Synchronous Multi-Directional Motion Encoding in Magnetic ... · of the nuclear magnetic resonance signal, and applying relevant mechanical wave theory, a map of the complex shear

Synchronous Multi-Directional Motion Encoding

in Magnetic Resonance Elastography

BY

DAVID A. BURNS

B.S., University of Illinois, Urbana-Champaign, 2010

THESIS

Submitted as partial fulfillment of the requirements

for the degree of Master of Science of Mechanical Engineering

in the Graduate College of the

University of Illinois at Chicago, 2014

Chicago, Illinois

Defense Committee:

Dieter Klatt, Chair and Advisor

Thomas J. Royston

Michael J. Scott

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ACKNOWLEDGMENTS

I would like to thank the members of my thesis defense committee; Dr. Dieter Klatt, Dr. Thomas

Royston, and Dr. Michael J. Scott; for their support and assistance.

Additionally, I am sincerely thankful to the following individuals for their tireless support during the

course of my research: Steve Kearney, Spencer Brinker, Altaf Khan, Dr. Weiguo Li, Dr. Andrew Larson,

Dr. Daniel Procissi, Sol Misener, Mark Brown, Andrew Gordon, and Dr. Kaya Yasar.

Much of the background explanation in this work follows the direction of Dr. Dieter Klatt’s “MR

Elastography” and “Advances in MR Elastography” courses, held at the University of Illinois at Chicago

(Fall 2013 – Spring 2014). In completing this thesis, I am greatly indebted to his intensive and succinct

coverage of, and patient guidance through, an enormous body of knowledge.

DAB

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TABLE OF CONTENTS

CHAPTER PAGE

I. INTRODUCTION……………………………………………………………………... 1

II. CONCEPTUAL FRAMEWORK……………………………………………………… 3

A. Magnetic Resonance Imaging Background……………………………………….. 3

1. Nuclear Magnetic Resonance…………………………………………….. 3

2. Magnetic Field Gradients………………………………………………… 5

3. Signal Echoes…………………………………………………………….. 6

4. Pulse Sequence Diagrams………………………………………………… 8

B. Mechanical Wave Theory Background…………………………………………… 13

1. Equation of Motion of a Continuum…………………………………….... 13

2. Mechanical Stiffness Retrieval…………………………………………… 16

C. Magnetic Resonance Elastography Background………………………………….. 18

1. Phase Accumulation……………………………………………………… 19

2. Magnetic Resonance Elastography Pulse Sequence……………………… 27

3. Data Sampling……………………………………………………………. 30

III. MOTIVATION………………………………………………………………………... 32

A. Three-dimensional Encoding……………………………………………………… 32

B. Selective spectral Displacement Projection - Magnetic Resonance Elastography... 32

C. SampLe Interval Modulation – Magnetic Resonance Elastography……………… 38

IV. METHODS…………………………………………………………………………….. 44

A. Spin Echo Programming…………………………………………………………... 44

B. Magnetic Resonance Elastography Programming……………………………….... 46

C. Selective spectral Displacement Projection Programming………………………... 53

D. SampLe Interval Modulation Programming………………………………………. 54

E. Frequency Correction……………………………………………………………... 55

V. EXPERIMENTAL SETUP……………………………………………………………. 59

A. Physical Vibration Source………………………………………………………… 59

B. Experimental Gel Sample…………………………………………………………. 60

C. Sequence Parameters……………………………………………………………… 62

VI. RESULTS……………………………………………………………………………… 65

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TABLE OF CONTENTS (continued)

CHAPTER PAGE

VII. DISCUSSION…………………………………………………………………………. 70

A. Limitations………………………………………………………………………… 70

B. Conclusion………………………………………………………………………… 71

CITED LITERATURE………………………………………………………………… 72

VITA…………………………………………………………………………………... 74

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LIST OF TABLES

TABLE PAGE

I. TIME-DOMAIN FUNCTIONS AND THEIR FREQUENCY-DOMAIN

COUNTERPARTS, OBTAINED VIA THE FOURIER TRANSFORM………… 15

II. SELECTIVE SPECTRAL DISPLACEMENT PROJECTIN – MAGNETIC

RESONANCE ELASTOGRAPHY PARAMETERS, INCLUDING MOTION-

ENCODING GRADIENT FREQUENCY AND NUMBER OF CYCLES FOR

EACH PROJECTION……………………………………………………………... 35

III. SELECTIVE SPECTRAL DISPLACEMENT PROJECTIN – MAGNETIC

RESONANCE ELASTOGRAPHY PARAMETERS COMPATIBLE WITH

TRAPEZOIDAL MOTION-ENCODING GRADIENTS, INCLUDING

FREQUENCY AND NUMBER OF CYCLES FOR EACH PROJECTION……... 37

IV. MOTION-ENCODING GRADIENT START TIMES FOR SAMPLE

INTERVAL MODULATION – MAGNETIC RESONANCE ELASTOGRAPY

WITH VIBRATION FREQUENCY AND NUMBER OF OFFSETS,

WITHOUT AND WITH CONSIDERATION OF THE SYMMETRY AND

PERIODICITY OF MOTION-ENCODING GRADIENTS………........................ 42

V. MAGNETIC RESONANCE IMAGING PARAMETERS USED FOR ALL

MAGNETIC RESONANCE ELASTOGRAPHY SCANS……………………….. 62

VI. MAGNETIC RESONANCE ELASTOGRAPHY PARAMETERS FOR ONE-

DIMENSIONAL SCANS IN THE SLICE, READ, AND PHASE

DIRECTIONS, AND FOR THE SAMPLE INTERVAL MODULATION

SCAN……………………………………………………………………………… 63

VII. MAGNETIC RESONANCE ELASTOGRAPHY PARAMETERS FOR ONE-

DIMENSIONAL SCANS IN THE SLICE, READ, AND PHASE

DIRECTIONS, AND FOR THE SELECTIVE SPECTRAL

DISPLACEMENT PROJECTION SCAN………………………………………... 64

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LIST OF FIGURES

FIGURE PAGE

1. The response of magnetic spin vectors with no external field, with an external magnetic

field applied, during excitation by an RF pulse, and emitting a Free Induction Decay

signal during relaxation …………………………………………………………………… 5

2. The state of a macroscopic magnetization vector after excitation by a 90-degree RF

pulse, after magnetization vectors have dephased due to local field inhomogeneities at

one-half echo time, after application of a 180-degree refocusing RF pulse, and after

magnetization vectors have rephrased to produce a signal echo at echo time……………... 7

3. The Nuclear Magnetic Resonance signal intensity decay during Free Induction Decay,

during a Spin Echo, and during a Gradient Echo………………………………………….. 8

4. The slice, read, and phase directions as they relate to a specimen in a Magnetic

Resonance Imaging scanner, to the acquired k-space data, and to an image produced

through the inverse two-dimensional Fourier Transform of such k-space data …………… 9

5. A Gradient Echo Magnetic Resonance Imaging pulse sequence with echo time defined…. 11

6. A Spin Echo Magnetic Resonance Imaging pulse sequence with both half-echo times

defined……………………………………………………………………………………… 13

7. A zeroth-moment-nulled function K(t) with two periods of oscillation and amplitude

of A. ……………………………………………………………………………………….. 20

8. The encoding efficiency of sinusoidal motion-encoding gradients for vibration

frequencies surrounding the motion-encoding gradient frequency. ……………………… 23

9. A realistic trapezoidal motion-encoding gradient with finite ramp up, ramp down, and

flat top times. ……………………………………………………………………………… 24

10. A Gradient Echo Magnetic Resonance Elastography pulse sequence diagram with echo

time defined. ………………………………………………………………………………. 27

11. A Spin Echo Magnetic Resonance Elastography pulse sequence diagram with both

half-echo times and the motion-encoding gradient gap defined. ………………………….. 29

12. Motion-encoding gradients applied at different starting phases of the physical vibration

to produce different snapshots of the propagating wave. …………………………………. 31

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LIST OF FIGURES (continued)

FIGURE PAGE

13. Pulse sequence diagram for a Selective spectral Displacement Projection – Magnetic

Resonance Elastography sequence with motion-encoding gradient frequencies of

500, 1000, and 1500 Hertz and cycle numbers of 1, 2, and 3 cycles, respectively. ………. 35

14. Pulse sequence diagram for a Selective spectral Displacement Projection – Magnetic

Resonance Elastography sequence with motion-encoding gradient frequencies of

500, 1000, and 1500 Hertz and cycle numbers of 3, 6, and 9 cycles, respectively. ………. 36

15. Pulse sequence diagram for a SampLe Interval Modulation – Magnetic Resonance

Elastography sequence with all motion-encoding gradient offsets superimposed. ……….. 41

16. Each of eight motion-encoding gradient start times in a SampLe Interval Modulation -

Magnetic Resonance Elastography sequence depicted in real time and in discretized

time. ……………………………………………………………………………………….. 43

17. One sGRAD_PULSE_SIN structure with Amplitude, Ramp Up Time, Ramp Down

Time, and Duration defined. ……………………………………………………………… 47

18. One period of a sinusoidal motion-encoding gradient composed of two

sGRAD_PULSE_SIN structures with opposite amplitudes. ……………………………… 48

19. One sGRAD_PULSE structure with Amplitude, Ramp Up Time, Ramp Down Time,

and Duration defined. ……………………………………………………………………… 49

20. One period of a trapezoidal motion-encoding gradient composed of two

sGRAD_PULSE structures with opposite amplitudes. ……………………………………. 50

21. Quarter-period motion-encoding gradient pulses defined for the leading end of a

sinusoidal motion-encoding gradient, the trailing end of a sinusoidal motion-encoding

gradient, and either end of a trapezoidal gradient. ………………………………………… 51

22. One period of a flow-compensated motion-encoding gradient composed of two

quarter-pulses and one half-pulse for sinusoidal and trapezoidal motion-encoding

gradients. …………………………………………………………………………………... 51

23. Applied frequency versus input frequency for motion-encoding gradients when using

fSDSRoundUpGRT() and fSDSRoundDownGRT() rounding functions. ………………… 57

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LIST OF FIGURES (continued)

FIGURE PAGE

24. Applied frequency versus input frequency for motion-encoding gradients when

alternating use of both fSDSRoundUpGRT() and fSDSRoundDownGRT() rounding

functions. …………………………………………………………………………………... 57

25. Maximum erroneous frequency trend lines for motion-encoding gradients when using

fSDSRoundUpGRT(), fSDSRoundDownGRT(), and alternating both rounding

functions……………………………………………………………………………………. 58

26. Experimental setup to produce physical vibration in test sample. ………………………… 59

27. Schematic for experimental inhomogeneous gel sample. …………………………………. 60

28. Orientation of vibration actuation for experimental gel sample as it relates to slice

direction in scanner. ………………………………………………………………………. 61

29. Acquired displacement images for SampLe Interval Modulation – Magnetic Resonance

Elastography sequence in slice, read, and phase directions, and for one-dimensional

Magnetic Resonance Elastography sequences in slice, read, and phase directions. ………. 66

30. Calculated complex shear modulus images using curl-operator inversion method for

SampLe Interval Modulation – Magnetic Resonance Elastography and one-dimensional

Magnetic Resonance Elastography. ……………………………………………………….. 67

31. Acquired displacement images for Selective spectral Displacement Projection –

Magnetic Resonance Elastography sequence in slice, read, and phase directions,

and for one-dimensional Magnetic Resonance Elastography sequences in slice,

read, and phase directions. ………………………………………………………………... 68

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LIST OF ABBREVIATIONS

1D One-dimensional

3D Three-dimensional

ADC Analog-to-Digital Converter

.CPP File extension for programs designed using the C++ language

FID Free Induction Decay

FOV Field of View

GE Gradient Echo

IDEA Integrated Development Environment for Applications (Siemens Software)

MRE Magnetic Resonance Elastography

MRI Magnetic Resonance Imaging

NMR Nuclear Magnetic Resonance

RF Radio Frequency

RTE Real-Time Event

RTEB Real-Time Event Block

SDP Spectral selective Displacement Projection

SE Spin Echo

SLIM SampLe Interval Modulation

TE Echo Time

TE/2 One-half Echo Time

TTL Transistor-transistor logic

TR Repetition Time

x

SUMMARY

Magnetic Resonance Elastography is a non-invasive method of acquiring biological tissue

mechanical stiffness data. Such data is useful in the detection of a number of diseases and medical

conditions. By applying a physical vibration to a tissue sample, encoding this displacement into the phase

of the nuclear magnetic resonance signal, and applying relevant mechanical wave theory, a map of the

complex shear modulus of a tissue can be produced.

