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Development of a desktop high-resolution MRI for microflow visualization Sahebjavaher, Ramin 2008

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Development of a Desktop High-Resolution MRI for Microflow Visualization by Ramin Sahebjavaher  BASc., The University of British Columbia, 2006  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  Master of Applied Science in THE FACULTY OF GRADUATE STUDIES (Electrical and Computer Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) October 2008 © Ramin Sahebjavaher, 2008  Abstract Research in lab-on-a-chip (LOC) technology involving microfluidics is a growing field aiming at the development of miniaturized biomedical systems with rich functionality. In order to design effective LOC microfluidic systems, the flow fields and the fluids inside LOC devices need to be carefully characterized. High-resolution magnetic resonance imaging (MRI) offers a powerful non-intrusive technology for this application. In this thesis, the design and implementation of a prototype for a desktop high-resolution MRI instrument, consisting of a magnet, gradient coils, gradient amplifiers, and radio frequency (RF) electronics, is presented. To reduce the size and cost of this MRI instrument, a permanent magnetic configuration with a magnetic flux density of 0.6 T is designed with off-the-shelf NdFeB permanent magnets. The coils of the triaxial gradient module are developed using a novel lithography technique. This gradient module is capable of generating gradient fields as high as 2.83 T/m with custom made current amplifiers. The radio frequency (RF) probe is integrated with the gradient module and is connected to the RF electronics which are made using off-the-shelf components. Pulse sequences and signal processing for acquiring static images and velocity profiles are described. The performance of this instrument in terms of static and dynamic image resolution are presented. As a preliminary test, the velocity profile of water flowing inside a small tube was measured with a nominal resolution of 40 µm. The instrument is designed for a static resolution of better than 30 µm and a velocity resolution better than 50 µm/s. Improvements to the current instrument in addition to theoretical limitations are also detailed.  ii  Table of Contents Abstract.......................................................................................................................................... ii Table of Contents ......................................................................................................................... iii List of Tables ............................................................................................................................... vii List of Figures............................................................................................................................. viii Acknowledgements .................................................................................................................... xix Dedication .................................................................................................................................... xx Chapter 1 - Introduction .............................................................................................................. 1 1.1.1 Motivation.......................................................................................................................... 1 1.1.2 Objective ............................................................................................................................ 7 1.2 MRI Fluid Measurement at Microscale - Prior Art .............................................................. 9 1.3 Thesis Overview ................................................................................................................. 11 1.4 Principles of Magnetic Resonance Imaging ....................................................................... 11 1.4.1 Magnetic Dipole Moment ............................................................................................ 11 1.4.2 Bulk Magnetization...................................................................................................... 12 1.4.3 Bloch Equations ........................................................................................................... 14 1.4.4 Excitation ..................................................................................................................... 18 1.4.5 Detection of Magnetic Resonance Signal .................................................................... 21 1.4.6 Gradient Fields............................................................................................................. 22 1.4.7 The Signal Equation..................................................................................................... 25 1.4.8 Imaging Principles ....................................................................................................... 26  iii  1.4.9 Flow Imaging ............................................................................................................... 33 1.4.10 Limitations of MRI Techniques................................................................................. 41 Chapter 2 - Components of the High-Resolution MRI............................................................ 47 2.1 Magnet ................................................................................................................................ 47 2.1.1 Magnet Requirements .................................................................................................. 47 2.1.2 Magnet Design ............................................................................................................. 48 2.1.3 Magnet Characterization.............................................................................................. 76 2.1.4 Summary of Magnets................................................................................................... 87 2.2 Gradient Coils ..................................................................................................................... 88 2.2.1 Gradient Coil Requirements ........................................................................................ 88 2.2.2 Gradient Coil Design ................................................................................................... 89 2.2.3 Gradient Coil Characterization .................................................................................. 110 2.2.4 Summary of Gradient Coils ....................................................................................... 122 2.3 Gradient Amplifiers .......................................................................................................... 123 2.3.1 Gradient Amplifier Requirements.............................................................................. 123 2.3.2 Gradient Amplifier Design ........................................................................................ 124 2.3.3 Gradient Amplifier Construction ............................................................................... 142 2.3.4 Characterization and Testing of the Gradient Amplifiers.......................................... 146 2.3.5 Summary of the Gradient Amplifiers ........................................................................ 151 2.4 RF Electronics................................................................................................................... 153 2.4.1 RF Electronics Requirements .................................................................................... 153 2.4.2 RF Design .................................................................................................................. 154 2.4.3 Characterization of RF Electronics ............................................................................ 164 iv  2.4.4 Summary of the RF Electronics................................................................................. 172 Chapter 3 - Overall System Design ......................................................................................... 173 3.1 Overall Component Layout and Interconnections ............................................................ 173 3.2 Data Acquisition Software................................................................................................ 175 3.3 Frequency Shift Compensation and Signal Acquisition ................................................... 179 Chapter 4 - Micro MRI Static Imaging and Velocity Visualization .................................... 185 4.1 Pulse Sequence for Static Imaging ................................................................................... 185 4.2 Sample Static Image ......................................................................................................... 190 4.3 Pulse Sequence for Velocity Imaging............................................................................... 192 4.4 Sample Velocity Visualization Measurements ................................................................. 195 Chapter 5 - Conclusion and Future Work.............................................................................. 200 5.1 Conclusion ........................................................................................................................ 200 5.2 Future Work ...................................................................................................................... 201 5.2.1 Elimination of Noise in the Experimental Setup ....................................................... 201 5.2.2 Design of Optimized Pulse Sequences ...................................................................... 202 5.2.3 Improvements of the Magnet ..................................................................................... 203 5.2.4 Improvements for Gradient Coils and Amplifiers ..................................................... 204 5.2.5 Improvements of the RF Electronics ......................................................................... 204 Bibliography .............................................................................................................................. 206 Appendix A – Gradients........................................................................................................... 214 A.1 Gradient Fabrication Process ........................................................................................... 214  v  Appendix B – Gradient Amplifiers ......................................................................................... 219 B.1 Circuit Schematics of the Gradient Amplifiers ................................................................ 219 B.2 Printed Circuit Board (PCB) Layout of the Gradient Amplifiers .................................... 227 B.3 Gradient Amplifier Bill of Material (BOM) .................................................................... 235 Appendix C - RF ....................................................................................................................... 238 C.1 Impedance Matching Setup.............................................................................................. 238 C.2 Selection of Gains in the Receiver ................................................................................... 240 Appendix D - Pulse Sequence Design ...................................................................................... 245 D1. Static Imaging .................................................................................................................. 245 D2. Velocity Imaging.............................................................................................................. 249  vi  List of Tables Table 1. Advantages and disadvantages of micro PIV and micro MRI fluid visualization instruments at micro scale....................................................................................................... 6 Table 2. Advantages and disadvantages of magnets used for MRI. ............................................. 50 Table 3. Properties of commonly available hard magnetic material [44]..................................... 53 Table 4. Properties of commonly available soft magnetic material.............................................. 54 Table 5. Clinical whole-body MRI permanent magnets available as of 2008.............................. 60 Table 6. Electrical and magnetic circuit analogy.......................................................................... 64 Table 7. Specification of the magnet based on results from simulations...................................... 73 Table 8. Specification of the magnet based on results from measurements. ................................ 77 Table 9. Specification of the gradient coils. The clearance between the gradients is 9.5 mm and the total insert thickness is 16 mm. (number of turns n = 3) .............................................. 106 Table 10. Measured electrical properties of the gradient coils including connecting wires....... 110 Table 11. Measured gradient sensitivity and strength of the triaxial gradient module............... 116 Table 12. Required and measured specification of the gradient amplifiers and the compensation coil amplifier....................................................................................................................... 152 Table 13. Properties of the RF probe. ......................................................................................... 165 Table 14. Properties of the RF transmitter.................................................................................. 167 Table 15. Properties of the RF receiver. ..................................................................................... 170 Table 16. Specification of the RF receiver. ................................................................................ 244  vii  List of Figures Figure 1. Simplest form of a sample microfluidic microchannel with one inlet and one outlet. (dimensions are in mm) .......................................................................................................... 2 Figure 2. Velocity profile of a Newtonian fluid passing through the microchannel. ..................... 2 Figure 3. Schematic view of a typical micro PIV setup. ................................................................ 3 Figure 4. Artistic rendition of the futuristic MRI instrument for visualization of flow in microfluidic chips (courtesy of V. Dokhani). ......................................................................... 8 Figure 1. Direction of bulk magnetization and external field (a) at equilibrium and (b) after it is disturbed................................................................................................................................ 14 Figure 2. Relaxation bulk magnetization after a 90° excitation onto the x-axis. In this simulation T1 = 40 ms and T2 = T2* = 10 ms. (B0 directed along in z-direction) .................................. 16 Figure 3. The three components of the bulk magnetization after a 90° excitation onto the x-axis. ............................................................................................................................................... 17 Figure 4. Effect of T2 (and T2*) on the phase coherence of the magnetic resonance signal. The three numbered arrows indicate the magnetization of three spins. The spin marked as 1 experiences a slightly lower magnetic field, which also makes it precess slower, compared to spins marked as 2 and 3. In (a) all spins are in phase and result in a large Mxy. As time passes (a  b  c) the spins go out of phase and their net magnitude Mxy is reduced. (B0 is  pointing into the page) .......................................................................................................... 18 Figure 5. Effective magnitude (a) and the corresponding flip angle (b) of the bulk magnetization in the rotating frame of reference.......................................................................................... 19 Figure 6. Spin Echo: starting with 90y° flip (a) the T2* decay from in-homogeneities of the  viii  magnet de-phase the signal (b). At time τ the 180x° pulse is applied (c) and the phases are rotated about the x-axis (d). The spins start to rephrase (e) and form an echo (f) at time 2τ with a reduced amplitude due to the intrinsic T1 and T2 relaxation times. (B0 is pointing into the page)................................................................................................................................ 21 Figure 7. Ideally, when no gradient field is applied (a) a single frequency is present. When a gradient is applied (b) a spread of frequencies that corresponding to the shape of the sample will be present. ...................................................................................................................... 24 Figure 8. In phase encoding (a) k(t) stays constant during acquisition of the MR signal whereas in frequency encoding (b) k(t) varies during sampling......................................................... 28 Figure 9. Spin echo (SE) imaging sequence. ................................................................................ 29 Figure 10. A standard SE imaging sequence is shown in (a) with its trajectory in k-space (b). An inverse 2D Fourier transform of the k-space (c) yields and image (d). ................................ 32 Figure 11. Gradient recalled echo (GRE) imaging sequence ....................................................... 33 Figure 12. Inversion saturation SE sequence for 1D flow imaging. ............................................. 34 Figure 13. Simplified SE for 1D imaging of fast flow. ................................................................ 35 Figure 14. When spins do not have any dynamic components, a uni-polar gradient is used to encode their position (a). By applying a bi-polar gradient (b), phase accumulation from stationary components is eliminated and non-stationary spins will have a velocity dependent phase shift. ............................................................................................................................ 38 Figure 15. Gradient recalled echo (GRE) velocity encoding imaging sequence. The experiment is repeated twice: first using VEG1 and then VEG2. ............................................................... 40 Figure 16. Definition of the coordinate system. ........................................................................... 43 Figure 17. Magnetic materials are characterized by their intrinsic and normal B-H hysteresis  ix  curves. ................................................................................................................................... 52 Figure 18. C-shaped permanent magnet configurations. The leakage is highest for (a) and the field is highest for (c). ........................................................................................................... 55 Figure 19. H-shaped permanent magnet configurations. The leakage is highest for (a) and the field is highest for (c). ........................................................................................................... 57 Figure 20. Halbach permanent magnet configurations. Ideally a hard magnetic material is magnetized as seen in (a). Due to practical limitations, the magnetization pattern is approximated by discrete magnets (b-e). Materials with high magnetic permeability can be used to in crease homogeneity (d). ....................................................................................... 59 Figure 21. A permanent magnet can be modeled as a voltage source Fm with an internal resistance Rm. ........................................................................................................................ 65 Figure 22. C-shaped (a) and H-shaped (b) permanent magnet configurations chosen for this work and their corresponding equivalent magnetic circuit. ........................................................... 67 Figure 23. The fringe field and region of saturation of the C-shaped magnetic configuration along with its effective gap is shown.................................................................................... 68 Figure 24. Final design of the C-shaped permanent magnet. ....................................................... 71 Figure 25. Final design of the H-shaped permanent magnet. ....................................................... 72 Figure 26. Distribution of magnetic flux density in the final C-shape design (a) and the leakage and fringing effects (b) are shown. ....................................................................................... 74 Figure 27. Distribution of magnetic flux density in the final H-shape design (a) and the leakage and fringing effects (b) are shown. ....................................................................................... 75 Figure 28. Manufactured C-shaped (a) and H-shaped (b) magnets. ............................................. 76 Figure 29. Magnetic flux density along the gap of the C-shaped magnet (a). The solid line  x  represents simulation values and the crosses show measurement data points. The variation of the magnetic flux density along the center of the x and y axes are shown in (b). ............ 80 Figure 30. Magnetic flux density along the gap of the H-shaped magnet (a). The solid line represents corrected simulation values and the crosses show measurement data points. The variation of the magnetic flux density along X and Y axes in the middle of the magnet (Z = 0 mm) is shown in as contour lines (b). ........................................................................ 82 Figure 31. Fluctuation of the magnetic flux density and the room temperature in 6 hours (a). The temperature coefficient is -707 ppm/°C (b). ......................................................................... 85 Figure 32. Effect of flow of current in the B0 compensation coils. ............................................. 87 Figure 33. The direction of the field generated by the gradient coils (a); location of the gradient coils and the probe (b); The gradient field in x, y, and z directions are shown in (c), (d) and (e), respectively..................................................................................................................... 91 Figure 34. Maxwell gradient coil.................................................................................................. 92 Figure 35. Rectangular loop gradient coil..................................................................................... 93 Figure 36. The positioning of the three gradient coils: Gx, Gy and Gz.......................................... 97 Figure 37. Simulated current path configurations for the Maxwell gradient................................ 