Magnetic resonance fingerprinting

Bart Tukker

July 14, 2017

Bachelor Thesis Mathematics, Physics and Astronomy

Supervisors: dr. Rudolf Sprik, dr. Chris Stolk

Korteweg-de Vries Instituut voor Wiskunde

Faculteit der Natuurwetenschappen, Wiskunde en Informatica

Universiteit van Amsterdam

Abstract

Magnetic resonance fingerprinting (MRF) is a new approach to quantitative magneticresonance imaging proposed in the paper “Magnetic Resonance Fingerprinting” fromNature by Duerk et al.. MRF uses a random RF pulse sequence, a dictionary of pre-determined signal evolutions for different physical parameters and a pattern matchingalgorithm to determine the parameters of a measured signal. The goal of this the-sis is to determine whether a random pulse sequence works better than a non-randompulse sequence for MRF. Also two pattern matching algorithms are compared with eachother. The first pattern matching algorithm uses the correlation and the second oneuses the least squares. MRF is implemented for three different pulse sequences andthe two matching algorithms for 50 pulses and 500 pulses where the parameters T1, T2and off-resonance frequency δω are estimated. In the first pulse sequence are the flipangles, axis angles and repetition times choosen constant. The second pulse sequencehas random flip angles, a constant axis angle and random repetiton times. In the lastpulse sequence are the flip angles, axis angles and repetiton times chosen randomly. Theimplementation is done for the single single voxel case and with a known proton density.The pulse sequences and matching algorithms are compared with each other by doingmultiple simulation with noise and using statistics to quantify how well the parametersare estimated. For 50 pulses no pulse sequence performs significantly better than theother. When the amount of pulses is increased to 500, the random pulse sequence withrandom axis angles performs better than the other two. Least squares estimates theparameters best when the proton density is known.

Title: Magnetic resonance fingerprintingAuthor: Bart Tukker, 10797025Supervisors: dr. Rudolf Sprik, dr. Chris StolkSecond graders: dr. Tom Hijmans, prof. dr. Rob StevensonDate: July 14, 2017

Korteweg-de Vries Instituut voor WiskundeUniversiteit van AmsterdamScience Park 904, 1098 XH Amsterdamhttp://www.science.uva.nl/math

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Contents

1. Introduction 4

2. Magnetic dipole moment in a magnetic field 62.1. Proton spin in a magnetic field . . . . . . . . . . . . . . . . . . . . . . . . 62.2. Larmor precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3. Solution of a magnetic dipole moment in a static magnetic field . . . . . . 8

3. Magnetization in a magnetic field 103.1. Magnetization of non-interacting protons . . . . . . . . . . . . . . . . . . 103.2. Longitudinal relaxation time . . . . . . . . . . . . . . . . . . . . . . . . . 123.3. Transverse relaxation time . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4. Bloch equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4. RF pulse sequences 174.1. RF pulse and RF pulse sequence . . . . . . . . . . . . . . . . . . . . . . . 174.2. The spin echo and T2 measurement . . . . . . . . . . . . . . . . . . . . . . 184.3. Inversion recovery and T1 measurement . . . . . . . . . . . . . . . . . . . 18

5. Magnetic resonance fingerprinting and its implementation 225.1. Magnetic resonance fingerprinting . . . . . . . . . . . . . . . . . . . . . . . 225.2. Equivalence of the correlation and least squares . . . . . . . . . . . . . . . 245.3. Implementation of MRF . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.4. Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6. Results of the MRF simulations 306.1. Non-random pulse sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 336.2. Random pulse sequence with a constant axis angle . . . . . . . . . . . . . 376.3. Random pulse sequence with random axis angles . . . . . . . . . . . . . . 396.4. Overview results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 40

7. Conclusion 43

8. Popular summary 44

A. Matlab code 46

B. Figures of the results 50

Bibliografie 63

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1. Introduction

Magnetic resonance imaging (MRI) is an imaging technique that uses nuclear spin res-onance to form an image. MRI is noninvase and relative safe since it uses magneticfields and no ionizing radiation unlike other imaging techniques such as a CT scan orradiography that uses X-rays. That is why MRI is widely used in the medical field as adiagnostic tool and for medical research.

An additional benefit of MRI is that it can use multiple physical parameters such asthe relaxation times of nuclear spins and the proton density to form an image. Sincethese parameters are characteristic for different materials, it can be used to distinguishdifferent type of tissues in an image. However, in practice the images that are createdwith MRI are essentially qualitative where only the relative values of the parametersare used for imaging. With quantitative MRI additional informaton can be obtained byestimating the spatial variation of the physical parameters. This additional informationcan help in the discrimination of different types of tissues. However, the quantitativeMRI methods that are used typically provide information for just a single parameterat a time. Moreover, the imaging can be really slow and is highly sensitive to theimperfections of the system.

In a paper from Nature by Duerk et al. [5] a novel approach to quantitative imagingis proposed. They call this method magnetic resonance fingerprinting (MRF). Thistechnique is based on the idea that for different parameter values the measured signalevolution is unique, like a fingerprint. With MRF a dictionary of predetermined signalevolutions for different parameter values is made. Then the parameters of a measuredsignal are simultaneously determined by matching the signal evolution with a signalevolution from the dictionary.

Because this technique seems very promising, in this thesis is investigated whethera random pulse sequence performs better than a non-random pulse sequence for MRF.Two different pattern matching algorithms are used for this and are also compared witheach other. Due to the complexity of MRF, it is too much for a bachelor thesis toanalyse this generally. That is why the choice is made to analyse the pulse sequencesand pattern matching algorithms by implementing it. The pulse sequences and matchingalgorithms are compared with each other by doing multiple simulation with noise andusing statistics to quantify how well the parameters are estimated.

To be able to understand how MRF works, first is the basic theory of MRI studied.These are covered in the next three chapters. In chapter 2 is explained how a singleproton spin behaves in a magnetic field. This motion is captured by differential equationswhich will also be solved. In chapter 3 are the relaxation times of the nuclear spinsintroduced. They are used to derive the differential equations that describe how anensemble of spins behaves in a static external magnetic field. These differential equations

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are known as the Bloch equations and will also be solved. In chapter 4 is the RF pulsesequence introduced and how it is used to determine the relaxation times. These threechapters about the basic theory of MRI are written by using the books from Brown etal. [2], Constantinides [3] and Oldendorf et al. [7].

The second half of the thesis is about MRF. In chapter 5 is described how MRF works,how it is implemented and what statistics is going to be used for the simulations. Inchapter 6 are the results shown how well the parameters are estimated for three differentpulse sequences and two different matching algorithms. This is done by doing multiplesimulations and using statistics to quantify how well it performs. Then the results arediscussed and explained.

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2. Magnetic dipole moment in amagnetic field

To be able to understand how MRI uses the motions of a nuclear spin in a magneticfield to obtain the information for the imaging, first is studied how the magnetic dipolemoment behaves in a magnetic field. In this chapter the magnetic dipole moment of asingle proton spin is introduced while the proton interactions with its surroundings areignored. The motion of a magnetic dipole moment is described by differental equations.With this the Larmor precession formula is derived which tells with which frequencya magnetic dipole moment precess around the magnetic field. Then the solutions ofthe differential equations are found. The obtained equations are important for the nextchapter where they are going to be used to derive the Bloch equations and its solutionsincluding the proton spin interactions with its surroundings.

2.1. Proton spin in a magnetic field

The proton spin can be viewed as a small circulating current. This circulating is as-sociated with a magnetic dipole moment. The magnetic dipole moment ~µ is definedas

~µ ≡ I∫d~a = I~a, (2.1)

where I is the current and ~a the vector area. The magnetic dipole moment can bewritten in terms of its spin angular momentum ~J

~µ = γ ~J, (2.2)

where γ is the proportionallity constant. This constant is called the gyromagnetic ratio.The gyromagnetic ratio is determined experimentally and depends on the nuclues orparticle. The gyromagnetic ratio for the proton is found to be

γ = 2.675× 108rad/s/T. (2.3)

The torque N on a current distribution in a constant magnetic field ~B in terms of themagnetic dipole moment is

~N = ~µ× ~B. (2.4)

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The derivation of this formula can be found in the book of Griffiths [6]. If the torque isnonzero, then the change in the total angular momentum ~J is

~J

dt= ~N. (2.5)

By combining all these expressions, the following formula is obtained:

d~µ

dt= γ~µ× ~B. (2.6)

By the right-hand rule ~µ rotates clockwise around the magnetic field, which is shown infigure 2.1. Since d~µ

dt is perpendicular to ~µ, its magnitude remains constant

dµ

dt= 0. (2.7)

Figure 2.1.: Clockwise precession of a proton’s magnetic dipole moment ~µ around themagnetic field ~B. Figure comes from Brown et al. [2].

2.2. Larmor precession

By looking at figure 2.1, the following formula can be constructed:

|d~µ| = µ sin θ|dφ|. (2.8)

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One also has

|d~µ| = γ|~µ× ~B|dt = γµB sin θdt (2.9)

This gives the equality γB|dt| = |dφ|. With this the Larmor precession formula isobtained

ω ≡ |dφdt| = γB. (2.10)

So if the magnetic dipole moment is not perfectly aligned with a static magnetic field,the magnetic moment precesses around the magnetic field with the Larmor frequency.

2.3. Solution of a magnetic dipole moment in a staticmagnetic field

Let x, y and z be unit vectors parallel to the x, y and z-axis respectively. Then one canwrite ~µ(t) as µx(t)x + µy(t)y + µz(t)z, where µx(t), µy(t) and µz(t) are the x, y and

z-component of ~µ respectively. Suppose ~B = B0z. Then equation (2.6) can be writtenas

d~µ

dt= γ[µx(t)x+ µy(t)y + µz(t)z]×B0z

= γ[−µx(t)y + µy(t)x]B0.(2.11)

By splitting it in its components, the following three differential equations are obtained:

dµxdt

= γB0µy(t) = ω0µy(t), (2.12)

dµydt

= −γB0µx(t) = −ω0µx(t), (2.13)

dµzdt

= 0, (2.14)

where ω0 is the Larmor frequency. To find the solutions for (2.12) and (2.13), take thesecond derivative

d2µxdt2

= −ω20µx(t),

d2µydt2

= −ω20µy(t).

The solutions for these two second-order differential equations are

µx(t) = a cos(ω0t) + b sin(ω0t),

µy(t) = c cos(ω0t) + d sin(ω0t),

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where a, b, c and d are constants. To satisy their inital condition, a must be µx(0)and c must be µy(0). Putting these two solutions back into the first-order differentialequations, one gets

dµxdt

= ω0[−µx(0) sin(ω0t) + b cos(ω0t)]

= ω0[µy(0) cos(ω0t) + d sin(ω0t)],

dµydt

= ω0[−µy(0) sin(ω0t) + d cos(ω0t)]

= −ω0[µx(0) cos(ω0t) + b sin(ω0t)].