Conventionally, three-dimensional displacement data requires three subsequent Magnetic

Resonance Elastography scans. The novel techniques of Selective spectral Displacement Projection and

SampLe Interval Modulation, recently developed at the University of Illinois at Chicago, encode three-

dimensional displacement data in a single scan, thereby decreasing the total required time by a factor of

three. Such an improvement is relevant to a clinical application, where the delay between subsequent

scans can include alterations to the physiological state and bulk motion of the experimental target,

potentially resulting in inaccurate three-dimensional data. Further, the rapid data acquisition may improve

clinical acceptance of MRE.

In this study, Selective spectral Displacement Projection and SampLe Interval Modulation pulse

sequences were designed using the previously-untested system of a Bruker 7T Clinscan MRI Scanner

operating with Siemens IDEA VB15 software. The novel sequences were tested with scans of an

inhomogeneous EcoFlex gel sample. The acquired displacement images and calculated shear modulus

elastograms of the novel sequences were found similar to those of related conventional Magnetic

Resonance Elastography scans, verifying the effectiveness of the novel sequences. Additionally, adding to

the existing implementations of the novel sequences, this study serves as further evidence that these

sequences can be implemented successfully independent of the modality.

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I. INTRODUCTION

One of the most prevalent diagnostic techniques available to doctors is simple manual palpation.

By touching and deforming a tissue, the mechanical properties of the tissue are being tested. A tissue’s

stiffness can, in many cases, be related to its health. Several illnesses have been shown to correlate to a

change in an affected tissue’s mechanical stiffness. One well-defined case of this effect is found in human

diffuse liver disease. As the disease progresses, increasing liver fibrosis has been shown to cause an

increase in the liver’s mechanical stiffness, boosting both shear and Young’s moduli [1] [2].

Traditionally, a tissue’s stiffness is tested through tactile methods: the physical touching and

manipulation of a tissue by a doctor. This method has several drawbacks. Palpation cannot access regions

deep in the body, making this technique unusable for many organs. Furthermore, the method is

subjective; its accuracy is highly dependent on the experience and judgment of the doctor, and offers no

quantitative results.

Elastography is the mapping of the mechanical properties of a material. Magnetic Resonance

Elastography (MRE) is the acquisition of this mechanical data through the non-invasive technique of

Magnetic Resonance Imaging (MRI). Through the application of both forced vibrations and carefully

selected oscillating magnetic gradients to a tissue during an MRI scan, physical shear waves propagating

through the tissue can be detected and measured. These waves can be analyzed through relevant

mechanical models to calculate mechanical stiffness values for the tissue. In short, MRE allows for the

quantitative measurement of the mechanical properties of a material, giving doctors another indicator for

a patient’s health.

MRE requires three distinct steps. The first is the introduction of a physical vibration to a target

material. The second is the measurement of this vibration as it propagates through the material using

MRI. Finally, the spatio-temporal characteristics of this vibration are used to calculate the mechanical

properties of the material.

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When a physical vibration is applied to the target material, the protons embedded in the material’s

atoms are displaced through harmonic oscillation. MRE uses an MRI machine as a tool to visualize this

displacement. The movement must be encoded into the Nuclear Magnetic Resonance (NMR) signal of the

protons.

Accurate mechanical stiffness analysis requires three-dimensional displacement data, but

conventional MRE scans acquire one direction of displacement data at a time. To acquire a fully three-

dimensional data set using conventional MRE, three subsequent scans must be performed, which is a

lengthy process and risks inaccuracy due to potential changes in the physiological state and bulk motion

of the target between scans.

Two novel techniques for acquiring three directions of displacement data simultaneously are

implemented in this study: Selective spectral Displacement Projection (SDP) – MRE, introduced by

Yasar et al. [3], and SampLe Interval Modulation (SLIM) – MRE, introduced by Klatt et al [4]. These

novel techniques have previously been introduced and tested using Agilent MRI software on an animal

scanner [3] [4], and Siemens IDEA VB17 on a 3T human scanner [5], but the modality-independence of

the novel MRE techniques has not yet been verified for all systems.

The purpose of this study is to apply the novel three-dimensional MRE techniques of SDP-MRE

and SLIM-MRE to a previously-untested Siemens IDEA VB15 pulse sequencing system on a 7T Bruker

Clinscan MRI scanner. By comparing the displacement images and calculated mechanical stiffness maps

generated by the novel techniques to those of conventional MRE sequences, the successful

implementation of the novel three-dimensional MRE sequences as time-saving innovations is verified.

Additionally, success in applying the novel techniques to another MRI pulse sequencing software further

confirms their modality-independence.

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II. CONCEPTUAL FRAMEWORK

A. Magnetic Resonance Imaging Background

1. Nuclear Magnetic Resonance

NMR is a process dependent on the naturally-occurring spin of protons. Because protons

are charged particles, their spin produces an associated magnetic moment. The collection of protons

spinning in a tissue produces a measureable macroscopic magnetization vector. The dynamics of this

macroscopic magnetization vector are described by the Bloch Equations, summarized in equations 1.1,

1.2, and 1.3:

(1.1)

(1.2)

(1.3)

where M(t) is the macroscopic magnetization vector with x-, y-, and z-direction

components of Mx(t), My(t), and Mz(t), respectively, B(t) is the magnetic field experienced by the

magnetization vector, γ is the Gyromagnetic Ratio (a property unique to the nucleus), T2 is the transversal

(x- and y-direction) relaxation time, T1 is the longitudinal (z-direction) relaxation time, and M0 is the

macroscopic magnetization at thermal equilibrium [6].

One important result of the Bloch equation is that when a magnetic field is applied to a

tissue, the material’s naturally misaligned magnetization vectors tend to align along and undergo

precession about the direction of the field. In the presence of a magnetic field vector B, nuclear spins

tend to undergo precession about the B axis with the “Larmor Frequency”, given by equation 1.4:

(1.4)

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where ω is the Larmor Frequency [6].

An NMR signal is generated by first exciting the magnetization vectors with a radio-

frequency (RF) pulse. This pulse, when produced with a frequency equal to the Larmor Frequency and in

a direction perpendicular to the applied magnetic field vector, causes the target magnetization vectors to

misalign with the applied magnetic field vector (or “tilt”) by an angle dependent on the duration and

amplitude of the RF pulse, as given in equation 2:

(2)

where α is the flip angle of the magnetization vector, B1 is the strength of the magnetic

field produced by the RF pulse, and τRF is the duration of the RF pulse [6].

After the application of the RF pulse, the magnetization vectors will “relax” by returning

to their positions before the RF pulse, that is, aligned with the applied magnetic field B0. This relaxation

produces an RF signal, known as the Free Induction Decay (FID), which can be detected by MRI

equipment. Each magnetization vector produces a signal with a magnitude and phase.

Figure 1 displays the macroscopic magnetization vector of a material as it is placed in an

applied magnetic field, excited with an RF pulse, and allowed to relax to its original state, producing a

FID signal.

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Figure 1. The response of magnetic spin vectors with no external field (a), with an external

magnetic field applied (b), during excitation by an RF pulse (c), and emitting an FID signal during

relaxation (d).

2. Magnetic Field Gradients

MRI records the phase and magnitude of NMR signals emitted by precessing

magnetization vectors after being excited by an RF pulse to produce a spatially-resolved image of the

target material. Equation 1 shows that the precession frequency of a nuclear spin is directly proportional

to the strength of the applied magnetic field. By changing the magnitude of the magnetic field vector, the

frequency of nuclear spin precession can be controlled, and thus the phase of the nuclear spin vector can

be controlled. This process is performed in conventional MRI using magnetic field gradients. When

applied within the permanent magnetic field vector B0, a magnetic field gradient vector K alters the total

magnetic field strength. Magnetic field gradients produce a magnetic field with a spatially-dependent

magnitude, as described by equation 3.1:

(3.1)

where B(r) is the spatially-dependent total magnetic field vector, B0 is the permanent

magnetic field vector, K is the magnetic field gradient vector, and r is the spatial position vector.

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The Larmor Frequency of a nuclear spin can be controlled through the strength of the

magnetic field gradient and the position of the nuclear spin. Equations 1.4 and 3.1 can be combined to

form equation 3.2, which gives the Larmor Frequency ω of nuclear spins within a permanent magnetic

field B0 and magnetic field gradient K.

(3.2)

3. Signal Echoes

The FID signal produced by a material as its nuclear spins relax after RF excitation is

very brief, often on the order of microseconds. This is due to the existence of magnetic field

inhomogeneities, which tend to cause nuclear spins in close proximity to undergo precession with slightly

different frequencies, resulting in a larger distribution of nuclear spin vectors and, thus, a smaller

macroscopic magnetization vector. To remove this inhomogeneous dephasing, it is necessary to produce

an “echo” of the signal, occurring later than the original FID signal and with lower amplitude.

One way to produce an NMR signal echo is to apply a second RF pulse after the initial

excitation RF pulse. This second pulse, known as the “refocusing pulse”, is conventionally calibrated via

equation 2 to produce a flip angle of 180° (π radians). The refocusing pulse has the effect of inverting the

magnetization vectors. This inversion causes a phenomenon known as the “spin echo” of an FID, as

summarized in Figure 2.

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Figure 2. The state of a macroscopic magnetization vector after excitation by a 90-degree RF pulse

(a), after magnetization vectors have dephased due to local field inhomogeneities at time TE/2 (b),

after application of a 180-degree refocusing RF pulse (c), and after magnetization vectors have

rephrased to produce a signal echo at time TE (d).

After an excitation pulse, the affected magnetization vectors are rotated away from the

permanent magnetic field vector, where they proceed to begin relaxation while undergoing precession

about the permanent field vector. The decay of the subsequent NMR signal is caused by two effects: static

effects, dephasing caused by magnetic field inhomogeneities, and spin-spin interactions, the relaxation

effects of interactions between magnetic spins. The former causes some magnetization vectors to go out

of phase with each other, the effect of which is reversed by inverting the magnetization vectors with a

180° refocusing RF pulse. This allows magnetization vectors dephased by magnetic field inhomogeneities

to rephase, producing a spin echo signal at echo time (TE). The relaxation effects due to spin-spin

interactions, however, cannot be reversed in this way.

