx(τ ), Gy(τ )]T. Denoting: k(t) = γ 2π Z t 0 G(τ )dτ (1.1)

where k(t) = [kx(t), ky(t)]T, it is possible to write the equation that describes the

signal acquired by a homogeneous receiver coil: s(t) =

Z

R

Mxy(r)e−i2πk(t)·rdr (1.2)

The MRI signal equation (1.2) states that the acquired signal is the Fourier transform of the transverse magnetization, evaluated at the spatial frequency location k(t). Thus the signal acquired in MRI corresponds to a certain profile in the so called k-space.

The temporal sequence of the gradient waveforms and RF pulses is called pulse sequence. Several pulse sequences have been developed, allowing to generate different image contrasts. An example of cartesian gradient echo sequence is shown in Fig. 1.1. The ky and kxlocations are encoded with two different strategies. After the excitation

of one slice, a trapezoidal Gy waveform is played out to reach a certain ky location.

This procedure is called phase encoding. Then a trapezoidal Gx waveform is used to

encode the kx location, generating an echo signal that is acquired by the RF receiver

coil. This is referred to as frequency encoding.

This pulse sequence is repeated to fill the k-space matrix, by changing only the Gy

waveform (and as a consequence the value ky). The number of repetitions depends

FOVx FOV y Δx Δy 1/FOVx 1/FOV y 1/Δx 1/ Δ y

Image Domain k-Space

Figure 1.2: Relationships between k-space dimensions and geometrical dimensions. Green dots repre-sent the discretization grid used in the k-space and in the image domain.

and the spatial resolution. The relationships between the k-space dimensions and these geometrical dimensions are shown in Fig.1.2. A not appropriate coverage of the k-space can lead to image smoothing or image aliasing, as shown in Fig.1.3.

In the case of a three dimensional (3D) cartesian MRI acquisition, two phase encod-ing directions (ky and kz) and a frequency encoding direction (kx) are present.

Image D

omain

k-S

pac

e

Full coverage Low Resolution (ky) Equispaced

Undersampling (ky)

Figure 1.3: Effect of a not appropriate k-space coverage. Scan time reduction can be achieved by ac-quiring only a subset of points in the k-space, leading to smoothing or aliasing. Green dots represent the points acquired in the k-space.

**1.2**

**Accelerated MRI**

Cartesian MRI requires the repetition of the pulse sequence several times, in order to fill appropriately the k-space matrix. This procedure can be time consuming, especially for high resolution 3D acquisitions or dynamic acquisitions (cine MRI).

A long scan time can hinder the application of some recently introduced advanced techniques. Furthermore, longer acquisitions are more prone to movement artifacts, that can result in a worse image quality. Last but not least, shorter scans increase the patient comfort.

For these reasons, over the years, there have been great efforts from the scientific community to accelerate MRI acquisitions. In this section I will introduce some com-monly used acceleration techniques.

1.2.1 Non-Cartesian MRI

MRI offers a unique capability to generate acquisition trajectories in the k-space. In fact, as stated in Eq. 1.1, the k-space locations are proportional to the temporal integral of the gradient waveforms. Thus it is possible to acquire non-Cartesian trajectories by appropriately modifying the Gx, Gy and Gz waveforms. Several trajectories have been

proposed, both for 2D MRI [1–5] and for 3D MRI [6–10].

Radial trajectories

Radial trajectories[1] and their variants [4] are a simple extension of Cartesian MRI. They consist on several radial spokes acquired with trapezoidal gradient waveforms. Each spoke is rotated by a certain angle, in order to appropriately cover the k-space. Their extension to 3D imaging can be obtained by using a stack of stars trajectory or fully 3D radials trajectories, like VIPR [7].

Spiral trajectories

Scan time reduction can be effectively achieved using spiral trajectories [2, 3]. They exploit a long readout window to cover a greater portion of the k-space, with respect to Cartesian or radial MRI. The basic 2D spiral trajectory can be easily parametrized in polar coordinates: |k| = αθ (1.3) kx ky kx ky kx ky kz kx ky kz a) b) c) d)

Figure 1.4: Examples of radial trajectories. a) 2D radial trajectory. b) 2D PROPELLER trajectory. c) 3D stack of stars trajectory: several radial trajectories are phase encoded along thekzdirection. d) 3D radial trajectory (VIPR).

1 interleave 2 interleaves 3 interleaves
kx kx kx
ky
ky
ky
kx
ky
θ
**|k|**

Figure 1.5: Examples of spiral trajectories.

where |k| is the distance from the center of the k-space and θ is the azimuthal angle. The constant α is tuned to respect the Nyquist theorem and to avoid undersampling:

α = Ns

2πF OV where Nsis the total number of spiral interleaves.

The generation of Gxand Gy waveforms, needed in the case of spirals, is not simple

as in the case of radials. Several algorithms have been proposed in the past to generate such trajectories, respecting hardware constraints on maximum gradient amplitude and maximum slew rate [2, 3, 11–13].

As in the case of radials, it is possible to realize a simple 3D version by using phase encoding along the kz direction [6]. Other 3D trajectories have been proposed in the

past, like stack of cones [6, 14], distributed spirals [10] or FLORET [9]. 1.2.2 Parallel Imaging

The scan time reduction achievable with non-Cartesian MRI is due to a more efficient space sampling. It is also possible to reduce the scan time by undersampling the k-space. This results in aliasing due to the violation of the Nyquist theorem (last column in Fig. 1.3).

