Synchronous Multi-Directional Motion Encoding. in Magnetic Resonance Elastography. DAVID A. BURNS B.S., University of Illinois, Urbana-Champaign, 2010

Transcription

1 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

2 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 ii

3 TABLE OF CONTENTS CHAPTER PAGE I. INTRODUCTION... 1 II. CONCEPTUAL FRAMEWORK 3 A. Magnetic Resonance Imaging Background Nuclear Magnetic Resonance Magnetic Field Gradients 5 3. Signal Echoes Pulse Sequence Diagrams 8 B. Mechanical Wave Theory Background Equation of Motion of a Continuum Mechanical Stiffness Retrieval 16 C. Magnetic Resonance Elastography Background Phase Accumulation Magnetic Resonance Elastography Pulse Sequence Data Sampling. 30 III. MOTIVATION A. Three-dimensional Encoding 32 B. Selective spectral Displacement Projection - Magnetic Resonance Elastography C. SampLe Interval Modulation Magnetic Resonance Elastography 38 IV. METHODS.. 44 A. Spin Echo Programming B. Magnetic Resonance Elastography Programming C. Selective spectral Displacement Projection Programming D. SampLe Interval Modulation Programming. 54 E. Frequency Correction V. EXPERIMENTAL SETUP. 59 A. Physical Vibration Source 59 B. Experimental Gel Sample. 60 C. Sequence Parameters 62 VI. RESULTS 65 iii

4 TABLE OF CONTENTS (continued) CHAPTER PAGE VII. DISCUSSION. 70 A. Limitations 70 B. Conclusion 71 CITED LITERATURE 72 VITA iv

5 LIST OF TABLES TABLE PAGE I. TIME-DOMAIN FUNCTIONS AND THEIR FREQUENCY-DOMAIN COUNTERPARTS, OBTAINED VIA THE FOURIER TRANSFORM 15 II. III. IV. SELECTIVE SPECTRAL DISPLACEMENT PROJECTIN MAGNETIC RESONANCE ELASTOGRAPHY PARAMETERS, INCLUDING MOTION- ENCODING GRADIENT FREQUENCY AND NUMBER OF CYCLES FOR EACH PROJECTION SELECTIVE SPECTRAL DISPLACEMENT PROJECTIN MAGNETIC RESONANCE ELASTOGRAPHY PARAMETERS COMPATIBLE WITH TRAPEZOIDAL MOTION-ENCODING GRADIENTS, INCLUDING FREQUENCY AND NUMBER OF CYCLES FOR EACH PROJECTION 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 V. MAGNETIC RESONANCE IMAGING PARAMETERS USED FOR ALL MAGNETIC RESONANCE ELASTOGRAPHY SCANS.. 62 VI. VII. MAGNETIC RESONANCE ELASTOGRAPHY PARAMETERS FOR ONE- DIMENSIONAL SCANS IN THE SLICE, READ, AND PHASE DIRECTIONS, AND FOR THE SAMPLE INTERVAL MODULATION SCAN 63 MAGNETIC RESONANCE ELASTOGRAPHY PARAMETERS FOR ONE- DIMENSIONAL SCANS IN THE SLICE, READ, AND PHASE DIRECTIONS, AND FOR THE SELECTIVE SPECTRAL DISPLACEMENT PROJECTION SCAN v

6 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 The Nuclear Magnetic Resonance signal intensity decay during Free Induction Decay, during a Spin Echo, and during a Gradient Echo 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 A Spin Echo Magnetic Resonance Imaging pulse sequence with both half-echo times defined A zeroth-moment-nulled function K(t) with two periods of oscillation and amplitude of A The encoding efficiency of sinusoidal motion-encoding gradients for vibration frequencies surrounding the motion-encoding gradient frequency A realistic trapezoidal motion-encoding gradient with finite ramp up, ramp down, and flat top times A Gradient Echo Magnetic Resonance Elastography pulse sequence diagram with echo time defined A Spin Echo Magnetic Resonance Elastography pulse sequence diagram with both half-echo times and the motion-encoding gradient gap defined Motion-encoding gradients applied at different starting phases of the physical vibration to produce different snapshots of the propagating wave.. 31 vi

7 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 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 Pulse sequence diagram for a SampLe Interval Modulation Magnetic Resonance Elastography sequence with all motion-encoding gradient offsets superimposed 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 One sgrad_pulse_sin structure with Amplitude, Ramp Up Time, Ramp Down Time, and Duration defined One period of a sinusoidal motion-encoding gradient composed of two sgrad_pulse_sin structures with opposite amplitudes One sgrad_pulse structure with Amplitude, Ramp Up Time, Ramp Down Time, and Duration defined One period of a trapezoidal motion-encoding gradient composed of two sgrad_pulse structures with opposite amplitudes 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 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 Applied frequency versus input frequency for motion-encoding gradients when using fsdsroundupgrt() and fsdsrounddowngrt() rounding functions. 57 vii

8 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 Maximum erroneous frequency trend lines for motion-encoding gradients when using fsdsroundupgrt(), fsdsrounddowngrt(), and alternating both rounding functions Experimental setup to produce physical vibration in test sample Schematic for experimental inhomogeneous gel sample Orientation of vibration actuation for experimental gel sample as it relates to slice direction in scanner 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 Calculated complex shear modulus images using curl-operator inversion method for SampLe Interval Modulation Magnetic Resonance Elastography and one-dimensional Magnetic Resonance Elastography 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 viii

9 LIST OF ABBREVIATIONS 1D 3D ADC.CPP FID FOV GE One-dimensional Three-dimensional Analog-to-Digital Converter File extension for programs designed using the C++ language Free Induction Decay Field of View Gradient Echo IDEA Integrated Development Environment for Applications (Siemens Software) MRE MRI Magnetic Resonance Elastography Magnetic Resonance Imaging NMR Nuclear Magnetic Resonance RF RTE Radio Frequency Real-Time Event RTEB Real-Time Event Block SDP SE Spectral selective Displacement Projection Spin Echo SLIM SampLe Interval Modulation TE TE/2 TTL TR Echo Time One-half Echo Time Transistor-transistor logic Repetition Time ix

10 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 threedimensional 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. x

11 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. 1

12 2 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 threedimensional 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.