Another means of producing an NMR signal echo is through the use of magnetic

gradients rather than a refocusing RF pulse. Using the “gradient echo” technique involves the application

of a dephasing magnetic gradient immediately following the excitation RF pulse, followed by a

rephrasing magnetic gradient to generate a signal echo. After the dephasing gradient selectively changes

the frequencies of precession for the nuclear spin vectors, forcing the decay of the NMR signal, the

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rephrasing gradient reverses this effect by giving opposite precession frequencies to the nuclear spin

vectors, resulting in the rephrasing of the magnetization vectors. The time between excitation pulse and

generated echo is known as the echo time, TE. The gradient echo process produces an NMR signal echo

faster than the spin echo process, but does not remove the dephasing effect due to magnetic field

inhomogeneities.

Figure 3 compares the NMR signal characteristics of FID, spin echo, and gradient echo

processes. The signal decay due to both static field inhomogeneities and spin-spin interactions is known

as T2* decay, whereas the signal decay due to static field inhomogeneities alone is known as T2 decay.

The forced decay used to produce a gradient echo is known as TG decay.

Figure 3. The NMR signal intensity decay during Free Induction Decay (a), during a Spin Echo (b),

and during a Gradient Echo (c).

4. Pulse Sequence Diagrams

The acquisition of an image using MRI is possible through the coordinated operation of

magnetic gradients and RF coils. The precise timing of these elements is controlled by a piece of software

called a pulse sequence. A pulse sequence, when input the specific parameters of a desired scan, outputs

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directions to the scanner of when, how long, and with what strength to apply magnetic gradient and RF

pulses. Magnetic gradient pulses can be applied in three mutually orthogonal directions, known as “slice”,

“read”, and “phase”. Acquired images are always displayed in the read-phase plane, with subsequent

images aligned along the slice direction. By convention, the spatial directions corresponding to slice,

phase, and read are z, x, and y, respectively. Figure 4 depicts the directions as they are conventionally

defined in a scanner and image. By default, the slice direction is parallel to the permanent magnetic field

vector, but this is not required.

MRI scanners acquire data in complex-valued “k-space”, which is the two-dimensional

Fourier Transform of the image being measured. The two-dimensional Fourier Transform is given in

equation 4.1. An example of k-space data, which is in the frequency domain, is given in Figure 4b. To

convert raw k-space data into an image, the inverse two-dimensional Fourier Transform is used, as given

by equation 4.2. An example of the result is the image given in Figure 4c.

Figure 4. The slice, read, and phase directions as they relate to a specimen in an MRI scanner (a), to the acquired k-space data (b), and to an image produced through the inverse two-dimensional Fourier Transform of such k-space data (c).

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(4.1)

(4.2)

MT* is the spatially-dependent transverse magnetization vector, S is the k-space-

dependent acquired signal, and kx and ky are the spatial frequencies in x and y directions, respectively.

Figure 5 is a “pulse sequence diagram”, a means of displaying the relative timing of the

magnetic field gradients, RF elements, and associated equipment signals for a specific pulse sequence.

The horizontal axis is time, whereas each of the rows represents a different element-controlling signal.

The pulse sequence diagram depicts a single echo, which corresponds to a single pixel-wide “line” of

image data. This sequence must be repeated for each line required to produce an image of the desired

resolution. The time required for all MRI components during the acquisition of a single line of k-space

data is called Repetition Time (TR).

Figure 5 depicts a Gradient Echo (GE) pulse sequence. The top row describes the RF

signal. In a GE sequence, one RF pulse per echo is required. This excitation pulse usually produces a flip

angle of 90° or less via equation 2.

The second, third, and fourth rows depict the magnetic field gradients for the slice, phase,

and read directions, respectively. Three slice direction pulses per echo are required. The first pulse, the

“slice select” gradient, is positive in polarity and coincides with the excitation RF pulse. The purpose of

this pulse is to spatially select the slice being excited with the RF pulse via equation 3.2. The second slice

direction gradient, the “slice rephase” gradient, is negative in polarity and occurs immediately after the

slice select gradient. This second gradient has the purpose of re-aligning the phase of the precession of the

nuclear spins within the slice. The third pulse, the “spoiler” gradient, occurs after the gradient echo. This

gradient destroys any residual phase alignment in the slice direction. The purpose of the spoiler is to

rapidly dephase the nuclear spins to prepare for another repetition of the pulse sequence

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The third row depicts the phase direction magnetic field gradient. Two phase direction

pulses per echo are required. The first pulse, occurring directly after the slice select gradient, is the “phase

encode” gradient. This pulse encodes the phase-direction spatial location of each nuclear spin into its

precession phase. By convention, phase-direction encoding usually corresponds to the x-direction location

of the affected nuclear spins. After a scan has been completed, the phase of each NMR signal can be used

to assign its phase-direction spatial location. For each echo, a unique amplitude of phase encoding

gradient is used. Each line of k-space data corresponds to a unique precession phase, indicating a unique

phase-direction spatial location. In the pulse sequence diagram, this is represented by the phase encode

gradient appearing with multiple amplitudes visible. The second phase direction pulse is the “phase

rewind” gradient. This gradient is identical to the phase encode gradient, except that it has an opposite

amplitude. The phase rewind gradient eliminates the phase encoding of the nuclear spins in preparation

for the next phase encoding gradient.

Figure 5. A Gradient Echo MRI pulse sequence with Echo Time TE defined.

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The fourth row depicts the read direction magnetic field gradient. Two read direction

pulses per echo are required. The first pulse, coinciding with the phase encode gradient, is the “read

dephase” gradient. This pulse rapidly removes the alignments of the nuclear spins’ precession phases in

the read direction, which conventionally corresponds to the y-direction location of the affected nuclear

spins. Following the read dephase pulse is a gradient pulse of longer duration and smaller amplitude. This

pulse, the “readout” gradient, has three purposes. First, it rephrases the frequency-direction phases of the

nuclear spins’ precession, producing the gradient echo signal halfway through its duration from which the

NMR data can be extracted. In addition, the readout gradient encodes the read-direction spatial location of

each nuclear spin in its precession frequency via equation 3.2. After a scan has been completed, the

frequency of each NMR signal can be used to assign its read-direction spatial location. Finally, after the

gradient echo has occurred, the second half of the readout gradient removes any residual phase encoding,

in preparation for the next echo.

The fifth row depicts the Analog-to-Digital Converter (ADC) signal. Only one ADC

pulse is required per echo: to turn on the ADC when acquiring the NMR signal via the RF receiver.

The sixth row depicts the NMR signal produced by the target material. Immediately

following the excitation pulse, the material produces a signal, which decreases in amplitude

exponentially. The gradient echo is visible after the application of the readout gradient. The peak signal

occurs halfway through the readout gradient at Echo Time TE.

Figure 6 is the pulse sequence diagram for a Spin Echo (SE) sequence. The main

difference between a SE sequence and a GE sequence is the 180° refocusing RF pulse required to

generate a spin echo. This pulse is visible on the first row of Figure 6. The refocusing pulse occurs at a

time halfway between the excitation RF pulse and the signal echo, called TE/2. Due to the inversion of

magnetization vectors caused by the refocusing pulse, the polarities of both the phase rewind and readout

gradients are reversed from their values in the GE sequence. Additionally, a slice-direction gradient

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associated with the refocusing RF pulse is required. This pulse, called the “slice refocus” gradient, is

visible on the second row of Figure 6. Its amplitude is set so that the 180° RF pulse applies only to the

desired slice via equation 3.2.

B. Mechanical Wave Theory Background

1. Equation of Motion of a Continuum

The stiffness of a material is directly related to the wave properties of a mechanical wave

propagating through it. The deflection caused by a mechanical wave is related to the material’s resistance

Figure 6. A Spin Echo MRI pulse sequence with both half-echo times defined.

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to shear and longitudinal deformation. When the material is considered to be linearly elastic and isotropic,

this relationship is summarized by the Navier equation, as derived by Chandrasekharaiah et al. [7] in

equation 5.1:

(5.1)

where ρ is the density of the material, u is the three-dimensional deflection vector of the

wave as it propagates through the material, is the divergence operator, Δ is the Laplace operator,

defined by equation 5.2, and λ and μ are the first and second Lamé Parameters, respectively. Each of the

variables in equation 5.1 are spatially-dependent.

(5.2)

The two Lamé Parameters are used to quantify the mechanical stiffness of a material: the

second Lamé Parameter, μ, is also referred to as the shear modulus of the material; the first Lamé

Parameter, λ, is a factor with magnitude correlated to a material’s resistance to compression (longitudinal)

waves. The Lamé Parameters and Young’s Modulus E are related by equation 6.

(6)

Biological tissues are generally regarded as viscoelastic materials. Therefore, it makes

sense to convert equation 5.1 to the frequency domain to account for the frequency-dependent

characteristics of the Lamé Parameters. The Fourier Transform, given in equation 7, is used to convert a

temporal function to the frequency domain [8].

(7)

F(ω) is the frequency-domain function corresponding to time-domain function f(t) and i

is the imaginary unit. The Fourier transform allows frequency-domain counterparts to be found for most

15

time-domain functions. Table I gives a number of frequency-domain counterparts to useful time-domain

functions, computed using equation 7.

Table I

Time-Domain functions and their Frequency-Domain counterparts, obtained via the Fourier

Transform of equation 7

Time-Domain function Frequency-Domain function

Applying the Fourier Transform pairs in table I allows the conversion of equation 5.1 to

the frequency domain. This yields equation 8:

(8)

where ω is the angular frequency of the mechanical vibration, GL and GS are the complex

frequency-dependent moduli of the first and second Lamé Parameters, respectively, and U is the

frequency-dependent deflection vector. The real parts of the complex moduli are related to a material’s

tendency to store energy, and the imaginary parts are related to its damping characteristics. Each of the

variables in equation 8 are spatially-dependent.

16

2. Mechanical Stiffness Retrieval

The retrieval of mechanical properties from equation 8 can be performed with or without

consideration of the first Lamé Parameter complex modulus GL. Sinkus et al. demonstrated a technique

involving the vector curl operator [9]. The curl operator can be used to define vector Q:

(9)

which simplifies equation 8 to equation 10, effectively removing the longitudinal wave

contribution to the Navier equation.

(10)

Equation 10 can be used to solve for the complex shear modulus GS using the method of

Least-Squares solution to the overdetermined problem, resulting in equation 11:

(11)

where GS and Q are both functions of space and angular frequency ω, and superscript T

and -1 denote the transpose and inverse of a vector, respectively.

Equation 11 provides a method for obtaining the complex shear modulus GS from

displacement data, but if compression waves are not ignored, the solution can follow the method

presented by Oliphant et al. [10], as follows. Rearranging equation 8 yields equation 12:

17

or,

where:

(12)

where U1, U2, and U3 are the first, second, and third orthogonal components of deflection

vector U, respectively, and x1, x2, and x3 are the first, second, and third orthogonal spatial directions,

respectively. Equation 12 is solved using the method of Least-Squares solution to the overdetermined

problem to yield equation 13.