This implies that b = µy(0) and d = −µx(0). By putting these two expressions into(2.12) and (2.13), the following solutions are obtained:

µx(t) = µx(0) cos(ω0t) + µy(0) sin(ω0t), (2.15)

µy(t) = µy(0) cos(ω0t)− µx(0) sin(ω0t), (2.16)

µz(t) = µz(0), (2.17)

for a magnetic field ~B = B0z.The solution can also be written in a complex form. Define µ+(t) as µx(t) + iµy(t).

By writing out µx(t) + iµy(t), one obtains

µx(t) + iµy(t) = µx(0) cos(ω0t) + µy(0) sin(ω0t) + iµy(0) cos(ω0t)− iµx(0) sin(ω0t)

= µx(0)[cos(ω0t)− i sin(ω0t)] + iµy(0)[cos(ω0t)− i sin(ω0t)]

= µx(0)[cos(−ω0t) + i sin(−ω0t)] + iµy(0)[cos(−ω0t) + i sin(−ω0t)]

= [µx(0) + iµy(0)]e−iω0t.

So the following formula is obtained:

µ+(t) = µ+(0)e−iω0t. (2.18)

With this expression, it can be easily seen that the magnetic dipole moment in the xy-plane rotates with the Larmor frequency clockwise and that its magnitude is constant.

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3. Magnetization in a magnetic field

In the last chapter the magnetic dipole moment of a single proton spin in pressenceof a magnetic field is analysed. However, the interactions of the proton spin with itssurrounding are ignored. The proton spin interactions with its neighboring atoms causesa change of the local magnetic field. This leads an ensemble of spins precessing atdifferent frequencies than the Larmor frequency. Protons can also exchange spin energywith its surroundings due to these proton interactions. The quantum physical descriptioncan be found in the books of Brown et al. [2] and Constantinides [3].

In this chapter the magnetization is introduced. Its motion in a magnetic field withthe effects of the proton interactions is described by a differential equation known asthe Bloch equations. The solution of the Bloch equations will also be derived in thischapter. The The dynamics of the spin system is characterized by two constants whichare the longitudinal relaxation time T1 and the transverse relaxation time T2. Sincethese relaxation times are characteristic for a material, these two parameters are veryimportant for MRI to differentiate tissues from each other.

3.1. Magnetization of non-interacting protons

The magnetization ~M is defined as the sum over the magnetic moments ~µi in a volumeV divided by this volume

~M =1

V

∑i

~µi. (3.1)

If the interactions of the protons with their environment is neglected, the derivative ofthe magnetization with respect to the time is

1

V

∑i

d~µidt

=γ

V

∑i

~µi × ~Bext,

where equation (2.6) is used. The differential equation for the magnetization is then

d ~M

dt= γ ~M × ~Bext. (3.2)

If this expression is written in its x-, y-, and z-component and set ~Bext as B0z, then

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it becomes the same as the equations for a single spin:

dMx

dt= γB0My(t) = ω0My(t), (3.3)

dMy

dt= −γB0Mx(t) = −ω0Mx(t), (3.4)

dMz

dt= 0. (3.5)

Their solutions are derived in the last chapter, which are

Mx(t) = Mx(0) cos(ω0t) +My(0) sin(ω0t), (3.6)

My(t) = My(0) cos(ω0t)−Mx(0) sin(ω0t), (3.7)

Mz(t) = Mz(0). (3.8)

From equation (2.18) can be seen that in a reference frame that rotates the xy-planewith the Larmor frequency ω0, the solutions of the magnetization then becomes

Mx(t) = Mx(0), (3.9)

My(t) = My(0), (3.10)

Mz(t) = Mz(0). (3.11)

So the derivative of the magnetization with respect to the time in this reference frame is

d ~M

dt= 0. (3.12)

It is useful to split the magnetization in terms of the parallel and the perpendicularcomponent with respect to the external field. If ~Bext is B0z, then the parallel componentof the magnetization is

M‖ = Mz, (3.13)

and the perpendicular component is

~M⊥ = Mxx+Myy. (3.14)

If these two equations are put into (3.2), one gets the differential equations when theproton interactions are ignored:

dM‖

dt= 0, (3.15)

d ~M⊥dt

= γ ~M⊥ × ~Bext. (3.16)

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3.2. Longitudinal relaxation time

It is energetically beneficial for the protons to align its spins with the external magneticfield. The spins can transfer their excess energy by interacting with neighboring atomsand radiating electromagnetic fields. The magnitude of the magnetization in this equi-librium state is proportional to the external field. For an external field ~Bext = B0z,let the magnitude of the magnetization in this equilibrium state be M0. Due to theproton interactions with its suroundings, the change of the longitudinal magnetizationis proportional to the difference M0−Mz. The differential equation for the longitudinalmagnetization when the static external field ~Bext = B0z is applied, then becomes

dMz

dt=

1

T1(M0 −Mz), (3.17)

where T1 is the longitudinal relaxation time. T1 depends on the environment of theprotons. The longitudinal relaxation time is very important for MRI because MRI usesthis parameter to differentiate different types of tissues.

To obtain the solution of this differential equation, replace Mz with M∗z + M0. Thedifferential equation then becomes

dM∗z +M0

dt=dM∗zdt

= − 1

T1M∗z .

The solution for this equation is

Mz(t)∗ = ce−t/T1 ,

where c is a constant. If Mz(t)∗ is replaced with Mz(t) − M0, then the longitudinal

magnetization at t = 0 is

Mz(0) = M0 + c.

To satisfy the inital condition, c must be Mz(0) −M0. So the solution of differentialequation (3.17) is

Mz(t) = M0 + (Mz(0)−M0)e−t/T1 . (3.18)

This equations states that when the magnetization is tipped away from its equilibriumstate, the longitudinal magnetization regrows exponentially from the initial value, Mz(0),to the equilibrium value, M0, with a rate that depends on T1.

3.3. Transverse relaxation time

Interactions of spins with their neighboring atoms cause a change of the local magneticfields. The local magnetic field that a spin experiences is a combinations of the appliedexternal field and the field of their neighbours. The variations in local fields causes spinsto precess at different frequencies. The individual spins tend to fan out in time. This is

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Figure 3.1.: The upper pictures shows the dephasing of a set of spins after tipping themin the transverse plane. The lower pictures shows the corresponding mag-netization vector of the spins. This figure comes from Brown et al. [2].

referred to as dephasing. The net magnetization vector decreases in magnitude becauseof the dephasing. This effect is illustrated in figure 3.1.

This decay in transverse magnetization adds an additional term to the differentialequation and becomes

d ~M⊥dt

= γ ~M⊥ × ~Bext −1

T2~M⊥, (3.19)

where T2 is the transverse relaxation time. Like the longitudinal relaxation time T1, T2depends on the environment of the spins and is used in MRI to differentiate differenttype of tissues.

In a reference frame that rotates with the Lamour frequency, differential equation(3.19) becomes

d ~M⊥dt

= − 1

T2~M⊥. (3.20)

This has the solution

~M⊥(t) = ~M⊥(0)e−t/T2 . (3.21)

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If the magnitude of the transverse magnetization is defined as M⊥ ≡ | ~M⊥|, the solutionbecomes

M⊥(t) = M⊥(0)e−t/T2 , (3.22)

which describes the decay of the transverse magnetization in a non-rotating referenceframe. From the solution can be seen that the decay is exponential with a rate dependson T2 and the transverse magnetization goes to zero when the time goes to infinity.

In practice, the external magnetic field is not perfectly homogeneous. This adds anadditional dephasing of the magnetization. The additional decay is characterized withdecay time T

′2. The overal relaxation time T ∗2 is defined as

1

T ∗2=

1

T2+

1

T′2

. (3.23)

In the next chapter, it will be shown that the loss of transverse magnetization due to T′2

is recoverable. The intrinsic losses due to T2 are not recoverable, since they are causedby local, random, time-dependent field variations.

3.4. Bloch equations

By combining the differential equation of the longitudinal magnetization (3.17) and thetransverse magnetization (3.19), the differential equation of the magnetization becomes

d ~M

dt= γ ~M × ~Bext +

1

T1(M0 −Mz)z −

1

T2~M⊥. (3.24)

This differential equation is known as the Bloch equation. The Bloch equation is veryimportant for MRI to determine the relaxation times T1 and T2.

Suppose ~Bext = B0z. By splitting the Bloch equation into its x-, y- and z-component,the differential equations becomes

dMx

dt= ω0My −

Mx

T2, (3.25)

dMy

dt= −ω0Mx −

Mx

T2, (3.26)

dMz

dt=M0 −Mz

T1, (3.27)

where ω0 ≡ γB0 is the Larmor frequency. The differential equation of the z-componentis the same as equation (3.17). Its solution has already been derived, which is equation(3.18). To solve the other two equations, replaceMx with µxe

−t/T2 andMy with µye−t/T2 .

The differential equation of the x-component then becomes

dµxe−t/T2

dt= e−t/T2

dµxdt− µxe

−t/T2

T2,

= ω0µye−t/T2 − µxe

−t/T2

T2.

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After canceling the terms µxe−t/T2T2

and dividing both sides by e−t/T2 , the equation be-comes

dµxdt

= ω0µy.

Similarly for the y-component, it becomes

dµydt

= −ω0µx.

Their solutions were already found in chapter 2, which are

µx(t) = µx(0) cos(ω0t) + µy(0) sin(ω0t),

µy(t) = µy(0) cos(ω0t)− µx(0) sin(ω0t).

Changing µx back into Mxet/T2 gives

Mx(t)et/T2 = Mx(0)e0/T2 cos(ω0t) +My(0)e0/T2 sin(ω0t),

My(t)et/T2 = My(0)e0/T2 cos(ω0t)−Mx(0)e0/T2 sin(ω0t).

Then the complete set of solutions of the Bloch equations are

Mx(t) = e−t/T2 [Mx(0) cos(ω0t) +My(0) sin(ω0t)], (3.28)

My(t) = e−t/T2 [My(0) cos(ω0t)−Mx(0) sin(ω0t)], (3.29)

Mz(t) = M0 + (Mz(0)−M0)e−t/T1 . (3.30)

In the reference frame that rotates xy-plane with the Larmor frequency, the solutionsare

Mx(t) = e−t/T2Mx(0), (3.31)

My(t) = e−t/T2My(0), (3.32)

Mz(t) = M0 + (Mz(0)−M0)e−t/T1 . (3.33)

When t is taken to infinity, the magnetization goes to the equilibrium value M0z. Impor-tant to note is that the longitudinal and transverse magnetizations behaves independentfrom each other, so the magnitude of the magnetization, | ~M |, does not necessarily havea fixed length. In figure 3.2 the trajectory for an initial magnetization that lies in thetransverse plane can be seen.