99 Figure 38. Simulated current path configurations for the “rectangular loop” gradient. ............... 99 Figure 39. Simulation results of Gz: (a) shows the magnetic field contours; (b) and (c) show the gradient field strength and linearity. ................................................................................... 101 Figure 40. Simulation results of Gy: (a) shows the magnetic field contours; (b) and (c) show the gradient field strength and linearity. ................................................................................... 102 Figure 41. Simulation of the transient response of the Gz gradients in the presence of conductive and non-conductive poles. The induced currents are shown in the poles in plot (a); the  xi  transient response of the labelled point is shown in (b) with driving currents that have rise times of 10, 50 and 100 microseconds................................................................................ 104 Figure 42. Dimensions of the final design of the three gradient coils are shown. All the conductors have the same width of 1.5 mm and height of 0.25 mm. ................................. 105 Figure 43. Mask (a) and the assembly (b) of the gradient coils.................................................. 107 Figure 44. Photograph of the fabricated (a) and the assembled (b) gradient coils (only the top piece of the insert is shown)................................................................................................ 108 Figure 45. Top view of the insert module with integrated probe and gradients ......................... 109 Figure 46. Pulse sequence used to verify the strength of the gradient coils. .............................. 111 Figure 47. Bandwidth of the MR signal for different gradient field strength are shown for the three gradients Gx, Gy and Gz in (a), (b) and (c), respectively............................................ 113 Figure 48. Gradient strength vs the input current of the gradients Gx, Gy and Gz are shown in (a), (b) and (c) respectively. ...................................................................................................... 115 Figure 49. Magnetic flux detected by a pickup coil inside the probe area during a gradient pulse. ............................................................................................................................................. 118 Figure 50. Pulse sequence to detect the effect of eddy currents. ................................................ 119 Figure 51. FID signal immediately after the removal of the gradient pulse. (Gz = 0.28 T/m) ... 120 Figure 52. FID signal immediately after the removal of the gradient pulse. (delay = 25 µs)..... 121 Figure 53. The duty cycle is defined as the ratio of the ON time of the pulse to its period (= repetition time, TR)............................................................................................................. 124 Figure 54. The schematics of the three stages of the gradient amplifier: the power stage, the driver of the power stage and the current controller is shown............................................ 126 Figure 55. Simplified schematic of the power stage................................................................... 128  xii  Figure 56. Simplified schematic of the driver stage. .................................................................. 129 Figure 57. Simplified schematic of the current controller. ......................................................... 130 Figure 58. Schematic of the current monitor. ............................................................................. 131 Figure 59. Screenshot of the final values obtained by simulation. ............................................. 133 Figure 60. Simulated steady-state output current versus the input voltage. The input voltage range is set to the maximum range of the DAQ (= ±10 V). The maximum output current is ±20 A. ................................................................................................................................. 134 Figure 61. Simulation of a 10 kHz sinusoidal input with ±5 V amplitude (top) resulting in a sinusoidal output of amplitude ±10 A (bottom).................................................................. 135 Figure 62. Simulation of a +20 A pulse (bottom) for +10 V input (top). The rise and fall times indicates that this amplifier exceeds the performance requirements necessary for very fast gradient switching............................................................................................................... 136 Figure 63. Simulation of the steady-state power dissipation (top) of the most critical components of the power stage for as full range ±10 V input (bottom). ................................................ 137 Figure 64. Simulation of the power dissipation (top) of the most critical components of the power stage for a 50 kHz ±10 A sine wave (bottom). The average dissipated power of the combined resistors and BJTs of the power stage is shown as the bold line saturating at about 195 W for each amplifier. ................................................................................................... 138 Figure 65. The top view shows the schematic layout of the BJTs and the power resistors on the heat sink. The three cross-sectional views show the details of the mounting of the BJTs and resistors. .............................................................................................................................. 140 Figure 66. Lumped model of the power dissipation of the power stage..................................... 141 Figure 67. CAD drawing of the assembled power stage and the PCBs of all four amplifiers. The  xiii  overall dimensions are also indicated. ................................................................................ 143 Figure 68. Photos of the heat sink with the BJTs mounted on top (a) and resistors mounted on the bottom (b). The PC cooling fans are attached on the top side (c)....................................... 144 Figure 69. Photo of the assembled gradient amplifiers and the B0 compensation amplifier (top). An angled view of the amplifier and the power supply is shown in the bottom photo. ..... 145 Figure 70. Measured response of the amplifiers for sinusoidal input (a) 1 A at ±30 kHz and (b) 10 A at ±10 kHz.................................................................................................................. 146 Figure 71. Measured frequency response of the amplifier into a 1.0 Ohm load is shown for current levels of ±1 A (= 2 App) and ±10 A (= 20 App). The -3 dB bandwidth (at 90° phase shift) is approximately 180 kHz which exceeds the performance requirements. ............... 148 Figure 72. Measured response of the gradient amplifiers to a step input of (a) +10 A and (b) -10 A with the Gz coil attached. The rise and fall times are below 10 µs which exceed the performance requirements. ................................................................................................. 149 Figure 73. Measured transient response of the gradient amplifier to the rising edge of a +15 A pulse with the Gz coil attached attached. The Voltage at the load and its current are shown. The overshoot and the small oscillations can be further minimized by tuning the P-I current controller. ............................................................................................................................ 150 Figure 74. Measured response of the amplifiers to a step input of (a) +5 A and (b) -5 A with the B0 compensation coil. The rise and fall times are below 10 µs which by far exceed the performance requirements of this amplifier........................................................................ 151 Figure 75. Orientation of the varying magnetic field (B1) in Solenoidal and Helmholtz coils inside a static magnetic field B0.......................................................................................... 155 Figure 76. Impedance matching of the RF probe is achieved by adjusting the two capacitors.. 156  xiv  Figure 77. Coil winding machine capable of winding 20 µm wires onto 100 µm micro capillary tubes. ................................................................................................................................... 157 Figure 78. The CAD drawing of the enclosure for the RF probe along with the orientation of the magnetic fields is shown in (a). The machined enclosure and the components of the probe are shown in (b) and (c). ..................................................................................................... 158 Figure 79. Pulse shaping of the excitation signal (bottom plot) is achieved by modulating a carrier sine wave (middle plot) with a pulse shaping signal (top plot). .............................. 159 Figure 80. Architecture of the transmitter with pulse shaping capability and LNA protection.. 161 Figure 81. Architecture of the receiver with direct quadrature phase-sensitive detection......... 162 Figure 82. Architecture of the overall RF system for excitation and detection of MR signals. 163 Figure 83. The RF electronics box implemented with off-the-shelf components. .................... 164 Figure 84. Scope screenshot of the tuned and matched probe. The probe is tuned to 25.2 MHz and matched to 50 Ohms..................................................................................................... 165 Figure 85. Output power of the power amplifier versus the input power. The amplifier has a gain of 30.4 dB and a 1-dB compression point of 29.9 dBm. .................................................... 166 Figure 86. Scope screenshots of the pulse shaping for a rectangular (a) and sinc (b) pulse....... 168 Figure 87. Amplitude of the FID signal versus the pulse duration of rectangular shape. The 90° flip is achieved for a pulse duration of about 24 µs. ........................................................... 169 Figure 88. 10 kHz sample outputs of the both quadrature channels form the receiver for -100 dBm input signal. The output waveform is averaged 16 times. The SNR of the non-averaged signal is 18 dB..................................................................................................................... 170 Figure 89. The dead-time of the receiver for a sample free induction decay MR signal is about 100 µs.................................................................................................................................. 171  xv  Figure 90. Schematic of the overall system indicating the connections between system components. ........................................................................................................................ 174 Figure 91. Experimental setup of the microfluidic visualization instrument with all the components connected. ....................................................................................................... 175 Figure 92. The synchronization between the DAQs is achieved by physically interconnecting them as shown..................................................................................................................... 177 Figure 93. Flow chart of the LabVIEW program that synchronizes the two DAQs to output six analog waveforms read from a data file. The LabVIEW program is in slave mode and controlled by a MATLAB program (master)...................................................................... 178 Figure 94. The LabVIEW program with a user friendly GUI in the front panel........................ 179 Figure 95. Flow chart of the MATLAB program (master) that controls the analog outputs via the LabVIEW program (slave). ................................................................................................ 181 Figure 96. The performance of the B0 compensation in terms of the shift of the magnetic resonance signal as a function of time is shown in the top figure with the reference is set to zero Hz. The error in the frequency shift is within ±150 Hz. The current inside the compensation coil is also shown as a function of time in the bottom plot. The peak of the error at time 6 min occurred because a lab chair was purposely brought close (50 cm) to the MRI setup............................................................................................................................ 182 Figure 97. The performance of the B0 compensation is shown in terms of the shift of the magnetic resonance signal as a function of time in the top figure with the reference changing by ±2 kHz. The error in the frequency shift is within ±150 Hz. The current inside the compensation coil is also shown as a function of time in the bottom plot. .................. 183 Figure 98. The coordinate system adopted in this work is shown for the sample. The field-of-  xvi  view (FOV) is also defined in the cross-sectional view of the sample. .............................. 186 Figure 99. Basic spin echo (SE) pulse used for 2D imaging. ..................................................... 187 Figure 100. A sample k-space showing the points for which the raw data is stored. The recorded MR signal from each phase encoding step is stored in a horizontal line in k-space that is vertically offset from the center depending on the area of the phase encoding pulse. ....... 190 Figure 101. Sample static image featuring the (cropped to 128x128 pixels) cross-sectional view of the tube (inner diameter = 1.67 mm). TE/TR = 5/102 ms; NEX = 128; FOVP = 15 mm; FOVF = 58 mm.................................................................................................................... 191 Figure 102. The basic modified spin echo (SE) pulse for phase sensitive velocity encoding. ... 192 Figure 103. Simplified SE for 1D imaging of fast flow (a) and the sequencing diagram used for the instrument (b)................................................................................................................ 194 Figure 104. A reduction in the amplitude of the MR echo is observed as the flow rate is increased in the tube. The reduction in echo amplitude is observed for flow rates above 100 ml/hr, and it is semi-linear for flows up to 2,300 ml/hr. ...................................................................... 196 Figure 105. The 1D flow imaging experiment for a flow rate of 393 ml/hr (Vmean = 50 mm/s) in (a) and a flow rate of 786 ml/hr (Vmean = 100 mm/s) in (b). The residual plots in the bottom plot of (a) and (b) are an indication of the velocity profile in the tube............................... 198 Figure 106. The velocity profile inside a 1.67 mm tube at flow rates of 393 ml/hr (Vmean = 50 mm/s) and 786 ml/hr (Vmean = 100 mm/s). The velocity profiles fit well with an expected parabolic profile of laminar flow. ........................................................................ 199 Figure 107. The assembly of the designed gradient coils (left) and its mask (right).................. 214 Figure 108. Fabrication process of the gradient coils is represented schematically from a crosssectional point of view. For more information refer to the text.......................................... 216  xvii  Figure 109. Photos of the fabrication steps. For more info refer to the text............................... 217 Figure 110. Final CAD drawing of the four-layered PCB design of two of the gradient amplifiers. ........................................................................................................................... 228 Figure 111. Lumped model of the RF probe............................................................................... 238 Figure 112. Test setup to measure reflections of the probe (a) and the resulting signals on the oscilloscope (b). .................................................................................................................. 240 Figure 113. The components of the receiver are shown in (a), and the signal path of the MR signal is shown in (b). ......................................................................................................... 241 Figure 114. Basic spin echo (SE) pulse used for 2D imaging. ................................................... 245 Figure 115. A sample spin echo (SE) pulse used for 2D imaging in this work.......................... 249 Figure 116. The basic modified spin echo (SE) pulse for phase sensitive velocity encoding. ... 250  xviii  Acknowledgements I am most grateful to both of my advisors Dr. Boris Stoeber and Dr. Konrad Walus for their guidance and encouragement that they have provided throughout the duration of my M.A.Sc. None of this work could have been accomplished without the countless productive meetings (often me randomly showing up at their offices) we had. It has been a privilege working with both them on such an interesting and challenging project. I would like to thank all the technicians and machinists especially Markus Fengler, Glenn Jolly and Bernhard Nimmervoll who have always given me excellent practical advice. My deepest thanks to: Dr. Walter Hardy for the many meetings and practical help he gave me to get my first NMR signal; Dr. Piotr Kozlowsky, Dr. Alex Rauscher and Dr. Alex MacKay for their extensive help with my many MRI questions; and Dr. Carl Michal and his students for teaching me how to tune and match my RF probe, and providing useful tips for my receiver design. A special thank you to my brother Dariush who has significantly contributed, with his hard and outstanding work, to the design and implementation of the gradient amplifiers. Thank you to all my friends and lab mates (Iman Mansour, Mahmood Valizadeh, Hamed Valizadeh, Pirooz Darabi, Amir Entezarmahdi, Vahid Bazargan, Reynald Hoskinson, Larsen Dittrich, Daniel Buescher, Holger Knell, Benjamin Mustin, Stefan Hampl, and many of whom I failed to mention) for their help and encouragement. A big thank you to Veis Dokhani for the futuristic sketch of the proposed instrument. And finally, and most importantly, I would like to thank Shaghayegh Khayambashi for her love and encouragement that pushed me to always perform at my best.  xix  Dedication  This thesis is dedicated to my parents who, with their endless inspiration, guidance and support, have taught me to lead my life to achieve beyond what I think I am capable of.  xx  Chapter 1 - Introduction 1.1.1 Motivation Research in lab-on-a-chip (LOC) technology involving microfluidics is a rapidly growing field aiming at the development of miniaturized biomedical systems with enabling functionality. These integrated devices can handle and manipulate minute amounts of biological samples to perform various tasks on a single microchip. These tasks range from sample preparation to detection of chemical or biological agents including DNA sequencing [1-3]. Conventionally, these experiments are performed in a lab requiring a considerably larger sample volume and longer analysis time. Characterization of fluid flow at these length scales is an essential step in the design of LOC devices. The behaviour of fluids at the microscale can differ from 'macrofluidic' behavior in that factors such as surface tension and viscosity start to dominate the system [4]. The presence of cells, biological molecules or other substances typically transported through lab-on-a-chip devices results in very complex fluid flows with unique properties [1], especially when these objects reach the dimensions of their flow environment. In order to design effective LOC microfluidic systems, these fluids need to be visualized for careful characterization at these length scales. In the simplest form, a LOC device consists of a microchannel, an inlet and an outlet as shown in Figure 1. Many different microfabrication methods have been developed for such integrated systems [3]; Soft lithography is one of the least expensive fabrication techniques which typically employs materials such as Polydimethylsiloxane (PDMS) for microstructures, that is bonded on a glass or silicon substrate [3].  1  Figure 1. Simplest form of a sample microfluidic microchannel with one inlet and one outlet. (dimensions are in mm) The fluid sample that is injected from the inlet port, passes through the microchannel, and exits from the outlet port as shown in Figure 2. The objective of this work is to visualize the flow field (i.e., the velocity profile) inside the microchannel, and to extract rheological information, such as viscosity, from the flow profile.  Figure 2. Velocity profile of a Newtonian fluid passing through the microchannel. Conventionally, instruments such as micro particle image Velocimetry (micro PIV) are used for visualization of flow in microfluidic structures [5-8]. A schematic of a typical micro PIV setup is  2  shown in Figure 3. First, small seed particles are added to the fluid of interest which would follow the flow. The measurement is performed under a microscope that is usually equipped with two pulsed laser sources and a high-resolution camera. The lasers are pulsed consecutively with a certain delay while the camera is recording the images. A correlation between consecutive images reveals a two dimensional (2D) velocity profile indicating the magnitude and direction of flow at various points in the microfluidic channel. More recent developments include a scanning micro PIVs which can create pseudo three dimensional (3D) images of the flow by combining several 2D slices [6]. The velocity profile can then be extracted by signal processing techniques [7].  