(13)

Equation 13 can be used to solve for both the complex shear modulus GS and the complex

first Lamé Parameter modulus GL.

If only two-dimensional data is available, the further assumption must be made that the

material is incompressible and that deflection occurs in a single direction. This simplification leads to the

reduced form of equation 8 given in equation 14.

(14)

18

Equation 14 decouples the spatial directions and the complex shear modulus can be

solved for each component of the deflection vector. This is possible through the scalar Helmholz

Inversion shown in equation 15:

(15)

where i = 1, 2, or 3.

Equations 11, 13, and 15 represent three different methods of obtaining complex shear

modulus values from physical deflection vectors, all being conventional techniques used in today’s MRE

research.

C. Magnetic Resonance Elastography Background

MRE combines the technique of MRI with mechanical wave theory to evaluate the stiffness properties of

a given tissue. To generate an elastogram, the following steps are taken:

1. A mechanical vibration is applied to a tissue

2. The displacement caused by the mechanical wave is encoded into the phase of the NMR signal during an

MRI scan

3. The resulting displacement map is used to determine mechanical stiffness properties of the tissue

Mechanical vibration application and the calculation of mechanical stiffness properties are

generally performed in MRE with standardized techniques. This study deals primarily with the particulars

of encoding motion into the phase of an NMR signal, and the bulk of the work herein is towards this end.

19

1. Phase Accumulation

Equation 1.4 indicates that the precession frequency of magnetic spins depends on the

magnetic field strength. When magnetic field gradients are applied for a finite time, the phase shift in the

NMR signal is given by equation 16:

(16)

where φ is the phase shift of the NMR signal, K(t) is the time-dependent magnetic field

gradient vector, u(t) is the deflection vector of the nuclear spins, and τK is the duration of application of

the magnetic field gradient [11]. Equation 16 demonstrates that the physical movement of a nuclear spin

causes a phase shift in the NMR signal. In MRE, the magnetic fields are varied in a controlled manner to

ensure that oscillating nuclear spins experience a position-dependent Larmor Frequency.

For the case of an oscillating magnetic field gradient vector, the duration of application

τK can be rewritten in the form of equation 17.

(17)

Here, τ is the period of oscillation and n is the number of cycles of oscillation applied.

To convert a standard MRI sequence to an MRE sequence, oscillating Motion-Encoding

Gradients (MEGs) must be added. These gradients have the effect of encoding physical motion into the

phase of the magnetization vectors. To avoid encoding constant displacement into the NMR phase, an

MEG must obey the “0th-moment-nulled” condition: it must be an oscillating function which, over any

integer number of periods, has equal positive and negative areas. This condition is summarized by

equation 18:

(18)

20

where τ is the period of the gradient’s polarity-switching oscillation and N is any integer.

Such a gradient function is shown in Figure 7 with N = 2 and an amplitude of A, and with a single period

of oscillation described by equation (19).

for

(19)

Figure 7. A zeroth-moment-nulled function K(t) with two periods of oscillation and amplitude of A.

The process described by equation 16 allows intra-voxel coherent motion to be encoded

into the phase of the NMR signal. Conventionally, two types of MEGs are used: sinusoidal and

trapezoidal pulses.

Sinusoidal gradient pulses provide a magnetic field gradient which is spatially constant

and shaped like a sinusoidal function in time, as described by equation 20:

21

(20)

where the amplitude, frequency, and initial phase (with respect to an arbitrary but fixed

time point) of the gradient waveform are represented by K0, ωK, and , respectively. Considering the

physical motion of nuclear spins as harmonic oscillation yields equation 21:

(21)

where the amplitude, frequency, and initial phase of the physical vibration are

represented by Yn, ωn, and , respectively.

The phase accumulated by a nuclear spin undergoing harmonic motion in a magnetic

field gradient is given by equation 16. This equation is adapted to account for an arbitrary start time of the

magnetic field gradient K, yielding equation 22:

(22)

where s is the start time of the magnetic field gradient. To obey the 0th-moment-nulled

condition of equation 18, the duration of magnetic field gradient application τK is taken to be an integer

multiple of the gradient’s period of oscillation, as in equation 23:

(23)

where q is an integer number of cycles of the magnetic field gradient with angular

frequency .

When the magnetic field gradient takes the form of a sinusoidal function as in equation

20 with a phase of zero at its onset, which is provided by the condition: , then equation 22 can

be solved with the method used by Muthupillai et al. [12] for the accumulated phase given by equation

24:

22

where

and

for

(24)

Equation 24 indicates that the accumulated phase in the NMR signal is proportional to the

number of applied cycles of the magnetic field gradient q, the amplitude of the magnetic field gradient K0

and the physical vibration amplitude Yn. Additionally, the sinusoidal function in equation 24 involving

MEG start time s gives the accumulated phase a spatial dependence of equal frequency to that of the

physical vibration. The spatially-dependent phase accumulation is therefore a “snapshot” (a single

moment in time) of the physical wave with a certain phase offset as it propagates through the

target material.

In equation 24, the factor ξ is called the “encoding efficiency”, as it represents the ratio of

phase accumulation amplitude to physical vibration amplitude. As equation 24 indicates, the encoding

efficiency exists for two cases: matching or mismatched MEG and physical vibration frequencies. For

matching MEG and vibration frequencies, the encoding efficiency is linearly proportional to the strength

and number of cycles of the MEG, and inversely proportional to the vibration frequency. For mismatched

MEG and vibration frequencies, the encoding efficiency has a sinusoidal dependence on the product of

the number of MEG cycles, pi, and the ratio of vibration to MEG frequencies. This results in an encoding

efficiency function as it appears in Figure 8.

Figure 8 demonstrates that, when sinusoidal MEGs are used, no vibration is encoded for

frequencies that obey equation 25, the “filter condition” [13]:

23

when

(except: ) (25)

where N is any integer.

Figure 8. The encoding efficiency of sinusoidal motion-encoding gradients for vibration frequencies

surrounding the motion-encoding gradient frequency.

Trapezoidal gradient pulses provide a magnetic field gradient shaped like a trapezoidal

function, oscillating regularly between positive and negative amplitude. Figure 9 is an example of such a

function.

24

Figure 9. A realistic trapezoidal motion-encoding gradient with finite ramp up, ramp down, and

flat top times.

A practical trapezoidal MEG must have finite rise and fall times, but for the evaluation of

a trapezoidal encoding efficiency, the rise times are considered instantaneous. With this simplification,

the trapezoidal MEG takes the form of equation 26.

(26)

where τK is the MEG period of oscillation.

To obtain the trapezoidal encoding efficiency, equation 22 can be solved for the

accumulated phase when the magnetic field gradient takes the form of equation 26. When the magnetic

25

field gradient obeys the 0th-moment-nulled condition as in 23, the accumulated phase takes the form of

equation 27:

where

and

for

(27)

Equation 27 gives the encoding efficiency for trapezoidal MEGs for two cases: matched

and mismatched MEG and vibration frequencies. For matched frequencies, the trapezoidal encoding

efficiency is identical to the sinusoidal encoding efficiency besides a factor of

. This is the primary

benefit to using a trapezoidal MEG instead of a sinusoidal MEG. For mismatched MEG and vibration

frequencies, however, the trapezoidal encoding efficiency function has sinusoidal and trapezoidal factors.

The trapezoidal encoding efficiency can be solved for several cases using L’Hôpital’s Rule [14], shown in

equation 28:

(28)

where f(x) and g(x) represent functions of independent variable x, and f’(x) and g’(x)

represent the first derivatives with respect to x of f(x) and g(x), respectively. Using L’Hôpital’s Rule, the

trapezoidal encoding efficiency of equation 27 can be solved for several cases, as shown in equations 29,

30, and 31:

when

even integer (29)

when

odd integer (30)

26

when

any integer (31)

The trapezoidal encoding efficiency is similar to the sinusoidal encoding efficiency for

mismatched MEG and vibration frequencies, besides one major difference. Equation 30 demonstrates the

trapezoidal encoding efficiency is non-zero for cases where the ratio of vibration to MEG frequency is an

odd integer. Unlike sinusoidal MEGs, trapezoidal MEGs encode vibration frequencies of higher odd-

integer harmonics.

Additionally, MEGs can be flow-compensated. Flow-compensated MEGs are designed to

resist encoding constant-velocity motion. This is useful for situations where encoding of vibration is

desired, but constant-velocity flow (such as the flow of blood through veins) should be suppressed.

An MEG is flow-compensating if it is 1st-moment-nulled. As with the 0

th-moment-nulling

condition of equation 18, a function K(t) is said to be 1st-moment-nulled if it obeys equation 32:

(32)

where τ is the period of the gradient’s polarity-switching oscillation and N is any integer.

A flow-compensated MEG must be 0th- (equation 18) and 1

st-moment-nulled (equation

32). One function that fills both conditions is a cosine-shaped function, like the one described by equation

33:

(33)

where K0 and ωK are the amplitude and frequency of the function, respectively.

Effectively, a cosine-shaped MEG is flow-compensated, thereby suppressing constant-velocity flow

during motion encoding.

27

2. Magnetic Resonance Elastography Pulse Sequence

To construct an MRE pulse sequences, MEGs are added to an MRI pulse sequence. To

encode motion, the MEGs must be active after the excitation RF pulse and before the ADC readout

signal. Additionally, for SE sequences, the MEGs cannot be active during the refocusing RF pulse, as this

would result in an incorrect location of the refocusing event.

Figure 10 is a Gradient Echo Magnetic Resonance Elastography (GE MRE) pulse

sequence diagram for motion encoding in the slice direction, using three cycles of a sinusoidal MEG.

Figure 10. A Gradient Echo Magnetic Resonance Elastography pulse sequence diagram with Echo

Time TE defined.

28

The one-dimensional motion encoding is performed by the sinusoidal MEG visible on the

second row of Figure 10, corresponding to the slice direction magnetic field gradient. The effect of this

gradient is to encode the slice direction motion (out-of-plane with respect to the magnitude image

obtained by the sequence) into the phase of the generated NMR signal. Because only a single direction of

motion is encoded, this type of sequence is further referred to as One-Dimensional Magnetic Resonance

Elastography (1D MRE). The physical vibration of the sample is visible in the last raw of Figure 10.

A similar 1D MRE pulse sequence can be made for either of the other two directions by

applying MEG along a different directional magnetic field gradient. In this manner, a phase-direction or

read-direction 1D MRE sequence is attainable. Each 1D MRE sequence encodes a single direction of

deflection into the phase of the NMR signal.

Figure 11 is a Spin Echo MRE (SE MRE) pulse sequence diagram for a motion encoding

gradient in the slice direction, using four cycles of a sinusoidal MEG.

As in the GE MRE pulse sequence diagram of Figure 10, the SE MRE pulse sequence

diagram in Figure 11 includes a sinusoidal MEG on the second row, indicating one-dimensional motion

encoding in the slice direction. Unlike the GE MRE sequence, however, SE MRE requires a 180°

refocusing RF pulse to generate the spin echo. The MEG cannot be active during this RF pulse, as it

would cause the refocusing event to occur in an incorrect location. As a result, the MEG is split so that

even portions occur before and after the refocusing RF pulse. For clarity, the portions of the MEG before

and after the refocusing RF pulse will be referred to as the “first run MEG” and “second run MEG”,

respectively.