The Bloch equations in the non-rotating reference frame can be simplified by usingthe complex representation

M+(t) = Mx(t) + iMy(t). (3.34)

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Figure 3.2.: The trajectory of the magnetization for an initial magnetization that lies inthe transverse plane along the y-axis. It shows the regrowth of longitudinalmagnetization and the decay of the transverse magnetization which happensindependently from each other. This figure comes from Brown et al. [2].

Writing out Mx and My gives

Mx(t) + iMy(t) = e−t/T2 [(Mx(0) + iMy(0)) cos(ω0t) + (My(0)− iMx(0)) sin(ω0t)]

= e−t/T2 [(Mx(0) + iMy(0)) cos(ω0t)− i(Mx(0) + iMy(0)) sin(ω0t)]

= e−t/T2(Mx(0) + iMy(0))[cos(−ω0t) + i sin(−ω0t)]

= e−t/T2−iω0t(Mx(0) + iMy(0))

Then the following equation is obtained:

M+(t) = M+(0)e−t/T2−iω0t. (3.35)

With this equation it can be easily seen by looking at the complex exponent e−iω0t

that the magnetization in the xy-plane rotates with the Larmor frequency. Since themagnitude of M+(t) is the same as the transverse magnetization M⊥ and does notchange by the complex exponent, the solution can be written as M⊥(t) = M⊥(0)e−t/T2 ,which agrees with (3.22). By writing the Bloch equations in in a longitudinal andtransverse component with respect to the external magnetic field, we following solutionsare obtained:

M‖(t) = M0 + (M‖(0)−M0)e−t/T1 , (3.36)

M⊥(t) = M⊥(0)e−t/T2 , (3.37)

where M⊥(t) = | ~M⊥(t)|.

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4. RF pulse sequences

The solutions of the Bloch equations can be used to rotate the magnetization away fromits equilibrium state by applying an additonal magnetic field. In this chapter it will beshown how this is done with RF pulses. There will also be two basic RF pulse sequencesintroduced to see how an RF pulse sequence work. The first one is the spin echo methodwhich is used to determine T2. The second one is the inversion recovery method whichis used to determine T1.

4.1. RF pulse and RF pulse sequence

An RF pulse rotates the magnetization by applying an external magnetic field ~B1 fora finite time. In the reference frame that rotates the transverse plane with the Larmorfrequency, the following equations tell how the magnetization changes by the RF pulse:

M‖(t) = M1 + (M‖(0)−M1)e−t/T1 , (4.1)

M+(t) = e−t/T2−iω1tM+(0), (4.2)

where M1 is the corresponding equilibrium magnetization to the external field ~B1.If the RF pulse is turned on for a very short duration of time, the effect of the relaxation

times T1 and T2 can be ignored. That is because typical values of ω1 = γB1 are muchlarger than 1/T1 and 1/T2. This reduces the equations to

M‖(t) = M‖(0), (4.3)

M+(t) = e−iω1tM+(0). (4.4)

Thus, if an RF pulse is applied for a short amount of time τ , then the RF pulse causesa rotation of the magnetization with an angle of ωτ radians around ~B1.

If the applied magnetic field ~B1 is perpendicular to ~B0 and rotates the magnetizationby an angle 90 degrees, then the RF pulse is called a π/2-pulse. A rotation of 180 degreesis called a π-pulse.

A series of RF pulses performed one after another is called and RF pulse sequence.The time between RF pulses is called the repetition time TR. Between the RF pulsesthe signal can be measured and be used to determine T1 or T2. However, the coils thatmeasures the magnetization only pick up the rotating magnetization. So the measuredsignal contains only the transverse component of the magnetization. In Brown et al. [2]can be read how this works in more detail.

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4.2. The spin echo and T2 measurement

In practice, the external magnetic field is not perfectly uniform. This causes additionaldephasing of the transverse magnetization characterized with time constant T

′2. T

′2 can

often be so small that 1/T′2 dominates 1/T2. Fortunately, the loss in magnetization

caused by T′2 is recoverable. This can be done with an RF pulse sequence called the spin

echo method.The spin echo method consists of two RF pulses. First a π/2-pulse is applied and is

followed by a π-pulse. Suppose a π/2- pulse is applied at t = 0 about the x-axis in therotating reference frame that rotates with the Larmor frequency. Then the magnetizationpoints along the y-axis. The individual spins at different positions ~r experience differentmagnetic fields that is not excatly equal to B0. The spins rotate with different frequenciesand start to dephase. The accumulated phase φ of a spin relative to the y-axis is

φ(~r, t) = −γ∆B(~r)t, (4.5)

where ∆B(~r) is the difference in magnetic field between the field a spin at position ~rexperiences and the external field B0. At time t = τ the phase is −γ∆B(~r)τ . If at t = τa π-pulse is applied about the y-axis, a positive phase becomes negative, and vice versa

φ(~r, τ+) = −φ(~r, τ−) = γ∆B(~r)τ. (4.6)

Then for t > τ the accumulated phase is

φ(~r, t) = φ(~r, τ+)− γ∆B(~r)(t− τ)

= −γ∆B(~r)(t− 2τ)

= −γ∆B(~r)(t− TE),

(4.7)

with the echo time TE is defined as 2τ . Since the rate of phase accumaltion of each spinremains unchanged, at time TE all spins has phase φ = 0, regardless of their positionsand the fields they experience. In figure 4.1 you can see what how the spin echo is formedby the spin echo method.

Additional echoes can be formed by applying π-pulses at times t = (2n − 1)τ withn ∈ N. This causes to form echoes at t = 2nτ = nTE . By measuring the signal at theecho’s, the transverse relaxation time T2 can be determined. In figure 4.2 is the pulsesequence diagram of the repeated spin echo method and the corresponding signal shown.In this figure can be seen that at the times t = 2nτ an echo is formed. Also a signal isdrawn when the dephasing effect of T

′2 is removed.

4.3. Inversion recovery and T1 measurement

The measurement of T1 can be done through inversion recovery. However, unlike the spinecho method that just needs a single experiment, multiple experiments are required todetermine T1. Inversion recovery starts with the application of an π-pulse which inverts

18

Figure 4.1.: The formation of an echo by the spin echo method. Figure comes fromBrown et al. [2].

the magnetization. Then at time t = TI a π/2pulse is applied. In figure 4.3 you can seethe pulse sequence diagram of inversion recovery.

If at t = 0 a π-pulse is applied, the magnetization after the RF pulse is

Mz(0+) = −M0 (4.8)

The magnetization at 0 < t < TI by using the solution of the z-component of the Blochequation (3.30) is

Mz(t) = M0 + (−M0 −M0)e−t/T1 = M0(1− 2M0)e

−t/T1 . (4.9)

At time t = TI an π/2-pulse is applied. The magnetization after at t > TI is

M⊥(t) = |M0(1− 2M0)e−TI/T1 |e−(t−TI)/T ∗2 . (4.10)

19

Figure 4.2.: The pulse sequence diagram for multiple spin echoes and the correspondingsignal. At times t = 2nτ , with n ∈ N, an echo is formed. At the echoes thesignal has the same magnitude as a signal without the dephase effect of T

′2.

Figure comes from Brown et al. [2].

Figure 4.3.: The pulse sequence diagram for the inversion recovery. It consists of aninitial π-pulse followed by a π/2-pulse after time TI .

This shows that after the magnetization has been rotated in the transverse plane, themagnitude of the initial magnetization depends on TI . If a π/2-pulse is applied at time

TI = T1 ln 2, (4.11)

the factor (1−2M0)e−TI/T1 becomes zero. So T1 can be found by varying TI until a zero

in the signal is measured. The graph in figure 4.4 shows the transverse magnetizationright after t = TI as function of TI .

20

Figure 4.4.: The magnitude of the transverse magnetization for an inversion recovery attime TI as a function of TI . For TI = T1 ln 2 the magnitude is zero. Thisfigure comes from Brown et al. [2].

21

5. Magnetic resonance fingerprintingand its implementation

In practise, the images that are created with MRI are essentially qualitative where onlythe relative values of the parameters are used for imaging. With quantitative MRI thespatial variation of the parameter values are also estimated. This provides additionalinformation which can help in the discrimination of different types of tissues. However,the imaging with quatitative MRI is very slow and often only one parameter can beestimated at a time.

In a paper from nature of Duerk et al. [5] another approach of quantitative imaging isproposed. They call this method magnetic resonance fingerprinting (MRF). The differ-ence of MRF compared to traditional MRI is that MRF uses a random pulse sequence,a dictionary of predetermined signal evolutions for different known values of the param-eters, and a pattern matching algorithm. By matching an observed signal with one fromthe dictionary, all parameters are determined quantitatively at the same time.

In this chapter is first explained how MRF works. The mathematical notation forMRF follows closely the paper from Davies et al. [4]. Then it is shown how MRF isimplemented. Two pattern matching algorithms are used in the implementation. Thefirst one uses the correlation and the second uses least squares. Statistics is used toquantify how well MRF performs.

Due to the time constraint, only the single voxel case is implemented. In the paperfrom Nature can be read how it is done for three-dimensions. MRI in more dimensioncan be read in the books of Brown et al. and Constantinides.

5.1. Magnetic resonance fingerprinting

MRF uses a random pulse sequence. The random pulse sequence starts with an in-version pulse. Let N be the amount of pulses after the initial inversion pulse in thepulse sequence. Then the random pulse sequence consists of RF pulses with flip anglesα = {α1, ..., αN} and repetition times TR = {TR1, ..., TRN} where each flip angle andrepetition time is chosen randomly. In figure 5.1 is a pulse sequence diagram of a ran-dom pulse sequence illustrated. The random pulse sequence can also have axis anglesφ = {φ1, ..., φN} where the angles are chosen randomly. They indicate that the n-th RFpulse rotates the magnetization around axis y = x tan(φn) for n = 1, ..., N .

After each pulse the signal MX + iMy is measured. Let the time between the pulseand the measurement be TS, with TS < min{TR1, ..., TRN}. Let the measured signalevolution be X ∈ CN , where the i-th component of X is the signal measured afterthe i-th RF pulse for i = 1, ..., N . X depends on parameters T1, T2, off-resonance δω

22

Figure 5.1.: An RF pulse sequence for random repetition times TR1, ..., TRN and ran-dom flip angles α1, ..., αN .

and proton density ρ. δω is the difference between the frequency a proton spin precesswith and the Larmor frequency. A mapping of the parameters can be made from theparameters to X. For a chosen random pulse sequence with flip angles α, axis angles φ,repetition times TR, and sample time TS, let B be the parameter map such that

X = ρB(θ), (5.1)

where θ = {T1, T2, δω} is the parameter set of the parameters T1, T2 and δω. ρ ismultiplied with B because the signal is proportional to the proton density.