Parallel imaging (PI) [15, 16] can be used to remove aliasing artifacts, by exploiting multi coil acquisitions. The signal acquired from different RF receiver coils is

modu-Stack of Spirals Spherically Distributed FLORET Spirals

lated by the coil sensitivity map Sc. Thus each coil acquires a signal scthat is different

from the one acquired from the other coils:

sc(t) =

Z

R

Sc(r)Mxy(r)e−i2πk(t)·rdr (1.4)

The different information content of the signals acquired from different coils is ex-ploited in PI. For example, the SENSE algorithm [17] reconstructs the image by assum-ing the aliased coil images xcas a linear transformation of the original discrete image

x0:

xc = Ecx0+ η

where Ecis the encoding matrix of the coil c (that includes the coil sensitivity weighting

and the aliasing model) and η represents acquisition noise. The aliasing is then removed by solving the following equation:

xu = (EHC−1E)−1EHC−1xa

where E = [E1, E2, . . . , Ec]T, xa = [x1, x2, . . . , xc]T and C is the noise

covari-ance matrix. Several other methods have been proposed for PI MRI [17–21]. A com-plete description of PI methods is beyond the aim of this chapter.

1.2.3 Compressed Sensing MRI

The theory of Compressed Sensing (CS) has been recently introduced [22–25]. It al-lows to recover sparse signals from a small number of measurements. Its application to accelerated MRI has been proposed in [26, 27], and takes advantage of the image sparsity in some transformed domains, like the Wavelet domain. Since its introduc-tion, there has been a great interest in the scientific community in developing new reconstruction algorithms and acquisition strategies able to improve the reconstruction accuracy [26, 28–35].

Mathematically the acquisition is described as a linear operation: y = Ax0+ η

where x0 = [x1, x2, . . . , xN] ∈ CN is the sparse or compressible signal, A ∈ Cm×N

is an acquisition matrix and η represents an additive noise. In the case of CS-MRI x0 represents the original image, whilst the acquisition matrix A corresponds to the

Fourier transform, evaluated on the chosen location of the k-space. This is usually referred to as undersampled Fourier transform.

The aim of CS is to recover a faithful estimate of the original signal, starting from few projections (when m N ). In order to be effective the following constraints on the signal and on the acquisition matrix should be respected:

• sparsity: the signal x0should be sparse or compressible, i.e. characterized by only

few non zero coefficients. More generally, the sparsity of the signal is exploited
in an orthonormal transformation domain Ψ ∈ CN ×N_{, for example the Wavelet}

Wavelet Domain Finite Differences Domain

Figure 1.7: Sparsity of MRI images. Example of Wavelet sparsity and Finite Differences sparsity.

• inchoerence: the acquisition matrix A should be inchoerent. The coherence µ of the matrix can be evaluated as the maximum side-lobe peak of the Point Spread Function (PSF):

P SF = AHA µ = max

i6=j P SF (i, j)

Here AH is the Hermitian transpose of A. A low level of coherence guarantees that aliasing artefacts, that result from the undersampling, are noise-like (Fig. 1.8). If these constraints are respected, the reconstruction of the signal can be performed by solving the following convex optimization problem:

minimize

x kΨxk1 subject to kAx − yk

2

2 < ξ (1.5)

where ξ represents an estimation of the noise level. The two functions k·k1 and k·k2

correspond to the `1 and `2 norms defined as:

kxk1 =
N
X
j=1
|xj|
kxk2 =
N
X
j=1
|xj|2
1_{2}

Equation 1.5 imposes sparsity of the image after transformation in the Ψ domain constraining data consistency. Usually an unconstrained formulation of this problem is used:

minimize

x kAx − yk

2

2+ λkΨxk1 (1.6)

The regularization parameter λ trade off between the data consistency and the spar-sity of the reconstructed image. In the remaining parts of this section I describe some commonly used techniques for CS-MRI.

Equispaced undersampling S ampling pa tt er n A liasing ar tifac ts 1D random undersampling 2D random undersampling

Figure 1.8: Effect of the sampling pattern on the aliasing artefacts. Equispaced undersampling leads to coherent repetitions of the image. Random undersampling reduces the coherence of aliasing arte-facts. Exploiting randomness in more than one dimension can further reduce the artifacts coherence.

Sparsity

MRI images are compressible in several domains. For example it is possible to exploit sparsity in the Wavelet domain, or the finite difference domain (Fig. 1.7).

The compressibility of MRI images can be even higher in the case of multi-dimensional acquisitions, like in the case of dynamic MRI. In fact the correlations between different images can be exploited. Temporal transformations, like the Fourier Transform, can be used to take advantage of temporal redundancy [31,32,35–37]. The basic assumption in this case is that only a small portion of the image contains dynamic information, whilst the remaining part of the image is static. Temporal transformations can be used also in combination with spatial transformations, like the above mentioned Wavelet transform.