29

The splitting of the MEG is done to minimize TE. Because the refocusing pulse must

occur at exactly TE/2, it is most efficient to apply exactly half of the MEG cycles as the first run MEG,

and the other half of the cycles as the second run MEG. Additionally, care must be taken to ensure that

the MEG provides continuous motion encoding before and after the refocusing pulse. Because the

refocusing RF pulse inverts the magnetization vectors, the second run MEG must also be inverted to

continue encoding vibration into the phase of the signal. This is visible in the MEG of Figure 11. The

second run MEG is applied with a reversed polarity to that of the first run MEG: in the case of a

sinusoidal function, this is equivalent to a phase offset of π radians. Without this reversal of MEG

polarity, the second run MEG would remove the accumulated phase of the first run MEG. Additionally,

care must be taken to ensure that the second run MEG is initiated at a point where the physical vibration

Figure 11. A Spin Echo Magnetic Resonance Elastography pulse sequence diagram with both half-echo times and the motion-encoding gradient gap defined.

30

is at the same phase as it was when the first run MEG was stopped. This is ensured by requiring the time

span between the end of the first run MEG and beginning of the second run MEG, referred to as the

“MEG Gap”, is an integer multiple of periods of the physical vibration. This condition is expressed in

Figure 11.

3. Data Sampling

The Fourier Transform, shown in equation 7 in its continuous form, is also useful in the

analysis of discrete functions. The discrete Fourier Transform is used to decompose a function into its

spectral components. Among other uses, this technique is used in conventional MRE to remove any

constant-displacement signal from a deflection vector, which would otherwise cause error in the

calculation of stiffness properties.

The discrete Fourier transform is useful for converting a signal in the time domain to one

in the frequency domain. To acquire a time-depended displacement vector using MRE, several scans are

performed, each with the MEG applied at a different phases of the physical vibration. Conventionally, this

is done by shifting the starting time of the MEG relative to the physical vibration, though in theory, it

would also be possible to vary the phase of the MEG at its onset for a fixed onset time to achieve the

same effect. This will result in multiple displacement vector images, each representing a unique

“snapshot” of the wave as it prorogates the material, as demonstrated in Figure 12. Together, these

snapshots represent a time-dependent displacement field.

Conventionally, a time-dependent displacement vector is given full temporal resolution

by acquiring N snapshots each with an MEG starting time shifted by Δt from the previous MEG starting

time, that add up to a complete period of the physical vibration. This allows the discrete Fourier transform

[15] to provide spectral data at each of the discrete frequencies, multiples of Δf, given in equation 34.

for

(34)

31

The discrete Fourier transform, when applied to a time-dependent displacement vector

u(t), yields a frequency-dependent displacement vector U(f). The discrete frequency corresponding to n =

0 in equation 34 is the constant-displacement signal, which does not contribute towards the calculation of

stiffness of a material. By discarding the value of U(f) at this frequency, the non-oscillating signal is

effectively removed from the displacement vector.

Figure 12. Motion-encoding gradients applied at different starting phases of the physical vibration

to produce different snapshots of the propagating wave.

In conventional 1D MRE, the discrete frequency corresponding to n = 1 in equation 34,

the base sampling frequency, is the only spectral data point of U(f) required to calculate the stiffness of

the material.

32

III. MOTIVATION

A. Three-Dimensional Encoding

Realistically, waves must propagate through any medium in three dimensions. When motion is

encoded in only a single direction, two-thirds of the wave’s behavior is lost. To encode three spatial

directions using 1D MRE, three individual scans must be completed consecutively, with the MEG applied

in one of the three directions during each scan. This has two major drawbacks. First, the time required to

acquire 3D data this way is three times longer than a single 1D MRE scan. Additionally, there is danger

of the target changing between single-directional scans, especially when the target is a biological system.

This would result in inaccuracy between results of different directions for the same target.

Two schemes for three-dimensional MRE (3D MRE) will be implemented in this study. The first,

developed by Yasar et al., is Spectral selective Displacement Projection – MRE (SDP-MRE) [3]. This

modality exploits the filter condition of equation 24 to simultaneously encode three independent

directions of displacement values using a multifrequency vibration. The second 3D MRE technique,

developed by Klatt et al., is SampLe Interval Modulation – MRE (SLIM-MRE) [4]. This modality utilizes

a monofrequency vibration and simultaneously encodes three independent directions of displacement

values by using different sample intervals, resulting in each direction encoding as a different apparent

frequency. In both SDP-MRE and SLIM-MRE, the three individual components of the displacement

vector can be obtained through decomposition of the generated NMR phase signal using a discrete

Fourier transform. Both techniques have not previously been applied to the Siemens IDEA VB15 system

used for this study.

B. Selective spectral Displacement Projection – Magnetic Resonance Elastography

SDP-MRE encodes three directions of displacement simultaneously by exploiting the filter

condition of equation 25.

33

The phase accumulation of an oscillating magnetization vector within a magnetic field gradient,

given by equation 22, can be adapted for the case where three orthogonal MEGs are applied

simultaneously [3]. This yields equation 35:

(35)

where s is the (simultaneous) start time of the three MEGs K1, K2, and K3, T is the duration of the

MEGs, and , , and represent the NMR signal phase accumulated by the MEGs corresponding to j

= 1, 2, and 3, respectively. It is further assumed that the physical vibration u(t) takes the form of the

superposition of three sinusoidal waveforms, each taking the form of equation 21, and each of the

orthogonal MEG components take the form of a sinusoidal waveform [3]. The accumulated phase of

equation 35 is therefore solved to yield equation 36:

with

(36)

where physical vibration amplitude , encoding efficiency , and phase offset are

specified for each physical vibration frequency component q and encoding direction j. In equation 36, fq

represents the frequency of the physical vibration’s qth component, and

, τj, and Nj are the amplitude,

period of oscillation, and integer number of cycles of the MEG in the jth direction, respectively.

34

As with equation 25, the filter condition can be acquired for equation 36. For the case of

mismatched MEG and vibration frequencies, no phase is accumulated for vibration frequencies

corresponding to an encoding efficiency of zero, which yields the filter condition of equation 37:

when

(37)

for all integers n besides n = Nj. In equation 37, T represents the total MEG duration [13]. The

reciprocal of T is the “base frequency” of the SDP-MRE experiment. Equation 37 enforces the additional

constraint of SDP-MRE that each of the three MEGs must be applied with a period of oscillation and

number of cycles that have a product equal to T. For each MEG frequency, equation 37 demonstrates that

all vibration frequencies that are integer multiples of the base frequency are not encoded, besides the case

where the MEG and vibration frequencies match. Through careful selection of the MEG frequencies and

number of cycles for each of the three MEG projections, it is possible for each direction to encode a

single frequency and filter all others.

Figure 13 is an SDP-MRE pulse sequence diagram based on a GE sequence. In this case, the

MEG frequencies and number of cycles are summarized in Table II. With these parameters, the base

frequency is 500 Hz. Equation 34 is used to show that, for the sequence parameters in Table II, 8

snapshots are required for the discrete Fourier transform to generate enough resolution for all three

discrete frequencies: 500, 1000, and 1500 Hz. The discrete Fourier transform of the acquired time-

dependent displacement vector provides direction-specific displacement vectors at each discrete

frequency.

35

Table II

SDP-MRE Parameters of Figure 13, including MEG frequency and number of cycles for each

projection

MEG Projection MEG Frequency MEG Number of Cycles

Slice 500 Hz 1

Read 1000 Hz 2

Phase 1500 Hz 3

Figure 13. Pulse sequence diagram for an SDP-MRE GE sequence with MEG frequencies of 500, 1000, and 1500 Hz and cycle numbers of 1, 2, and 3 cycles, respectively.

36

While the selection of SDP-MRE parameters, rather than being arbitrary, must follow the

constraints set by the filter condition, it is possible to boost the encoding efficiency of the MEGs once

values of MEG frequencies and number of cycles have been chosen. In the matching MEG and vibration

frequency case of equation 36, it is clear that encoding efficiency increases with MEG number of cycles.

By multiplying the number of cycles of each MEG component by the same integer, the encoding

efficiency can be increased without altering the ratios necessary to fulfill the filter condition. Such an

increase is made on the pulse sequence of Figure 13 by multiplying the number of MEGs by 3, as shown

in Figure 14.

Figure 14. Pulse sequence diagram for an SDP-MRE sequence with MEG frequencies of 500, 1000,

and 1500 Hz and cycle numbers of 3, 6, and 9 cycles, respectively.

37

The use of trapezoidal MEGs is possible with SDP-MRE, but one additional constraint must be

considered. Equation 30 demonstrates that, unlike sinusoidal MEGs, trapezoidal MEGs encode vibration

at frequencies that are odd integer multiples of the MEG frequency. With this consideration, trapezoidal

MEGs in an SDP-MRE sequence must be carefully chosen so that each of the three MEG frequencies

encodes only one of the three physical vibration frequencies. The example in Table II would not be

compatible for SDP-MRE with trapezoidal MEGs, as the 500 Hz MEG would encode both 500 Hz and

1500 Hz physical vibration frequencies. An example of trapezoidal-MEG-compatible SDP-MRE

parameters is given in Table III, which has a base frequency of 250 Hz.

Table III:

SDP-MRE Parameters compatible with trapezoidal MEGs, including MEG frequency and number

of cycles for each projection

MEG Projection MEG Frequency MEG Number of Cycles

Slice 750 Hz 3

Read 1000 Hz 4

Phase 1250 Hz 5

38

C. SampLe Interval Modulation – Magnetic Resonance Elastography

SLIM-MRE encodes three directions of displacement simultaneously through the use of different

sample intervals for each MEG. This causes each orthogonal displacement projection to encode with a

different “apparent frequency”, which can be retrieved through the use of the discrete Fourier transform.

In an MRE experiment, the sample interval is the time span by which the MEG’s start time is

shifted between successive snapshot acquisitions. The total number of snapshots, also called the number

of “offsets”, denoted by N, and the sample interval, denoted by Δt, dictate the discrete frequencies Δf that

are given values in the discrete Fourier transform of the displacement vector according to equation 34

(repeated below).

for

(34) [repeated]

In conventional 1D MRE, Δt is defined so that the MEG is shifted over one complete period of

the physical vibration with frequency f, as in equation 38.

(38)

When combined with equation 34, this constraint guarantees that the physical vibration frequency

will correspond to the first discrete frequency of the discrete Fourier transform of the displacement vector,

that is, with n = 1. In conventional 1D MRE, all other discrete frequencies are discarded.

The accumulated phase generated by an MEG, given by equation 16, can be extended to include

three different start times for the three directional MEGs, given in equation 39:

(39)

39

where subscripts j = 1, 2, and 3 correspond to the slice, read, and phase directions, respectively,

and Kj and uj stand for direction-specific MEG and displacement projection, respectively [4]. Assuming

sinusoidal-shaped MEGs and displacement functions as governed by equations 20 and 21, respectively,

equation 39 can be solved [4]. Adding the additional constraint that each directional MEG starts with zero

phase simplifies the solution to equation 40:

where

(40)

where ξj, Yj, sj, , Kj0, and qj are the direction-specific encoding efficiency, physical vibration

amplitude, phase offset, MEG amplitude, and number of MEG cycles, respectively. By using different

sampling intervals, Δtj, for each direction, it is possible to encode different directions with different

apparent frequencies. This yields the direction-specific sampling intervals of equation 41:

(41)

and the direction-specific MEG start times of equation 42:

for (42)

where N is the number of offsets, f is the physical vibration frequency, and subscript j = 1, 2, and

3 corresponds to the slice, read, and phase directions, respectively. Equation 42 demonstrates that the

MEGs are shifted over one, two, and three times the vibration frequency for slice, read, and phase

direction encoding, respectively [4]. Equation 42 is used to evaluate the discrete form of equation 39,

which yields equation 43.