With this random pulse sequence a dictionary can be built. The dictionary consistsof predetermined signal evolutions for different values of parameters T1, T2, and δω. For

a discrete set of parameter sets θ(k) = {T (k)1 , T

(k)2 , δω(k)} with k = 1, ...,K, define the

dictionary as D = {D1, ..., DK} with

Dk = B(θk) ∈ CN , (5.2)

where ρ is set to 1. It is not essential to use a dictionary for MRF, but the benefit of adictionary is that the signal evolutions can be determined in advance and the dictionarycan be used multiple times to determine the parameters in each voxel. It is also not arequirement to use a random pulse sequence for MRF, but it is an easy way to have adictionary with signal evolutions that are distinct from each other.

After the dictionary is generated, a signal is observed. Let the observed signal be

d = ρB(θ) ∈ CN , (5.3)

where ρ and θ are unknown. To find the the parameters of the observed signal, a patternmatching algorithm is used. The pattern matching algorithm matches the observed signalwith a signal Dk from the dictionary D. One pattern matching algorithm is done withthe correlation. For x, y ∈ CN define the inner product as 〈x, y〉 =

∑Ni=1 xiyi and the

norm as ||x|| =√〈x, x〉. Then define the correlation between the observed signal d and

signal Dk from the dictionary as

Cor(d,Dk) =|〈d,Dk〉|||d|| ||Dk||

. (5.4)

23

From the Cauchy-Schwarz inequality it is known that

|〈d,Dk〉| ≤ ||d|| ||Dk||.

So the correlatation is less than or equal to 1. To find the signal from the dictionarythat corresponds to the observed signal best, the correlation is maximized over k:

k = arg maxk

Cor(d,Dk). (5.5)

Then the parameters of the observed signal are θ(k) = {T (k)1 , T

(k)2 , δω(k)} and

ρ =|〈d,Dk〉|||Dk||2

. (5.6)

Another pattern matching algorithm uses the least squares. However, for the leastsquares to work, it is assumed that the proton density of the observed signal is known.For the single voxel case this is not a problem since the proton density can be determinedby looking at the equilibrium magnetization M0. For an observed signal d with protondensity ρ and signal Dk from the dictionary, define the least squares as

LS(d,Dk) = ||dρ−Dk||2. (5.7)

The value is non-negative and is zero if dρ = Dk. The signal that represents the observed

signal best is found by minimizing the least squares over k:

k = arg mink

LS(d,Dk). (5.8)

Then the matched parameters of the observed signal are θ(k) = {T (k)1 , T

(k)2 , δω(k)}.

Because the dictionary is made for a discrete set of parameters, there is a discretizationerror on the estimated parameters. But there are also errors in the estimation of theparameters due to the noise on the observed signal. Part of the noise is due to theexperimental equipment, but the main noise source is the pressence of strong RF pulsesthat are used to make a three-dimensional image. This noise is more like a backgroundsignal and is assumed to be statisctically uncorrelated for a single voxel.

5.2. Equivalence of the correlation and least squares

In this section will be shown that for a certain proton density ρ, the least squares andthe correlation matching algorithms are equivalent. Physically, ρ is real valued and non-negative, but it is common in MRI to allow this parameter to absorb additional phaseterms [4]. Therefore, ρ is often allowed to be complex valued, which will be assumed inthis section. In the last section the least squares is defined as

LS(d,Dk) = ||dρ−Dk||2,

24

where d is the observed signal and Dk = B(θk) the signal from the dictionary forparameter set θk. To show the equivalence of the two matching algoritms, multiply theleast squares with |ρ|2 and replace Dk with B(θk). Then by writing this out:

||d− ρB||2 = 〈d− ρB, d− ρB〉= 〈d, d〉 − 2Re(ρ〈B, d〉) + |ρ|2〈B,B〉.

(5.9)

Let δ be a disturbance on ρ. Then |ρ+ δ|2 = |ρ|2 + 2Re(ρδ) + |δ|2. If ρ is replaced withρ+ δ in equation (5.9), the additional terms are

−2Re(δ〈B, d〉) + 2Re(ρδ)〈B,B〉+ |δ|2〈B,B〉. (5.10)

For a stable ρ, this expression must be equal to zero. Assuming δ is small enough suchthat the last term can be neglected, equation (5.10) is zero when

ρ =〈B, d〉||B||2

. (5.11)

If the absolute value is taken, then this is the same as the ρ in equation (5.6) thatis calculated with the correlation. Putting this condition in equation (5.9), the leastsquares becomes

||d− 〈B, d〉||B||2

B||2 = ||d||2 − 2Re(〈B, 〈B, d〉||B||2

d〉) +∣∣∣〈B, d〉||B||2

∣∣∣2||B||2= ||d||2 − 2Re(

〈B, d〉||B||2

〈B, d〉) +|〈B, d〉|2

||B||2

= ||d||2 − |〈B, d〉|2

||B||2.

With the least squares matching algorithm, the parameters that correspond best withthe observed signal d is found by minimalizing this expression over k:

||d||2 − |〈B(θ(k)), d〉|2

||B(θ(k))||2. (5.12)

Divide this expression by ||d||2 and it becomes the same as maximizing this equationover k:

|〈B(θ(k)), d〉|||d|| ||B(θ(k))||

, (5.13)

which is the same as the correlation.

5.3. Implementation of MRF

To asses how well MRF performs, MRF is implemented for the single voxel case. In thisimplementation the proton density ρ is set to 1, so only the parameters T1, T2 and δωwill be estimated for the observed signals.

25

Let ~M(t) = (Mx(t),My(t),Mz(t))T be the magnetization with ~M = (0, 0,M0)

T be themagnetization in equilibrium state. Let the reference frame be rotating with the Larmorfrequency in the transverse plane. Assume for now that the off-resonance δω = 0. Thenby using the solutions of the Bloch equations, the magnetization for t > 0 can be writtenas

~M(t) =

Mx(0)e−t/T2

My(0)e−t/T2

M0 + (Mz(0)−M0)e−t/T1)

=

e−t/T2 0 0

0 e−t/T2 0

0 0 e−t/T1

Mx(0)My(0)Mz(0)

+

00

M0(1− e−t/T1)

.

If the magnetization does not spin with the Larmor frequency, then the magnetizationrotates with an angle of φ = 2πδωt around the z-axis. The rotation matrix

Rz(φ) =

cos(φ) − sin(φ) 0sin(φ) cos(φ) 0

0 0 1

(5.14)

rotates a three dimensional vector around the z-axis with an angle of φ radians. Foroff-resonance δω, the magnetization becomes

~M(t) = Rz(φ(t))

[e−t/T2 0 0

0 e−t/T2 0

0 0 e−t/T1

Mx(0)My(0)Mz(0)

+

00

M0(1− e−t/T1)

]

= Rz(φ(t))

e−t/T2 0 0

0 e−t/T2 0

0 0 e−t/T1

Mx(0)My(0)Mz(0)

+

00

M0(1− e−t/T1)

,

where φ(t) = 2πδωt.For easier notation, define matrices A and B as

Aθ(t) = Rz(2πδωt)

e−t/T2 0 0

0 e−t/T2 0

0 0 e−t/T1

, (5.15)

Bθ(t) =

00

M0(1− e−t/T1)

, (5.16)

where θ = {T1, T2, δω} is the parameter set to show the dependence of the matrices onthe parameters.

An RF pulse that rotates the magnetization with α radians around the y-axis can becalculated with the rotation matrix

Ry(α) =

cos(α) 0 − sin(α)0 1 0

− sin(α) 0 cos(α)

. (5.17)

26

An RF pulse that rotates the magnetization with α radians around the axis y = x tan(φ)can be calculated with the rotation matrix

R(α, φ) = Rz(φ)Ry(α)Rz(−φ). (5.18)

By putting these together, the pulse sequence can be simulated. Let the randompulse sequence consist of N RF pulses with flip angles α = {α1, ..., αN}, axis anglesφ = {φ1, ..., φN} and repetition times TR = {TR1, ..., TRN}. Let the sample timebe TS. The pulse sequence begins with an inversion pulse. For parameter set θ, themagnetization right after the inversion pulse at t = 0 is

~Mθ(0+) =

00−M0

.

Then at the time just before TR1 the magnetization is

~Mθ(TR−1 ) = Aθ(TR1) ∗ ~Mθ(0

+) +Bθ(TR1).

Then RF pulse with an angle of α1 and around axis y = x tan(φ1) is applied and themagnetization right after the RF pulse is

~Mθ(TR+1 ) = R(α1, φ1) ∗ ~Mθ(TR

−1 ).

After time TS the signal Mx + iMy is read

~Mθ(TR1 + TS) = Aθ(TS) ∗ ~Mθ(TR+1 ) +Bθ(TS). (5.19)

The magnetization at the times when the signal is read can be calculated by

~Mθ(τn + TS) =

Aθ(TS) ∗ [R(αn, φn) ∗Aθ(TRn − TS) ∗ ~Mθ(τn−1 + TS) +Bθ(TRn − TS)] +Bθ(TS),

(5.20)

where τn =n∑i=1

TRi and n = 2, ..., N . Then for signal Dk ∈ CN from the dictionary, the

n-th component is equal to

Dk,n = Mx,θ(τn + TS) + iMy,θ(τn + TS). (5.21)

To simulate noise on the observed signal, a disturbance δ ~M is added at the measure-ments of the signal. Since it is assumed that the background noise is uncorrelated, thenoise is normal distributed. The probability density of the normal distribution is

f(x) =1

σ√

2πe−

(x−µ)2

2σ2 , (5.22)

27

where µ is the mean and σ the standard deviation. Let a value generated from thenormal distribution be notated as

X ∼ N (µ, σ2). (5.23)

Then let the magnitude of the disturbance be normal distributed with mean µ = 0 anda magnitude that is proportional to M0:

δM(σ) ∼M0|N (0, σ2)|. (5.24)

The uniform distribution is used to rotate ~M randomly in any direction of the three-dimensional space. The probability density of the uniform distribution with minimumvalue a and maximum value b is

f(x) =

{1b−a a ≤ x ≤ b,0 otherwise

. (5.25)

Let a value from the uniform distribution with minimum value a and maximum value bbe notated as

X ∼ U(a, b). (5.26)

Then the disturbance is generated with this expression:

δ ~M(σ) ∼ Rz(2π ∗ U(0, 1)) Ry(π ∗ U(0, 1))

00

M0|N (0, σ)|

, (5.27)

where Rz and Ry are the rotation matrices from formula (5.14) and (5.17) respectively.So this formula generates a magnetization with an azimuthal angle uniformly distributedbetween 0 and π radians, a polar angle uniformly distributed between 0 and 2π radians,and a normal distributed magnitude multiplied by M0 with mean µ = 0. Then the n-thcomponent of the observed signal d ∈ CN for parameter set θ is

dn = Mx,θ(τn + TS) + δMx(σ) + iMy,θ(τn + TS) + iδMy(σ), (5.28)

where τn =n∑i=1

TRi.