Figure 3. Schematic view of a typical micro PIV setup. This technique produces high resolution images (sub micron) at relatively high speeds (delays of tens of millisecond). For the measurement to be successful, however, the seed particles need to be carefully chosen such they are small enough, so that their presence does not significantly  3  affect the flow pattern, and large enough that the particle are visible by the optics either by scattering or fluorescence detection mechanisms [7]. The size of the seed particle also needs to be large enough so that the Brownian motion does not introduce significant errors in the measured velocity profiles. As the seed particles affect the flow to some extent, it is necessary to average the velocity fields to reduce theses errors. In addition, this technique requires that the microfluidic structure and the sample to be transparent (at least partially) to the lasers so that the particles can be detected by the camera. This is usually the case when transparent materials such as PDMS or glass are used for the microfluidic structure. If, however, opaque samples or material are used for fabrication of microfluidic structures, then the micro PIV measurements are not possible. For structures made out of silicon it is possible to use infrared light sources as silicon is semi-transparent to infrared light, at the expense of reduced resolution [8]. For opaque materials the only other option for visualizing flow is to use X-ray sources and specialized detectors [9]. The main limitation using this technique is that the imaging plane is not well defined. In other words the resulting image is an average value of the signal at different depths which again limits the resolution. The objective of this work is to overcome the limitations posed by micro PIV by using magnetic resonance imaging (MRI) techniques. MRI is a non-invasive imaging method that uses magnetic fields and radiofrequency (RF) pulses to interact with nuclear spins in order to collect information regarding the sample under investigation. Nuclear spins such as hydrogen (H1, also referred to as proton) are almost always present in fluid samples (e.g., water and oil) used in microfluidics experiments. It is possible to encode spatial and temporal information into the spins to detect diffusion, relaxation, velocity, acceleration (and even higher order dynamics) of a fluidic sample [10-12]. Through a Fourier 4  transform and further signal processing the encoded information are presented to the researcher in form of a simple 1D plot, or 2D and even 3D images. MRI flow imaging is commonly used for blood flow imaging for clinical diagnosis and is referred to as MRI angiography (MRA) [1215]. For more information regarding the principles of magnetic resonance and velocity encoding techniques refer to Section 1.4 of this work. MRI is commonly used in steady or periodic flow due to limitations in temporal resolution (long acquisition times) of conventional MR imaging, however, recent advances in encoding and detection schemes and coil design have enabled fast flow imaging [16] of even turbulent flows [16, 17]. MRI has advantages in terms of being noninvasive (as it does not require injection of seed particles that would affect the flow) and not being adversely affected by optical opaqueness (many multiphase mixtures are optically opaque). A disadvantage of MRI technique is that the resolution cannot be much less than one-hundredth of the sample size due to signal-to-noise constraints [18]. In most MRI machines used for microscopy the spatial resolution is limited to about 10 µm which is far worse than the resolution of better than 1 µm obtained by micro PIV measurements. However, it turns out that MRI is very sensitive to detecting motion more accurate than conventional techniques. This is because the motion measurements techniques are not subject to the constraints of static imaging. Bonn et al. [18] report that sample displacements of less than 1 µm and velocities as small as 10 µm/s can be measured in MRI facilities in which the static imaging resolution cannot be better than 1 mm. This accuracy of flow measurement makes MRI particularly suitable for fluid mechanics research at the micron scale. Disadvantages of MRI techniques in general include experimental complexity, cost, and the large size of conventional superconducting magnets. For these reason the microfluidic research using MRI has been limited to researchers who have access to specialized MRI facilities. One important limitation of MRI is that ferromagnetic materials  5  cannot be present in the microfluidic structure as it will distort the magnetic field. The advantages and disadvantages of conventional and MRI fluid visualization instruments are summarized in Table 1. Table 1. Advantages and disadvantages of micro PIV and micro MRI fluid visualization instruments at micro scale. Advantages  Disadvantages Requires seed particles  micro PIV (conventional)  High spatial resolution (< 1 µm) Only transparent structures Fast acquisitions (< 10 ms) Inherently 2D No seed particle Long acquisitions Additional information (diffusion,  micro MRI  No ferromagnetic material relaxation time, etc)  (proposed method)  Complexity High spatial resolution (<10s µm) Cost High sensitivity to flow(<10s µm/s) Size 3D  In summary, available instrumentation, such as micro PIV, have limited velocity resolution, are inherently intrusive techniques that affect the flow with the required seed particles, and require a transparent microfluidic channel and fluidic samples. High-resolution magnetic resonance imaging (MRI) is a powerful technology that is able to characterize fluid flow in 3-dimensions, is  6  non-intrusive, and does not require transparent channels or fluidic samples.  1.1.2 Objective The objective of this work is to develop a desktop MRI fluid visualization instrument that is inexpensive so that it can be used by engineers and researchers working in the microfluidic research. The most expensive and massive component of an MRI instrument is the main magnet. Conventionally, superconductive magnets are used which, including their accessories (refrigeration), occupy a complete room. The rooms are usually specifically designed to withstand the large weight of the magnet and to shield the large stray magnetic fields. To reduce the cost and size (desktop) of the proposed instrument it was decided to utilize a permanent magnetic configuration. Permanent magnets are very compact, have relatively small stray magnetic fields, and do not require refrigeration. For more information regarding magnet types and design of permanent magnets refer to Chapter 2.1. The envisioned instrument would be placed in a corner of a research lab. The researcher would insert a sample microfluidics chip inside the proposed instrument, setup the desired information encoding scheme and press start. After the acquisition, the images can be post-processed to highlight the desired information and be displayed to the researcher. An artistic rendition of the proposed instrument is shown in Figure 4. The magnet is inside a magnetically shielded and thermally insulated enclosure on a workbench. The instrumentation to control the flow is attached to the side of this enclosure. The electronics and the computer are located underneath the desk. The user would place the chip in an insert that is used to accurately position the chip inside the magnet. The insert is also used to hold and transfer all the tubes and cables from the chip to the instrumentation outside of the magnet enclosure.  7  Figure 4. Artistic rendition of the futuristic MRI instrument for visualization of flow in microfluidic chips (courtesy of V. Dokhani). The proposed instrument will be an important research tool for scientists and engineers working in the growing area of microfluidic research and development including biomedical microsystems for bioanalysis and drug discovery applications for label-free flow imaging. This instrument has further applications in imaging of cells and other biological organisms within the channels and will provide direct benefits to the biotech industry.  8  1.2 MRI Fluid Measurement at Microscale - Prior Art To this date, no magnetic resonance imaging instrument has been designed solely for the purpose of measuring of flow fields. Most research groups have conducted velocity encoded imaging experiments either on clinical or general purpose imaging machines without any major modifications to the standard hardware [5, 10, 16-22]. This is especially true for velocity imaging of fluids in ‘macroscale’ and also clinical imaging of blood flow. For imaging at the microscale, however, there are a limited number of research groups who have modified parts of a standard MRI machine to optimize their performance for flow imaging. For example Sun et al. [21] have constructed a thermostatted glass rheometer that is optimized for use in a MRI machine as a ‘plug-and-play’ module. Rheological tests on aqueous solutions of sucrose and xanthan gum were performed, which were in good agreement with measured results by classical rheometer. They used a 2 T Oxford Instruments superconducting magnet and a general purpose Bruker console. The gradient coils were custom made which were driven by off-the-shelf Techron gradient amplifiers. They also had a custom made quadrature birdcage-type RF coil. A maximum fluid speed on 275 mm/s was measured inside a 4 mm diameter tube. The spatial resolution was 39 x 39 µm2 need the wall of the tube. A quantitative velocity resolution was not reported. Bosshard [19] reports very fast imaging of velocity profiles of tubes 4.8 mm in diameter at 200 frames per second (fps) at the expense of lower spatial resolutions of 1 mm x 2 mm in a 2D image. Using conventional imaging with an acquisition time of approximately 13 min, resolutions of 10 x 10 µm2 can be attained. The maximum velocity in the tube was 1120 mm/s; the velocity resolution is not reported. In this research, a 4.7 T commercial Bruker/GE  9  superconducting system was used. Only the sample holder was custom made. Perlo et al. [23] have measured the velocity profile of a fluid flowing in small tubes in a highly in-homogeneous magnetic field of a single sided magnetic configuration with a field of 0.23 T. They show a 1D plot of velocity for a tube of 3 mm with a spatial resolution of 0.3 mm and a 2D velocity profile inside a 6 mm diameter tube with a resolution of 0.5mm. The maximum velocity at the center was 20 mm/s; the velocity resolution was not reported. The spectrometer was custom made from off-the-shelf components. Uludag et al. [24] made a tubular heated rheometer to determine viscosity over a range of shear rates of melted polymers. They use commercial General Electric systems with TecMag spectrometers, a 0.6 T Oxford superconducting magnet and water-cooled gradient amplifiers. Velocities up to 36 mm/s were tested in a tube with 12.7 mm in diameter. A spatial resolution of 0.25 mm and a velocity resolution of 0.9 mm/s are reported. Goloshevsky et al. [25, 26] have made integrated planar gradient coils with an RF coil using microfabrication techniques. They utilized a 0.6 T superconducting magnet with commercial spectrometer and gradient amplifiers. The 1D velocity profile of a 1.02 mm tube with a nominal spatial resolution of 10 µm and a velocity resolution of 0.13 mm/s is presented. The maximum velocity inside the tube was of 51.5 mm/s. Callaghan [11, 27, 28] is probably the most prominent researcher in fluid mechanics who has used and developed spectroscopic NMR and MRI imaging techniques to make rheological measurements of fluids. He and his research group have made a number of setups using commercial magnets, gradients and RF electronics to measure diffusion and flow fields in macroscale and microscale. In his papers and books he suggest that 10 µm spatial resolutions are  10  possible with small samples and probes.  1.3 Thesis Overview The rest of this thesis is organized as follows: In Chapter 1.4, the principles of magnetic resonance imaging are briefly reviewed. Chapter 2 is broken down into four sections that describe the design, construction and characterization of the main components of the proposed MRI instrument. The main components are: the magnet (section 2.1), gradient coils (section 2.2), gradient amplifiers (section 2.3) and the radio frequency (RF) electronics (section 2.4). The overall system design and synchronization of the main components is detailed in Chapter 3. Pulse sequence design for static and velocity imaging, including preliminary velocimetry results obtained using the proposed instruments, are presented in Chapter 4. The conclusion and future work are covered in Chapter 5.  1.4 Principles of Magnetic Resonance Imaging The flowing section gives a semi-classical description of the pulsed magnetic resonance phenomenon. It also includes static and velocity encoded imaging sequences and a discussion of the limitations for MRI techniques for static and velocity imaging. 1.4.1 Magnetic Dipole Moment Fundamental properties of nuclear particles are mass, charge and spin. Fermionic spin is a vector quantity that can have values of ±½ [14]. The angular momentum  11  S = ℏI  (1)  is proportional to the net nuclear spin I = ±½ and the reduced Plank’s constant ћ = 1.055·10-34 J·s [29]. The magnetic dipole moment  µ = γS  (2)  can be calculated from the angular momentum [29] using the gyromagnetic ratio γ = 2π·(42.5764 MHz/T) for a proton (H1). Therefore, the magnetic dipole moment can be re-written as  µ = γℏI .  (3)  In atoms with a zero net nuclear spin (paired spins), no magnetic dipole moment exists. For atoms that have a non-zero net nuclear spin (un-paired spins), such as hydrogen, a magnetic dipole exists that interacts with an externally applied magnetic field B0 [30]. The collective interaction of magnetic dipole moments with an external magnetic field gives rise to the magnetic resonance signal. 1.4.2 Bulk Magnetization  At equilibrium, dipole moments either align (low-energy state) or anti-align (high-energy state) with an externally applied magnetic field [30]. In a large sample of hydrogen for example, not all dipole moments align with the applied external magnetic field. The bulk magnetization is the net dipole moment of a sample at equilibrium is given by M 0,V = ∑ V µ i .  (4)  12  Each aligned dipole moment cancels out an anti-aligned dipole in the sample. The bulk magnetization is zero if there is an equal number of aligned and anti-aligned dipole moments in a sample. Only with a non-zero magnetization is the nuclear magnetic resonance (NMR) signal detectable. The ratio of aligned versus anti-aligned dipole moment is given by the Boltzmann statistics for the ratio of aligned (low-energy) dipole moments n+ to the anti-aligned (highenergy) dipole moments n- for an external magnetic field B0 [T],  −  n =e n+  −  γ ℏ B0 k BT  (5)  the Boltzmann constant kB = 1.381·10-23 J/K, and the absolute temperature T [K] [30]. It follows that with an external magnetic field of 0.6 T at room temperature, for every 106 anti-aligned dipole moments there are approximately 106 + 4 aligned dipole moments. It is this excess of only four dipole moments out of every million that results in a non-zero bulk magnetization that can be used for the detection of a magnetic resonance (MR) signal. This is the main reason why nuclear magnetic resonance techniques have inherently low sensitivity. The magnetization of a sample with N hydrogen atoms can be derived as [29]  M 0,V =  N γ 2 ℏ 2 B0 . 4k B T  (6)  At thermal equilibrium, M0 and B0 point the same direction. It is customary to assign the direction of B0 to the z-axis as shown in Figure 1. If the bulk magnetization is disturbed out its equilibrium condition, precessional behaviour of the magnetization is observed. This means that the bulk magnetization will tend to recover to its thermal equilibrium condition (i.e., align with 13  the magnetic field). During this recovery, energy is dissipated into surrounding environment which is discussed in the next section.  z  B0  B0  Precession  z  M0 M  Mz  My  y  y  Mx  x  x  (a)  Mxy  (b)  Figure 1. Direction of bulk magnetization and external field (a) at equilibrium and (b) after it is disturbed. 1.4.3 Bloch Equations  The precessional dynamics of the magnetization is defined by the Bloch equation [31]  (M z − M 0 )k M x i + M y j dM = M × γB − − , dt T1 T2  (7)  where i, j and k are unit vectors in the Cartesian coordinate system. The first term describes the path the bulk magnetization follows after a disturbance (excitation) in order to align with the external magnetic field B0. It also includes the precessional frequency  ω0 = γ B0 ,  (8)  which is also called the Larmor frequency. The Larmor frequency is the frequency at which the  14  bulk magnetization precesses about the axis of the external magnetic field. For example, an external magnetic field B0 = 0.6 T results in a precessional frequency of ω0 ~ 25 MHz. The other two terms in (7) refer to the relaxation processes. The second term refers to the longitudinal (Mz) relaxation towards thermal equilibrium, which is characterized by the T1 (spinlattice) time constant. T1 depends on the sample material, temperature, and the magnetic field strength [32]. The last term refers to the transverse magnetization (Mxy) decay characterized by the T2 (spinspin) time constant. The T2 relaxation corresponds to decoherence of the transverse bulk magnetization. Detailed description and analysis of Bloch equation and the relaxation mechanisms are covered elsewhere [28-32]. In practice, another time constant called the T2* is also considered with (7) 1 1 = + γ∆B0 . * T2 T2  (9)  It takes into account non-ideal conditions, especially the non-homogeneity of the static magnetic field ∆B0. Assuming a sample with bulk magnetization of M0 in a homogenous magnetic field along the zaxis, if M0 is excited out of equilibrium onto the x-axis, the precessional path would be similar to the path shown by the Bloch simulations shown in Figure 2.  15  Mz [A.U.]  1  0 1  1  M  y  0  [A .U .]  0  -1  -1  M  .] A.U [ x  Figure 2. Relaxation bulk magnetization after a 90° excitation onto the x-axis. In this simulation T1 = 40 ms and T2 = T2* = 10 ms. (B0 directed along in z-direction) Starting from the x-axis, the bulk magnetization vector M0 starts precessing about the z-axis (parallel to the magnetic field) at the Larmor frequency ω0. The transverse bulk magnetizations  Mx, My decay according to the spin-spin time constant T2 = 10 ms while the longitudinal relaxation Mz recovers at a slower rate T1 = 40 ms as shown in Figure 3. After about 5·T1 = 200 ms thermal equilibrium is reached and the magnetization vector points towards the external magnetic field B0.  16  Mx [A.U.]  1 0.5 0 -0.5 -1 0  0.01  0.02  0.03  0.04  0.05  0.06  0.07  0.08  0.09  0.1  0  0.01  0.02  0.03  0.04  0.05  0.06  0.07  0.08  0.09  0.1  0  0.01  0.02  0.03  0.04  0.05  0.06  0.07  0.08  0.09  0.1  My [A.U.]  1 0.5 0 -0.5 -1  Mz [A.U.]  1 0.8 0.6 0.4 0.2 0  Time [s]  Figure 3. The three components of the bulk magnetization after a 90° excitation onto the xaxis. The T2* time constant is the dominant factor in determining the received signal decay. Basically, the spins of the sample experience slightly different magnetic fields at various locations of the magnet. This results in some spins precessing faster than others. After the initial excitation, all the spins are more or less in phase resulting in a large bulk transverse magnetization. A snap shot after the initial excitation is shown in Figure 4a. After a short time delay however, the spins start to acquire phase shift which results in a reduced overall bulk transverse magnetization (see Figure 4b and c). In other words, the phase coherence of the nuclei is lost.  17  1.4.4 Excitation  t = t0  t = t2 y  t = t1  y  y Mxy 3  2 1  Mxy x  (a)  Mxy  3 2  3  1  2  1  x (b)  x (c)  Figure 4. Effect of T2 (and T2*) on the phase coherence of the magnetic resonance signal. The three numbered arrows indicate the magnetization of three spins. The spin marked as 1 experiences a slightly lower magnetic field, which also makes it precess slower, compared to spins marked as 2 and 3. In (a) all spins are in phase and result in a large Mxy. As time passes (a b c) the spins go out of phase and their net magnitude Mxy is reduced. (B0 is pointing into the page)  To excite (flip) the bulk magnetization from its equilibrium position, a time varying magnetic field B1 is applied that is perpendicular to the main static magnetic field B0. The frequency of B1 is chosen to be the Larmor frequency ω0 (hence the term resonance) and the amplitude is usually two to three orders of magnitude less than the main magnetic field B0. Due to the high frequency of the excitation signal, it is called the radiofrequency (RF) signal. The circularly polarized component of the bulk magnetization  B1 (t) = B1 (cos(ω0 t) i − sin (ω0 t ) j )  (10)  corresponds to the projection of the path shown in Figure 2 onto the x-y plane. By controlling the amplitude, phase and duration of the pulse, the magnetization can be excited to almost any arbitrary direction. In the rotating frame (about the z-axis), the excitation signal can be  18  interpreted as changing the effective magnetic field [29]  ω Beff = (B0 − ) k + B1 (t) i γ  (11)  experienced by the sample. The rotating frame is a coordinate system that rotates about the z-axis (parallel to the magnetic field B0) with a rotational speed ω. For example, the excitation field B1 which is a function of ω according to (10) appears stationary in the rotating frame shown Figure 5a. The static magnetic field B0 is along the z-axis with excitation field B1 applied along the xaxis. The effective excitation field Beff is shown in Figure 5a. According to (11), if ω is very close to ω0 the effective magnetic field would almost exclusively be in the direction of excitation field B1, even thought its amplitude is only a fraction of that of B0. The magnetization is flipped about the effective magnetic field Beff according to the left hand rule (if the left-hand thumb is aligned in the direction of Beff, the spins are flipped in the direction of other fingers). For example, a 90° excitation on applied on the x-axis flips the magnetization onto the y-axis when the frequency is set at the Larmor frequency as shown in Figure 5b.  z  z  Flip Angle θ  M0 B0-ω/γ Beff y  x  y  x  B1 ω ≠ ω0  B1 =Beff ω = ω0  (a)  (b)  Figure 5. Effective magnitude (a) and the corresponding flip angle (b) of the bulk magnetization in the rotating frame of reference. 19  For a “hard” rectangular RF pulse, a 90° flip is achieved when B1 is applied on the x-axis at resonance (Larmor) frequency for a specific duration so that the magnetization is excited onto the y-axis. The required time,  t 90o  π θ = = 2, ω γB1  (12)  typically ranges from 10 µs to 1 ms for a 90° flip angle depending on the excitation coil and the RF electronics. A rectangular pulse (in the time domain) is commonly used to excite the whole sample. To excitation a specific region of the sample, various RF excitation pulses exist [15]. The RF pulse sequence used for imaging is explained later in this work. One way to recover the T2* de-coherence of the magnetic resonance signal mentioned in section 1.4.3 is to apply a second RF excitation, the 180° pulse, to rephase the spins [33] as shown in Figure 6. This is accomplished by flipping all the spins either about the x-axis or the y-axis after a short time delay τ (shorter than T2) after the initial 90° pulse. After 2τ, the spins rephase and the result is a spin echo as shown in Figure 6.  