40

where

(43)

Equation 43 demonstrates that each offset acquisition produces an NMR signal phase with

contributions from all three directions, each with a different apparent frequency. As with 1D MRE, the

NMR signal phase vector can be divided by the encoding efficiency of equation 43 to yield the

displacement vector. The discrete Fourier transform decomposes the displacement vector into discrete

frequencies via equation 34. Each directional displacement projection, having been acquired with a

unique apparent frequency, is stored independently in discrete frequencies corresponding to n = 1, 2, and

3 for slice-, read-, and phase-direction displacement, respectively. It is clear from equation 34 that the

number of offsets N must be at least 8 for three direction-dependent discrete frequencies to appear in the

discrete Fourier transform of the displacement vector.

Figure 15 depicts a pulse sequence diagram for a SLIM-MRE experiment, with MEGs from all of

the offsets superimposed into one diagram.

To minimize the increase in TE required by the increased sample intervals of SLIM-MRE, the

periodicity of harmonic waveforms is taken into account [4]. The MEG start times can be condensed

through the use of the modulo function, as in equation 44.

for (44)

Furthermore, the symmetry of harmonic waveforms is taken into account by recognizing that, for

N offsets divided evenly over period of oscillation 1/f, equation 45 must hold.

41

Figure 15. Pulse sequence diagram for a SLIM-MRE sequence with all MEG offsets superimposed.

(45)

Therefore, any MEG start time above the period of oscillation 1/f can be reduced by reversing the

polarity of the MEG amplitude and increasing the phase by π.

Table IV lists MEG start times for a SLIM-MRE sequence, both without (a) and with (b)

consideration of periodicity/symmetry, for vibration frequency f. MEGs with reversed amplitude

polarities are marked with (-).

It is apparent from Table IV that the consideration of symmetry and periodicity, shown in the

rows marked with (b), requires a much smaller increase in TE than otherwise. The maximum start time is

for number of offsets N and vibration frequency f, whereas without symmetry and periodicity

consideration this value is, at highest,

.

42

Table IV

MEG Start Times for SLIM-MRE with vibration frequency f and number of offsets N, without (a)

and with (b) consideration of the symmetry and periodicity of MEGs.

Offset n Slice MEG

start time (a)

Read MEG

start time (a)

Phase MEG

start time (a)

Slice MEG

start time (b)

Slice MEG

start time (b)

Slice MEG

start time (b)

0 0 0 0 0 0 0

1

2

3

4

5

6

7

Figure 16, devised by Klatt et. al [4], gives a visual representation of the apparent frequency

modulation of SLIM-MRE. Figure 16a represents the timing of the eight offsets for each MEG in real

time domain t. Figure 16b represents the timing of the eight offsets for each MEG in discretized time

domain t’, due to each offset sampling simultaneously for all three MEGs, as defined by equation 43.

43

Figure 16. Each of eight MEG start times in a SLIM-MRE sequence depicted in real time (a) and

discretized time (b).

44

IV. METHODS

A. Spin Echo Programming

The MRE pulse sequences are programmed using Siemens Integrated Development Environment

for Applications (IDEA) VB15 software for use on a Bruker 7T Clinscan MRI Scanner. MiniFLASH

pulse sequences, distributed with IDEA VB15 software, are used as the base sequence to be upgraded to

MRE sequences [16].

IDEA software is designed to simplify pulse sequence programming and testing. Pulse sequences

in IDEA are in the form of one or more C++-language files. IDEA-specific syntax calls for each pulse

sequence to be a distinct .CPP file containing five functions: fSEQInit, fSEQPrep, fSEQCheck, fSEQRun,

and fSEQRunKernel. Each of these functions performs a specific task in calculating and generating the

various outputs required to produce a pulse sequence on an MRI scanner: fSEQInit declares the limits on

user-defined parameters, fSEQPrep calculates the required timing for the pulse sequence by defining the

shape and duration of each of the RF / gradient pulses, fSEQCheck guarantees that the sequence does not

violate safety protocols, fSEQRun defines the loops over which the pulse sequence will be repeated, and

fSEQRunKernel (which is called by fSEQRun) sends the output signals to the MRI scanner that produce

RF and magnetic gradient pulses.

MiniFLASH is a very basic GE sequence. For each line of data acquired, MiniFLASH generates

a pulse sequence as seen in the pulse sequence diagram of Figure 5. To upgrade this sequence to a SE

sequence, thereby converting it to “MiniSE”, the following steps must be taken:

1. Add a 180° Refocusing RF pulse centered about time TE/2.

2. Add a Slice Refocus gradient pulse coinciding with the Refocusing RF pulse

3. Reverse the polarity of the Phase Rewind and Readout gradient pulses

45

4. Ensure that the readout gradient is centered about time TE, a time occurring exactly TE/2 after the center

of the RF pulse.

The addition of RF/Gradient pulses to a sequence using IDEA is performed in three steps. First,

the pulse is declared as an IDEA-specific C++ structure, either an sRF_PULSE or sGRAD_PULSE, for

RF or magnetic gradient pulse, respectively. This declaration must be made “globally”, that is, outside of

the four required functions in the .CPP file. Each structure defining a pulse is called a Real-Time Event

(RTE). Next, the RTE is prepared in the fSEQPrep function through IDEA-specific method calls that

define the pulse’s shape, duration, and amplitude. These parameters can be calculated or obtained through

user input in fSEQInit. Additionally, fSEQPrep organizes the timing of all RTEs. This guarantees that no

two pulses in the same direction are applied at the same time, and that the assigned values of TE and TR

are sufficient for a consistent pulse sequence. Finally, the RTE is applied in fSEQRunKernel within a

method call known as a Real-Time Event Block (RTEB). The RTEB is a list of all of the RTEs to be

applied during the pulse sequence along with their corresponding start times and directions of application.

As fSEQRunKernel is inside a loop within fSEQRun, the new pulse can be applied as many times as

required for the entire scan.

The above procedure can be used to include a refocusing RF pulse and a slice refocus gradient in

the MiniFLASH pulse sequence. The reversal of the phase rewind and readout gradient pulses’

amplitudes is performed during these pulses’ definition in fSEQPrep.

To guarantee that the timing conditions are fulfilled, two calculations must be made. First, the

minimum time required for all pulses between the midpoint of the excitation RF pulse and the midpoint of

the refocusing RF pulse is determined. Next, the minimum time required for all pulses between the

midpoint of the refocusing RF pulse and the midpoint of the ADC event is determined. The greater of

these two times is taken as the minimum required TE/2 for the pulse sequence, and is multiplied by two to

determine the minimum required TE. Finally, the excitation RF pulse, refocusing RF pulse, and ADC

46

event are given start times that guarantee their midpoints hit t = 0, TE/2, and TE, respectively. Thus, the

timing of the sequence is aligned.

These steps create a sequence, MiniSE, which generates a pulse sequence for each line of data

required identical to that of Figure 6.

B. Magnetic Resonance Elastography Programming

To upgrade MiniFLASH or MiniSE to an MRE sequence, the following steps must be taken:

1. Guarantee there is enough time between non-MEG gradients to apply MEGs without any gradient

overlap.

a. For GE sequences, MEGs are applied between the RF excitation pulse and the Readout gradient pulse.

b. For SE sequences, MEGs are applied in two locations: between the RF excitation pulse and the RF

refocusing pulse, and also between the RF refocusing pulse and the Readout gradient pulse.

2. Add oscillating MEGs to pulse sequence

3. For SE sequences, ensure that the MEG Gap is an integer multiple of half the vibration period.

4. Add an external trigger event to initiate the physical vibration

An MRE sequence requires several new user-controlled parameters. For each MEG, the user

specifies the frequency, amplitude, and number of cycles to be applied, as well as whether the gradient

pulses are sinusoidal or trapezoidal in shape, and flow-compensated or not flow-compensated.

Furthermore, the number of offsets acquired to resolve a time-dependent displacement function, as well as

the frequency corresponding to the time period over which the offsets are shifted, are given as user-

defined parameters.

User-controlled parameters can be added to a sequence by defining a global variable for each

parameter and assigning a value to it within the fSEQInit function. User control of these variables is made

convenient with the Siemens-supplied Parameter Map function, designed by Maxim Zaitsev [16]. Within

47

fSEQInit, user-controlled values are added to the new variables, and within fSEQPrep, these values are

used to prepare the various RF and gradient pulses, or rejected if found invalid for the sequence.

MEGs are declared as gradient pulses with the method used previously. For sinusoidal MEGs,

there is a specialized IDEA-specific C++ structure, sGRAD_PULSE_SIN, which is shaped like a positive

half-period of a sinusoidal function with an extended “flat top” time, as shown in Figure 17. Public

methods allow the ramp up time, duration, ramp down time, and amplitude to be defined, with each

parameter as described in Figure 17.

Figure 17. One sGRAD_PULSE_SIN structure with Amplitude, Ramp Up Time, Ramp Down

Time, and Duration defined.

48

A single period of a sinusoidal MEG is composed of two sGRAD_PULSE_SIN structures. By

assigning the ramp up time, duration, and ramp down time all equal to a quarter of the intended MEG

period, a sinusoidal half-pulse is generated. When one half-pulse is placed immediately before a second

half-pulse with opposite amplitude, one complete sinusoidal MEG period is generated, as shown in Figure

18.

Figure 18. One period of a sinusoidal MEG composed of two sGRAD_PULSE_SIN structures with

opposite amplitudes.

Similarly, a trapezoidal MEG can be constructed using the sGRAD_PULSE structure, which is

shaped like the function in Figure 19.

49

Figure 19. One sGRAD_PULSE structure with Amplitude, Ramp Up Time, Ramp Down Time, and

Duration defined.

To construct a trapezoidal MEG, two sGRAD_PULSE structures are used. Unlike sinusoidal

pulses, the ramp up and ramp down times of a trapezoidal pulse should be as close to zero as possible, so

that the NMR signal phase encoded by the MEG approximates to equation 27. The minimum ramp up and

ramp down times are defined by the system. Therefore, each trapezoidal half-pulse is assigned with ramp

up and ramp down times equal to the system’s minimum, and duration equal to the difference between

half of the intended MEG period and the assigned ramp down time. When one half-pulse is placed

immediately before a second half-pulse with opposite amplitude, one complete trapezoidal MEG period is

generated, as shown in Figure 20.

50

Figure 20. One period of a trapezoidal MEG composed of two sGRAD_PULSE structures with

opposite amplitudes.