5.4. Statistics

In the implementation of MRF, statistics is used to quantify how well MRF performswhen noise is added to the observed signal. This is done by simulating the signal multipletimes and calculating statistical quantities on the found parameters.

28

Let S be the amount of simulations. Then MRF finds parameter values (T11 , ..., T1S ),(T21 , ..., T2S ) and (δω1, ..., δωS). Let x = (x1, ..., xS) with

xs =

T1sT2sδωs

, (5.29)

for s = 1, ..., S. Then the sample mean of x is

x =

T1T2δω

=1

S

S∑s=1

T1sT2sδωs

=1

S

S∑s=1

xs. (5.30)

The sample covariance matrix of x is defined as

C(x) =1

S − 1

S∑s=1

(xs − x)(xs − x)T . (5.31)

For i, j = 1, 2, 3, the (i, j)-th element of the covariance matrix is given by

Ci,j(x) =1

S − 1

S∑s=1

(xi,s − xi)(xj,s − xj).

The covariance matrix is a symmetric matrix with on the diagonal the variances and onthe off-diagonal the covariances. The standard deviation of x is the square root of thevariance:

σ(x) =

√C1,1√C2,2√C3,3

. (5.32)

According to Taylor [8], the parameters of the observed signal can be estimated asT1T2δω

= x± σ(x) =

T1T2δω

±√C1,1√

C2,2√C3,3

, (5.33)

where the standard deviation indicates the error of estimation.

29

6. Results of the MRF simulations

Three different pulse sequences are compared with each other in the simulations to seewhether a random pulse sequence works better than a non-random pulse sequence. Thefirst one is the non-random pulse sequence which has a constant flip angle, axis angleand repetition time. The second one is the random pulse sequence with random flipangles and repetition times, but a constant axis angle. The last pulse sequence is therandom pulse sequence with random flip angles, random repetition times and randomaxis angles. Both the correlation algorithm and the least squares algorithm are used forevery pulse sequence to see whether one performs better than the other. The simulationsare done with a sequence length of 50 pulses and 500 pulses.

In the simulations M0 is set to 1 and sampling time TS at 5.8 ms. All pulse sequencesuses a dictionary with signal evolutions for the same parameters. The dictionary isbuild up as follow. The parameter T1 ranges from 30 ms to 300 ms in steps of 30 ms, T2ranges from 10 ms to 100 ms in steps of 10 ms, and δω from -5 Hz to 5 Hz in steps of1 Hz. The dictionary contains the signal evolutions of all the possible parameter valuecombinations. So the dictionary is a grid that consists of 10 ∗ 10 ∗ 11 = 1100 signalevolutions. The dictionary is kept small such that the amount of memory to store thedictionary on a computer is limited, but also to keep the simulation time low so thatmany simulations can be done in a reasonable time.

For all experiments, the observed signal has parameters T1 = 165 ms, T2 = 55 ms andδω = 1.5 Hz and noise δ ~M(σ) with σ = 0.5 . Normally, the parameters of the observedsignal is not exact the same as one from the dictionary. That is why the parameters arechosen in-between two values from the dictionary. This will give a discretization errorof 15 ms for T1, 5 ms for T2 and 0.5 Hz for δω. This is why the dictionary is chosen witha constant stepsize such that the discretization error is constant.

To show how much the signal is disturbed by the noise, in figure 6.1 is a signal evolutionwith noise and a signal evolution with the same parameters but without noise shownfor some random pulse sequence. In the two upper pictures is the magnetization of thex-component shown and in the lower two the y-component of the magnetization. Thepictures show only snapshots of the magnetization that is measured at the sample timeTS after each pulse. In these pictures can be seen that the signal with noise is quitedifferent from the signal without noise.

Now, let an observed signal be simulated and matched with a signal from the dictionaryfor some pulse sequence. In figure 6.2 is the value of the matching shown for differentparameters. In the left pictures are the values for the correlation shown and in the rightpictures the values for the least squares. In each picture, one parameter varies while theother two remain fixed. In the blue figures T1 is varied, in the red pictures T2 and in thegreen pictures δω is varied. The parameters vary from all the dictionary values. The

30

Figure 6.1.: A signal evolution without noise and a signal evolution with noise are shownfor some random pulse sequence. Both signals have the same parameters.The noise is δ ~M(0.5). In the upper two pictures the x-component of themagnetization is shown and in the lower two pictures the y-component.

31

correlation algorithm finds the maximum whereas the least squares finds the minimum.Due to the noise on the observed signal, the extremum is not exactly at the true value.In the pictures can be seen that the extrema lies mostly around the true value of theobserved signal. The pictures also shows that values of the matchings goes monotonicto the extremum and then goes monotonic in the other direction. This happens forall three pulse sequences. Because of this, the true extremum point, if the dictionarywas continuous, could be found by fitting a curve through the points. Furthermore,this shows that increasing the density of the dictionary does not cause the matchingalgorithm to match the observed signal with completely different parameters.

Figure 6.2.: The matching algorithms calculate the value of the matching with differentsignals from the dictionary. The observed signal has parameter values T1 =165 ms, T2 = 55 ms and δω = 1.5 Hz. In the left pictures are the valuesfor the correlation shown and in the right pictures the values for the leastsquares. In each picture, one parameter varies while the other two remainfixed. In the blue pictures is T1 varied, in the red pictures T2 and in thegreen pictures is δω varied. The parameters vary from all the dictionaryvalues. The correlation algorithm finds the maximum whereas the leastsquares finds the minimum.

32

6.1. Non-random pulse sequence

For the simulations is the non-random pulse sequence used where the flip angles, axisangles and repetition times are taken constant. The flip angles are set to 45 degrees, theaxis angle to 0 degrees and the repetition time to 15 ms. These values tend to give methe best results. The amount of pulses is set to 50.

Figure 6.3.: The x-component of a few signal evolutions from the dictionary for the non-random pulse sequence. In the blue graphs T1 is varied, in the green graphsT2 is varied and in the red graphs δω is varied. The black lines have thesmallest value of the varying parameter and the brightest colors the largestvalue.

33

Figure 6.4.: The y-component of a few signal evolutions from the dictionary for the non-random pulse sequence. In the blue graphs T1 is varied, in the green graphsT2 is varied and in the red graphs δω is varied. The black lines have thesmallest value of the varying parameter and the brightest colors the largestvalue.

In figure 6.3 is the x-component of a few signal evolutions from the dictionary shownand in figure 6.4 the y-component of the same signal evolutions. The pictures show onlysnapshots of the magnetization that is measured at the sample time TS after each pulse.In each picture, one parameter varies while the other two remain fixed. The fixed valuesare T1 = 150 ms, T2 = 50 ms and δω = 2 Hz. In the blue graphs T1 is varied from 30ms to 270 ms in steps of 60 ms. In the green graphs T2 is varied from 10 ms to 90 ms in

34

steps of 20 ms. And in the red graphs δω is varied from -4 Hz to 4 Hz in steps of 2 Hz.The black lines have the smallest value of the varying parameter and the brightest colorsthe largest value. In the pictures can be seen that different signals evolves a bit differentfrom each other. An important observation is that they all go to an equilibrium valueafter about 20 pulses. Also the symmetry around δω = 0 in the x-component of thesignal can be seen. This is due to the fact that for an axis angle of 0 degrees, the axisaround which the magnetization is rotated by the RF pulses points along the y-axis.

Now the observed signal is simulated and matched with one from the dictionary. Foreach matching algorithm this is simulated for 20 times. In figure 6.5 is shown whichparameter values are chosen for each simulation. In the upper pictures the signals arematched with the correlation algorithm and in the lower pictures with the least squaresalgorithm. Each simulation is indicated with a circle that differs in size. The red crossis the real value of the observed signal. In the left pictures are the matched parametersT1 and T2 shown for each simulation, in the pictures in the middle are the matchedparameters T1 and δω shown, and in the right pictures are the matched parametersT2 and δω shown. In these pictures can be seen that the matched parameters are quitewidely spread. But they lie around the true value and not biased to one side for example.

From these simulations the mean and the covariance matrix of the matched parameterscan be calculated for the correlation and the least squares. The mean and the covariancematrix are

µ =

180671.8

, C =

3410.5 284.2 12.6284.2 832.6 −12.212.6 −12.2 3.3

(correlation)

µ =

17160.51.5

, C =

2851.6 273.2 9.5273.2 531.3 −10.39.5 −10.3 1.9

(least squares).

Then the estimated parameter values with error are

(T1, T2, δω) = (180± 58.4ms, 67± 28.8ms, 1.8± 1.82Hz) (correlation),

(T1, T2, δω) = (171± 53.4ms, 60.5± 23.1ms, 1.5± 1.4Hz) (least squares),

where the errors are calculated by taking the square root of the diagonal elements ofthe covariance matrix. The errors in the estimation of the parameters are quite large.For T1 the error is about 4 times the discretization error, for T2 about 5 times and forδω about 3 times for both the correlation and the least squares. There is not much of adifference between the correlation and least squares, but least squares has a bit smallererror and estimates T2 and δω better than the correlation.

The same is going to be done but with 500 pulses. In figure 6.6 are the matchedparameters shown for 20 simulations where in the upper pictures the correlation is usedand in the lower pictures the least squares. The pictures are quite similar to the pictureswith 50 pulses. The biggest difference is that with the least squares T1 and especiallyδω are estimated much more accurately.

35

Figure 6.5.: The parameter matching for the non-random pulse sequence with 50 pulses.This is done for 20 simulations and each simulation is indicated with a circlethat differs in size. In the upper three pictures the matching is done withthe correlation and in the lower three pictures with the least squares. Thered cross is the real value of the parameters of the observed signal.

The mean and the covariance matrix for these found parameters are

µ =

18458.51.6

, C =

1998.9 −633.2 36.9−633.2 676.6 −13.3

36.9 −13.3 1.4

(correlation),

µ =

16561.51.3

, C =

236.8 39.5 0.0539.5 350.3 −7.320.05 −7.32 0.33

(least squares).

The estimated parameter values with error are

(T1, T2, δω) = (184± 44.7ms, 58.5± 26ms, 1.6± 1.19Hz) (correlation),

(T1, T2, δω) = (165± 15.4ms, 61.5± 18.7ms, 1.3± 0.58Hz) (least squares).

The correlation does not perform much better than with 50 pulses. However, the leastsquares has a much lower error of T1 and δω, which is already seen in the figure 6.6.These errors are almost as small as the discretization error.