Incoherent MRI acquisition

The scan time reduction achieved with CS-MRI is due to the reduction of the number of profiles acquired in the k-space. The respect of the incoherence constraint guaran-tees that the aliasing artefacts due to undersampling are noise-like. In this case it is possible to perform a successful CS reconstruction. A low level of coherence is typi-cally achieved by randomly selecting the phase encoding profiles in the k-space. In 3D MRI the level of coherence can be further reduced by choosing randomly the two phase encoding locations. A common strategy in this case is to choose the k-space locations according to a random Poisson disk distribution [38] (Fig. 1.8). In the case of dynamic MRI acquisition, the sampling mask is usually changed randomly in time. This

strat-ˆ
*x*
*ˆx = arg min*
*x* *Ax* *y*2
2
+ *x*_{1}
*x0*
ˆ
*x*
*y = Ax*_{0}+
*x*0 *C*
*N*
*, A* *Cm N*
*, m << N*
*PSF = A A*H
*z = A y = PSF x0*
H
x
Adjoint Reconstruction
Aliasing
Undersampled
acquisition
Compressed Sensing
Reconstruction

Figure 1.9: Summary of CS-MRI. The starting image is acquired in the k-space with an incoherent sampling pattern. This undersampling causes the appearance of noise-like aliasing artifacts. A faithful estimate of the image is obtained by solving a convex optimization problem that enforces data consistency and sparsity in a linear domainΨ (e.g. Wavelet domain).

egy adds temporal incoherence that results in noise-like aliasing artefacts in temporal transformed domains.

Non-cartesian trajectories have also been proposed for CS-MRI. In fact they gen-erate less coherent aliasing artefacts respect to cartesian MRI [39–41]. In this case random or pseudo-random [42] rotation angles between the acquisition spokes may be used to further reduce the acquisition coherence.

Non linear reconstruction

CS reconstruction is performed by solving the non linear problem described in Equa-tion 1.6. This is a more computaEqua-tionally convenient form of EquaEqua-tion 1.5 for large scale problems like MRI images reconstruction. It takes the form of a `1 regularized

least squares problem, which tries to get a sparse solution for the least squares data consistency problem.

Other solutions have been proposed in the past, exploiting more than one sparsity domain. A common used strategy consists in minimizing also the Total Variation of the reconstructed image:

minimize

x kAx − yk

2

2+ λΨkΨxk1+ λT VT V(x) (1.7)

where the Total Variation is defined as:

T V(x) = N X i,j=1 q xi+1,j − xi,j 2 + xi,j+1− xi,j 2 (1.8) In the case of multi-coil acquisitions the knowledge of the coil sensitivity maps S can be used to further reduce the acquisition time. Assuming the signal model described by Equation 1.4, PI can be used along with CS reconstruction [20, 43, 44]:

minimize x kASx − yk 2 2+ Nreg X i=1 λiRi(x) (1.9)

where Ri(x) represents a suitable sparsity inducing regularization term, λi is the

corresponding regularization parameter and Nreg is the number of regularization terms.

Usually the encoding operator E is used in Eq. 1.9 (E = AS).

Equation 1.9 represents a general formulation for CS-PI-MRI. Several algorithms have been proposed to solve this problem. For example gradient based methods have been proposed, like the Iterative Soft Thresholding Algorithm (ISTA [45]), the Fast It-erative Soft Thresholding Algorithm (FISTA [45, 46]) and the Fast Composed Splitting Algorithm (FCSA [47,48]). Other methods, based on the Alternating Direction Method of Multipliers (ADMM [49, 50]), are also usually used [51, 52].

### CHAPTER

## 2

**Variable Density Randomized Stack of Spirals**

**(VDR-SoS) for Compressed Sensing MRI**

The content of this chapter is described in the paper:

Giuseppe Valvano, Nicola Martini, Luigi Landini, and Maria Filomena Santarelli. Variable density randomized stack of spirals (VDR-SoS) for compressive sensing MRI. Magnetic Resonance in Medicine. 2016; 76:59-69. DOI: 10.1002/mrm.25847.

Copyright (2016) Wiley. Used with permission from Magnetic Resonance in Medicine, Wiley.

### C

OMPRESSED SENSING(CS) MRI has become a state-of-the-art technique inac-celerated MRI. A successful CS reconstruction requires the incoherence of the aliasing artefacts arising from undersampling. A 3D variable density random-ized stack of spirals (VDR-SoS), able to minimize the coherence of aliasing artefacts, is presented hereafter. The method provides a 3D pseudo-random sampling pattern with variable density across all the directions of the k-space. This study demonstrates that it is possible to achieve incoherence performances similar to 3D spiral trajectories with-out requiring 3D gridding during the reconstruction, reducing the overall computational cost of non-cartesian CS-MRI.

The chapter is organized as following:

• The design of the trajectory is covered in Section 2.2. The base 2D trajectory is described in Section 2.2.1, then a simple method to randomize a standard stack of spirals is presented in Section 2.2.2. Finally the proposed VDR-SoS trajectory is

described in section 2.2.3.

• Section 2.3.2 describes the simulation experiments carried on to assess the perfor-mances of the proposed trajectory.

• Section 2.3.3 provides details on the comparison of the proposed trajectory with 3D spirals trajectories.

• Sections 2.4 and 2.5 present and discuss the results of this study.

**2.1**

**Introduction**

In CS-MRI scan time reduction is achieved with a k-space undersampling by generating non coherent aliasing artefacts, which are then removed solving a convex optimization problem, that enforces data consistency and sparsity in a given transformation domain. Non coherent aliasing artefacts are theoretically produced by a random selection of the k-space locations. Since a pure random sampling is not feasible due to hardware constraints, pseudo-random undersampling is performed skipping some phase encoding profiles of the k-space (in Cartesian MRI), or using non-cartesian trajectories. Several methods have been proposed in the past to create a random 2D pattern [26, 53] for 2D MRI, but they require ad hoc algorithms for the calculation of the gradient waveforms [12]. It is desirable that the undersampling be performed with a variable density, i.e. sampling more densely the central region of the k-space [25, 54]. Therefore variable density spirals (VDSs) [8] are good candidates for CS purposes. They have been used in CS-MRI because of their ability to generate aliasing artefacts that are less coherent than those generated by standard spirals [26, 55].