To construct flow-compensated MEGs, quarter-pulses must be combined with the existing half-

pulses. A sinusoidal quarter-pulse can be generated by constructing an sGRAD_PULSE_SIN structure

with one ramp time equal to the minimum rise time of the system, and the other ramp time equal to the

difference between a quarter of the intended MEG period and the minimum system ramp time. A

trapezoidal quarter-pulse can be generated by constructing an sGRAD_PULSE structure with ramp up

and ramp down times equal to the minimum system ramp time and the duration equal to the difference

between a quarter of the intended MEG period and the assigned ramp down time. Both sinusoidal and

trapezoidal quarter-pulses are shown in Figure 21.

To construct a full period of a flow-compensated MEG, two quarter-pulses are applied on either

side of a half-pulse of opposite amplitude, as in Figure 22.

51

Figure 21. Quarter-period MEG pulses defined for the leading end of a sinusoidal MEG (a), the

trailing end of a sinusoidal MEG (b), and either end of a trapezoidal MEG (c).

Figure 22. One period of a flow-compensated MEG composed of two quarter-pulses and one half-

pulse for sinusoidal (a) and trapezoidal (b) MEGs.

52

To ensure that a sequence gives proper time for MEGs to be applied, a simple calculation is

made. The total time required by the MEGs for each acquisition line is the product of the MEG period and

number of MEG cycles, which are both user-defined parameters. This total is added to the calculation of

the minimum required TE. In the case of a SE sequence, half of the total MEG time is added to both the

calculation of the TE/2 before the refocusing pulse and the TE/2 after the refocusing pulse.

MEGs differ from most other gradient pulses in that MEGs are applied an arbitrary number of

times, namely, the number of cycles specified by the user. To accomplish this, a loop is created within the

fSEQRunKernel function, repeating the application of both halves of an MEG as many times as required.

For each instance of the loop, the MEG half-pulses are given a greater start time, so that no one period of

the MEG overlaps another.

For SE MRE sequences, special care must be taken to ensure that the time between the end of the

first run MEG and the start of the second run MEG, called the MEG Gap, must be an integer number of

half-periods of the vibration frequency. Within fSEQPrep, the minimum possible MEG Gap is computed

for the given value of TE as the time between the end of the first run MEG and the end of the slice

refocus gradient. Next, the smallest multiple of the vibration period that is greater than the minimum

possible MEG Gap is determined. This value is stored as the actual MEG Gap, and the start time of the

second run MEG is set as the sum of the end of the first run MEG and the actual MEG Gap.

To initiate the physical vibration, an external signal must be sent from the scanner to the

equipment producing the vibration. This is programmed into an MRE pulse sequence using an RTE called

sSYNC_EXTTRIGGER. This structure, when applied by a pulse sequence, produces a signal not to a

magnetic gradient or RF coil, but instead to an output port located on the scanner. This signal can be

directed to any external equipment.

Applying an sSYNC_EXTTRIGGER structure is achieved similarly to all other RTEs: the

structure must be defined globally, prepared within fSEQPrep, and applied within fSEQRunKernel. When

53

preparing an sSYNC_EXTTRIGGER structure, two parameters must be defined: the external port to

which the signal should be sent, and the duration of the RTE. The standard values for these parameters are

0 (the first output port) and 10 (the minimum acceptable duration in microseconds), respectively. Using

these parameters, every time this RTE is applied within fSEQRunKernel, a 10-microsecond transistor-

transistor logic (TTL) signal is produced on the first output port.

The above methods convert MiniFLASH and MiniSE to GE 1D MRE and SE 1D MRE

sequences, respectively.

C. Selective spectral Displacement Projection Programming

To upgrade 1D MRE sequences to SDP-MRE, the following steps must be taken:

1. Three simultaneous MEGs must be defined and applied, each with independent frequencies, number of

cycles, and amplitudes

2. For a given set of MEG frequencies, optimal number of cycles, number of offsets, and base frequency

must be calculated.

Defining and applying three simultaneous MEGs is achieved identically to the 1D case, except

performed in triplicate. Each parameter must be individually defined for slice-, read-, and phase-direction

MEGs.

To achieve proper SDP multi-directional motion encoding, each MEG’s combination of

frequency and number of cycles must selectively filter all other MEG frequencies by the process of

equation 37. To determine optimal SDP parameters, an additional calculation must be performed within

fSEQPrep. First, the greatest common divisor of the three user-defined MEG frequencies is calculated.

This is achieved by looping over all integers between zero and the smallest given MEG frequency,

inclusive, and saving the largest of these integers that divides evenly into all three given MEG frequencies

without remainder. This value is the base frequency of the SDP sequence. The minimum number of

54

cycles for each MEG is calculated by dividing each MEG frequency by the base frequency. If this

algorithm reports that the minimum number of cycles for any MEG is greater than 50, which is much too

large for any practical MRE sequence, the function exits and reports that the chosen MEG frequencies are

not SDP-compatible. Finally, the user can select an integer “MEG cycle factor”, which is multiplied by

the number of minimum cycles for each MEG, to boost MEG cycles without breaking the SDP

conditions.

The above methods convert MiniFLASH and MiniSE to SDP-MRE GE and SDP-MRE SE

sequences, respectively.

D. SampLe Interval Modulation Programming

To upgrade MiniFLASH and MiniSE to SLIM-MRE sequences, the following steps must be taken:

1. Three simultaneous MEGs must be defined and applied, all with identical frequencies

2. Each MEG is applied with a different sampling interval, resulting in different start times, as displayed in

Table IV

3. To account for the periodicity and symmetry of harmonic MEGs, the polarity of the MEGs is switched for

some offsets, as displayed in Table IV

Defining and applying three simultaneous MEGs is achieved identically to the 1D case, except

performed in triplicate. Each parameter must be individually defined for slice-, read-, and phase-direction

MEGs, except frequency, which for SLIM-MRE is identical in all directions.

To coordinate the starting times and polarities for each MEG during each offset, six arrays with

eight members each are defined globally. Next, in fSEQPrep, the arrays are filled with values. Three of

the arrays contain starting times for each of the three directional MEGs for each of eight offsets. The

other three arrays contain values of 0 or 1, corresponding to positive- or negative-polarity MEGs for each

of eight offsets. The values stored in these six arrays correspond to the starting time and polarity

55

assignments found in Table IVb. Finally, within fSEQRunKernel, the active offset being applied

corresponds to one of the eight members in each of the six arrays, which define the starting time and

polarity for each of the three directional MEGs.

E. Frequency Correction

One condition of IDEA pulse sequence programming is the system’s temporal resolution of 10

microseconds. All magnetic gradient pulses must have ramp up, duration, and ramp down times that are

integer multiples of 10 microseconds. This generates error when arbitrary MEG frequencies are used.

When the user of an MRE sequence inputs an MEG frequency, the quarter-frequency in

microseconds is calculated in fSEQPrep. Because this quarter-frequency time is used for the ramp up and

ramp down times of the MEG, it must be rounded to the nearest integer multiple of 10 microseconds. This

is performed using either the fSDSRoundUpGRT or fSDRoundDownGRT, which round the function up

or down to the nearest integer multiple of 10, respectively. These rounded values are used to prepare the

MEG pulses. If the quarter-frequency of the MEG is not already an integer multiple of 10 microseconds,

this rounding induces error in the actual applied frequency of the MEG. This error is considerable for

large values of MEG frequency. Since each quarter-period of the MEG must be rounded to the nearest 10-

microsecond multiple, each period has a maximum error of 40 microseconds. Thus, the maximum percent

error when fSDSRoundUpGRT is used follows equation 46:

(46)

and the maximum percent error when fSDSRoundDownGRT is used follows equation 47:

(47)

where MPE is the maximum percent error generated when rounding quarter-period values for

frequency f in Hertz.

56

To minimize this error, fSDSRoundUpGRT and fSDSRoundDownGRT are both used during

each MEG preparation. By preparing each MEG with half its quarter-pulse durations rounded using

fSDSRoundUpGRT and fSDSRoundDownGRT, the maximum percent error is decreased by nearly a

factor of 2, following equation 48.

(48)

Figures 23, 24, and 25 display the effect of the rounding error and its partial correction. Figure 23

displays the actual MEG frequencies generated for intended frequencies between 1 and 5000 Hz when

fSDSRoundUpGRT or fSDSRoundDownGRT are used alone. Figure 24 displays the actual MEG

frequencies generated for intended frequencies between 1 and 5000 Hz when partial correction is

performed by alternating fSDSRoundUpGRT and fSDSRoundDownGRT. Figure 25 displays the

bounding lines of erroneous frequencies for the cases described in figures 23 and 24, demonstrating that

the partial correction of Figure 24 limits the erroneous frequencies to a smaller bandwidth about the

intended frequency.

57

Figure 23. Applied frequency versus input frequency for MEGs when using fSDSRoundUpGRT()

and fSDSRoundDownGRT() rounding functions.

Figure 24. Applied frequency versus input frequency for MEGs when alternating use of both

fSDSRoundUpGRT() and fSDSRoundDownGRT() rounding functions.

58

Figure 25. Maximum erroneous frequency trend lines for MEGs when using fSDSRoundUpGRT(),

fSDSRoundDownGRT(), and alternating both rounding functions.

59

V. EXPERIMENTAL SETUP

A. Physical Vibration Source

To produce the vibration necessary for an MRE experiment, the experimental setup of Figure 26 is

used.

Figure 26. Experimental setup to produce physical vibration in test sample.

Four pieces of equipment are illustrated in Figure 26:

1. Agilent 33250A 80MHz Function / Arbitrary Waveform Generator

2. Elenco XP-581A DC Variable Voltage Supply

3. Yamaha P3500S Power Amplifier

4. PI Ceramic P-887.91 Stack Multilayer Piezoelectric Actuator

An external trigger event in an MRE pulse sequence generates a TTL signal that is fed into the

function generator. This signal triggers the function generator to output a voltage waveform matching the

60

shape of the desired vibration, which is fed into the power amplifier. The vibration waveform is amplified

to 40 Volts peak-to-peak. This amplified signal is given a DC offset of 20 Volts by the variable voltage

supply, guaranteeing that the voltage signal never drops below zero Volts. Finally, the amplified and

offset vibration waveform is fed into the piezoelectric actuator, which vibrates with frequency equal to the

input voltage waveform. The actuator vibrates the sample with a desired frequency, allowing an MRE

experiment to take place.

B. Experimental Gel Sample

To test the MRE sequences, an inhomogeneous gel sample is produced. An inhomogeneous

design is chosen to guarantee three-dimensional wave characteristics. For the purpose of validating the

efficacy of the 3D MRE sequences, the specific stiffness of the inner and outer gels are not relevant; it is

more important that the gels are of different stiffness, thereby producing 3D wave patterns caused by

refraction and reflection. The cylindrical gel sample followed the general format of Figure 27.

Figure 27. Schematic for experimental inhomogeneous gel sample.

61

The container is a hollow Garolite cylinder of length 50 mm, inner diameter 32 mm and wall

thickness of approximately 1 mm. One end of the cylinder is closed, and a threaded hole is bored into the

closed end to attach to the actuator. The actuation of the sample occurs mainly parallel to the slice

direction, as displayed in Figure 28.

Figure 28. Orientation of vibration actuation for experimental gel sample as it relates to slice

direction in scanner.

The gel sample is composed of two masses of Smooth-On EcoFlex 00-10 Silicone rubber with

slightly different densities. The inner gel sample, of irregular shape and approximate diameter of 25 mm,

is first produced using EcoFlex 00-10 with standard density of 1040 kg/m3 [17]. After curing for 24

hours, this inner mass is placed in the cylinder container and surrounded with EcoFlex of lower density,

produced by mixing silicone oil with the EcoFlex 00-10 reagents at a volume ratio of 1:9 before curing.