36

Figure 6.6.: The parameter matching for the non-random pulse sequence with 500 pulses.This is done for 20 simulations and each simulation is indicated with a circlethat differs in size. In the upper three pictures the matching is done withthe correlation and in the lower three pictures with the least squares. Thered cross is the real value of the parameters of the observed signal.

For the next two pulse sequences, the pictures of the dictionary and the parameterestimation of the correlation and least squares are omitted. This is done because thepictures takes up a lot of space which reduces the readability of the text. These pictures,together with the the pictures from the non-random pulse sequence, are put in appendixB. The pictures are put next together so that they can be easier compared with eachother.

6.2. Random pulse sequence with a constant axis angle

In this section the simulation are done with the random pulse sequence with a constantaxis angle. The flip angles are chosen from a uniform distribution between 30 and 60degrees, the axis angles is set to zero and the repetition times are chosen from a uniformdistribution between 10.5 and 14 ms. Let the pulse sequence consist of 50 pulses. Infigures B.3 and B.4 from appendix B are the x-component and y-component respectively

37

shown for a few signal evolutions from the dictionary. The value of the parameters arethe same as in pictures of the last dictionary. The signals look quite similar to the signalsof the non-random pulse sequence, but they keep oscillating a little bit. There is stilla symmetry in the x-component of the signal around δω = 0. The expectation is thatthe random pulse sequence with constant axis angle performs similar to the non-randompulse sequence.

Now, let the observed signal be simulated and matched with a signal from the dic-tionary for 20 times. The mean and the covariance matrix of the parameters of thesesimulations are

µ =

183532

, C =

3022.1 22.1 26.822.1 516.8 −6.826.8 −6.8 2.2

(correlation),

µ =

172.5581.8

, C =

3114.5 142.1 23.7142.1 258.9 −0.423.7 −0.4 1.1

(least squares).

The estimated values with error of the parameters are

(T1, T2, δω) = (183± 55ms, 53± 22.7ms, 2± 1.49Hz) (correlation),

(T1, T2, δω) = (172.5± 55.8ms, 58± 16.1ms, 1.8± 1.05Hz) (least squares).

The errors are quite big for almost all parameters except δω for the least squares. Theleast squares performs a bit better than the correlation for all parameters because themean of T1 is much closer to the real value of the observed signal and the error in T2 isquite smaller than the error with the correlation.

The matching is also done for 500 pulses for 20 simulations:

µ =

18061.51.8

, C =

2084.2 −868.4 37.9−868.4 602.9 −19.7

37.9 −19.7 1.9

(correlation),

µ =

17156.51.65

, C =

483.2 −64.7 4.58−64.7 318.7 −6.024.58 −6.02 0.45

(least squares).

The estimated values with error of the parameters are

(T1, T2, δω) = (180± 45.7ms, 61.5± 24.6ms, 1.8± 1.36Hz) (correlation),

(T1, T2, δω) = (171± 22ms, 56.5± 17.9ms, 1.65± 0.67Hz) (least squares).

The correlation does not perform better with 500 pulses than 50 pulses, but the leastsquares have an improvement in the error of T1 and δω. These two errors are about assmall as the discretization error. There is no improvement in the estimation of T2 withthe least squares.

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6.3. Random pulse sequence with random axis angles

The random pulse sequenced that is used for the next simulations has the same flipangles and repetition times as the last random pulse sequence, but the axis angles arechosen randomly from a uniform distribution between 0 and 360 degrees. Let the amountof pulses be 50. In figures B.5 and B.6 from appendix B are a few signals evolutionsfrom the dictionary shown with the same setup as the figures of the dictionaries of thelast two pulse sequences. In these pictures can be seen that the signals behaves quite abit different from the signals of the last two pulse sequences. The signals do not go toan equilibrium and there is no symmetry anymore around δω = 0.

The mean and the covariance matrix of the parameters for these simulations is

µ =

175.561.51.2

, C =

1541.8 −572.4 −3.8−572.4 661.8 1−3.8 1 2.2

(correlation),

µ =

175.559

1.35

, C =

973.4 −4.74 6.4−4.74 241.1 0.89

6.4 0.89 1.61

(least squares).

The estimated values with error of the parameters are

(T1, T2, δω) = (175.5± 39.3ms, 61.5± 25.7ms, 1.2± 1.47Hz) (correlation),

(T1, T2, δω) = (175.5± 31.2ms, 59± 15.5ms, 1.35± 1.27Hz) (least squares).

This pulse sequence has errors about two to three times the discretization error. In thisrandom pulse sequence least squares performs also better than the correlation.

The mean and the covariance matrix of the parameters for 20 simulations with 500pulses are

µ =

163.558.51.4

, C =

613.4 −41.8 0.63−41.8 66.1 0.110.63 0.11 0.25

(correlation),

µ =

166.557

1.55

, C =

329.2 52.1 1.552.1 53.7 −0.371.5 −0.37 0.26

(least squares).

The estimated values with error of the parameters are

(T1, T2, δω) = (163.5± 24.8ms, 58.5± 8.1ms, 1.4± 0.5Hz) (correlation),

(T1, T2, δω) = (166.5± 18.1ms, 57± 7.3ms, 1.55± 0.51Hz) (least squares).

This is a big improvement for both the correlation and the least squares. All parame-ters are estimated acurately and all errors are about as big as the discretization error,especially δω. Again, least squares performs better than the correlation.

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6.4. Overview results and discussion

In table 6.1 are the results shown what the estimated parameters and their errors are forall three pulse sequences, for 50 pulses and 500 pulses, and for the correlation and theleast squares algorithms. In this table the true value of the parameter of the observedsignal is subtracted from the esimated parameter value and this is indicated with ∆.

This table shows that the non-random pulse sequence performs about as good asthe random pulse sequence with a constant axis angle for both the correlation and theleast squares and for both amount of pulses. When the correlation is used, these twopulse sequences do not perform better when the amount of pulses are increased to 500.This is expected from the pictures of the dictionary which can be seen in appendix B.After about 20 pulses the signal evolution of the non-random pulse sequence goes to anequilibrium value and the signal evolution of the random pulse sequence with a constantaxis angle oscillates slightly about an equilibrium value. So increasing the length ofthe pulse sequence does not help differentiating the signal evolutions from each other.Because the correlation normalizes the magnitude of magnetization, all signals looks thesame since they are essentially straight lines with different magnitudes. So the correlationkeep having problems in matching the parameters when the amount of pulses is increasedfor these two pulse sequences. The pulse sequences do perform better when the leastsquares is used. Because least squares keeps the magnitude of the signal in account,the different signal evolutions from the dictionary can be distinguished from each other.By increasing the length of the pulse sequences, the magnitude of the observed signalaverages out. This is why knowing the magnitude of the signal helps significantly indetermining the parameters.

Pulse sequence Pulses Matching alg. ∆T1(ms) ∆T2(ms) ∆δω(Hz)

Non-random 50 Correlation 15± 58.4 12± 28.8 1.8± 1.82Constant axis angle 50 Correlation 18± 55 −2± 22.7 0.5± 1.49Random axis angle 50 Correlation 10.5± 39.3 6.5± 25.7 −0.3± 1.47

Non-random 50 Least squares 6± 53.4 5.5± 23.1 0± 1.4Constant axis angle 50 Least squares 7.5± 55.8 3± 16.1 0.3± 1.05Random axis angle 50 Least squares 10.5± 31.2 4± 15.5 −0.15± 1.27

Non-random 500 Correlation 19± 44.7 3.5± 26 0.1± 1.19Constant axis angle 500 Correlation 15± 45.7 6.5± 24.6 0.3± 1.36Random axis angle 500 Correlation −1.5± 24.8 3.5± 8.1 −0.1± 0.5

Non-random 500 Least squares 0± 15.4 6.5± 18.7 −0.2± 0.58Constant axis angle 500 Least squares 6± 22 1.5± 17.9 0.15± 0.67Random axis angle 500 Least squares 1.5± 18.1 2± 7.3 0.5± 0.51

Table 6.1.: The difference in estimated parameter value and the true value of the ob-served signal with their corresponding error are shown in this table for allperformed simulations. The difference is calculated by subtracting the truevalue from the esimated parameter value and this is indicated with ∆.

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The random pulse sequence with random axis angles has a significant improvementin the estimation and error of the parameters when the amount of pulses is increased.This is because the signal evolutions from the dictionary do not go to an equilibriumvalue and keep showing different patterns from each other. So the results shows thatwhen the pulse sequence is long enough, the random pulse sequence with random axisangles becomes better than the random pulse sequence with a constant axis angle andthe non-random pulse sequence.

In the table can be seen that for the random pulse sequence with random axis anglesand 500 pulses, the error is almost as small as the discretization error. The discretizationerror is 15 ms for T1, 5 ms for T2 and 1 Hz for δω. In this case the error can probablybe further reduced by increasing the density of the dictionary.

The mean of T1 is consistently larger than the true value because the signal evolutionsfrom the dictionary with a larger value of T1 are very similar to each other in boththe x-component and the y-component of the signal. So the chances are higher thatthe observed signal is more similar to a larger value of the parameter than a smallervalue. T2 and δω are also estimated consistently too large for the non-random pulsesequence and the random pulse sequence with a constant axis angle. This is probablybecause while the magnitude of the signal increases for a larger parameter, the relativeincreasement in magnitude decreases. So the signals becomes relatively more similar.For the random pulse sequence with random axis angles, T2 is also consistently estimatedtoo big. The reason is less obvious, but it is probably because the signal evolutions ofthe y-component in the dictionary are more simular to each other for larger values of T2.

For all pulse sequences there is a trend that the least squares performs better than thecorrelation. This is probably because the proton density is known, which is now set at 1.Then the correlation and least squares are not equivalent because 〈B,d〉||B||2 is normally not

equal to 1 in the simulations. Least squares takes the magnitude in account whereas thecorrelation normalizes it. This extra information seems to help to match the observedsignal better. So if the proton density is known, then least squares performs probablybetter than the correlation.