In the case of 3D CS-MRI, the sampling pattern should be ideally variable in each of the three directions of the k-space. In the 3D Cartesian case, the undersampling is usu-ally obtained by randomly skipping some lines in the two phase encoding directions. In this way, the randomized variable density design is done in only two directions of the k-space. Non-cartesian 3D trajectories are well suited for CS-MRI [9, 10, 56], due to the incoherence of aliasing artefacts and their ability to generate 3D variable den-sity sampling patterns. However, for a CS reconstruction, it is desirable to avoid the computational burden of 3D gridding, needed for their reconstruction. In fact, since CS reconstruction algorithms are iterative, reducing the computation cost of each it-eration can significantly reduce the overall reconstruction time. Trajectories that can be reconstructed using 2D gridding followed by an inverse Fourier transform along the third direction of the k-space are better candidates for CS-MRI. Stacks of spirals (SoS), which belong to this class of trajectories [8], have been used in CS-MRI [41, 57, 58].

The acquisition trajectory proposed in this chapter consists in an extension of SoS trajectories. It is able to generate a 3D pseudo-random sampling pattern having a sam-pling density that varies across all the directions of the k-space. Contrarily to 3D spirals trajectories, it can be easily used in CS-MRI, since it does not require 3D gridding for the reconstruction.

**2.2**

**Trajectory Design**

From here, I assume that each spiral lies in a kx − ky plane and the phase encoding,

necessary for the SoS trajectory, is performed in the kz direction.

2.2.1 Base 2D Trajectory: Variable Density Spiral

For a given field of view (F OVxy) and a certain number of spiral interleaves Ns, the

trajectory of a fully sampled euclidean spiral in the polar coordinate system is the fol-lowing [2]:

|ke| =

Ns

2πF OVxy

θ (2.1)

where |ke| is the distance of the trajectory from the center of the k-space and θ is

the azimuthal angle. The subscript e was used here to distinguish the trajectory of the euclidean spiral from the VDS trajectory. The trajectory of a VDS is described by the following expression [3, 8]:

|k| = Ns

2πF OVxyρ(|k|, θ)

θ (2.2)

The density function ρ(|k|, θ) reduces the effective F OV at the location (|k|, θ) of the k-space. In this study the density function is the following:

ρ(|k|, θ) = exp
−|ke|
2
k2
c
= exp
−α
2_{θ}2
k2
c
(2.3)
where α = Ns/2πF OVxy. The parameter kcregulates the amount of undersampling

of the spiral. In fact, the effective F OV used in correspondence of k = kc is reduced

to the 36.79% of the nominal F OVxy.

With this formulation, the density function ρ(|k|, θ) depends only on θ, and the gradient waveforms can be calculated as described in [3]. Some examples of the above described trajectories and the corresponding gradient waveforms are shown in Fig. 2.1. 2.2.2 Randomized Stack of Spirals: Rotations and Random Delays

The standard 3D stack of VDSs (SoVDS) trajectory consists in the repetition of the same spiral trajectory for each kx− ky plane. The energy of the PSF of this trajectory is

totally contained in the slice corresponding to z = 0 (Fig. 2.2). In this case, no energy dispersion occurs in the other slices. This results in a relatively high side-lobe peak in the PSF.

A convenient way to spread the PSF energy across the other slices consists in ro-tating by a random angle the trajectory of each kx − ky slice encoding plane. In this

manner, it is possible to achieve a through-plane randomization without changing the in-plane sampling pattern.

To reduce the acquisition coherence it is also desirable to achieve a randomization within each slice encoding. With this aim I propose to use the strategy of random delays: for each interleave, a random location is picked to keep the gradient strength unchanged for a fixed time duration. After this short period, the gradients are released

and the original waveform is played out (left panel in Fig. 2.3). This change results in a randomization of the VDS trajectory only in the undersampled region of the k-space (right panel in Fig. 2.3). By choosing appropriately the temporal position and duration of the delays it is possible to ensure that spiral interleaves do not cross each other. In all the experiments carried on in this study, I used delays with a duration of two gradient dwell times, inserted with uniform probability distribution between 45% and 88% of the total duration of the gradient waveform (shaded area in the left panel of 2.3). This setup for random delays was chosen carrying on some pilot experiments. As shown in Fig. 2.4 this strategies together can reduce the coherence of the PSF, when compared to the standard stack of VDSs.

I hereafter refer to the stack of VDSs with random rotations and random delays as randomized SoS(R-SoS).

2.2.3 VDR-SoS

It is desirable that the sampling density be variable across all the directions of the k-space. Thus, the density function in Eq. 2.2 should also depend on kz. For this purpose

I propose to design different VDSs for different kx−kyplanes by modifying the density

function of Eq. 2.3 as follows:
ρ(θ, kz) = exp
− α
2_{θ}2
kc(kz)2
where kc(kz) = exp
−k
2
z
k2
zc
(2.4)
The optimization of the density function is beyond the aim of this study. My aim in
this study is to demonstrate how a variable density approach improves the
reconstruc-tion quality and to propose a simple method to achieve this kind of trajectory without
requiring a 3D spiral.