20  t = t0  t = τ-  t = t1  y  After of 90y° Pulse  y  y  Mxy  Mxy 3 2  3  2 1  Mxy  2  3  1  x  Before 180x° Pulse 1  x  (a)  x  (b)  t = τ+  (c)  t = t3  y  After 180x° Pulse  t = 2τ y  y  Spin Echo  3 3  1 2  x 3  x  1 2  2 1  Mxy x  Mxy  Mxy  (d)  (e)  (f)  Figure 6. Spin Echo: starting with 90y° flip (a) the T2* decay from in-homogeneities of the magnet de-phase the signal (b). At time τ the 180x° pulse is applied (c) and the phases are rotated about the x-axis (d). The spins start to rephrase (e) and form an echo (f) at time 2τ with a reduced amplitude due to the intrinsic T1 and T2 relaxation times. (B0 is pointing into the page) Excitation sequences need to be separated far enough apart in time to allow the longitudinal magnetization to align with the main static magnetic field. The repetition time TR is commonly chosen to be 3-5 times the expected value of T1.  1.4.5 Detection of Magnetic Resonance Signal To detect the MR signal, a coil perpendicular to the main magnetic field B0 is used to pick-up the  21  changing magnetic flux produced after the transverse (Mxy) bulk magnetization is excited out of equilibrium. The magnetic flux of the transverse bulk magnetization ΦM  ∫  ΦM =  d 3r B Receiver (r ) ⋅ M (r, t)  (13)  Sample  can be calculated, where r [m] is the position of the magnetization vectors, Breceiver(r) [T/A] is the magnetic flux density generated by the coil at position r per unit current, and M(r,t) is the transverse component of the bulk magnetization. The change of the magnetic flux  emf = Vreceiver = −  (14)  d ΦM dt  induces a voltage in the receiver coil. From (13) and (14) the voltage in the receiver coil can be re-written as  Vreceiver = −  d 3 ∫ d r B Receiver (r) ⋅ M(r, t = 0) dt Sample  (15)  where the voltage induced in receiver coil Vreceiver [V] has the maximum amplitude immediately after a 90° excitation (at t = 0 , Mxy is maximum).The signal decays as a function of time according to the relaxation times. The signal is demodulated and further processed in the receiver and sent to the computer to generate an image as will be explained later. 1.4.6 Gradient Fields According to (8), the frequency of the MR signal depends on the magnetic flux density in a linear fashion. Under a linearly varying main magnetic field B0, the sample experiences a different magnetic field as a function of location, which in turn results in location-dependent Larmor frequencies. The corresponding MR signals of different frequencies now correspond to  22  different locations in space. The function of the gradient field is to modulate the main magnetic field linearly in one direction in space. Usually three gradients are employed to encode the signal in 3-dimensions. The resulting flux density (16)  B = B 0,z + G r ⋅ r  of any point in space depends on the three gradients  Gr =  dBz dBz dBz + + dx dy dz  (17)  that are applied. Gradient fields have units of [T/m] and are generated by gradient coils. An example of the use of a gradient applied over a disc shaped object is shown in Figure 7. When no gradients are applied, the sample experiences a homogenous field; therefore, the resulting MR signal ideally contains a single frequency as shown in Figure 7a. If a gradient is applied over the object, however, the received MR signal has a frequency spread that corresponds to the projection shape of the object as shown in Figure 7b.  23  Bz  (a)  ω  ω0  B0 Sample  Spatial Position, r  Signal ω  Bz  (b)  ω0+γG(r)r  B0+Gr·r Sample  Spatial Position, r  Signal  Figure 7. Ideally, when no gradient field is applied (a) a single frequency is present. When a gradient is applied (b) a spread of frequencies that corresponding to the shape of the sample will be present.  The applied gradients change the local Larmor frequency and, thus, encode spatial position into the frequency of the detected MR signal. The precessional frequency with the application of gradient is (18)  ω (r, t) = γ (B0 + G(t) r) . The phase difference in the MR signal relative to the signal at r = 0 can be calculated as  24   t0  φ (r, t0 ) = r.  γ ∫ G(t′)dt′  .    0   (19)  The term inside the brackets t0  (20)  k (t0 ) = γ ∫ G (t′)dt′ 0  is a function of the applied gradients and time. The vector quantity k is the reciprocal space vector with units of m-1. By applying an adequate sequence of gradients with correct timing, a sample can be encoded to highlight hydrogen density, T1 relaxation, T2 relaxation, mass diffusion and even mass transport (i.e., flow) in three dimensions. Another application of the gradients is for selective excitation of the sample. To excite a crosssectional slice of certain thickness for example, a gradient field is applied along with a RF excitation of a specific bandwidth. To excite slice, the frequency component of the excitation pulse needs to be of rectangular shape. A rectangular shape in the frequency domain corresponds to a Sinc (sin x/x) in time domain. The slice thickness  ∆z =  BWSinc γGSlice  (21)  depends on the amplitude of the slice selection gradient GSlice and the bandwidth of the selective RF pulse BWSinc. Various types of pulses exist which are discussed in detail in [15]. The use of selective excitation is further discussed in the following sections. 1.4.7 The Signal Equation For an object (sample to be imaged) with a spatially varying magnetization density of M(r) the  25  MR signal before the detection coil  SignalMR (t) = ∫ M ( r′ ) e  − j ( ω0 t + φ)  d3r′  (22)  r  includes contributions of all precessing transverse magnetizations in the volume (assuming the relaxation terms are ignored). The term with ω0 can be removed by setting the reference frequency of the demodulator to be at baseband ω0. The signal equation can then be re-written in terms of k  SignalMR (k (t) ) = ∫ M ( r′ ) e− j r′ ⋅ k (t)d3r′ .  (23)  r  According to (19) and (22) the signal of a slice of thickness ∆z is − j k x (t)x + k y (t)y   SignalSlice (k (t) ) = ∆z ∫ ∫ M ( x, y ) e  dxdy .  (24)  x y  The signal equation has the form of a Fourier transform (FT), thus an image I(x,y) can be formed by taking the inverse FT of the signal equation  I(x, y) = FT−1 (SignalSlice (k (t) ) ) , I(x, y) = ∆z ∫ ∫ SignalSlice (k (t) ) ⋅ e  j k x (t)x + k y (t)y  (25)  dk x dk y ,  (26)  kx ky  I(x, y) = M ( x, y ) ∆z .  (27)  The cross-sectional 2D image is a representation of the spin density of a slice of the sample. 1.4.8 Imaging Principles Two implementations of spatial encoding can be derived from (26) namely phase encoding and  26  frequency encoding [12]. Frequency encoding includes the application of a constant gradient over the sample resulting in MR signal of different frequencies corresponding to the projection of the sample, as shown in Figure 7. Phase encoding involves a gradient is applied only for a short time t to introduce phase shifts (28)  Φ (r, t) = γ (B0 + G(t) r) t  in the MR signal. The phase encoding gradient is usually assigned to the y-axis while the frequency encoding gradient along the x-axis. A sequencing diagram of the RF and gradient pulses is shown in Figure 8 as a function of time. For example, in Figure 8a the 90° pulse is applied first followed by the phase encoding gradient. The time the MR signal is sampled is indicated by the data acquisition plot. In phase encoding the gradient (phase gradients, Gphase) is applied before the sampling the MR signal starts, as shown in Figure 8a. During the acquisition of the MR signal Gphase = 0 [T/m], thus according to (20) ky(t) = kphase(t) does not change during the sampling window. This ensures that each sample of the MR signal will have the same phase encoding kphase as seen in Figure 8a. When the phase gradient is applied, the phase of the MR signal changes linearly according to (20). This experiment is repeated for different durations or amplitudes for the phase encoding gradient so that each experiment has a different phase encoding. In frequency encoding the gradient is applied while the MR signal is sampled as shown in Figure 8b. Again the, 90 RF pulse is applied first, then after a time delay the frequency encoding gradient is applied for a specific duration. According to (20), kfrequency(t) changes linearly during the acquisition of the MR signal, this ensures that each sampled signal has a slightly different phase increment associated with it. In principle, the phase and frequency encoding techniques are  27  the same. The only difference between the frequency encoding and phase encoding is that kx(t) =  kfrequency(t) is different for each sample of the MR signal as shown in Figure 8b. In other words, the frequency encoding gradient alters the local magnetic field according to (18), which results in a spatial variation in the Larmor frequencies of the magnetization vectors of the sample.  90°  RF Gphase (a)  kphase(t)  Data Acquisition  90°  time  RF Gfrequency (b)  kfrequecy(t)  Data Acquisition 128 or 256 Samples of the MR Signal  Figure 8. In phase encoding (a) k(t) stays constant during acquisition of the MR signal whereas in frequency encoding (b) k(t) varies during sampling.  28  According to (26), a two dimensional inverse Fourier transform of the MR signal would yield an image. To generate a meaningful image, the MR signal needs to be sampled at specific k(t) locations. In practice, the data is collected on a Cartesian grid in k-space for computational efficiency. One of the earliest pulse sequences used to cover the complete k-space is called the spin-warp imaging sequence [34, 35]; where phase and frequency encoding are used. The more common spin echo pulse sequence for a 2D slice is Figure 9.  TE 90°  180°  RF GSlice  GPhase  GFrequency Data Acquisition  Tacq  Figure 9. Spin echo (SE) imaging sequence. In the spin echo (SE) imaging sequence, a 180° pulse is applied after the initial 90° pulse after a time delay TE/2; thus the MR signal forms at time TE. The selective 90° pulse is applied along with a slice-selection gradient to excite the nuclei in a slice. This is necessary to generate a crosssectional slice of the sample. After the end of the RF pulse, the slice-selection gradient is  29  reversed for a half of the time of the RF pulse to rephase all the spins of the selected slice. The frequency encoding gradient is applied during data acquisition for the same duration as the data acquisition. Gfrequency is also applied before the 180° pulse for a duration of half the acquisition time. The first part of the frequency encoding gradient de-phases the nuclei so that the second part of the frequency encoding gradient, which works in conjunction of the 180° pulse, rephases the spin exactly at the time TE as explained previously (see Figure 6). The entire horizontal line in k-space is sampled in a single repetition of the SE pulse sequence which is the frequencyencoding implementation discussed before. The phase encoding gradient is applied between the 90° and 180° pulses. At each repetition of the SE pulse sequence the phase gradient is applied with different amplitude to increment the phase. This results in vertical shifts in k-space at each repetition. This is the phase-encoding implementation mentioned before. After several repetitions the SE waveform (usually 128-512), the complete k-space is covered. The simplified trajectory in 2D k-space is shown in Figure 10b for a simplified SE pulse sequence in Figure 10a. Consider the first iteration of this pulse sequence when the phase encoding gradient is zero. The point 0 (in Figure 10a and Figure 10b) is the time just before the de-phasing pulse of the frequency encoding gradients is applied. The de-phasing pulse of the frequency encoding gradient, marked as 4, shifts the current position in k-space horizontally. Thus, at the end of the of both pulses before the 180° pulse, the position in k-space has moved to the point marked as 3,4 that lies on the horizontal ky = 0. After the 180° pulse, the current position in k-space moves to 6 by the trajectory marked as 5 in Figure 10b. Next, when the frequency encoding gradient is applied between points 6 and 7 while the MR signal is sampled, as shown in Figure 10a. The sampled MR signal corresponds to a single horizontal line (from 6 to 7) in k-space as shown in Figure 10b at ky = 0. Now, the iteration restarts with a different 30  phase encoding step. Consider an iteration in which the phase encoding gradient is maximum. The point 0 (in Figure 10a and Figure 10b) is the time just before the phase and frequency encoding gradients are applied. The phase encoding gradient marked with as number 1 shifts the current position in the k-space vertically (upwards); The de-phasing pulse of the frequency encoding gradient, marked as 4, shifts the current position in k-space horizontally. Thus, at the end of the of both pulses before the 180° pulse, the position in k-space has moved to the point marked as 1,4. After the 180° pulse, the current position in k-space moves to 6 by the trajectory marked as 5 in Figure 10b. Next, when the frequency encoding gradient is applied between point 6 and 7 while the MR signal is digitized, as shown in Figure 10a. The digitized MR signal corresponds to a single horizontal line (from 6 to 7) in k-space as shown in Figure 10b at ky =  kPhase,max. If this pulse seqcuence is repeated for different phase encoding steps, the entire k-space can be covered as shown in Figure 10c. A 2D inverse Fourier transform of the k-space would yield a cross-sectional static image of the sample (Figure 10d). For more information on various implementations to cover the k-space, refer to the book by Bernstein et al. [15].  31  180  90  5  ky = kPhase Individual Sample of the MR Signal  RF  6  7  6  7  GSlice 1 2 3  5  GPhase 7  0 4  6  7  kx = kFrequency  6  GFrequency  0 5  3,4  x 2,4  Data Acquisition  5 1,4  (a)  (b)  ky = kPhase  kx = kFrequency  (c)  IMAGE (d)  Figure 10. A standard SE imaging sequence is shown in (a) with its trajectory in k-space (b). An inverse 2D Fourier transform of the k-space (c) yields and image (d). Another commonly used pulse sequence is the gradient recalled echo (GRE) sequence as shown in Figure 11. The primary difference between SE and GRE is that the MR echo in the SE sequence is formed by a 180° pulse while in the GRE sequence it is formed by a gradient echo. A  32  gradient echo forms whenever spins are de-phased by applying a gradient for time t and then rephase by applying the gradient in the reverse direction for 2t.  TE 90° RF  GSlice  GPhase  GFrequency  MR Signal Tacq  Figure 11. Gradient recalled echo (GRE) imaging sequence  1.4.9 Flow Imaging Two commonly used techniques exits to encode velocity information: time-of-flight (TOF) and phase-contrast (PC). These methods are extensively described in [15, 29] and are mostly developed for the clinical sector to detect blood flow for magnetic resonance angiography (MRA) applications. In one implementation of the TOF technique, an additional 180° pulse is applied with a certain 33  delay tInversion before the imaging sequence. A sample inversion SE pulse sequence to visualize flow in one dimension is shown in Figure 12. For example, if a pulse sequence with tInversion = 0.6·T1 is used [30], the stationary spins do not generate a MR signal. The nonstationary spins, however, experience the inversion pulse with a slightly different timing compared to stationary spins. Thus moving spins generated an MR signal proportional to their velocity. A phase encoding can be added the sequence shown in Figure 12 along with slice selection gradients and selective RF pulses to generate a TOF flow encoded two dimensional image.  TR  180° 90°  180°  180°  RF GFrequency Data Acquisition tInversion  Figure 12. Inversion saturation SE sequence for 1D flow imaging. An even simpler TOF pulse sequence to visualize fast flows is shown in Figure 13. This pulse sequence is a one-dimensional SE pulse sequence with no slice selection. Stationary spins that experience both the 90° and 180° pulses generated a strong spin echo pulse. Moving spins that flow fast enough, exit the excitation/detection area and do not experience one of both the RF pulses. In that case, a reduction of the spin echo signal is observed semi-proportional to the flow.  34  The pulse sequence shown in Figure 13 is repeated twice, once when all the spins are stationary and once when the spins are moving. A difference between the magnitudes of their Fourier transform is an indication of the velocity profile inside the sample. This technique is used as a proof-of-concept test of the experimental setup of this work.  TR  180°  90°  90°  RF GFrequency Data Acquisition Figure 13. Simplified SE for 1D imaging of fast flow.  Phase contrast methods are widely used because they are much more sensitive to slow flows. For this reason, phase contrast (PC) techniques need to be implemented in this setup to measure the flow rates that are more commonly encountered in microfluidics chips. Unlike TOF based techniques that are dependent on relaxation effects, signal drop out in regions or complex flow, phase-based techniques are self calibrating since phase is directly proportional to motion, and is independent of other parameters, such as relaxation rates, geometry of the sample, and inflow effects [12]. The primary disadvantage of PC techniques is that, when the induced phase change is greater than 2π, phase warp occurs and causes aliasing artifacts in the velocity encoded  35  images. This problem, however, can be compensated to some extent by carefully adjusting the pulse sequence and also through post processing the images. As mentioned in sections 1.4.6 and 1.4.9, phase information is used to encode position. If position is generalized and is let to be also a function of time, then phase encoding can be extended to include higher order motions such as velocity. Assuming a flow with constant velocity, the position of the nuclei of the sample are given as (29)  r(t ) = r0 (τ) + v0 (τ) ⋅ (t − τ)  The coefficients represent r0(τ) and v0(τ) position and velocity at time τ. Referring to (19), the phase of the MR sample can be rewritten as t  (30)  φMR (r, t) = γ ∫ G (t′) ⋅ r(t′) dt′ 0 t t   ′ ′ φMR (r, t) = γ r0 (τ) ∫ G(t )dt + v 0 (τ) ∫ G (t′) ⋅ (t′ − τ) dt′ 0 0    (31)  which can be simplified using n-th moment of the gradient pulse sequence t  (32)  Mn (t ,τ ) = ∫ G (t′) ⋅ (t′ − τ)(n) dt′ 0  to read  φMR (r, t) = γ [r0 (τ) ⋅ M 0 (t, τ) + v 0 (τ) ⋅ M1 (t, τ)] .  (33)  The phase (34)  φMR (t) = φr (t, τ) + φv (t, τ)  36  is the combination of the phase accumulation due to static position components and also dynamic components (only velocity here). Assuming a uni-polar gradient similar to the one depicted in Figure 14a is applied (i.e., M0(t,τ) ≠ 0). According to (33), if velocity or higher order phase terms are negligible, the measured phase φMR is a measure for position. If the higher order terms, such as φv , are not small in comparison to φr then errors appear in the image as ghosts and blurriness. Now assume that a bi-polar gradient which has positive and negative gradient lobes with equal areas is used (i.e., M0(t,τ) = 0 and M1(t,τ) ≠ 0). As shown in Figure 14b the phase developed during the positive lobe cancels out the phase developed during the negative lobe for stationary spins. For moving spins, however, the developed phase does not cancel out because they experience a different gradient, and therefore a different phase shift, due to their change in location. The developed phase is an indication of the velocity of the spins. The mathematics to generate a velocity encoded image are presented next.  37  Uni-polar Gradient  Bi-polar Gradient 2x tgradient Gmax t  t  -Gmax Φr  Φr t  t  Φv t  (a)  (b)  Figure 14. When spins do not have any dynamic components, a uni-polar gradient is used to encode their position (a). By applying a bi-polar gradient (b), phase accumulation from stationary components is eliminated and non-stationary spins will have a velocity dependent phase shift. For a rectangular bi-polar gradient with amplitude Gmax and total pulse duration of tgradient the phase change  φMR (r, t) = − γ v0 (τ) G max t  2  (35)  Gradient  of a nucleus moving with speed v0 follows from (31). The maximum velocity  Vmax ≤  π  (36)  γ G max t 2  Gradient  38  can be calculated from the above equation noting that the allowable phase change for unambiguous measurement has to be –π < ΦMR < π. Ideally, if a bi-polar velocity encoding gradient is applied along one of the spatial axes, the velocity component in the same direction can be estimated [12, 29] in a single experiment. In practice, this is not feasible due to errors induced by eddy currents and other sources of phase error. Therefore, two acquisitions each with a different first moment M1(t) are applied to make accurate velocity estimates. If the phases of the reconstructed images are subtracted the velocity image is acquired. The subtraction process also removes the common mode errors due to eddy currents and other sources of error. This technique is known as the phase contract (PC) MR. The basic concept of PC imaging is to superimpose the velocity-encoding gradients onto the basic imaging sequences, such as SE and GRE, that were described in 1.4.8 and to reconstruct a series of velocity encoded images [30]. Various velocity encoding gradient (VEG) configurations and encoding technique exist to encode velocity in all spatial direction which are covered in [15]. An example velocity encoding sequence is shown in Figure 15 where a GRE imaging sequence is used along with a velocityencoded gradient in a single direction. First, a slice is excited by the 90° pulse and the slice selection gradient, Gslice. The phase and frequency encoding are exactly as explained for the standard GRE imaging. Two velocity encoding gradients (VEG) were applied in the slice selection direction to detect flow perpendicular to the cross-sectional slice. In the first iteration of this pulse sequence, VEG1 is applied and the MR data is recorded for a single phase encoding step. Then, for the same phase encoding step, VEG2 is applied and the data is recorded. This process is repeated until all the phase encoding steps are complete and the entire k-space can be covered.  39  90 RF  GSlice  GPhase  GFrequency  MR Signal  Velocity Encoding Gradient 1 Velocity Encoding Gradient 2  Figure 15. Gradient recalled echo (GRE) velocity encoding imaging sequence. The experiment is repeated twice: first using VEG1 and then VEG2.  40  The signal equation  φ  − j k x (t)x + k y (t)y  − j v  Signal2D−Slice−Velocity (k (t) ) = ∫ ∫ M ( x, y ) e  e  dxdy  (37)  x y  for a velocity encoded image is derived from (24), (34). After taking an inverse 2D Fourier transform the resulting image becomes  I(x, y) = FT−1 (Signal2D−Slice−Velocity (k (t)) ) , I(x, y) = M ( x, y ) e  φv  −j  (38) (39)  .  