In table 6.2 are the covariances of the parameters shown for all three pulse sequences,for 50 pulses and 500 pulses, and for the correlation and the least squares algorithms.The covariance is scaled by dividing it by the stepsize of the parameters in the dictionary.So the covariance of T1 and T2 is divided by 30 ∗ 10 = 300, the covariance of T1 andδω by 30 ∗ 1 = 30, and the covariance of T2 and δω is divided by 10 ∗ 1 = 10. It seemsfrom the table that there is no particular connection in the covariance between differentamount of pulses or matching alrgorithms. The only remarkable observation is that thecovariance for both the non-random pulse sequence and the random pulse sequence withconstant axis angle when the correlation is used, has significant values for all parametercombinations with the same signs. This is probably because the signal evolutions fromthe dictionary for these pulse sequences go to an equilibrium after about 20 pulses. Inthe pictures of the dictionaries can be seen that a larger T1 has a smaller magnitudein the x-component of the signal than a smaller T1. And larger values of T2 and δωhave a larger magnitude in the x-component of the signal than smaller values of T2

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Pulse sequence Pulses Matching alg. T1, T2 T1, δω T2, δω

Non-random 50 Correlation 0.947 0.42 −1.22Constant axis angle 50 Correlation 0.074 0.893 −0.68Random axis angle 50 Correlation −1.908 −0.127 0.1

Non-random 50 Least squares 0.911 0.317 −1.03Constant axis angle 50 Least squares 0.474 0.79 −0.04Random axis angle 50 Least squares −0.016 0.213 0.089

Non-random 500 Correlation −2.111 1.23 −1.33Constant axis angle 500 Correlation −2.895 1.263 −1.97Random axis angle 500 Correlation −0.139 0.021 0.011

Non-random 500 Least squares 0.132 0.002 −0.732Constant axis angle 500 Least squares −0.216 0.153 −0.602Random axis angle 500 Least squares 0.174 0.05 −0.037

Table 6.2.: The covariance of the estimated parameters for all performed simulations.The covariances of different parameters are scaled with each other by dividingthe covariance by the stepsize of the parameters in the dictionary.

and δω. Because the correlation does not take the magnitude of the signal in account,it becomes more likely that the observed signal is matched with a signal of differentmagnitude. When the observed signal is matched with a signal that has a small signalin the x-component, it is more likely going to be matched with a large value of T1 and asmall value of T2 and δω are chosen. But if the observed signal is matched with a signalthat has a large x-component, it is more likely going to be matched with a small valueof T1 and a large value of T2 and δω. Because T1 is chosen small while T2 or δω arechosen large or vice versa, the covariance of T1 and T2 or δω is negative. In the sameway a large T2 and large δω are chosen together and a smal T2 and small δω are chosentogether. So the covariance of T2 and δω is positive. It seems that this effect begins todominate only when signal is in equilibrium for a certain amount of pulses.

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7. Conclusion

In this thesis it is shown whether a random pulse sequence with constant or randomaxis angles performs better than a non-random sequence for MRF and also whether theleast squares matching algorithm or the correlation matching algorithm performs better.Table 6.1 shows that for 50 pulses one pulse sequence does not perform better than theother. If the amount of pulsed is increased to 500, MRF estimates the parameters bestwith the smallest errors when the random pulse sequence with random axis angles is used.When the amount of pulses is increased to 500, the non-random pulse sequence and therandom pulse sequence with a constant axis angle perform only better when the leastsquares is used. Furthermore, the least squares algorithm determines the parametersbetter than the correlation for all three pulse sequences and for both 50 pulses and 500pulses in the case that the proton density is known. There is also a bias of the estimationin parameters to the larger values due to the fact that the signal evolutions from thedictionary are relatively more similar for larger values of the parameters.

For further research the same simulations as in this thesis can be done, but with moresimulations, a denser dictionary and a larger range of values of the parameters. For thisa big enough memory is needed to store the dictionary and a fast enough computer todo the simulations in a reasonable time. From figure 6.2 one sees that it is not neededto look through every signal from the dictionary to find the match. So the matchingalgorithm can be made more efficient to decrease the time of the simulations to somedegree. From the results and the pictures of the dictionaries is concluded that it is bestwhen different signal evolutions have very different patterns from each other. So a goodway to improve MRF is to design an RF pulse sequence such that signal evolutions fordifferent parameters looks as different as possible from each other. Another step couldbe implementing MRF for a higher dimensional case. Not only a good understandingof MRI imaging in more dimensions is needed, MRF in higher dimensions brings alsoadditional problems as the noise from al the other voxels and undersampling. In thepapers by Duerk et al. and Davies et al. can be read more about this.

It is worth to do more research in MRF because of the benefit that it determinesthe spatial variations of the parameters quantitatively which is really useful for identi-fying materials. MRF has especially much potential when deep learning AI are moredeveloped. These AI can learn to recognize patterns and become better at it with everysimulation. There is already research in deep learning for MRI such as in the paper ofArdon et al. [1]. Due to the nature of MRF that matches two signal patterns with eachother, it seems like a logical step to replace the pattern matching algorithm in MRF witha deep learning AI. Deep learning AI could also be potentially used to design RF pulsesequences. With the constant improvement in technology and hardware, MRF will alsolikely keep developing and improving.

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8. Popular summary

Magnetic resonance imaging (MRI) is an imaging technique that uses magnetic fieldsto make pictures of the inside of an object without opening it. In the figure below is apicture of the head that is made with MRI. In such pictures, the different tissues can bedistinguished from each other.

Figure 8.1.: A picture of the insides of the head made with MRI. The different types oftissues can be seen. The pictures comes from https://nl.wikipedia.org/

MRI-scanner.

The benefit of MRI is that it does not use ionizing radiation unlike other imagingtechniques such as CT scan or radiography that uses X-rays. Ionizing radiation damagesthe DNA which can lead to cancer. That is why MRI is widely used in the medical fieldas a diagnostic tool and for medical research.

The working of MRI is based on the working of the nuclear spin. An atomic nucleus hasa nuclear spin which acts like a tiny magnet. Normally, the nuclear spins are randomlyorientated. But when in pressence of a strong external magnetic field, the spins alignwith the magnetic field. Just like a compass needle that aligns with earth’s magneticfield. The speed how fast the spins align with the magnetic field depends on the materialwhere the spins reside in. This dependence can be used to differentiate different typesof materials.

To measure the signal of the spins, MRI uses an sequence of RF pulses. An RF pulsesrotates the spins away from their alignment with the magnetic field. If the RF pulses

44

are chosen cleverly, the type of material can be identified for each position in the objectby measuring the response signal of the spins by this RF pulse sequence. If every typeof material is given a different color, a picture can be made in which the different typeof materials can be distinguished from each other.

A method to identify the materials is with magnetic resonance fingerprinting (MRF).MRF measures the signal of the spins of known materials when some RF pulse sequenceis used. Like the uniqueness of a fingerprint, the measured signal of the spins is unique fordifferent materials. Then an unknown object can be identified by matching its measeredsignal with one of the known signals. This can be done for every position of the unknownobject to create a full image.

45

A. Matlab code

The main codes that are used for the simulations consists of generating the pulse se-quence, simulating a signal, generating the dictionary, and the matching algorithmscorrelation and least squares.

This function generates the random pulse sequence with repetition times, TRrandom,flip angles, fliprandom, and axis angles, axisrandom, for n pulses. By using seeds, thesame random pulse sequence will be generated.

func t i on [ TRrandom , f l iprandom ] = randomsequence (n)stream1 = RandStream ( ’ mrg32k3a ’ , ’SEED’ , 1 ) ;stream2 = RandStream ( ’ mrg32k3a ’ , ’SEED’ , 2 ) ;stream3 = RandStream ( ’ mrg32k3a ’ , ’SEED’ , 3 ) ;ang1 = ( 3 0 . 0 / 1 8 0 . 0 )∗ pi ; % min r o t a t i o n ang le in rad iansang2 = ( 6 0 . 0 / 1 8 0 . 0 )∗ pi ; % max r o t a t i o n ang le in rad iansaa = ang2−ang1 ;f l iprandom = rand ( stream2 , [ n , 1 ] ) ∗ aa+ang1 ;axisrandom = rand ( stream3 , [ n , 1 ] ) ∗ 2 ∗ pi ;TRmax = 14 ; % ms .TRmin = 1 0 . 5 ; % ms .TRrandom = rand ( stream1 , [ n , 1 ] ) ∗ (TRmax−TRmin)+TRmin ;end

Rz is the rotation matrix that rotates a three-dimensional vector with an angle of phiradians around the z-axis.

f unc t i on Rz = zro t ( phi )Rz = [ cos ( phi ) −s i n ( phi ) 0 ; s i n ( phi ) cos ( phi ) 0 ; 0 0 1 ] ;end

Ry is the rotation matrix that rotates a three-dimensional vector with an angle of phiradians around the y-axis.

f unc t i on Ry = yrot ( phi )Ry = [ cos ( phi ) 0 s i n ( phi ) ; 0 1 0 ; −s i n ( phi ) 0 cos ( phi ) ] ;end

46

Rth rotates a three-dimensional vector with an angle of φ radians around the axisy = x tan(theta).

f unc t i on Rth = throt ( phi , theta )Rz = zro t (− theta ) ;Rx = xrot ( phi ) ;Rth = inv (Rz)∗Rx∗Rz ;end

This functions generates matrices Afp ∈ R3×3 and Bfp ∈ R3 to calculate the mag-netization in a reference frame that rotates with the Larmor frequency ~M(t + T ) =Afp ∗M(t) +Bfp after time T for parameters T1 = T1, T2 = T2 and df = δω.

f unc t i on [ Afp , Bfp ] = f r e e p r e c e s s (T, T1 , T2 , df )phi = 2∗ pi ∗df ∗T/1000 ; % rad ians .E1 = exp(−T/T1 ) ;E2 = exp(−T/T2 ) ;Afp = zro t ( phi ) ∗ [ E2 0 0 ;0 E2 0 ;0 0 E1 ] ;Bfp = [ 0 0 1−E1 ] ’ ;end

This function simulates the signal evolution for parameters T1 = T1, T2 = T2 anddf = δω when a random pulse sequence with repetition times, TRrandom, flip angles,fliprandom, and axis angles, axisrandom. After each RF pulse at sample time, TS, thesignal is measured. Noise is added from a normal distribution to the x- and y-componentof the measured signal.

f unc t i on M = Simrandomseq (T1 , T2 , TS, df , TRrandom , f l iprandom , axisrandom , no i s e )M = ze ro s (3 , s i z e (TRrandom,1 )+1) ;M( : , 1 ) = [ 0 ; 0 ; −1]; % i n v e r s i o n pu l s e app l i ed to [ 0 ; 0 ; 1 ][ Ats , Bts ] = f r e e p r e c e s s (TS, T1 , T2 , df ) ;M3 = Ats∗M(: ,1)+ Bts ;K = s i z e (TRrandom , 1 ) ;f o r k=1:K

f l i p = throt ( f l iprandom ( k ) , axis irandom ( k ) ) ;[ Atr , Btr ] = f r e e p r e c e s s (TRrandom( k)−TS, T1 , T2 , df ) ;M3 = Atr∗M3+Btr ;M3 = Ats∗ f l i p ∗M3+Bts ;d i s turbance = zro t (2∗ pi ∗ rand )∗ yrot ( p i ∗ rand ) ∗ [ 0 ; 0 ; normrnd (0 , no i s e ) ] ;