After curing for 24 hours, the sample is ready for testing.

62

C. Sequence Parameters

The MRE sequences were tested using a 7T Bruker ClinScan MRI Scanner operated using

Siemens IDEA VB15 pulse sequencing software. To obtain the maximum amount of signal, SE

sequences were used. Sinusoidal, non-flow-compensating MEGs were used for all results presented in

this work, although other MEG shapes were implemented as part of this thesis as well. The general MRI

parameters used are collected in Table V. The MRE-specific parameters used for the SLIM-MRE scan

and associated 1D MRE scans are collected in Table VI, and those of the SDP-MRE scan and associated

1D MRE scans are collected in Table VII.

Table V

MRI Parameters used for all MRE scans

Parameter Value

TR 1000 ms

Field of View (FOV) 32 mm (read) x 32 mm (phase) x 10 mm (slice)

Image Resolution 64 pixels (read) x 64 pixels (phase)

Number of Slices 20

Slice Thickness 0.5 mm

Flip Angle 90°

63

Table VI

MRE Parameters for 1D MRE scans in the Slice (a), Read (b), and Phase (c) directions, and for the

SLIM-MRE scan (d)

1D MRE –

Slice (a)

1D MRE – Read

(b)

1D MRE – Phase

(c)

SLIM-MRE

(d)

Slice MEG Frequency 1000 Hz N/A N/A 1000 Hz

Slice MEG Number of

Cycles

3 N/A N/A 3

Slice MEG Amplitude 250 mT/m N/A N/A 250 mT/m

Read MEG Frequency N/A 1000 Hz N/A 1000 Hz

Read MEG Number of

Cycles

N/A 3 N/A 3

Read MEG Amplitude N/A 250 mT/m N/A 250 mT/m

Phase MEG Frequency N/A N/A 1000 Hz 1000 Hz

Phase MEG Number of

Cycles

N/A N/A 3 3

Phase MEG Amplitude N/A N/A 250 mT/m 250 mT/m

Number of Offsets 8 8 8 8

TE 12.99 ms 12.99 ms 12.99 ms 11.96 ms

Vibration Frequency 1000 Hz 1000 Hz 1000 Hz 1000 Hz

64

Table VII

MRE Parameters for 1D MRE scans in the Slice (a), Read (b), and Phase (c) directions, and for the

SDP-MRE scan (d)

1D MRE –

Slice (a)

1D MRE – Read

(b)

1D MRE – Phase

(c)

SDP-MRE

(d)

Slice MEG Frequency 1000 Hz N/A N/A 1000 Hz

Slice MEG Number of

Cycles

3 N/A N/A 3

Slice MEG Amplitude 250 mT/m N/A N/A 250 mT/m

Read MEG Frequency N/A 2000 Hz N/A 2000 Hz

Read MEG Number of

Cycles

N/A 6 N/A 6

Read MEG Amplitude N/A 250 mT/m N/A 250 mT/m

Phase MEG Frequency N/A N/A 3000 Hz 3000 Hz

Phase MEG Number of

Cycles

N/A N/A 9 9

Phase MEG Amplitude N/A N/A 250 mT/m 250 mT/m

Number of Offsets 8 8 8 8

TE 12.99 ms 12.99 ms 12.99 ms 13.52 ms

Vibration Frequency 1000 Hz 2000 Hz 3000 Hz Superposition

of 1000, 2000,

and 3000 Hz

65

VI. RESULTS

Figure 29 compares the temporally-resolved displacement images acquired from one SLIM-MRE

experiment (Figures 29a, 29b, and 29c) to those of three related 1D MRE experiments (Figures 29d, 29e,

and 29f). Twenty slices total were acquired, and one example slice (slice number 7), is displayed in

Figure 29. The real parts of the complex displacement images are shown. Of note, all wave images were

noise-filtered using a 4-pixel Butterworth lowpass filter of order 2 [8], and therefore may still contain the

contribution due to compression waves.

Figure 29 demonstrates that the SLIM-MRE sequence encodes similar displacement images to

that of 1D MRE sequences. Because the three SLIM-MRE displacement projections were acquired in

one-third the time of using 1D MRE, SLIM-MRE represents a significant acceleration in acquiring 3D

MRE data. The additional noise visible in the 1D MRE data is a product of the increased TE of the 1D

MRE sequences, which was not optimized for TE minimization.

66

Figure 29. Acquired displacement images for SLIM-MRE sequence in slice (a), read (b), and phase

(c) directions, and for 1D MRE sequences in slice (d), read (e), and phase (f) directions.

Figure 30 compares the calculated stiffness factor images obtained from SLIM-MRE (Figure 30a)

and 1D MRE (Figure 30b) sequences. It was found that the inversion technique not using the curl

operator, shown in equation 13, was unsuitable for processing of the acquired images. The compression

waves propagating through the sample cause substantial error in the calculation of shear modulus values

using this inversion technique. However, the technique that involves use of the curl operator, shown in

equation 11, removes the effects of compression waves in the displacement images, resulting in clearer

67

shear modulus images. This inversion technique is used to generate the images in Figure 30. Of the

twenty acquired slices, an example slice (number 7) was chosen for display in Figure 30. The real parts of

the complex modulus images are shown.

Figure 30. Calculated complex shear modulus images using curl-operator inversion method for

SLIM-MRE (a) and 1D MRE (b).

Figure 30 demonstrates that the elastograms generated by the SLIM-MRE sequence are

comparable to those of 1D MRE sequences. The slight difference in stiffness between the outer and inner

gel masses is visible. While the characteristics of the elastograms are similar, differences in magnitude are

most likely due to extra noise in the 1D MRE sequences, compounded by the multiple spatial derivatives

necessary during inversion.

68

Figure 31 compares the temporally-resolved displacement images acquired from one SDP-MRE

experiment (Figures 31a, 31b and 31c) to those of three related 1D MRE experiments (Figures 31d, 31e,

and 31f). Twenty slices total were acquired, and one example slice (number 7) is displayed in Figure 31.

The real parts of the complex displacement images are shown. Of note, a Butterworth bandpass filter

(with low and high filter limits of 4 and 32 pixels, respectively) was applied to all wave images for

filtering noise and the contribution due to compression waves.

Figure 31. Acquired displacement images for SDP-MRE sequence in slice (a), read (b), and phase

(c) directions, and for 1D MRE sequences in slice (d), read (e), and phase (f) directions.

69

Figure 31 demonstrates that the SDP-MRE sequence encodes similar displacement images to that

of 1D MRE sequences. Because the three SDP-MRE displacement projections were acquired in the time

it took to acquire one 1D MRE displacement projection, SDP-MRE represents a significant improvement

in acquiring 3D MRE data. One notable shortcoming, however, of the SDP-MRE displacement data is the

decreased intensity of displacement encoding as compared to that of 1D MRE sequences. This is due to

the fact that in the SDP-sequence, by simultaneously exciting three frequencies of vibration

simultaneously, a decreased amplitude of vibration for each single frequency is experienced. In effect, the

multifrequency vibration, applied with identical amplitude to the monofrequency vibration of 1D MRE,

exhibits one-third the amplitude for each frequency component than that of the monofrequency vibration.

The decreased amplitude results in less motion encoding, visible in the lower intensity of the SDP-MRE

displacement images as compared to the 1D MRE displacement images in Figure 31.

The usual 3D inversion techniques of equations 11 and 13 are unusable with our SDP-MRE data

set because they require the 3D displacement for each frequency. Still, the wave patterns depicted in

Figure 31 indicate a successful implementation of SDP-MRE.

70

VII. DISCUSSION

A. Limitations

The SLIM-MRE displacement images of Figure 29 resemble the associated 1D MRE images, but

several improvements could be made. The noise apparent in the 1D MRE images could be mitigated

through optimization of the 1D MRE sequence. Instead of shifting the MEG relative to the other gradient

and RF events in the 1D MRE sequence, all other events could be shifted relative the external trigger

event. This would allow for minimization of TE, resulting in much less noise. Furthermore, it was

observed for the inhomogeneous gel sample that vibration in the read and phase directions had amplitudes

several orders of magnitude smaller than that of vibration in the slice direction. To normalize the

vibration amplitudes, actuation can be performed in a non-axial direction. Such actuation would result in

increased vibration amplitude in non-slice directions, which would increase the amount of NMR phase

signal encoded in these directions.

The SDP-MRE displacement images of Figure 31 resemble the associated 1D MRE images, but

several improvements could be made. As with the 1D MRE images of Figure 29, the noise level apparent

in both the SDP-MRE and 1D MRE images of Figure 31 could be decreased by optimizing the sequences

for TE minimization. In addition, the SDP-MRE images exhibit decreased intensity as compared to the

1D MRE images. This is due to the decreased vibration amplitude inherent in multifrequency vibrations:

each frequency component ends up with a smaller amplitude than that of a monofrequency vibration. A

solution to this would be an increase in multifrequency vibration amplitude: either by increasing the

voltage sent to the piezoelectric actuator, or adding more actuators in series with the first to boost

displacement.

71

B. Conclusion

SLIM-MRE and SDP-MRE generate displacement images comparable to those of standard 1D

MRE sequences, and do so in one third of the time. Therefore, SLIM-MRE and SDP-MRE represent

significant time-saving 3D MRE techniques, and may improve the acceptance of MRE in clinical settings.

This work demonstrates the viability of SLIM-MRE and SDP-MRE implementation on a

Siemens/Bruker MRI system, adding to the existing deployment of these sequences demonstrated by

Klatt et al. [4] [5] and Yasar et al. [3]. This work serves as further verification that the new 3D motion-

encoding techniques of SLIM-MRE and SDP-MRE are modality-independent, equally viable on any MRI

operating system or scanner.

72

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74

VITA

NAME:

EDUCATION:

RESEARCH EXPERIENCE:

HONORS:

CERTIFICATION:

David Arthur Burns

B.S., Nuclear, Plasma, and Radiological Engineering, University

of Illinois, Urbana-Champaign, Illinois, 2010

M.S., Mechanical Engineering, University of Illinois at Chicago,

Chicago, Illinois, 2014

Department of Bioengineering, University of Illinois at Chicago,

Chicago, Illinois: Motion-Sensitive MRI Laboratory, 2014

Department of Nuclear, Plasma, and Radiological Engineering,

University of Illinois, Urbana-Champaign, Illinois: Center for

Plasma-Material Interactions, 2007-2010.

Roy A. Axford Undergraduate Scholarship, University of Illinois,

Urbana-Champaign, Illinois, 2009

George H. Miley LENR Undergraduate Scholarship, University

of Illinois, Urbana-Champaign, Illinois, 2009-2010

Chancellor’s Scholar, University of Illinois, Urbana-Champaign,

Illinois, 2006-2010

James Scholar, University of Illinois, Urbana-Champaign,

Illinois, 2006-2010

Dean’s List, University of Illinois, Urbana-Champaign, Illinois,

2006-2010

Siemens IDEA Pulse Sequence Programming Training, Cary,

North Carolina, 2014

U.S. Peace Corps Pre-Service Training, Loitokitok, Kenya, 2010

Documents

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