A velocity dependent phase shift appears in the image that is independent of the GRE sequence. The velocity parameter can be extracted by subtracting the phases  ∆φMR = ∡ [ I1 (r)] − ∡ [ I2 (r)] = φv,1 − φv,2 ∝ v0 (τ)  (40)  of the two images to find the velocity component (41)  ∆φMR ∝ v0 (τ) . Many other techniques exist to acquire velocity component in all spatial directions [12, 15]. 1.4.10 Limitations of MRI Techniques  The limitations of MRI techniques are in terms of materials that can be present in the sample to be imaged and the resolution limitations due to signal to noise ratio and the performance of the experimental instrumentation.  41  A central limitation for any magnetic resonance imaging instrument is that no ferromagnetic material can be present in or even close to the sample for proper imaging. In addition, the amount of conductive material in the sample volume must be limited for successful imaging. Due to thed pulse nature of MRI techniques, the magnetic field becomes distorted by the eddy currents that are generated in the conductive material close or inside the sample. The dominant and fundamental limiting factor for high resolution microscopy is the signal-tonoise ratio. Various formulas to estimate SNR, following a 90° flip, exist in the literature [36] the most accepted one is given by Abragam [37] 1  KηM0  µ0Qω0 VC  2 SNR =   , NFR  4k BTC ∆f   (42)  where K is a numerical factor close to unity that depends on the coil geometry, η is the filling factor, VC is the coil volume, Q is the quality factor, TC is the temperature of the probe, ∆f is the bandwidth of the receiver and NFR is the noise factor of the receiver (a measure of how much noise is introduced by the receiver itself). Assuming the instruments and the sample are at room temperature, using (6) the SNR is re-written as   K ⋅ n γ ℏ2µ 12   ω3 2 ⋅ V2 ⋅ Q 12  v 0 s  ⋅ 0 SNR =  1  4k BTR   NF ⋅ V 2 ⋅ ∆f 12     R c   (43)  The variables in the first bracket are mostly physical constants while the variables in the second bracket are defined by the properties of the instrument and components. The most important design variable that affects the SNR is the frequency term which is defined by the magnetic field strength. In other texts, even a higher dependence (ω07/4) has been suggested [38, 39] due to inter-dependence of variables of (43). Once the magnetic field is known, SNR can be optimized 42  by a coil with a small enough volume, Vc, to encapsulate the sample Vs. The smaller the sample, the less spins exist, and the smaller the signal becomes. The receiver has to have minimal noise factor NFR and the coil should have the largest practically feasible quality factor. The receiver bandwidth has to be chosen carefully to match the expected frequencies from the chosen pulse sequence. Another way to increase SNR is to average the received MR signal. SNR is proportional to the square root of the number of averages. The definition of a coordinate system for a fluidic sample that is flowing inside a tube with the diameter Di = 2·Ly is shown in Figure 16. The tube is along the x-axis (left of Figure 16) and the cross-sectional view (right hand side of Figure 16) is shown in the z-y plane.  Capillary Tube z z  Fluid  y  Flow  Lz  y  x Ly Di  Coordinates of the sample  Cross-sectional view  Figure 16. Definition of the coordinate system. For static imaging of a sample shown in Figure 16, the limiting factors in terms of positional accuracy can be calculated by differentiating the Larmor equation (18) as  x=  1 ω ( − B0 ) Gx γ  43  ∆x ≥  ∂x ∂x ∂x ∆B 0 + ∆G x + ∆ω ∂B0 ∂G x ∂ω  (44)  ∆G y 1 1 ∆B0 + L y + ∆ω Gy Gy γG y  (45)  ∆x ≥  For high resolution, (45) requires high field homogeneity, large gradients with good linearity, and sensitive distinction of signals that have comparable frequencies. The other fundamental limiting factor comes from the uncertainty principle [28] which states that  Tacq ≥  1 . ∆ω  (46)  This principle can be re-written as  ∆x ≥  1 γG x Tacq  (47)  where the acquisition time is Tacq. Equation (47) calls for large gradients with minimal acquisition times. For accuracies of better than tens of microns, diffusion also becomes a limiting factor which is given by [40] ∆x ≥  (48)  2DTacq  where a typical diffusion constant of water at room temperature is D = 2·10-9 m2/s. Equation (48) indicate that the acquisition time must be minimized so that diffusion is not the limiting factor of  44  spatial resolution. Another important point is that a trade-off that exists between (47) and (48) where the optimal value for the acquisition time can be calculated. The nominal resolution of the image  ∆x =  Di  (49)  Nsamples  is also dependent on the sample size, and the number of phase and frequency encoding steps,  Nsamples. A trade-off exists between the selected field-of-view that defines the largest sample that can be imaged and the resolution for a given imaging time. A higher resolution necessitates a smaller FOV for a given imaging time. The resolution for velocity encoded images, if the PC techniques are used, can be derived from (31) where the developed phase of a moving nucleus is (50)  ΦMR = −2π γ G x τ2 Vτ . The velocity accuracy ∂V ∂V ∆ΦMR + ∆G x ∂ΦMR ∂G x  (51)  Φ MR 1 ∆Φ MR + ∆G x 2 2πγG x τ 2πγG 2x τ2  (52)  ∆V =  ∆V =  is found by differentiating (50). To minimize the velocity errors the above equations indicate that large gradients with high linearity must be applied for a long pulse duration τ. Furthermore, it is necessary to increase the phase encoding steps such that the phase error ∆Φ MR is minimized. It is also worth mentioning the maximum velocities that can be detected using PC techniques in 45  MRI. One limitation is in terms of maximum phase shift found from (36) is given as  Vmax ≤  π , γ G x τ2  which can be corrected for to some extent in software [41]. The second limitation is related to the length of the detection coil LReceiver. If the flow is faster than  Vmax <  (53)  LReceiver TE  the excited slice moves out of the detection area and therefore the MR signal is lost. TE is the echo time.  46  Chapter 2 - Components of the High-Resolution MRI 2.1 Magnet The requirements, design, and characterization of the magnet are described in the following section.  2.1.1 Magnet Requirements Important characteristics of a magnet for magnetic resonance applications are: the magnetic field strength, homogeneity, temporal stability and access.  Field Strength: The signal-to-noise ratio, SNR, is proportional to the Larmor frequency as (54)  SNR ∝ ω0α where, ω0 = γB0 , which  increases rapidly as the magnetic flux density B0 is increased  (3/2 < α < 7/4).  Homogeneity: The decay of the MR signal is dominated by the T2* time constant  T2* ≈  1 γ∆B0  (55)  which has to be maximized by reducing the flux density variations ∆B0 in the volume of interest. This is equivalent to improving the homogeneity, ∆B0/B0 of the magnet. In addition, the accuracy of the spatial localization  ∆x hom ogeneity =  1 ∆B0 Gx  (56)  47  is dependent on the homogeneity as described in Chapter 1.4.  Temporal Stability: There are two reasons why the magnetic field needs to be stable. One is related to using averaging to increase the SNR of the MR signal from a large number of acquisition cycles. This is only possible if the magnetic field is stable over the whole imaging time. The second reason is that even if no averaging is employed, the image may get distorted if the magnetic field fluctuates for two consecutive but slightly different (e.g., phase incremented) pulse sequences.  Access: The gap or bore of the magnet has to be sufficiently large to accommodate the sample, gradient and RF coils.  2.1.2 Magnet Design Resistive, superconducting and permanents magnets are the three main types of magnets used for MRI.  Resistive magnets, or electromagnets, generate the magnetic field using a coil through which a current is driven. The magnetic flux return path can be air or magnetic material. Typical whole body MRI scanners have flux densities less than 0.4 T. A constant large amount of electrical power (typically 50 kW) is required to maintain the field. Most of the power is dissipated in the windings as heat necessitating elaborate cooling systems. The field homogeneity and temperature stability is relatively poor compared to superconducting magnets. This is because the variation of the field is limited by the current regulation of the power supply in an electromagnet; whereas the power supply is disconnected from a superconducting magnet after it is energized. Advantages of resistive magnets include the ability to be switched off easily, lower fringe fields compared to superconducting magnets and better temperature stability compared to permanent 48  magnets. Electromagnets have much lower capital and operating cost than superconducting magnets [13, 30]. In Superconducting magnets, the coils are made of superconducting materials which have zero resistance at cryogenic temperatures. As long as the windings are kept below their critical temperature (between 1.9 to 70 °K), the current flow producing the magnetic field continues without application of electrical power. These magnets can produce fields up to 11 T for animal and small sample imaging and between 0.5 T to 3.0 T for whole body human imaging. Superconducting magnets offer superb homogeneity and stability (temporal and temperature). The disadvantages include high capital and operating cost [13, 30] (due to cryogenic liquid boil off), large fringe fields (which requires a magnetically shielded room), and their large size and weight. In addition, the magnetic field cannot be turned off easily because of the large energy stored inside magnetic field.  Permanent magnets are constructed from magnetized hard magnetic material such as AlNiCo, Samarium-Cobalt and most recently Neodymium-Iron-Boron. Magnetic fields up to 0.4 T have been used for whole body imaging, but for imaging of smaller volumes fields up to 1.0 T for imaging have been demonstrated [42]. Permanent magnets are lower cost when compared to superconducting and resistive magnets in terms of capital and operating costs. They occupy less space and weigh less for low to mid fields compared to the other magnets including their accessories (refrigeration system for superconducting magnets and cooling system for electromagnets). The fringe fields are less than superconducting magnets thus no magnetically shielded room are required. On the other hand, permanent magnets have poor homogeneity and temperature stability as the magnets have a large temperature coefficient.  49  The choice of the MRI magnet is application dependent. If the highest field, best homogeneity and temporal stability with a large access are desired the superconducting magnets would be the only choice. The trade-off here is cost and the required space for installing the magnet. If cost is an issue and lower field, poorer homogeneity and stability can be tolerated, then an electromagnet is suitable when a variable field is desired. The trade-off here is the cost associated with operating the cooling unit and supplying the electrical power.  Permanent  magnets are chosen when a variable field is not necessary. In that case permanent magnets are preferred to electromagnets as they occupy less space, require no electrical energy to generate the magnetic field, cost less, and have a larger access. However, the performance in terms of temperature stability is poorer than other magnets. The advantage and disadvantages of the various magnets is summarized in Table 2.  Table 2. Advantages and disadvantages of magnets used for MRI. Superconducting Magnet  Resistive Magnet Advantages  • High magnetic fields • Variable field • Good homogeneity  • Very high magnetic fields • High homogeneity  Permanent Magnet • Low capital cost • No operating cost • Large access  • Superb temporal stability • Large access  Disadvantages  • High capital cost  • Very high capital cost  • High operating cost  • High operating cost  • Size  • Size • Not easily turned off  • Moderate magnetic fields [43] • Poor temperature stability • Poor homogeneity • Always ON  50  Permanent magnets are the ideal choice for this work because - the goal is to demonstrate a prototype of a desktop MRI instrument that is relatively  inexpensive. - the samples to be imaged (microfluidic chips) are of relatively small size. Therefore, it is possible to generate high fields with permanent magnets for smaller gap sizes. Therefore, the magnetic field for this work will be on permanent magnets. 2.1.2.1 Permanent magnets: magnetic materials The relationship between magnetic field strength (H) and magnetic flux density (B) is shown in terms of a hysteresis (BH) curve as shown in Figure 17. Magnetic materials are characterized by the second quadrant of the intrinsic hysteresis curve. In Figure 17, Br is the residual magnetic flux density, Hc is the coercivity field and Hci is the intrinsic coercivity.  51  B [T] al rm o N  Intrinsic  Br  Hci  H [A/m]  Hc  Figure 17. Magnetic materials are characterized by their intrinsic and normal B-H hysteresis curves. The two types of magnetic material are soft and hard magnetic materials. Hard magnetic materials have a much larger Hc ( > 700 kA/m) value than soft magnetic materials (Hc < 100 kA/m). Therefore, a hard magnetic material retains its magnetism after being magnetized, whereas a soft magnetic material looses most of its residual magnetism after the magnetizing field is removed. Permanent magnets are made of hard magnetic materials which retain their magnetization after being magnetized. A summary of permanent magnetic material and typical properties is listed in Table 3. The performance of permanent magnetic material is reported in terms of their energy per unit volume BHmax. As seen in Table 3, NdFeB materials have the highest energy density per unit volume compared to other materials. The temperature coefficient parallel to the magnetization direction of the hard magnetic material αparallel is given as a percentage per unit temperature. A high temperature coefficient results in poor temperature stability of a permanent magnet as it will become very sensitive to temperature fluctuations. Unfortunately, the NdFeB material has one of the highest temperature coefficients compared to  52  other permanent magnetic material.  Table 3. Properties of commonly available hard magnetic material [44]. Ferrite  AlNiCo  SmCo  NdFeB  (Ceramic 8)  (Alnico 5)  (REC-26)  (HS-42AV)  Br [T]  0.4  1.25  0.9  1.31  Hci [kA/m]  262  51  1,600  1,110  (BH)max [kJ/m3]  30  43.8  159  334  αparallel [%/°C]  -0.18  -0.02  -0.05  -0.12  Soft magnetic materials are used for the flux path of the magnetic configurations because of their high relative magnetic permeability µr. The relative magnetic permeability is the slope of the intrinsic curve of the magnetic material in the BH diagram. A steeper slope corresponds to a larger µr. The leakage of a magnet can be reduced by selecting a soft magnetic material with a high µr as described later. A summary of soft magnetic material suitable for application with static magnetic field with their typical magnetic properties is listed in Table 4. The saturation limit of each material is indicated by the Bsaturation value.  53  Table 4. Properties of commonly available soft magnetic material (Annealed 1 mm thick stamped sheet). [45]. Cobalt-Iron Alloy  Silicon-Iron Alloy  Nickel-Iron Alloy  Iron  (TRAFOPERM N3)  (PERMENORM 5000 V5)  Initial µr  1,000  1,000  9,000  1,000  Max µr  4,000  3,000  135,000  9,000  Hc [kA/m]  <0.05  0.02  0.004  0.14  Bsaturation [T]  2.2  2.03  1.55  2.35  (VACOFLUX 50)  Nickel-Iron alloys have extremely high relative permeability with high magnetic flux saturation limits which make them ideal as pole materials. The thickness of these materials is limited to around 2 mm and must be annealed at 1200 °C in presence of a pure hydrogen gas after machining [46]. These relatively expensive specialized materials may not be available in small quantities or in specific shapes. Low-carbon steel (or iron) is a much less expensive and readily available soft magnetic material and has a relative magnetic permeability of about 4,000 with a saturation limit of 2.2 T. 2.1.2.2 Geometric configurations of magnetic circuits using permanent magnets Two main types of permanent magnetic configurations exist: Iron-cored and Halbach. The Iron-cored magnetic configuration is a classical design where the magnetic flux passes through the permanent magnets, two opposing poles (making the air gap) and the yoke (flux return path). The yoke, and especially the poles, must be made of soft magnetic material preferably with high relative magnetic permeability. For MRI applications the C-shaped and H54  shaped magnets are used for their superior homogeneity and relatively open access. Various Cshaped configuration of the soft and hard magnetic material exist, a selection of which were simulated for the purposes of this work are depicted in Figure 18.  (a)  Bg  (b)  Air Gap  Soft Magnetic Material  Permanent Magnetic Material  (c)  High Magnetic Permeability Material  Figure 18. C-shaped permanent magnet configurations. The leakage is highest for (a) and the field is highest for (c). The direction of magnetization of the hard magnetic material (permanent magnets) is indicated by an arrow (tail: S, tip: N). The return path (i.e., the yoke) is made of soft magnetic material such as low carbon steel. The poles are also made of soft magnetic material with high permeability such as Nickel-Iron (49%) for increased homogeneity.  55  The magnet efficiency  nL =  φgap  (57)  φTotal  is a figure of merit for magnets where Φgap [Wb] is the flux traveling through the air gap and ΦTotal [Wb] is the total magnetic flux produced by the permanent magnets. The total magnetic flux can be calculated as explained later in this chapter. The magnet efficiency nL depends on a number of factors including the geometric configuration and the gap ratio of the gap width to the cross-sectional area. Magnetic efficiencies can only be estimated using empirical data which were not available to the author; therefore, the magnet efficiencies were estimated using simulations. In the C-shaped configurations in Figure 18, nL increases as poles are placed closer together and the air gap decreases. If the magnets are placed very close to the air gap, however, the homogeneity is compromised. To achieve highest magnetic flux density for a given gap size, configuration in Figure 18c is preferred. H-shaped magnets are very similar to C-shaped magnets except that the return path is symmetric which results in simpler overall mechanical design. The cases that were simulated for the purposes of this work are shown in Figure 19. The lowest magnetic flux density is generated by the configuration shown in Figure 19a; also this configuration has the lowest efficiency nL. The magnetic flux density is highest for Figure 19c. Configuration Figure 19b has comparable magnetic flux density to the one shown in Figure 19c, and has a much simpler mechanical design. The homogeneity of H-shaped configurations is higher than C-shaped ones because of their symmetry in design. Also due to their symmetry, H-shaped magnets are less susceptible to variations in room temperature as the geometrical changes due to temperature are symmetric. It  56  was observed in the simulations that the homogeneity of the configuration shown in Figure 19a was better than the homogeneity of the magnetic configuration shown in Figure 19b and Figure 19c, because the magnets are further away from the poles.  (a)  Bg  (b)  Air Gap  Soft Magnetic Material  Permanent Magnetic Material  (c)  High Magnetic Permeability Material  Figure 19. H-shaped permanent magnet configurations. The leakage is highest for (a) and the field is highest for (c). The other category of permanent magnetic configuration is the Halbach cylinder design (see Figure 20). To generate a dipole field only a special case of the Halbach is considered [47]. Basically, the magnetic material surrounds the air gap and is ideally magnetized in such a way to create a homogenous magnetic field in the air gap. In practice, the magnetization pattern is approximated by number of discrete permanent magnets. The greater the number of the permanent magnets, the better the approximation resembles the ideal case Figure 20a. Halbach  57  magnets have the potential to generate much higher fields than iron-cored magnets. The homogeneity is usually less than iron-cored magnets and access to the air gap is very limited. The discrete components can be a number of magnetized rods that extend out of the page (Figure 20-b, Figure 20-d) and are rotated to resemble the magnetization of that of an ideal Halbach pattern. Alternatively, the discrete components can be of shapes that do not extend all across the magnet (Figure 20-c, Figure 20-e). The discrete magnetized blocks are assembled in stacked rings to cover the out-of-page dimension. Simulations of Halbach designs were not performed in this work.  58  Bg  (a)  (b)  Air Gap  (c)  (d)  Non-Magnetic Material  Permanent Magnetic Material  (e)  High Magnetic Permeability Material  Figure 20. Halbach permanent magnet configurations. Ideally a hard magnetic material is magnetized as seen in (a). Due to practical limitations, the magnetization pattern is approximated by discrete magnets (b-e). Materials with high magnetic permeability can be used to in crease homogeneity (d).  59  2.1.2.3 Prior art on Permanent Magnets for MRI Applications A summary of available magnets from MRI manufacturers is given in Table 5 for comparison. Most whole-body imaging machines have a homogeneity between 5-50 ppm and a field of view of typically 400 mm in one dimension. The temperature stability is usually better than 0.1 ppm/hour.  Table 5. Clinical whole-body MRI permanent magnets available as of 2008. B [T]  Size [m x m x m]  Weight [tonnes]  Magnetom Concerto  0.2  2.3x2.1x1.6  11  Magnetom C  0.35  2.3x2.1x1.6  16  Signa Profile  0.2  2.3x2.1x1.6  10  Signa Ovation Excite  0.35  2.1x2.6x2.2  19  AIRIS Elite  0.3  0.8x1.1x0.73  15  AIRIS II  0.3  0.8x1.1x0.73  15  MRP-7000  0.3  NA  NA  0.35  2.1x1.4x2.1  17  Company  Product  Siemens  General Electric  Hitachi  Millennium Tech Inc. Virgo  In terms of the prior art on permanent magnets, there is great deal of information available regarding the design and the experimental results for magnets that are used for research purposes. This is not true, however, for magnets used in clinical MRI instruments manufactured by private companies. Therefore, the literature review presented here is mostly gathered form  60  academic searches and patents.  