13 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 M x (t), M y (t), and M z (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 M 0 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) 3

14 4 where ω is the Larmor Frequency [6]. An NMR signal is generated by first exciting the magnetization vectors with a radiofrequency (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, B 1 is the strength of the magnetic field produced by the RF pulse, and τ RF 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 B 0. 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.

15 5 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 B 0, 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, B 0 is the permanent magnetic field vector, K is the magnetic field gradient vector, and r is the spatial position vector.

16 6 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 B 0 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.

17 7 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

18 8 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

19 9 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).

20 10 (4.1) (4.2) M T * is the spatially-dependent transverse magnetization vector, S is the k-spacedependent acquired signal, and k x and k y 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

21 11 Figure 5. A Gradient Echo MRI pulse sequence with Echo Time TE defined. 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.

22 12 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

23 13 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. Figure 6. A Spin Echo MRI pulse sequence with both half-echo times defined. 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

24 14 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

25 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, G L and G S 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.

26 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 G L. 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 G S using the method of Least-Squares solution to the overdetermined problem, resulting in equation 11: (11) where G S 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 G S 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:

27 17 or, where: (12) where U 1, U 2, and U 3 are the first, second, and third orthogonal components of deflection vector U, respectively, and x 1, x 2, and x 3 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 G S and the complex first Lamé Parameter modulus G L. 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)

28 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.

29 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 τ K 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 τ K 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 0 th -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)

30 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:

31 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 K 0, ω K, 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 Y n, ω n, 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 0 th -moment-nulled condition of equation 18, the duration of magnetic field gradient application τ K is taken to be an integer multiple of the gradient s period of oscillation, as in equation 23: (23) frequency. where q is an integer number of cycles of the magnetic field gradient with angular 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:

32 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 K 0 and the physical vibration amplitude Y n. 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]:

33 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.

34 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 τ K 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

35 25 field gradient obeys the 0 th -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)

36 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 oddinteger 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 1 st -moment-nulled. As with the 0 th -moment-nulling condition of equation 18, a function K(t) is said to be 1 st -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 0 th - (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 K 0 and ω K 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.

37 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.

38 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.

39 29 Figure 11. A Spin Echo Magnetic Resonance Elastography pulse sequence diagram with both half-echo times and the motion-encoding gradient gap defined. 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

40 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 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)

41 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.

42 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

43 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 K 1, K 2, and K 3, 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, f q represents the frequency of the physical vibration s q th component, and, τ j, and N j are the amplitude, period of oscillation, and integer number of cycles of the MEG in the j th direction, respectively.

44 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 = N j. 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 timedependent displacement vector provides direction-specific displacement vectors at each discrete frequency.

45 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.

46 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.

47 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

48 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)

49 39 where subscripts j = 1, 2, and 3 correspond to the slice, read, and phase directions, respectively, and K j and u j 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 ξ j, Y j, s j,, K j 0, and q j are the direction-specific encoding efficiency, physical vibration amplitude, phase offset, MEG amplitude, and number of MEG cycles, respectively. By using different sampling intervals, Δt j, 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.

50 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.

51 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,.

52 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) 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.

53 Figure 16. Each of eight MEG start times in a SLIM-MRE sequence depicted in real time (a) and discretized time (b). 43

54 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 44

55 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

56 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 userdefined 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

57 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.

58 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.

59 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.

60 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 halfpulses. 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.

61 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 halfpulse for sinusoidal (a) and trapezoidal (b) MEGs.

62 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

63 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 transistortransistor 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

64 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

65 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.

66 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.

67 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.

68 Figure 25. Maximum erroneous frequency trend lines for MEGs when using fsdsroundupgrt(), fsdsrounddowngrt(), and alternating both rounding functions. 58

69 V. EXPERIMENTAL SETUP A. Physical Vibration Source used. To produce the vibration necessary for an MRE experiment, the experimental setup of Figure 26 is 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 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 59

70 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.

71 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 Silicone rubber with slightly different densities. The inner gel sample, of irregular shape and approximate diameter of 25 mm, is first produced using EcoFlex with standard density of 1040 kg/m 3 [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 reagents at a volume ratio of 1:9 before curing. After curing for 24 hours, the sample is ready for testing.

72 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 TR Field of View (FOV) Image Resolution Value 1000 ms 32 mm (read) x 32 mm (phase) x 10 mm (slice) 64 pixels (read) x 64 pixels (phase) Number of Slices 20 Slice Thickness 0.5 mm Flip Angle 90

73 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 TE ms ms ms ms Vibration Frequency 1000 Hz 1000 Hz 1000 Hz 1000 Hz

74 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 TE ms ms ms ms Vibration Frequency 1000 Hz 2000 Hz 3000 Hz Superposition of 1000, 2000, and 3000 Hz

75 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. 65

76 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

77 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.

78 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.