M( : , k+1) = M3+di s turbance ;end

47

This function generates a dictionary with I∗J∗K signal evolutions for T1 = {T1,1, ...T1,I},T2 = {T2,1, ...T2,J} and df = {δω1, ...δωK}. Pulse sequence with repetition times, TR,flip angles, flip, and axis angles, axis, are used.

func t i on d i c t i o n a r y = makedict ionary (T1 , T2 , df , TS,TR, f l i p , a x i s )I = length (T1 ) ;J = length (T2 ) ;K = length ( df ) ;n = length (TR) ;d i c t i o n a r y = ze ro s ( I ∗J∗K, n ) ;f o r k=1:K

f o r j =1:Jf o r l =1: I

M = Simrandomseq (T1( l ) ,T2( j ) ,TS, df ( k ) ,TR, f l i p , ax i s , 0 ) ;d i c t i o n a r y ( l +(j −1)∗ I+(k−1)∗ I ∗J , : ) = M( 1 , 2 : n+1) + 1 i ∗M( 2 , 2 : n+1);

endend

endend

This is the matching algorithm that uses the correlation. It matches an observed signal,value, with a signal from the dictionary. It immediately retrieves the parameters T1m,T2m and dfm. cor is the value that the correlation assigns to every parameter set.

f unc t i on [T1m,T2m, dfm , cor ] = maxinproduct (T1 , T2 , df , value , d i c t i o n a r y )max = 0 ;index = 1 ;nv = norm( value ) ;cor = ze ro s ( s i z e ( d i c t i onary , 1 ) , 1 ) ;% f i n d match by maximal is ing corf o r k=1: s i z e ( d i c t i onary , 1 )

data = d i c t i o n a r y (k , : ) ;cor (k , 1 ) = abs ( data∗value ’ ) / ( norm( data )∗nv ) ;i f cor (k , 1 ) > max

max = cor (k , 1 ) ;index = k ;

endend% look ing up the matching T1 , T2 and o f f−resonance f o r t h i s indexI = length (T1 ) ;J = length (T2 ) ;r = f l o o r ( ( index −1)/( I ∗J ) ) ;d f index = r +1;dfm = df ( r +1);

48

index = index − r ∗ I ∗J ;s = f l o o r ( ( index−1)/ I ) ;T2index = s +1;T2m = T2( s +1);index = index − s ∗ I ;T1index = index ;T1m = T1( index ) ;K = length ( df ) ;cor = reshape ( inproduct , I , J ,K) ;end

This is the matching algorithm that uses the least squares. It matches an observed signal,value, with a signal from the dictionary. It immediately retrieves the matched parametersT1m, T2m and dfm. ls is value the the least squares assigns to every parameter set.

f unc t i on [ T1index , T2index , df index , l s ] = minls (T1 , T2 , df , value , d i c t i o n a r y )min = i n f ;index = 1 ;l s = ze ro s ( s i z e ( d i c t i onary , 1 ) , 1 ) ;

% f i n d match by min ima l i s ing l sf o r k=1: s i z e ( d i c t i onary , 1 )

data = d i c t i o n a r y (k , : ) ;l s (k , 1 ) = norm( data−value )ˆ2/ s i z e ( value , 2 ) ;i f l s (k , 1 ) < min

min = l s (k , 1 ) ;index = k ;

endend% look ing up the matching T1 , T2 and o f f−resonance f o r t h i s indexI = length (T1 ) ;J = length (T2 ) ;r = f l o o r ( ( index −1)/( I ∗J ) ) ;d f index = r +1;dfm = df ( r +1);index = index − r ∗ I ∗J ;s = f l o o r ( ( index−1)/ I ) ;T2index = s +1;T2m = T2( s +1);index = index − s ∗ I ;T1index = index ;T1m = T1( index ) ;K = length ( df ) ;l s = reshape ( l s , I , J ,K) ;end

49

B. Figures of the results

To compare the figures more easily that are made in the results, the figures of thedictionary and the parameter estimation are put together. In the next six figures are thesignal evolutions for a few different parameter values from the dictionaries shown for 50pulses. The first two figures are the signal evolutions for the non-random pulse sequenceshown, in the next two figures the signal evolutions for the random pulse sequence withconstant axis angle, and in the last two figures are the signal evolutions for the randompulse sequence with random axis angles shown. In the first of figure of each pulsesequence is the x-component of the signal evolutions shown and in the second figure ofeach pulse sequence is the y-component of the signal evolutions shown. The picturesshow only snapshots of the magnetization that is measured at the sample time TS = 5.8ms after each pulse. In each picture, one parameter varies while the other two remainfixed. The fixed values are T1 = 150 ms, T2 = 50 ms and δω = 2 Hz. In the blue graphsT1 is varied from 30 ms to 270 ms in steps of 60 ms. In the green graphs T2 is variedfrom 10 ms to 90 ms in steps of 20 ms. And in the red graphs δω is varied from -4 Hz to4 Hz in steps of 2 Hz. The black lines have the smallest value of the varying parameterand the brightest colors the largest value.

In the next six figures is the parameter matching of the observed signal shown. Theparameter is matched 20 times for each matching algorithm and is indicated with acircle that differs in size. The red cross is the real value of the observed signal, whichis T1 = 165 ms, T2 = 55 ms and δω = 1.5 Hz. In the upper pictures the correlationis used and in the lower pictures the least squares is used. In the first two picturesthe non-random pulse sequence is used, in the next two pictures is the random pulsesequence with constant axis angle used, and in the last two pictures is the random pulsesequence with random axis angles used. In the first picture of each pulse sequence it isdone for 50 pulses and in the second picture for 500 pulses. In the left pictures are thematched parameters T1 and T2 shown for each simulation, the pictures in the middle thematched parameters T1 and δω, and in the right pictures are the matched parametersT2 and δω shown.

50

Figure B.1.: The x-component of a few signal evolutions from the dictionary for the non-random pulse sequence. In the blue graphs T1 is varied, in the green graphsT2 is varied and in the red graphs δω is varied. The black lines have thesmallest value of the varying parameter and the brightest colors the largestvalue.

51

Figure B.2.: The y-component of a few signal evolutions from the dictionary for the non-random pulse sequence. In the blue graphs T1 is varied, in the green graphsT2 is varied and in the red graphs δω is varied. The black lines have thesmallest value of the varying parameter and the brightest colors the largestvalue.

52

Figure B.3.: The x-component of a few signal evolutions from the dictionary for therandom pulse sequence with a constant axis angle. In the blue graphs T1 isvaried, in the green graphs T2 is varied and in the red graphs δω is varied.The black lines have the smallest value of the varying parameter and thebrightest colors the largest value.

53

Figure B.4.: The y-component of a few signal evolutions from the dictionary for therandom pulse sequence with a constant axis angle. In the blue graphs T1 isvaried, in the green graphs T2 is varied and in the red graphs δω is varied.The black lines have the smallest value of the varying parameter and thebrightest colors the largest value.

54

Figure B.5.: The x-component of a few signal evolutions from the dictionary for therandom pulse sequence with random axis angles. In the blue graphs T1 isvaried, in the green graphs T2 is varied and in the red graphs δω is varied.The black lines have the smallest value of the varying parameter and thebrightest colors the largest value.

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Figure B.6.: The y-component of a few signal evolutions from the dictionary for therandom pulse sequence with ranxom axis angles. In the blue graphs T1 isvaried, in the green graphs T2 is varied and in the red graphs δω is varied.The black lines have the smallest value of the varying parameter and thebrightest colors the largest value.

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Figure B.7.: The parameters is matched with the observed signal for the non-randompulse sequence with 50 pulses. This is done for 20 simulations and eachsimulation is indicated with a circle that differs in size. In the upper 3pictures this is done with the correlation and in the lower 3 pictures withthe least squares. The red cross is the real value of the parameters of theobserved signal.

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Figure B.8.: The parameters is matched with the observed signal for the non-randompulse sequence with 500 pulses. This is done for 20 simulations and eachsimulation is indicated with a circle that differs in size. In the upper 3pictures this is done with the correlation and in the lower 3 pictures withthe least squares. The red cross is the real value of the parameters of theobserved signal.

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Figure B.9.: The parameters is matched with the observed signal for the random pulsesequence with a constant axis angle and 50 pulses. This is done for 20simulations and each simulation is indicated with a circle that differs insize. In the upper 3 pictures this is done with the correlation and in thelower 3 pictures with the least squares. The red cross is the real value ofthe parameters of the observed signal.

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Figure B.10.: The parameters is matched with the observed signal for the random pulsesequence with a constant axis angle and 500 pulses. This is done for 20simulations and each simulation is indicated with a circle that differs insize. In the upper 3 pictures this is done with the correlation and in thelower 3 pictures with the least squares. The red cross is the real value ofthe parameters of the observed signal.

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Figure B.11.: The parameters is matched with the observed signa for the random pulsesequence with random axis angles and 50 pulses. This is done for 20simulations and each simulation is indicated with a circle that differs insize. In the upper 3 pictures this is done with the correlation and in thelower 3 pictures with the least squares. The red cross is the real value ofthe parameters of the observed signal.

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Figure B.12.: The parameters is matched with the observed signal for the random pulsesequence with random axis angles and 500 pulses. This is done for 20simulations and each simulation is indicated with a circle that differs insize. In the upper 3 pictures this is done with the correlation and in thelower 3 pictures with the least squares. The red cross is the real value ofthe parameters of the observed signal.

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Bibliography

[1] Ardon, R., Bertrand, H., Bloch, I. & Perrot, M, Classification of MRI data usingdeep learning and Gaussian process-bosed model selection, ArXiv, https://arxiv.org/pdf/1701.04355.pdf, January 2017.

[2] Brown, R. W., Cheng, Y. N., Haacke, E. M., Thompson, M. R. & Venkatesan, R.,Magnetic Resonance Imaging: Physical Principles and Sequence Design, 2nd edition,Wiley Blackwell, 2014.

[3] Constantinides, C., Magnetic Resonance Imaging: The Basics, CRC Press, 2014.

[4] Davies, M, Puy, G., Vandergheynst, P. & Wiaux, Y., A Compressed Sensing Frame-work for Magnetix Resonance Fingerprinting, Siam J. Imaging Sciences, 7(4), 2623-2656, December 2016.

[5] Duerk, J. L., Griswold, M. A., Gulani, V., Liu, K., Ma, D., Seiberlich, N. & Sunshine,J. L., Magnetic resonance fingerprinting, Nature, 495, 187-193, March 2013.

[6] Griffiths, D. J., Introduction to Electrodynamics, 4th edition, Pearson, 2014.

[7] Oldendorf, W. & Oldendorf W., Basics of Magnetic Resonance Imaging, MartinusNijhoff, 1988.

[8] Taylor, J. R., An Introduction to Error Analysis, 2nd edition, University ScienceBooks, 1997.

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