Permanent magnets for clinical applications: An iron-cored H-shaped magnet made of NdFeB material, for whole-body imaging, with a field of 0.2 T homogeneity of 30 ppm over 350mm diameter-spherical-volume (DSV) that weighs 9,000 kg is given by Miyamoto et al. [48]. Another H-shaped iron-cored low-field PM is given by Kose et al. [49] for imaging of the Trabecular Bone. The reported field is 0.21 T, 50 ppm homogeneity over 120mm DSV and a gap of 160 mm. The magnet weighs approximately 500 kg and is installed in a 2 m by 2 m space. An iron-cored PM for human head surgery is presented by Katznelson et al. [50]. This patent describes the design of a magnet with a field of 0.1 T, homogeneity of 10 ppm and a field-ofview (FOV) of 200 mm DSV. A patent filed by Lee et al. [51] for whole-body imaging apparatus describes a Halbach configuration with design objectives requiring minimum 0.1 T field, variation of over a DSV of 400 mm less than 0.5 mT and stability no worse than 5 ppm/s. Permanent magnets with smaller footprint (i.e., miniature magnetic resonance machines imaging [52]) are also used for imaging extremities in clinical setting, animal research, spectroscopy and flow.  Halbach designs: The design and construction of a 0.31 T Halbach NdFeB magnet consisting of multiple rings for applications in NMR spectroscopy is presented by Raich et al.[47]. The homogeneity is 700 ppm over 18 mm DSV with a gap of 73 mm. A Halbach configuration for spectroscopy application with a field of 0.6 T, homogeneity 42 ppm over an air gap region of 3 mm x 3 mm x 5 mm is presented by Moresi et al. [53]. The permanent magnet material is SmCo17 and the pole pieces are constructed out high permeability material (µr > 10,000). The magnet is a cylindrical in shape with a diameter of 0.09 m and length of 0.23 m. The temperature  61  stability of the magnet is a reported -300 ppm/°C. Another Halbach permanent magnet is made for imaging mice by Shirai et al. [42]. They report a magnetic field of 1.0 T with 10 ppm homogeneity over 30 mm DSV. The magnet is approximately 980 kg with dimensions of 0.58 m (W) x 0.48 m (H) x 0.52 m (L) for the magnet and gap of 24 cm (W) x 9 cm (H) x 48 cm (L). The temperature sensitivity of this magnet was -1200 ppm/°C which is compensated for using an NMR lock. Multiple Halbach magnets are described by Kumada et al. [43, 54, 55], which are not specifically designed for MRI applications, have demonstrated fields exceeding 6 T for a 2 mm gap. The magnets weigh approximately of 340 kg. The homogeneity is worse than 4000 ppm with little room for any field corrections.  Iron-cored designs: A desktop iron-cored C-shaped PM out of NdFeB magnetic blocks was designed by Wright et al. [56-58] with a magnetic field of 0.21 T with a 100 mm gap and homogeneity of 20 ppm over approximately 7 mm DSV. The approximate dimension of this magnet is 0.2 m (W) x 0.41 m (H) x 0.2 m (L). A H-type magnet for educational purposes with a field of 0.08 T, homogenous region of 10 mm DSV and a gap of 44 mm is made by Sahakian et  al.[59] using off-the-shelf NdFeB PM blocks. The size of the magnet is approximately 0.13 m (W) x 0.13 m (H) x 0.25 m (L). An H-shaped desktop PM for MRI purposes was designed by Kose et al. [49] with a field of 1.02 T, homogeneity of over 150 ppm 3 mm DSV and weight of 1.7 kg. The magnet was found useless due to the poor homogeneity performance. Permanent magnets have been also developed for the food industry one of which is described by Li et al. [60]. They report the design of an NdFeB H-type PM with a field of 1.3 T, homogeneity of 92 ppm over 80 mm with dimensions of 0.3 m (W) x 0.05 m (H) x 0.2 m (L). An iron-cored 1.0 T PM made out of NdFeB material is constructed by Haishi et al. [61]. The reported air gap is 61 mm with homogeneity of 12 ppm over a 20 mm DSV. The dimensions of the magnet are 62  0.62 m (W) x 0.75 m (H) x 0.75 m (L) and it weighs 1,400 kg. The pole pieces are made of silicon steel plates that have radial cuts on them to suppress eddy currents from the switching gradients. The reported temperature coefficient for the magnet is -1200 ppm/°C which makes imaging practically impossible if no corrections where made during the imaging sequence. This problem was rectified by implementing a NMR lock to correct for the shift in resonance frequency. As for the development of magnets, Podol’skii [62-64] compares magnetic configurations for different applications including the Halbach, iron-cored, and a hybrid called the novel magnet system (NMS). The results suggest that depending on the application different magnets are suitable. For example, the Halbach design is more weight efficient for higher fields but is more complex to manufacture and it is more sensitive to temperature variations. In a similar study [65] where Halbach and C-shaped iron-cored magnets are compared, the results suggests that ironcored weigh less compared to other magnetic configurations in applications requiring a large cross-sectional gap with a shorter gap length. In addition, the assembly and manufacture of ironcored magnets is significantly simpler than Halbach designs. A number of research groups [66-68] have focused on optimizing the pole shapes of the magnets to correct for coarse inhomogeneities. As a first prototype only simple pole shapes have been considered in this work. 2.1.2.4 Theory and Basic Calculations Two permanent magnetic configurations were designed and tested in this work. iron-cored permanent magnetic configurations were chosen because of their:  63  simpler design, construction and assembly larger access (more open design than Halbach magnets) lower cost (magnet are purchased off the shelf) Magnetic configurations are analyzed using their equivalent magnetic configuration. Detailed information about this section is described in elsewhere [69, 70]. Table 6 presents the electrical circuit analogy to the magnetic circuits.  Table 6. Electrical and magnetic circuit analogy. Electrical Circuit Parameter  Magnetic Circuit Parameter  Electromotive Force (Voltage), EMF  Magnetomotive Force (Fm), MMF  Resistance, R  Reluctance, R  Current, I  Flux, Φ  The flux  Φ=  Hℓ = B⋅A R  (58)  of a magnet is found by calculating the reluctance  R=  ℓ µ 0µ r A  (59)  where l and A refer to the length and cross-sectional area of the material. The magnetic permeability of free space µ0 (= 4π·10-7 N/A2) and relative magnetic permeability of the material  64  µr are constants. If the magnetic material has relatively linear characteristics (such as NdFeB material) in the second quadrant of its BH curve, it can be modeled as a voltage source with a series resistor as shown in Figure 21.  B [T] Am  Operating Point  Br  lm  Φt  Rm= lm/(µ0µmAm)  Bm  Bm  H [A/m] Hc  Fm= Hclm  Hm  Figure 21. A permanent magnet can be modeled as a voltage source Fm with an internal resistance Rm. The equivalent voltage and resistance of a permanent magnet can be calculated by knowing the material properties (the intrinsic BH curve) and the geometry. For example, a single NdFeB permanent magnetic block with an area, Am, of 51 mm x 51 mm, a length lm of 25.4 mm and coercive field Hc = 1100 kA/m can be modeled as a voltage source with an magnetomotive force of  Fm = H c l m = 110 ⋅103 ⋅ 25.4 ⋅10 −3 ≈ 2800 A ,  (60)  and a resistance of  lm 25.4 ⋅10 −3 Rm = = = 7.4 ⋅106 −7 −3 2 µ 0µ m A 4 π ⋅10 ⋅1.05 ⋅ (51 ⋅10 )  65  A Wb  (61)  where the typical relative magnetic permeability, µm, of a NdFeB magnet is 1.05. Therefore, the permanent magnet is modeled as  F = Fm − Φ t R m .  (62)  With this technique the magnetic configuration of interest can be modeled. The specific magnetic configurations chosen in this work is depicted in Figure 22 along with their equivalent magnetic circuits. The magnetic flux path is represented by a dotted line Φ. The flux Φt is the total magnetic flux generated by the PM. The reluctances Rm, Rc, Rg, and Rl refer to the reluctance of the magnet, the iron core, the air gap and the leakage, respectively. Ac refers to the crosssectional area of the iron core.  66  Φ  lg  lg  Ac  Rc  Rc  Rc  Fm Φt  Rm  Φg Φl  Rm  Rg  Rl  Φt Φ2  Φ1 Φg  Fm  Rg  (a)  Φl Rl  (b)  Figure 22. C-shaped (a) and H-shaped (b) permanent magnet configurations chosen for this work and their corresponding equivalent magnetic circuit.  The effective cross-sectional area of the air gap Ag-effective is larger than the nominal area Ag as depicted in Figure 23, because the magnetic flux lines fringe away from the magnetic poles. As a rule of thumb, the effective area of the gap is increased by the length of the gap as shown in Figure 23. Saturation occurs when the magnetic flux density exceed the saturation flux density of the soft magnetic material. For example, a region of the magnet where saturation typically occurs is shown in Figure 23. Leakage of the magnetic flux lines mainly occurs as a result of these saturations.  67  Leakage Ag-effective Fringing  Ag  lg /2  Bg  Region of Saturation  Figure 23. The fringe field and region of saturation of the C-shaped magnetic configuration along with its effective gap is shown. Using (59), and assuming the reluctance of the iron core, Rc, to be negligibly small, the magnetic flux density at the gap, for all configurations, can be calculated    1 Bg =µ 0 H c   1 A g-effective lg + µm n × A lm m  L     .    (63)  The leakage factor is complicated to model analytically and ranges between 2% - 91% according to Moskowitz [70]. Simulations of the selected configurations show a magnet efficiency between 40% - 85%. The C-shaped magnetic configuration similar to that shown in Figure 22a has a leakage factor about 47% according to simulations described later. According to (60) the magnetic flux density in the gap is then  68          1 1   = 1.29 T ,   Bg =Br = 1.47T  1  37 mm 2 7 mm   1 A g-effective lg  + µm  × 1.05    × A  + lm m  nL   0.47  51 mm  381 mm   (64)  , where the effective gap area is a square of length and width of 37 mm, the area of the magnet is a square of length and width of 51 mm , the gap length is 7 mm and the total magnet length is 381 mm. For an H-shaped magnet with a magnet efficiency of 68% (simulation is presented later) the magnetic flux density in the gap becomes    1 Bg =Br   1 A g-effective lg + µm n × A lm m  L        1  = 1.32T   = 0.59 T ,   1 (144 mm) 2 16.5 mm    0.68 × (127 mm) 2 + 51 mm 1.05      (65)  where the effective gap is a disc with a 144 mm diameter, the area of the magnet is a 127 mm diameter disc , the gap length is 16.5 mm and the total magnet length is 51 mm. 2.1.2.5 Simulation of the selected magnetic configurations To better estimate the maximum magnetic flux density and homogeneity inside the gap, selected designs were simulated in three dimensions using the AC/DC module of COMSOL MultiPhysics 3.3a. In addition multiple simulations were carried out to assess the sensitivity of a magnet to manufacturing tolerances. Pole plates out of material with high relative magnetic permeability were excluded in the final design to minimize cost. The final design of the Cshaped magnetic configuration is displayed in Figure 24 with the approximate dimensions. The 2” (W) x 2” (L) x 1” (H) permanent magnetic blocks are off-the-shelf NdFeB N50 grade 69  (strongest available) with a residual flux density of Br = 1.47 T (Applied Magnets, 1111 Summit Avenue Suite #8, Plano, TX 75074, USA). The soft magnetic material is low carbon steel (AISI 1010)1. The poles are designed to be adjustable to correct for the parallelism of the poles faces due to manufacturing tolerances. The top pole pivots in the horizontal direction and the bottom pole pivots in and out of the page. Similar adjustment mechanisms are also described in [71, 72]. The air gap is approximately 7.5 mm in length and has a nominal dimension of 30 mm by 30 mm. The non-magnetic enclosure (Aluminum T6061) holds the magnetic pieces in place.  1  Courtesy of George Clark for donating the soft magnetic material.  70  Adjustment Mechanism  7 mm  250 mm  mm 60  Soft Iron 280 mm  Permanent Magnet  Aluminum Frame  Figure 24. Final design of the C-shaped permanent magnet. The final design of the H-shaped magnet with approximate dimensions is shown in Figure 25. The 5” (D) x 1” (H) permanent magnetic discs were also purchased off-the-shelf from the same distributor. The material is NdFeB of grade N42 and a residual flux density of Br = 1.32 T. The return path is a low carbon steel plate (AISI 1018). The distance between the pole faces (i.e., the gap) and also the parallelism is adjustable using four bolts on the top plate. The gap is adjustable between 5 – 20 mm and is set to 16.5 mm throughout this work. The aluminum blocks on the side are used to support the force exerted by the adjustment bolts. The steel acts as both the return path and the mechanical structure. The magnets were drawn using SolidWorks and imported to the magnetostatics sub-module of COMSOL for steady-state conditions. The relative magnetic permeability of iron is set to 4000, 71  the residual magnetic flux density of the C-shaped magnets is 1.47 T and the residual magnetic flux density of the H-shaped permanent magnet is 1.32 T. The boundary conditions of the outer boundaries are set to ‘zero potential’ and the inner boundaries are set to continuity. A free mesh was used for the overall structure of the magnet and was refined to 0.2 mm on the pole faces. A steady state solver (GEMRES) was used.  16.5 mm  280 mm  130 mm  Adjustment Bolt  Aluminum Block  150  mm  Permanent Magnet  Soft Iron  Figure 25. Final design of the H-shaped permanent magnet.  The results of these magnets are summarized in Table 7. The magnetic field and homogeneity is 72  evaluated in the center of the gap.  Table 7. Specification of the magnet based on results from simulations. C-shaped  H-shaped  Gap [mm]  7.5  16.5  B0 [T]  1.282  0.595  ∆B/B0 [ppm]  50  50  Along x-axis [mm]  10  13  Dimensions [m x m x m]  0.28L x 0.25H x 0.07W  0.28L x 0.13H x 0.15W  nL  47%  68%  Sample simulation results showing the distribution of the magnetic flux for each magnet is depicted in Figure 26 and Figure 27. The leakage and fringing effects are also depicted. The fringing in the C-shaped magnet is relatively small at the air gap. However, regions with a magnetic flux density of higher than 2.2 T saturate the soft iron cores; the leakage in the Hshaped magnet is mainly due to the saturation of the core which, according to simulations, can be easily solved by choosing a thicker plate which was not done due to time constraints. In Figure 26a and Figure 27a the arrows represent the magnetic flux lines. The magnet is also coloured based on the absolute magnetic flux density which ranges from 0.2 T to 2 T. The region of saturation are indicated in the figures. In Figure 26b and Figure 27b the magnetic flux is shown in terms of isosurfaces at 100 mT and 200 mT which highlight the leakage and fringing of the magnetic field. The fringing is larger in the H-shaped magnet because the iron cores are  73  saturated.  Normal Magnetic Flux Density B [T] 2T  (a)  Saturated Region  0.2 T  Normal Magnetic Flux Density B [T] 0.3 T  (b)  0.05 T  Figure 26. Distribution of magnetic flux density in the final C-shape design (a) and the leakage and fringing effects (b) are shown.  74  Normal Magnetic Flux Density B [T]  Saturated Region  2T  (a)  0.2 T  Normal Magnetic Flux Density B [T] 0.3 T  (b)  0.05 T  Figure 27. Distribution of magnetic flux density in the final H-shape design (a) and the leakage and fringing effects (b) are shown.  75  2.1.3 Magnet Characterization A photo of each manufactured magnet is shown in Figure 28. The C-shaped is shown in Figure 28a before the magnet was re-machined to incorporate the adjustment bolts.  (a)  (b)  Figure 28. Manufactured C-shaped (a) and H-shaped (b) magnets.  76  The summary of the measured performance of both magnets is displayed in Table 8. The field and homogeneity were evaluated in the middle of the gap of each magnet.  Table 8. Specification of the magnet based on results from measurements. C-shaped  H-shaped  Gap [mm]  7.5  16.5  B0 [T]  1.205  0.589  ∆B/B0 [ppm]  500  150  along x-axis [mm]  10  13  nL  44%  67%  For measuring the field of the C-shaped magnet a Hall-effect gaussmeter (F. W. Bell 4000, Sypris Test & Measurement, 6120 Hanging Moss Road,Orlando, FL 32807 USA) with a digital resolution of 1 mT was used. The resolution of this commercial probe is not high enough to make meaningful plots (see Figure 29b) of the magnetic field in order of part-per-million (ppm) and an analog output was not available. Therefore, a hall effect (SH-400 model from F.W. Bell) component was used along with a homemade electronics to measure the magnetic flux density. The probe generates a voltage proportional to the magnetic flux density. Data was taken every 1.0 mm and measured voltage averaged by an Agilent 34401A digital multimeter for 2 seconds and read from the display. The magnetic flux density across the gap of the C-shaped magnet is shown in Figure 29a. The solid line represents the simulation and the crosses represent the measured data. The data from simulation was corrected by a scaling factor (x 1.028) to account  77  for a number of uncertainties including the magnetization of the permanent magnets and relative magnetic permeability of the iron poles. In addition, the hall-effect probe with homemade electronics had an offset about 2 mT which was compensated for during calibration. The calibration was performed by recoding the magnetic field at a few locations in the magnet using the commercial probe, and then recording the voltage of the hall-effect probe (with homemade electronics) at the same locations. An error of up to 5% was present with this calibration scheme as there were errors in placing the probes exactly in the same locations. In addition, the data sheet of this probe reports a non-linearity error of 1%. The homogeneity of the C-shaped magnet along the X and Y axes is plotted in Figure 29-B. The magnetic flux density varies by approximately ±1 mT over ±8 mm in the X and Y axes at the middle of the gap. The variation of the magnetic flux density from one pole face to the other (i.e., along the Z-axis) is negligible according to simulation. This was confirmed with the commercial gaussmeter. However, measurements using the homemade probe show variations up 10 mT along the Z-axis. Since the sensor was relatively large (2 mm) compared to the gap (7.5 mm) this error was attributed to small amount of magnetic material being present inside the homemade probe, which resulted in a measurement error as the probe came close to the pole faces. In general the measured values for the maximum magnetic flux density and the shape of the field inside the gap were very well explained by the simulation results. However, the estimated homogeneity is different from measured homogeneity by a factor of 10 for the C-shaped magnet. This error arises from manufacturing tolerances such as the air gaps in the magnetic circuit, surface finish of the pole faces, and the ability to adjust for the parallelism of the poles. Other researchers have reported an error factor of a 100 [53] for a Halbach design. The addition of high permeability magnetic plates as Alloy 49 would improve homogeneity by a factor between 2 and 3 depending on the  78  thickness of the plates. The magnet efficiency is also close to the expected value from simulation.  79  Magnetic Flux Density [T]  (a)  X Z  Pole Face Y Y  Distance Across the Gap Y [mm]  ∆B [mT] X 2.00  Field Variation in X-Direction Field Variation in Y-Direction  Pole Face  1.50  Y  1.00  0.50  (b) -9  -8  -7  -6  -5  -4  -3  -2  0.00 -1 0  1  2  3  4  5  6  7  8  9  X, Y [mm] -0.50  -1.00  -1.50  -2.00  Figure 29. Magnetic flux density along the gap of the C-shaped magnet (a). The solid line represents simulation values and the crosses show measurement data points. The variation of the magnetic flux density along the center of the x and y axes are shown in (b).  The magnetic flux density along the gap of the H-shaped magnet is shown in Figure 30a. The  80  Hall-effect probe used for this measurement was an F.W. Bell 6010 model with an analog output. The datasheet of this probe claims an absolute accuracy of better than 1 mT with an uncertainty of ±1%. Data was taken every 3 mm at 5 kS/s for a duration of one second. The solid line represents corrected (x 1.0125) data from simulation and the error bars represent the measured data. The homogeneity of the H-shaped magnet in the middle of the gap (Z = 0 mm) is shown in Figure 30b as contour lines. The measured variation of the magnetic flux density from one pole face to the other is negligible (as expected from simulation) for the region of interest and therefore not presented here. Again, the results from simulation match the measured data very well except for homogeneity which is different from measurements by a factor of 3. According to simulations performed in this work, the magnet efficiency of the H-shaped PM can be significantly increased (form 67% to 85%) if the iron plates are made thicker to prevent them from saturation.  81  Magnetic Flux Density [T]  (a)  X Z  Pole Face Y Y  Distance Across the Gap Y [mm]  Variation of magnetic Flux Density [mT] Y  X  (b)  Distance Across the Gap Y [mm]  Pole Face  Pole Face Outline  Distance Across the Gap X [mm]  Figure 30. Magnetic flux density along the gap of the H-shaped magnet (a). The solid line represents corrected simulation values and the crosses show measurement data points. The variation of the magnetic flux density along X and Y axes in the middle of the magnet (Z = 0 mm) is shown in as contour lines (b).  