The number of spiral arms used in the outer planes should be reduced to spend approximately the same readout time for each interleave and to further reduce the total

Euclidean Spiral VDS kc = 0.5 mm-1 -1 VDS kc = 0.33 mm Gr adient W a v ef or ms [mT /m] 2D T raject or y

Time [ms] Time [ms] Time [ms]

−0.4 −0.2 0 0. 2 0. 4 0.5 0 - 0.5 −0.5 0 0. 5 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 PSF x y, z = 0 x Profiles x z, y = 0 z z = 0 y = 0 x = 0 y = 0 −0.4 −0.2 0 0. 2 0. 4 0.5 0 - 0.5 −0.5 0 0. 5 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0

Figure 2.2: PSF of a standard stack of VDSs: all the energy of the PSF is located in the central slice. Data are shown in logarithmic scale.

scan time. With this purpose in mind, the following calculations were performed. Let kj

max be the maximum distance form its center reached by the spiral in the jth plane

(i.e., kj_{max} = 1/2∆x = 1/2∆y) and let θj

maxbe the corresponding azimuthal angle. To

reach the same distance from the center, the following condition must hold:
k_{max}j = k0_{max} ∀j

By combining this equation with Eq. 2.2 and Eq. 2.4, and setting k_{c}j = kc(kzj) it is

possible to obtain:
N_{s}jexp
N_{s}jk0_{max}
2πF OVxykjc
2
θj_{max}= N_{s}0exp
N_{s}0k_{max}0
2πF OVxykc0
2
θ0_{max} (2.5)
After the generation of the gradient waveforms for kz = 0, the only unknown terms

in Eq. 2.5 are θj

max, and the number of interleaves Nsj played out in the jth plane.

Considering that the number of spiral turns is related to its duration, the following heuristic equation was used for determining θj

max.

θj_{max} = θ0_{max}exp k

j c− kc0 √ 2kcj 2 (2.6) G [ mT /m] Time [ms]

Gradient Waveforms Standard VDS Random delayed

VDS

Figure 2.3: Random delays. The gradient waveforms are held for a certain period of time, resulting in a randomly perturbed VDS trajectory.

−0.4 −0.2 0 0. 2 0. 4 0.5 0 - 0.5 −0.5 0 0. 5 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 PSF x y, z = 0 x Profiles x z, y = 0 z z = 0 y = 0 x = 0 y = 0 −0.4 −0.2 0 0. 2 0. 4 0.5 0 - 0.5 −0.5 0 0. 5 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 SoVDS R-SoS SoVDS R-SoS

Figure 2.4: PSF of R-SOS. The side-lobe peak is reduced compared to the standard stack of VDSs. Data are shown in logarithmic scale.

After the determination of θ_{max}j it is possible to solve Eq. 2.5 numerically. I hereafter
refer to this procedure as dead time minimization.

An example of VDR-SoS trajectory is shown in Fig. 2.5. The effect of the dead time minimizationis shown in Fig. 2.5c-d: the number of spiral arms used in the outer planes of the k-space is drastically reduced, to achieve approximatively the same readout time for each interleave. A comparison of the PSF of the three methods is shown in Fig. 2.6.

**2.3**

**Methods**

The performances of the proposed trajectory were evaluated by means of numerical simulations. Firstly I evaluated the performances of the standard SoVDS, R-SoS and VDR-SoS using an analytical phantom. Then I tested VDR-SoS on a MRI dataset. Finally I compared VDR-SoS with other 3D spiral trajectories.

The reconstructions were performed in MATLAB (MathWorks, Natick, Massachusetts, USA) on a PC with a 2.54 GHz dual core processor and 8 Gb RAM.

2.3.1 Compressed Sensing reconstructions

The CS reconstructions described in the following paragraphs were performed by solv-ing the followsolv-ing problem:

minimize

x kAx − yk

2

2+ λΨkΨxk1+ λT VT V(x) (2.7)

where A is the encoding operator and y represents the acquired data. A corresponds to the product between the Fourier transform along the z direction and the non uniform Fourier operator [59]. Ψ is a sparsity domain and T V(x) is the total variation function. The algorithm used was FCSA [48]. Voronoi density compensation was used during the gridding operations [60]. The regularization parameters λΨ and λT V and the sparsity

domains are described in the following sections. 2.3.2 Comparison with SoS trajectory

Simulations on a numerical phantom

A Shepp-Logan digital phantom (dimension 256 × 256 × 64) was used for this first simulation experiment. The simulated F OVxy and F OVz were both 20 cm. All the

−0.51 0 0.5
2
3
4
5
6
kz
R
e
a
d
o
u
t
T
ime
p
e
r
in
te
rl
e
a
ve
[
ms]
−0..5 0 0..5
0
2
4
6
8
10
12
14
16
k_{z}
N
u
mb
e
r
o
f
in
te
rl
e
a
ve
s
p
e
r
sl
ice

No dead time minimization Dead time minimization

−0.4 −0.2 0 0. 2 0. 4 0.5 0 - 0.5 −0.5 0 0. 5 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 e) PSF x y, z = 0 x f) Profiles x z, y = 0 z z = 0 y = 0 x = 0 y = 0 −0.4 −0.2 0 0. 2 0. 4 0.5 0 - 0.5 −0.5 0 0. 5 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 SoVDS VDR SoS SoVDS VDR SoS Dead time minimization

kx ky kz ky kz kx = 0 VDR-SoS trajectory a) b) c) d)

Figure 2.5: VDR-SoS trajectory. a) example of VDR-SoS trajectory. b) example of VDR-SoS sampling pattern in a central slice (kx= 0). c) Readout time needed for each spiral interleave. d) number of interleaves per slice. Black solid lines: without dead time minimization. Red dashed lines: with dead time minimization. e-f) PSF of the trajectory and corresponding central profiles.