82  The homogeneity of the C-shaped magnet would result in a T2* decay of approximately 40 µs according to equation (55). An MR signal this short would be very challenging to pick-up due to practical limitations. After the application of the excitation signal the receiver goes through a recovery phase (> 50 µs in the experimental setup of this work) which would make detection of the MR signal with a short decay time very challenging. The more important challenge is the tuning of parameters such as frequency, amplitude, and pulse duration of the excitation, tuning and matching of the receiver probe, adjustment of the receiver gains with no MR signal present and at the same time seeking the most homogenous section of the air gap. For example, the carrier frequency can be estimated by knowing the magnetic flux density of the C-shaped magnet. Assuming a measurement accuracy of 1%, the uncertainty in the carrier frequency would be over 500 kHz. Depending on the RF electronics, if the carrier frequency is offset more than a certain amount (say 50 kHz) at Larmor frequency (about 51 MHz here), no MR signal would be observed even if T2* was much longer than the recovery time of the receiver. As a result of these challenges and time constraints the application of the C-shaped magnet for the purpose of making a desktop imaging device was not further pursued. The homogeneity of the H-shaped magnet and the large access is much better suited as a first prototype of a desktop fluid flow visualization instrument. In addition, the homogeneity would improve for smaller samples as the variation of the magnetic field across a smaller sample is less. The trade-off however is the observed amplitude of the MR signal as a smaller number of protons exist to contribute to the signal. In terms of stability, the variation of the magnetic flux density was observed along with a measurement of the temperature of the magnet. As the fluctuations in the field are rather small, the magnetic flux density was measured by recording the frequency shift of the MR signal. The 83  temperature fluctuations of the magnet along with its shift in resonance frequency were measured over a period of 6 hours, as shown in Figure 31a. The drift in the magnetic resonance shift correspond to the changes in temperature of the magnet. Note that the resonance frequency shift in Figure 31a was inverted about the x-axis for ease of comparison. The shift in magnetic resonance vs the temperature of the magnet is shown Figure 31b where the temperature coefficient of the magnet was -707 ppm/°C. Other groups have reported -300 ppm/°C [53] and 1200 ppm/°C [42]. The fluctuation due to temperature is mainly due to the large temperature coefficient of the NdFeB magnets (-0.12%/°C, Table 3) which suggest a maximum temperature coefficient of -1200 ppm/°C for the whole magnetic structure according to (63). The fluctuations of the magnetic field due to geometric changes of the magnet (because of variations in temperature) are two orders of magnitude lower in amplitude compared to the temperature coefficient of the permanent magnets.  84  Temperature of  magnet °C ] Magnet [° [°C]  Temperature of  22  21.8  21.6 1  2  3  4  5  6  1  2  3  4  5  6  4  NMR Frequency Shift shift [kHz] [kHz]  Resonance frequency  (a)  2  0  -2  -4 0  Time [Hours]  (b)  Figure 31. Fluctuation of the magnetic flux density and the room temperature in 6 hours (a). The temperature coefficient is -707 ppm/°C (b).  85  The temperature stability of this magnet is rather poor for MRI purposes and imaging would be not practical unless corrections are applied either to the magnetic field or the carrier frequency [73-75]. Compensation of the magnetic field in this work is applied to the magnetic field by the means of coils wound on the each of the poles of the magnets. The homogeneity is optimized when the distance between the two coils is equal to their diameter; this configuration is called the Helmholtz coil. The current I in both coils flow in the same direction. The magnetic flux density of a Helmholtz coil in the middle of the two coils is given by [76]  B = (4 ) 5  3  2  µ0 µ r N I. r  (66)  The radius r is equal to the radius of the disc shaped permanent magnet and µr is the relative magnetic permeability (= 1.05) of NdFeB material. The number of turns was arbitrarily chosen as 8. From (66) and the Larmor equation, the expected frequency shift per amperes for the described Helmholtz coil is 4.8 kHz/A. The recorded frequency shift of the magnetic resonance signal is shown in Figure 32 as a function of current inside the coils. The measured value for this coefficient (= 5.2 kHz/A) is reasonably close to the estimated value given by (66). A typical magnetic resonance shift during the whole day is about ±15 kHz; by regulating a current up to ±3 A in this coil it is possible to compensate for drift in temperature as it will be discussed in Chapter 3.  86  6  Frequency [kHz] = 5.17 x Current [A]  NMR Frequency Shift [kHz]  5  R 2 = 1.0  4  3  2  1  0 0  0.2  0.4  0.6  0.8  1  1.2  Current in B 0 Compensation Coil [A]  Figure 32. Effect of flow of current in the B0 compensation coils. The implementation and the sequencing of the disturbance compensation of the permanent magnet are further discussed in Chapter 3.  2.1.4 Summary of Magnets In summary, two permanent magnets (C- and H-shaped) were designed based analytical equations and simulations, constructed and characterized. The C-shaped magnet has a measured field of 1.205 T with a 7.5 mm gap and a homogeneity of 500 ppm over 10 mm. Due to the small gap and poor homogeneity, this magnet was not used for magnetic resonance imaging. The Hshaped magnet has a measured field of 0.589 T with a 16.5 mm gap and a homogeneity of 150 ppm over 13 mm. The measured temperature coefficient of this magnet is -707 ppm/°C; Helmholtz coils were added to this magnet which can be used to compensate the magnetic drift. The H-shaped magnet is used in this work for magnetic resonance imaging.  87  2.2 Gradient Coils The design and characterization of the gradient coils is described next.  2.2.1 Gradient Coil Requirements As the strength of the gradient field (Gx) is increased, the spatial resolution given by  ∆x =  ∆G x 1 1 ∆B0 + L x + ∆ω Gx Gx γG x  (67)  of the image is improved (see Chap 1.4). Linear gradient fields (i.e., small deviations, ∆Gx) are also necessary to avoid image distortions. To achieve a resolution of about 30 µm, a gradient of greater than 0.67 T/m with a linearity of 3% is needed assuming that Lx = 1 mm and ∆B0 = 20 µT. The resolution is also limited by the uncertainty principle as  ∆x ≥  1 . γG x Tacq  (68)  To improve resolution, both the acquisition time, Tacq, and the gradient field strength have to be maximized. However, increasing the acquisition time leads to greater spin diffusion and is related to the resolution as ∆x ≥  (69)  2DTacq  for the molecular diffusion constant D. To avoid problems encountered with diffusion, the acquisition time has to be minimized, which is in conflict with equation (68) where longer Tacq  88  are preferred. From equations (68) and (69), if the acquisition time Tacq is less than 5 ms, in order to achieve a resolution of approximately 30 µm, a gradient of at least 0.6 T/m is needed. In this case, the resolution limited to approximately 5 µm due to self-diffusion, assuming a typical selfdiffusion coefficient D ≈ 2.2·10-9 m2/s for water at room temperature [36]. To keep the acquisition time short, the duration of the applied gradient field has to be minimized which requires that that the gradients are switched rapidly. Typically the rise time needs to be better than 5% of the total duration of the gradient pulse. So, for a gradient pulse duration of 1 ms, a 100 µs rise time is typical. To obtain optimal performance in terms of velocity resolution (as derived in chapter 1.4)  ∆V =  Φ MR 1 ∆Φ MR + ∆G x , 2 2πγG x τ 2πγG 2x τ2  (70)  large gradients of high linearity that are pulsed for long durations are necessary. To obtain a velocity resolution of 50 µm/s, a gradient strength of 0.68 T/m and a linearity of better than 5% is needed if we assume a gradient pulse duration τ of 2 ms and maximum phase shift Φ = ±π. The variation in Φ is determined by the number of phase encoding steps (=128) which results in phase steps of Φ = π /128. Therefore, the requirements of the gradient coils are: -  High gradient field strength (> 0.68 T/m)  -  High linearity (< 3%)  -  Fast switching speed (< 100 µs).  2.2.2 Gradient Coil Design  89  The design and simulation of the gradient coils is described in the following section. 2.2.2.1 Basic Theory: Gradients The orientation of the gradient fields inside relative to the static magnetic field is shown in Figure 33. The main magnet generates a vertical magnetic field along the z-axis. The three gradient coils generate a linearly dependent magnetic field in the z-direction (Figure 33-b). For example, the gradient in the x-direction, Gx, generates a magnetic field in the z-direction that is linearly dependent on x as illustrated in Figure 33-c. Similarly, the gradients in y and z are illustrated in Figure 33-d and Figure 33-e respectively. In other words, the gradients are defined as  Gr =  dBz dBz dBz + + . dx dy dz  90  Bz,Gx  Gx x  Main Magnet  Bz,Gy  Gradient Coils B0  Gy  (a)  Gx.x  (c)  y  Gradient Coils Bz,G  z  Z Gz  Y Main Magnet X  z  Main Magnet Main Magnet  Gradient Coils  Gradient Coils  B0  B0  Gy.y  (b) RF Coil  (d)  Capillary Tube  Gradient Coils  Z Y  Gradient Coils  Z Main Magnet  Main Magnet  Y  X  X  Main Magnet Gradient Coils B0 Gz.z  (e)  Gradient Coils Z  Y Main Magnet X  Figure 33. The direction of the field generated by the gradient coils (a); location of the gradient coils and the probe (b); The gradient field in x, y, and z directions are shown in (c), (d) and (e), respectively. The gradient field along the z-axis, Gz, is generated using a Maxwell coil configuration. This configuration consists of two current carrying loops with a radius a and separation d. The electrical current is passed through the coils in opposite directions as illustrated in Figure 34.  91  Figure 34. Maxwell gradient coil. The magnetic field along the z-axis is [76]     2   µ nIa 1 1  , Bz = 0 − 3 3 2 2 2  2 2     d d  2    a − z  + a 2    a + z  + a        2     2    (71)  where µ0 is the permeability of free space (4π·10-7 N/A2), I is the electrical current and n is the number of turns for each loop. If the separation is chosen to be  d = 3 ⋅a ,  the gradient fields are linear up through the fourth power [76] and the magnetic field gradient of this configuration is found by taking the derivative of (71) with respect to z  Gz =  48 3 ⋅µ 0 nI 5  7 2 a2  ≈ 0.64  µ 0 nI . a2  (72)  92  This approximation is true for a imaginary sphere of 0.6a in radius in the center of this coil (at  z = 0 in Figure 34) for a linearity of 5% [77]. The gradient fields along the x and y axes are generated using the “rectangular loop” coils described by Anderson et al. [78]. Other configurations exist that are described elsewhere [13, 32, 76]. The rectangular loop configuration is suitable for this work due to its compact design. This design consists of four current carrying loops with a width w. The coils are located on two planes separated by 2z0, each holding two of the loops of length L. The centers of the two loops on each plane are separated by 2yc. This gradient configuration is illustrated in Figure 35.  Figure 35. Rectangular loop gradient coil. The gradient in x and y are very similar; the only difference is that they are rotated by 90° with respect to each other. The maximum gradient field with vanishing second order terms can be achieved by choosing w/z0 = 1.55 and yc/z0 = 1.19 [78]. The gradient field is estimated using  G x = G y ≈ 0.46  µ 0 nI . z 02  (73)  93  The gradient field is linear to within 5% for a imaginary cube of 0.4z0 on each side in the center of the coils (at z = 0 in Figure 35) [77]. The length of the coil L has to be long enough compared to the size of the gradient so that it can be approximated as being infinitely long. 2.2.2.2 Prior Art: Gradients Typical values for gradients range from 0.001 to 0.01 T/m for clinical imaging while for microscopy 1 T/m is more typical. Much larger gradients (up to 250 000 T/m) can be obtained over a very small volume near ferromagnetic tips of newly developed NMR force microscopes [77]. The literature review of this work was focused on gradients used for microscopy only. Callaghan et al. [79] describe the design of a Maxwell gradient coil with a sensitivity of 1.65 T/m/A which was successfully tested up to 40 T/m (at a current of 24 A) for diffusion measurements. The uniformity over a sphere of 2 mm and 3 mm in diameter is about 2.5% and 8% standard deviation, respectively. The coil consists of 23 turns of 0.36 mm diameter enameled copper that is potted in epoxy. The coil has an inductance of 15.3 µH and a resistance of 0.8 Ohms. The gradient rise and fall time are reported to be on the order of 50 µs. Wright et al. [56] report rectangular loop planar gradient coils with a sensitivity of 0.0045 T/m/A that were tested up to 0.045 T/m (at current of 10 A). The gradients are linear enough for imaging a region of 25.4 mm with a resolution of 100 µm. They are etched from 8 oz copper clad printed circuit boards. Another paper also by Wright et al. [80], reports the design of a Maxwell coil with elongated loops with a sensitivity of 1.26 T/m/A is reported. This gradient coil was tested up to 52 T/m (at a current of 41 A) for diffusion measurements and consists of 23 turns of 0.36 mm diameter enamel-coated wire. The uniformity of the gradient over a 2 mm diameter sample is ±2% standard deviation. The coil has an inductance of 21 µH and a resistance of 1.7 Ohms and  94  has a rise time about 125 µs. Casanova et al. [81] report two gradient coils of rectangular loop type each with 20 turns and a sensitivity of 0.02 T/m/A and 0.012 T/m/A; the gradients were used up to 0.4 T/m and 0.24 T/m (at a current of 20 A), respectively. The reported resistance is 0.8 Ohm and the rise time is about 50 µs. The linear region of the gradients is a sample of 6 x 8 mm2. Seeber et al. [77] report the design and testing of a triaxial gradient module. The z-gradient is a Maxwell pair and the transverse gradients (Gx and Gy) are rectangular loop gradients as described by Anderson et al. [78]. The Gz has a sensitivity of 0.095 T/m/A and was tested up to 9.4 T/m (at a current of 99 A) with an estimated linearity of 5% in a sphere with 1.5 mm radius. The Gx and Gy gradient have a sensitivity of 0.25 and 0.14 T/m/A and were tested up to 40 T/m (at a current of 160 A) and 20 T/m (at current of 142 A), respectively. The linearity of the gradients is estimated to be within 5% in a region of (600 µm)3. The gradient are made using a single turn of a gauge 24 copper wire (diameter = 0.51 mm) embedded in slots machined in printed circuit boards. The gradients have inductances in the order of 0.1 µH with a rise time of about 15 µs. Another triaxial gradient system is developed by Haishi et al.[61] where gradient sensitivities of Gz =17 T/m/A, Gx =12 T/m/A and Gy =11 T/m/A are reported. The gradients are used up to Gz = 0.026 T/m, Gx = 0.029 T/m and Gy= 0.028 T/m. Details on the linearity, electrical properties and the dynamic response of these gradient coils were not given. Shen et al. report gradients of Golay type [82] with sensitivities about 0.0005 T/m/A with a linearity of better than ±5.4% over a spherical region with a radius of 80 mm. The coils are made using 13 turns of 2.5 mm diameter copper wires. Details on electrical properties and the dynamic response of these gradient coils are not provided. Design and calibration of multilayer gradient coils of Maxwell type are described by Bowtell and Robyr [83]. The reported gradient has a 95  sensitivity of 1.73 T/m/A which was tested up to 52 T/m (current of 30 A). The gradient is linear within 5% over a standard 5 mm NMR tube. The coil consists of 120 turns of 0.224 mm diameter copper wire and has an inductance of 50 µH and a resistance of 1.8 Ohms. 2.2.2.3 Proposed Design of the gradient Module From equations (72) and (73) it follows that in order to satisfy the high gradient strength requirement, it is necessary to minimize the coil dimension and to pass as large an electrical current as possible through the coil while increasing the number of turns. The dimensions of the gradient coils are limited by the required homogeneity over the sample. The larger the gradient coils are compared to the sample, the more uniform the gradient fields become. To increase the strength of the gradients further, a choice between the number of turns of the coils and the maximum current has to be made. However, the larger the number of turns the higher the inductance of the gradient coils becomes. This limits the switching speed thus a low number of turns is preferred. In summary, the size of the gradient coils is chosen based on the size of the sample. The number of turns is chosen based on the desired gradient strength. Current of approximately 5-20 amperes are relatively easy to generate using off-the-shelf solid-state electronics. Therefore, the number of turns for the gradients of this work were chosen by taking this in consideration. The three described gradients are stacked on top of each other as shown in Figure 36. The Maxwell coil (Gz) is located at the two opposite sides of the gradient module (which is referred to as the insert in this work) and the other two rectangular loop gradient coils (Gx and Gy) are sandwiched in between the sample and the Gz gradient coils. Each x and y gradient coil consists of three turns and is made from a 0.01 inch thick sheet of copper with a width of 1.5 mm. Each  96  coil is rated for a 20 A peak current for a duty cycle of less than 10% assuming a maximum current density of 5 A/mm2 for copper. The current rating could be increased if the coils are cooled so that the heat generated by the resistive power dissipation is removed.  Figure 36. The positioning of the three gradient coils: Gx, Gy and Gz. A clearance (size of the gap) of 9.53 mm was chosen to fit the RF probe. The gradient coils are assembled on the outer faces of the enclosure that consists of two back-to-back 3/16 inch (4.76 mm) thick plates of polypropylene. The enclosure is further described in section 2.4 along with the integrated radio frequency probe. 2.2.2.4 Simulation Equations (72) and (73) assume that the size of the wire is relatively small to the overall dimensions of the coil such that it can be approximated as an infinitely long current path. To better understand the effects of current paths with a rectangular cross-section where the size of  97  the wire is relatively large compared to the overall dimensions of the gradient coils, the gradient fields were simulated in 2D in COMSOL MultiPhysics. The AC/DC module was utilized to perform static and transient electromagnetic simulations. The boundary conditions of the outer boundaries were set zero potential while the inner boundaries where set to continuity. The various simulated configurations for Gz (Figure 37) and for Gx or Gy (Figure 38) vary based on the possible choices for placement of the current paths as illustrated by the cross-sections in the figures. The simplest configuration (Figure 37a and Figure 38a) is a vertical stack such that the three turns can be assumed as one single loop with three times the current. A configuration where the coils are stacked horizontally is also considered and shown in Figure 37c and Figure 38c. Figure 37b and Figure 38b illustrate a configuration in which the gradient coils are interlaced. For example, the Gz coil in Figure 37b could be interlaced with the gradient Gx in Figure 38-b so that the order of the current paths from bottom to top would be: Gx1 Gz1 Gx2 Gz2 Gx3 Gz3. The horizontal spacing remains fixed in this configuration because the effective area of the current paths is considered. The alternative has also been considered in Figure 37d and Figure 38d where each loop is considered as a separate loop and is spaced as described by Anderson et al. This was also simulated for the configuration of interlaced gradient as illustrated in Figure 37e and Figure 38e.  98  z0  z0  (a)  (c)  z0  z0  (b)  (d)  Current into the page z0  Current out of the page  (e)  Sample area Effective current path  Figure 37. Simulated current path configurations for the Maxwell gradient.  (a)  (b)  z0  z0  (c)  z0  z0  (d)  Current into the page  z0  Current out of the page  (e)  Sample area Effective current path  Figure 38. Simulated current path configurations for the “rectangular loop” gradient.  99  The simulations were performed assuming a current of 20 A with the copper coils extending 30 mm into the page. The results of the best gradients based on simulated gradient strength linearity and ease of fabrication are chosen for this work and are presented next. For Gz, the configuration in Figure 37a is chosen while for Gx and Gy the configuration in Figure 38a were chosen. The results for the z gradient are shown in Figure 39. The contour plot of the magnetic flux density is shown in Figure 39a where an axial symmetry exists at x = 0 mm. The current paths are two sets of three turn loops that are positioned as previously described. The magnetic flux density in the z-direction was evaluated at x = 0.1 mm (G0), x = 1 mm (G1) and x = 2 mm (G2) for z = ±2.5 mm which is shown in Figure 39b. The gradient field strength is about 0.71 T/m with linearity better than 0.5% for a square of 2 mm on each side. The results for the y gradient are shown in Figure 40. The contour plot of the magnetic flux density is shown in Figure 40a. The current paths are two sets of three turn loops that are positioned as previously described. The magnetic flux density in the z-direction was evaluated at z = 0 mm (G0), z = 1 mm (G1) and z = 2 mm (G2) for y = ±2.5 mm which is shown in Figure 40b. The gradient field strength is about 1.2 T/m with linearity of about 3% for a square of 2 mm on each side. The results for the x gradient are very similar to the y gradient and therefore not presented here. The results show a gradient field of 0.85 T/m and linearity better than 3% for a square of 2 mm on each side for the y-gradient.  100  Magnetic Flux Density in z [mT] 30 mT  z [mm]  0  (a)  0 mT  x [mm]  G0 G1 5 mm  G2  -30 mT Axial Symmetry  (b)  Figure 39. Simulation results of Gz: (a) shows the magnetic field contours; (b) and (c) show the gradient field strength and linearity.  101  Magnetic Flux Density in z [mT] 20 mT 5 mm  z [mm]  (a) 0  y [mm]  0 mT  G0 G1 G2  -20 mT  (b)  Figure 40. Simulation results of Gy: (a) shows the magnetic field contours; (b) and (c) show the gradient field strength and linearity.  102  Transient simulations were performed for the Gz gradient coil with and without the presence a nearby conductive medium (the pole of the main magnet) which is shown in Figure 41a. In Figure 41a, the pole is coloured according to the induced eddy current in the out-of-plane direction which ranges from 1 A/m2 to 2 MA/m2. As seen in Figure 41a, the highest level of eddy currents exist near the surface of the pole faces (conductive) close to the gradient coils. The simulations for the transient response are shown in Figure 41b. The magnetic field was evaluated at a point marked in Figure 41a as a function of time. In the first case in Figure 41b, the rise time is set to 10 µs. When no conductive pole is present, the magnetic field rapidly increases to its steady-state value with a rise time defined by the input. When a conductive pole is present, eddy currents oppose the rapid increase of the magnetic field which leads to a longer rise time of the magnetic field. As the rise time of the input is increased, less eddy currents are generated and the magnetic field follows the input more closely (as seen in Figure 41b for the cases where the rise time is 50 µs and 100 µs). The rise time of the gradient coil is on the order of microseconds if no conductive medium is present; if the conductive medium is placed 2 mm from the coil the rise time ranges from 10 µs to 100 µs. As expected the rise time is increased by the conductive medium which would resist the change in flux due to generation of eddy currents.  