spirals were generated with a target resolution of 0.78 mm for a set of gradients with maximum strength of 18.42 mT/m and maximum slew rate of 120 mT/m/ms. The num-ber of interleaves was 16 for the R-SoS and for the SoVDS. The VDR-SoS trajectory was designed with 16 interleaves in its central kz slice, whilst in the other slices Eq.

lo g 1 0 (PSF ) lo g 1 0 (PSF ) x z SoVDS R-SoS VDR-SoS SoVDS R-SoS VDR-SoS

### PSF Profiles

Figure 2.6: Comparison of the PSFs obtained with the three methods.

density functions used in the simulations are summarized in Tab. 2.1. Although value of kzcis important for the design of the VDR-SoS trajectory, in this experiment I tested

only the case kzc = 1.12cm−1. The study of the effect of kzc on the reconstruction

quality is addressed in the following section. To observe the effect of the rotations, the random delays and the variable density design, I used the same set of rotation angles for each of the three compared trajectories. The undersampling factor R was defined as the ratio of the number of samples of a fully sampled SoS and the number of samples of the undersampled SoS. In this case the aim of the simulations was to demonstrate the effectiveness of the sampling pattern as a function of the actual number of samples. An additive Gaussian noise (σ = 0.01 ∗ max(|I0|)) was added to the simulated k-space

data.

The reconstruction was performed solving Eq. 2.7 (λΨ = 2e-3, λT V = 2e-2, 200

it-erations of the algorithm). The 3D wavelet domain (Daubechies 1, 6 levels) was used as sparsity domain. The anisotropic total variation [47] was used in these reconstructions. All of the reconstructions were performed for 10 random realizations.

Effect of kzcparameter

A fully sampled Cartesian dataset of a human knee (F OVxy = F OVz = 14.4 cm,

voxel size 1 × 1 × 1 mm) from a healthy volunteer was acquired on a 3T Philips Ingenia scanner (Philips Healthcare, Best, The Netherlands). This dataset was resampled on a fully sampled SoS composed by 30 interleaves for each slice and with a readout time of 5 ms. The resulting data set was used as reference for next experiments. In this case I compared SoVDS with three VDR-SoS trajectories corresponding to different levels of kz undersampling by changing the kzc parameter in Eq. 2.4 (parameters shown in

Tab. 2.2). The spiral readout time was 5 ms. In these experiments, the acceleration was defined as the ratio of the number of interleaves needed for the fully sampled SoS and the total number of interleaves used in the undersampled SoS.

The retrospectively undersampled datasets were reconstructed using the isotropic total variation and Daubechies 8 wavelets, six levels (λΨ = 5e-3, λT V = 1e-3, 100

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**2.3.**
**Methods**
Undersampling Factor (R)
3 3.5 4 4.5 5

SoS and R-SoS kc(cm−1) 3.90 3.18 2.70 2.32 2.05 kc0(cm−1) 7.20 5.82 4.80 4.17 3.65 VDR-SoS

kzc(cm−1) 1.12 1.12 1.12 1.12 1.12 Table 2.1: Parameters of the density function used in the simulations.

2x Acceleration 3x Acceleration 4x Acceleration
Ns0 kc(cm−1)
Total n. of
arms Ns0 kc(cm
−1_{)} Total n. of
arms Ns0 kc(cm
−1_{)} Total n. of
arms
SoVDS 15 4.50 2160 10 2.70 1440 8 2.10 1152
VDR-SoS
kzc= 4.5cm−1
19 6.50 2169 13 4.00 1432 10 2.80 1081
VDR-SoS
kzc= 3.5cm−1
21 7.70 2095 15 4.60 1420 12 3.20 1106
VDR-SoS
kzc= 2.5cm−1
29 15.50 2443 19 6.50 1408 14 4.00 995

Table 2.2: Parameters of the density function used in the simulations (knee dataset). Each arm had a readout time of 5 ms. The fully sampled reference required 30 interleaves of 5 ms each one, per plane (4320 interleaves).

Data Analysis

For each reconstruction, the results were compared with the fully sampled reference. The normalized root mean square error (nRMSE), averaged on all the realizations of the simulations, and the mean structure similarity index (mSSIM) [61] were used to asses the image quality:

nRM SE = 1 Ntot v u u t Ntot X i=1 |I(i) − I0(i)|2 max |I0| (2.8)

2.3.3 Comparison with 3D spirals trajectories

In this experiment I compared the proposed VDR-SoS trajectory with two trajecto-ries belonging to the class of 3D spirals: spherically distributed spirals (sDS) [10] and FLORET [62].

A fully sampled 3D dataset of a human brain (F OVxy = F OVz = 25.6 cm, voxel

size 1 × 1 × 1 mm) from a healthy volunteer was acquired on a 3T Philips Ingenia scanner (Philips Healthcare, Best, The Netherlands). The dataset was resampled on a three-fold accelerated VDR-SoS trajectory (256 slices, N0 = 29, kc0 = 5 cm−1,

kzc = 3 cm−1, 4366 interleaves in total) and on an sDS (256 slices, 17 interleaves per

slices, 4352, interleaves in total) and FLORET (two orthogonal hubs with starting angle of 45◦, 201 cones per hub, 11 arms per cone, 4442 total interleaves). The compared trajectories were designed with a dual density design in order to keep approximately the same total number of interleaves and the same readout time (5 ms). FLORET and sDS trajectories were designed using the source code from http://www.ismrm.org/ mri_unbound/. The CS reconstructions were performed solving Eq. 2.7 (λΨ =

1e-4, λT V = 5e-4, 300 iterations of the algorithm).

**2.4**

**Results**

2.4.1 Comparison with SoS trajectory

Simulations on a numerical phantom

The quality metrics resulting from the phantom simulations are shown in Fig. 2.7. The randomization used in R-SoS improved the reconstruction accuracy, when compared with SoVDS. However the reconstruction quality decreased when the undersampling factor increased. A not proper coverage of the central region of the k-space is respon-sible for these results. This effect was reduced exploiting a variable density design also across the kz direction. The proposed VDR-SoS, in fact, yielded the best results.

The reconstruction error was successfully reduced also for high values of R, and the mSSIM index revealed optimal matching with the reference image. An example of reconstruction in case of R = 5 is shown in Fig. 2.8.

The reconstruction quality of VDR-SoS remained good for all the tested undersam-pling factors. This behavior is mainly due to the high compressibility of the Shepp-Logan phantom. For this reason, I did not perform further simulations for finding the maximum achievable acceleration. A better insight into the limit on the acceleration

mSSIM 3 3.5 4 4.5 5 0.975 0.98 0.985 0.99 0.995 1 1.005 R SoVDS Rotations (R−SoS)

Rotations and Random Delays (R−SoS)
VDR−SoS
3 3.5 4 4.5 5
−5.4
−5.3
−5.2
−5.1
−5
−4.9
−4.8
−4.7
−4.6
Log_{10}(nRMSE)
R

Figure 2.7: Results of the simulations on the Shepp-Logan analytical phantom for various undersam-pling factorsR. A value of mSSIM close to 1 indicate excellent matching with the original image.

factor of VDR-SoS can be gained by observing its results on a realistic dataset, de-scribed in the following section.

Effect of kzcparameter

Fig. 2.9 shows the reconstruction results for various acceleration factors for the knee data set. VDR-SoS outperformed the SoVDS trajectory for all the tested values of the kzc parameter. The reconstruction quality increased when the value of kzc decreased.

Sagittal and axial views are shown in Fig. 2.10 and Fig. 2.11. Aliasing artefacts are better removed in case of VDR-SoS trajectories (especially in case of higher kz

undersampling, i.e. kzc = 2.5 cm−1). The image quality remained good also for a

three-fold acceleration with the lowest value of kzc. By reducing the value of kzc in

VDR-SoS and the number of interleaves in the outer planes of the k-space, more time is spent in the central zone of the k-space, allowing for a better reconstruction.

SoVDS R-SoS VDR-SoS

5x

E

R

R

OR

Figure 2.8: Results of the simulations on the Shepp-Logan analytical phantom (R = 5) and relative error maps. The error maps are shown on a 5x color scale, with respect to the images.

2 2.5 3 3.5 4 −2.1 −2 −1.9 −1.8 −1.7 −1.6 −1.5 Acceleration 2 2.5 3 3.5 4 0.97 0.975 0.98 0.985 0.99 0.995 1 Acceleration SoVDS VDR−SoS k zc= 4.5 cm −1 VDR−SoS k zc= 3.5 cm −1 VDR−SoS k zc= 2.5 cm −1 nRMSE mSSIM

Figure 2.9: Results of the simulations on the knee dataset.

**2x** **3x** **4x** **2x** **3x** **4x**
Acceleration

### 2x Error

SoVDS VDR-SoS kzc = 4.5 cm-1 VDR-SoS kzc = 3.5 cm -1 VDR-SoS kzc = 2.5 cm-1Figure 2.10: Results of the simulations on the knee dataset and corresponding error maps. Sagittal view. The error maps are shown on a 2x color scale.

**2x** **3x** **4x** **2x** **3x** **4x**
Acceleration

### 2x Error

SoVDS VDR-SoS kzc = 4.5 cm-1 VDR-SoS kzc = 3.5 cm-1 VDR-SoS kzc = 2.5 cm -1Figure 2.11: Results of the simulations on the knee dataset and corresponding error maps. Axial view. The error maps are shown on a 2x color scale.

2.4.2 Comparison with 3D spirals

The parameters setup used for the design of the compared trajectories resulted in the PSFs shown in Fig. 2.12. Although the spatial distribution of the PSFs was found different for the compared trajectories (Fig. 2.12a-b), the coherence was found similar (Fig. 2.12c).

Fig. 2.13 shows zero filling reconstructions for the described trajectories. The re-construction corresponding to the sDS trajectory resulted in more coherent aliasing artefacts with respect to FLORET and VDR-SoS trajectories (particularly visible in the coronal slice, second row in Fig. 2.13). FLORET trajectory resulted in slightly more coherent artefacts in the sagittal slice. VDR-SoS, on the contrary, caused the appear-ance of non coherent aliasing artefacts that spread equally in all the directions.

The results of the CS reconstructions are shown in Fig. 2.14. Both sDS and
FLO-RET trajectories show residual aliasing artefacts in the reconstruction (hollow red
ar-rows). The reconstruction corresponding to the VDR-SoS trajectory did not show
resid-ual artefacts.
FLORET
sDS VDR-SoS
y,
z=
0
0.5
-0.5
-0.5 _{x} 0.5
0.5
-0.5
-0.5 _{x} 0.5
0.5
-0.5
-0.5 _{x} 0.5
z,
y=
0
0.5
-0.5
-0.5 _{x} 0.5
0.5
-0.5
-0.5 _{x} 0.5
0.5
-0.5
-0.5 _{x} 0.5
0
-0.5 0.5 -0.5 0 0.5 -0.5 0 0.5
y z d
L
o
g 10
(PSF
)
0
-1
-2
-3
-4
-5
0
-1
-2
-3
-4
-5
0
-1
-2
-3
-4
-5
x = 0
z = 0
x = 0
y = 0
x = 0
y = 0
z = 0
a)
C)
b)

Figure 2.12: PSFs of the compared 3D spiral trajectories. a) PSFs in the central x − y slice. b) PSFs in the centralx − z slice. c) Profiles of the PSFs across the y, z and diagonal (d) directions. All the images are shown in logarithmic scale

This behaviour was not predicted by the analysis of the PSF. In fact, as also ob-served by Adcock and Roman [25], the PSF does not account for the true structure of the signal. VDR-SoS performed better compared with sDS and FLORET trajecto-ries, mainly because of its denser sampling of the center of the k-space, rather than to a lower coherence (Fig. 2.12). In fact, the compared trajectories were undersampled with a dual density design, while VDR-SoS was designed with multilevel sampling density strategy. The multilevel sampling density strategy is more able to respect the structure of the signal, allowing for better reconstruction [25].

sDS Reference

Zero Filling Reconstructions

FLORET VDR-SoS

Figure 2.13: Zero filling reconstructions obtained with the compared trajectories (3x acceleration).

**2.5**

**Discussion**

In this study I investigated the use of a randomized variable density trajectory design for 3D CS-MRI. A simple way to randomize a SoS trajectory was proposed and tested with numerical simulations. The R-SoS trajectory helped to improve the reconstruction ac-curacy (Fig. 2.7). However, this trajectory does not account for the energy distribution in the k-space. In fact, a better reconstruction was possible with the proposed VDR-SoS trajectory (Fig. 2.8). A denser sampling of the center of the k-space increased the reconstruction quality also in the case of a realistic dataset (Fig. 2.9).

The results of this study demonstrated that with the proposed VDR-SoS it is possible to achieve a reconstruction quality similar to the one obtained using 3D spirals

trajecto-sDS FLORET VDR-SoS

CS Reconstructions

Figure 2.14: CS reconstructions obtained with the compared trajectories (3x acceleration). Red hollow arrows indicate region of the images where the aliasing artefacts are not correctly removed.

ries, like FLORET and sDS (Fig. 2.14). The main advantage of VDR-SoS with respect
to 3D spirals is that it does not require 3D gridding for the reconstruction. In this way
it is possible to reduce the computational cost of each iteration in CS reconstruction
algorithms. Furthermore it is also possible to reduce the memory loading due to grid
oversampling necessary for gridding operations [59]. In fact the memory loading
nec-essary for 3D gridding, assuming a cubic image volume of dimensions N × N × N , is
O(f3_{N}3_{), where f is the grid oversampling ratio. VDR-SoS, on the contrary, requires}

N 2D gridding operations, resulting in memory loading of O(f2N3) if they are carried on in parallel.

The computational burden of 3D spiral trajectories can be a limiting factor in case of high resolution and multidimensional datasets like in the case of dynamic MRI. This

is particularly true since the acquisition is usually performed using phased array coils. The effect of multilevel variable density sampling strategies for CS-MRI has been widely studied [25,26,54,63]. It has been demonstrated that multilevel sampling strate-gies result in better image reconstructions. This is in agreement with the results ob-tained in this study. A sole analysis of the PSF is not sufficient to infer the final results of CS reconstructions. This was also noted by Adcock and Roman in [25], and it agrees with the results of this study. In fact it is possible to notice that there is not a big dif-ference in the coherence of the PSF corresponding to R-SoS and VDR-SoS in Fig. 2.6, although the final reconstruction quality is different.

Although the computational burden was reduced, when compared with 3D spirals, a challenge of VDR-SoS is the long reconstruction time. This is due to the combination of gridding operations with CS reconstruction (30-60 minutes using a non-optimized and non-parallelized MATLAB code). The long reconstruction time could be reduced with the parallelization of the reconstruction algorithm and with Graphical Processing Unit (GPU) implementations.

**2.6**

**Conclusions**

In this simulation study I proposed a new non-cartesian trajectory for CS-MRI. The tra-jectory is based on a stack of variable density spirals and allows to incoherently sample a 3D k-space with a 3D variable sampling density. Numerical simulations showed that the proposed VDR-SoS outperformed standard stack of spirals trajectories. Fur-thermore I demonstrated that with VDR-SoS it is possible to achieve a reconstruction quality similar to 3D spiral trajectories, without requiring 3D gridding.

### CHAPTER

## 3

**Random delayed spirals in Dynamic MRI**

The content of this chapter is described in the following conference proceeding: Giuseppe Valvano, Nicola Martini, Dante Chiappino, Luigi Landini, and Maria Filom-ena Santarelli, Random Delayed Spirals for Compressive Sensing Cine MRI,Proc. Intl. Soc. Mag. Reson. Med. 23 (2015) 3633.

### I

NCOHERENT acquisition strategies are a needed to reduce the strength of aliasingartefacts in CS-MRI. Temporal incoherence can also be exploited in dynamic MRI to improve the reconstruction accuracy. In 2D MRI it is not possible to achieve the pure random undersampling pattern needed for CS purposes, since hardware phys-ical constraints are present. However, smooth pseudo-random trajectories, like random delayed spirals, have been shown to be effective in reducing the acquisition coherence. In this chapter I describe the application of the random delayed spirals to 2D dy-namic MRI. The chapter is organized as following:

• Section 3.1 describes the application of random delays to 2D spiral dynamic MRI.

• Section 3.2 describes the simulations experiments carried on in this study.