103  Induced Eddy Current in Conductive Magnet Poles 2 MA/m2  Measured Point  z [mm]  (a)  0  20 mT  x [mm] 5 mm  1 A/m2  -20 mT  Axial Symmetry  (b)  Figure 41. Simulation of the transient response of the Gz gradients in the presence of conductive and non-conductive poles. The induced currents are shown in the poles in plot (a); the transient response of the labelled point is shown in (b) with driving currents that have rise times of 10, 50 and 100 microseconds. 104  The positions of the current paths of the final design of the three gradient coils are shown in Figure 42.  8.3 mm  7.2 mm  Gz 0.8 mm  B0  1.5 mm 12.6 mm 2.7 mm 6.4 mm  Gx  11.0 mm 2.3 mm 5.6 mm  Gy  Figure 42. Dimensions of the final design of the three gradient coils are shown. All the conductors have the same width of 1.5 mm and height of 0.25 mm. The performance parameters extracted form the results of the simulations of the gradient coils are listed in Table 9.  105  Table 9. Specification of the gradient coils. The clearance between the gradients is 9.5 mm and the total insert thickness is 16 mm. (number of turns n = 3) Gx  Gy  Gz  Gradient Sensitivity [T/m/A]  0.0425  0.055  0.0355  Gradient Strength [T/m] at I = 20 A  0.85  1.2  0.71  Linearity for a cube of 2 mm on each side [%]  < ±3  <±3  < ± 0.1  The gradient coils are etched from a sheet of copper using the fabrication technique described in Appendix A. The gradient coils are made from copper parts stacked on top of each other to make up the three turns. All the coils have four aligning holes which are used for accurate placement of all the coils (Gx, Gy and Gz) on both sides of the insert. The mask used for the fabrication of all the coils is shown in Figure 43-a and the assembly drawing in Figure 43-b; the electrical connections points of the gradient coils are also shown.  106  Gx-1  Gx-2  Gy-1  Gy-2  10 mm  Gx-3  (a)  Gz-1  Gy-3 Gz-2  Gz-3  Gz  Gy (b)  10 mm  Gx Figure 43. Mask (a) and the assembly (b) of the gradient coils. A photograph of each individual current path after the etching process is shown in Figure 44-a. The top and angled side views of the assembled current paths before the final epoxy coating are shown in Figure 44-b. The copper strips are used for soldering connection wires between the 107  coils on top and bottom side of the Insert.  (a)  (b)  Figure 44. Photograph of the fabricated (a) and the assembled (b) gradient coils (only the top piece of the insert is shown).  Photos of the completed insert with integrated triaxial gradient coils and the RF probe are shown in Figure 45.  108  Figure 45. Top view of the insert module with integrated probe and gradients  109  2.2.3 Gradient Coil Characterization The gradient coils are connected using a 1.2 m long twisted wire (AWG 18) which are used to make connections to the measuring equipment and later to the current amplifiers. The measured electrical properties of the gradient coils are listed in Table 9. The measurements were performed while the insert was located outside the main magnet.  Table 10. Measured electrical properties of the gradient coils including connecting wires. Gx  Gy  Gz  Resistance [Ohm]  0.384  0.392  0.374  Inductance [µH]  2.35  2.04  2.06  The electrical properties are within reasonable agreement with theory and other similar gradient coils summarized in the literature review. In order to verify the gradient field strength, the bandwidth of the MR signal of a disk shaped sample was measured for various gradients strengths for the Gx, Gy and Gz gradient coils. Ideally, the shape of the obtained spectrum would have an oval shape if the sample has a circular cross section. The cross-section of the sample for the Gy and Gz gradients are roughly circular while the shape of the sample for the Gx gradient would be of rectangular shape if a selective excitation was used. Since slice selection was not used for these experiments (due to time constraints), the shape of the spectrum does not necessarily represent the cross-sectional shape of the sample. However, knowing the dimensions of the imaging area and recording bandwidth of the spectrum for various input currents, the gradient sensitivity can be obtained. The sample is in a polyethylene tubing with an outer diameter OD = 2.39 mm and inner diameter ID = 1.67 mm. The sample is tab water diluted with CuS04 (40mM) to reduce the T1 time 110  constant to about 30 ms [53]. A spin echo pulse with TE = 6 ms and TR = 102 ms was used. The pulse sequence is shown in Figure 46. The seconds half of the spin echo pulse is recorded, averaged for 256 times. The data is later phase corrected and Fourier transformed in MATLAB (MathWorks).  TE 180°  90° RF Gx, Gy or Gz MR Signal  Figure 46. Pulse sequence used to verify the strength of the gradient coils. The spectrum for the Gx gradient for field strengths up to 0.121 T/m are illustrated in Figure 47a. As expected, the bandwidth of the MR signal is increased as the gradient field strength is increased. For the lowest current setting (I = 0.24 A) which would result in a bandwidth smaller than the homogeneity of the magnet and an increase in the echo amplitude is observed. This suggests that the gradient in this direction can be used later to actively shim the magnet by introducing a constant offset current in this gradient. Due to limitations of the bandwidth of the IF amplifier (~30 kHz) and poor SNR for larger gradients, the maximum gradient field that could be tested using this technique was limited to about 0.2 T/m for all the gradient coils. For higher field strengths, the gradients can be calibrated using a technique described by Wright et al. [80].  111  To calibrate the gradients for high gradient fields, a SE pulse sequence similar to that shown in Figure 46 is used where the RF pulse remains the same but the gradient pulse is different. Basically, immediately after the initial RF pulse a large gradient is applied in the positive direction, which is then followed by a low gradient field in the negative direction; the area of the low amplitude gradient is adjusted until the observed echo has maximum amplitude. By comparing the areas of the high and low amplitude gradient pulses, the strength of the high amplitude gradient can be estimated. The spectrums for the Gy (tested up to 0.161 T/m) and Gz (tested up to up to 0.150 T/m) gradients are illustrated in Figure 47-b and Figure 47-c, respectively.  112  1 0.00 A 0.24 A 0.47 A 0.71 A 0.94 A 1.18 A 1.41 A 1.65 A 1.88 A 2.12 A 2.35 A  0.9 0.8  (a)  Real FFT [Arb.]  0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -10  -8  -6  -4  -2  0  2  4  6  8  10  Frequency [kHz]  1 0.00 A 0.18 A 0.36 A 0.55 A 0.73 A 0.91 A 1.09 A 1.27 A 1.45 A 1.64 A 1.82 A  0.9 0.8  (b)  Real FFT [Arb.]  0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -10  -8  -6  -4  -2  0  2  4  6  8  10  Frequency [kHz]  1 0.00 A 0.09 A 0.19 A 0.28 A 0.38 A 0.47 A 0.56 A 0.66 A 0.75 A 0.84 A 0.94 A  0.9 0.8  (c)  Real FFT [Arb.]  0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -10  -8  -6  -4  -2  0  2  4  6  8  10  Frequency [kHz]  Figure 47. Bandwidth of the MR signal for different gradient field strength are shown for the three gradients Gx, Gy and Gz in (a), (b) and (c), respectively. Again, the bandwidth increases as the gradient strength is increased. The tests performed without  113  compensating for the drift in resonance frequency due to magnetic field disturbances caused by temperature fluctuations. The experiments typically take between 2 to 20 minutes depending on the number of current steps and the number of averages. A frequency shift between 50-400 Hz/min is typical. This partially explains the shift in resonance frequency observed in the spectrum of all the gradients in Figure 47. Another reason for this shift that the sample is not perfectly centered in the insert which would impose a linear shift in the spectrum as a function of field strength. The bandwidths of the MR signals are extracted from plots in Figure 47 and plotted versus the input current and shown in Figure 48. As expected, the bandwidth increases linearly with increased current (resulting in increased gradients). The slope of the each graph shows the sensitivity of the Gx, Gy and Gz gradient coils.  114  Gx vs. Input Current 0.14  (a)  Gradient Strength G x [T/m]  0.12  0.1  Gx [T/m] = 0.0511 x Ix [A] 2  R = 0.996 0.08  0.06 Gx [T/m/A]  0.04  Linear (Gx [T/m/A])  0.02  0 0.00  0.50  1.00  1.50  2.00  2.50  Current Ix [A]  Gy vs. Input Current 0.18  (b)  Gradient Strength G y [T/m]  0.16 0.14  Gy [T/m] = 0.0821 x Iy [A]  0.12  R2 = 0.999 0.1 0.08 Gy [T/m/A]  0.06 Linear (Gy [T/m/A])  0.04 0.02 0 0.00  0.25  0.50  0.75  1.00  1.25  1.50  1.75  2.00  Current Iy [A]  Gz vs. Input Current 0.18  (c)  Gradient Strength G z [T/m]  0.16 0.14 0.12  Gz [T/m] = 0.1415 x Iz [A] R2 = 0.996  0.1 0.08  Gz [T/m/A]  0.06 Linear (Gz [T/m/A])  0.04 0.02 0 0.00  0.20  0.40  0.60  0.80  1.00  Current Iz [A]  Figure 48. Gradient strength vs the input current of the gradients Gx, Gy and Gz are shown in (a), (b) and (c) respectively.  115  The estimated error in the measured sensitivity of the gradients can be as high as 25% due to uncertainties in the measured signal bandwidth (varies depending on the signal amplitude), inner diameter of the tube, and applied current. The measured values for sensitivity of the coils are reasonably close to that of the simulation given above for the Gx and Gy gradient coils considering the tolerances of the insert assembly and the measurement errors. However, the gradient sensitivity obtained for the Gz gradient is off by a factor of about 3 compared to the results obtained in simulations. The measured strength of the gradient field is significantly larger than expected from simulations because the coil is exceedingly close (< 2 mm) to the iron pole (with µr = 4000) of the main magnet, which was not considered in the simulations during the design stage. This was later verified by simulation which suggests a linearity of better than ±3%. Fortunately, the increased gradient strength is an advantage as discussed in the beginning of this section. The measured sensitivity and maximum gradient strength on the gradients is summarized in Table 11.  Table 11. Measured gradient sensitivity and strength of the triaxial gradient module. Gx  Gy  Gz  Gradient Sensitivity [T/m/A]  0.0511  0.0821  0.1415  Gradient Strength [T/m] at I = 20 A  1.02  1.64  2.83  There are two ways to measure the linearity of the gradients: -  A Hall-effect probe is used to measure the field at different positions, or  -  An MRI image is formed of a calibration phantom that would indirectly indicate the linearity of the gradients.  If the first technique is used, the size of the 116  Hall-effect probe has to be much smaller than  the region over which the gradients are uniform. Such small Hall-effect probe was not available for this study. As for the second technique, a cross-sectional image of a sample could not be taken due to time constraints. Therefore, the linearity of the gradient was not evaluated. To check whether eddy currents would pose a significant problem in this setup, first a pickup coil was used to measure the induced voltage in the sample area during the switching of the gradients. The induced voltage of the coil is shown for the rising and falling edges of the 15 A current pulse into the Gz gradient in Figure 49. The Gz gradient was the only gradient tested for eddy currents as this gradient is the closest coil to the pole face of the magnet with the largest expected eddy currents. This experiment was performed once the insert was located outside the main magnet and once when the insert was inside the magnet. No significant change in the shape of the pick up voltage was observed which suggests that eddy currents are relatively small.  117  15A step input  Rising edge  Current output Falling edge External coil  20 µs  20 µs  Figure 49. Magnetic flux detected by a pickup coil inside the probe area during a gradient pulse. To obtain a better sense of the effect of the eddy currents, an eddy current pulse sequence is used as illustrated in Figure 50. A gradient field is applied for a certain duration ( 5 ms) and then switched off. After a short time delay (Tdelay) a 90° RF pulse is applied and a Fourier transform of the resulting FID signal is recorded. This experiment is performed for various values of the delay and it is expected that as the delay becomes smaller, the spectrum of the FID signal becomes increasingly distorted. The time delay at which the FID signal starts to become distorted indicates the duration of the eddy currents inside the poles of the magnets [15, 84]. The longer the duration of the eddy currents, the slower the gradients can be applied. which is not desirable for high resolution microscopy.  118  Tdelay 90° RF Gx, Gy or Gz MR Signal  Figure 50. Pulse sequence to detect the effect of eddy currents. The recorded MR signal (with a dead time of 60 µs) was processed offline using MATLAB. The in-phase phase corrected MR signal and the resulting spectrum are shown in Figure 51 where Gz = 0.28 T/m. The experiment is performed for three time delays: 100 µs, 15 µs and 1 µs. The spectrum is not distorted for time delays larger than 100 µs.  119  x 10  -3  Real FFT [Arb]  I Channel [V]  20  Tdelay = 100 us 10  0 0  1  2  3  1 0.5 0 -20  20  x 10  Tdelay = 15 us 10  0 0  1  2  3  0 -20  -10  0  10  20  Frequency [kHz] Real FFT [Arb]  I Channel [V]  20  -3  Tdelay = 1 us 10  0 0  10  0.5  Time [ms] x 10  0  1  Time [ms]  20  -10  Frequency [kHz]  Time [ms]  -3  Real FFT [Arb]  I Channel [V]  Time [ms]  1  2  3  Time [ms]  1 0.5 0 -20  -10  0  10  20  Frequency [kHz]  Figure 51. FID signal immediately after the removal of the gradient pulse. (Gz = 0.28 T/m) For shorter time delays however, the spectrum becomes increasingly distorted. This is more clearly shown in Figure 52 where a fixed time delay of 25 µs is chosen while the gradient strength is increased from 0.28 T/m to 1.84 T/m.  120  x 10  -3  Real FFT [Arb]  I Channel [V]  20  Gz = 0.28 T/m 10  0 0  1  2  1 0 -1 -20  3  Gz = 0.85 T/m 10  0  I Channel [V]  0 20  x 10  1  10  20  2  1 0 -1 -20  3  -3  10  0 1  2  -10  0  10  20  Frequency [kHz]  Time [ms] Gz = 1.84 T/m  0  0  -3  Real FFT [Arb]  x 10  Real FFT [Arb]  I Channel [V]  20  -10  Frequency [kHz]  Time [ms]  3  1 0.5 0 -0.5 -1 -20  Time [ms]  -10  0  10  20  Frequency [kHz]  Figure 52. FID signal immediately after the removal of the gradient pulse. (delay = 25 µs) Experimental data suggests that for gradients of about 1 T/m the eddy currents last about 100 µs. This is still acceptable considering that practically no measures where taken to reduce the eddy currents using known techniques such as reducing the rise times and employing pre-emphasis techniques [80]. Another common technique to suppress eddy currents is to introduce a thin plate with slits between the poles and the gradients [42]. More sophisticated techniques exist to actively shield the gradients using more complex gradient coil [76]. Offline image processing techniques for reducing image distortions due to eddy currents also exist and are thoroughly discussed by Bernstein et al. [15].  121  2.2.4 Summary of Gradient Coils A triaxial gradient coil module with gradient strengths of Gx = 1.02 T/m, Gy = 1.6 T/m and Gz = 2.83 T/m has been designed, constructed and characterized. The linearity of each gradient is better than ±3% and the switching speeds of better than 20 µs have been achieved. Eddy currents would only adversely affect the signal when the z gradient is switched faster than 100 µs only for large gradients (> 0.6 T/m). The performance of this module would allow for imaging of a region of (2 mm)3 with a static image resolution of approximately 30 µm and a velocity resolution of about 50 µm/s.  122  2.3 Gradient Amplifiers The design and characterization of the amplifiers that drive the magnet compensation coil and the three gradient coils are described next. The design of amplifier was mainly based on data sheets found on manufacturers websites2, 3, 4, 5 and the microelectronics reference book by Sedra and Smith [85]. The design is also similar to a high current amplifier designed by an undergraduate student [86] for the purposes of this work.  2.3.1 Gradient Amplifier Requirements The amplifiers drive the desired current through the gradient coils. The designed gradient coils (refer to Chapter 2.2) require currents of up to 20 amperes to generate gradients of 0.68 T/m (required for a static resolution of 30 µm and velocity resolution of 50 µm/s). The gradients are driven in bi-polar mode meaning that the current is amplified for positive and negative pulses. The gradients also need to be switched with a settling time less than 100 µs. This requires that the gradient amplifier have a rise time less than 20 µs that is equivalent to a bandwidth (BW) of greater than 50 kHz. The gradients are applied for a short time for each repetition which is referred to as the duty cycle (see Figure 53), and the duty cycle is typically below 2%. Another requirement is that input range of the amplifier has to match the output voltage range of the data acquisition/generation card (= ±10 V).  2  ON Semiconductor (www.onsemi.com).  3  Texas Instruments (www.ti.com)  4  International Rectifier (www.irf.com)  5  Cirrus Logic, formerly know as Apex Power Technology (www.cirrus.com)  123  Current [A]  TR  D=  TON  TON TR  TOFF Time [s]  Figure 53. The duty cycle is defined as the ratio of the ON time of the pulse to its period (= repetition time, TR). Therefore, the requirements of the gradient amplifiers are: -  Output currents up to ±20 A  -  Fast switching speeds (rise time less than 20 µs, BW > 50 kHz)  -  Duty cycle of at least 5%  -  Input range of ±10 V.  2.3.2 Gradient Amplifier Design There are two power amplifier architectures: linear and switching. Switching amplifiers, though more efficient, are not considered appropriate for the application of this work mainly because of the high switching noise [87]. Typically, switching amplifiers operate in pulse-width-modulation (PWM) mode at the industry standard of 20 kHz, which is appropriate for applications with a bandwidth lower than 2 kHz. Even if an amplifier with a switching frequency of 200 kHz were to be made the bandwidth would be less than 20 kHz which is lower than what is needed to drive the gradient coils. More importantly, however, the higher switching noise would be picked up by the RF probe (considering that the gradient coils are millimetres away from the probe) and 124  significantly deteriorate the minuscule (typically between 0.2 to 10 µV in this work) MR signals. Therefore, a linear amplifier solution is chosen for this work even though it presents significant design challenge. Integrated high current OP amps (such as the PA52 by APEX Power technologies with peak output of 80 A) are commercially available for linear amplifiers but are expensive (about CAD 700 each) considering that four amplifiers are needed. Thus it was decided to design the amplifiers mainly from discrete components. Four amplifiers have been designed three of which are used to drive the gradient coils and one to drive the B0 compensation coil. All amplifiers are very similar in design except for the amplifier for the B0 compensation coil which is designed for ±5 A. Each amplifier is made of three stages as depicted in Figure 54. The power and driver stage are made from discrete components and the current controller is an operational amplifier (OP amp) that controls the response of the current in the load and corrects for any offsets and drifts from the power stage. The load is the gradient coil with a resistance of approximately R = 0.5 Ohms (with 0.35 Ohm gradient coils and 0.15 Ohm due to connections) and an inductance of approximately L = 3 µH. The maximum voltage at the load can be estimated (~13 V) at Imax of 20 A using  Vmax = R Imax + L  ∆I 20 = 0.5 ⋅ 20 + 3 ⋅10−6 = 13V ∆t 20 ⋅10−6  (74)  where ∆I is the change in current (20 A) and ∆t is the rise time (20 µs). A higher output voltage is desired as overdriving the voltage would result in faster rise times. A voltage compliancy of ±26 V for the amplifier is chosen which is twice the maximum voltage at the load. A significant voltage drop (typically 7 V) exists in the power stage of the amplifier and requires using a supply voltage approximately 33 V.  125  Input form DAQ Current Controller  Driver  Power Stage  ZL  L O A D  RSense  Figure 54. The schematics of the three stages of the gradient amplifier: the power stage, the driver of the power stage and the current controller is shown. The amplifier is designed to operate in class AB (i.e., push-pull) mode using Bipolar Junction Transistor (BJTs). Amplifier design with BJTs is preferred over Metal-Oxide-Field-Effect Transistors (MOSEFTS) for this work because they are lower cost and have simpler driving circuitry. The simplified schematic of the power stages is shown in Figure 55. The two complementary power BJTs (MJ11028 and MJ11029) are used to drive the current through the load. The power BJTs have a high DC gain (hfe> 400) with a high current capability (up to 50 A) and a maximum VCE voltage of 60V. The complementary BJTs are in fact Darlington pairs that increase the gain and also the input impedance of the BJTs. The load current measured as a voltage using a non-inductive current sense resistor (0.05 Ohm, 20 nH) and sent as a negative feedback to the current controller.  126  The VBE multiplier, consisting of a transistor and four resistors, biases the power BJTs to eliminate the cross-over distortion for input voltages (from the driver) less than the forward biasing voltage VBE2 = VBE3 (~0.65 V) of the BJTs. The degree of cross-over distortion is controlled by the selection of the R1 and R2 resistors. The VBE multiplier shifts the input (from the driver) by a voltage equal to  VCE1 = (1 +  R1 )VBE1 ≈ 4 × VBE1 . R2  (75)  Typically, the resistor values are selected to make VCE1 between 2.5V and 4V. One of the resistors (R1) is a potentiometer used to adjust the biasing. For more information about class AB amplifier design and biasing refer to microelectronics reference book by Sedra and Smith [85].  127  Vcc Biasing  Output Driver  Vcc MJ11028 VBE2  VCE1  R1  2N3904  R3  R3  R4 MJ11029  VBE1  R2  VBE2  Coil +  From Driver  R4  Coil -  Gradient Coil  RSense  Vee Vee To Current Controller  Figure 55. Simplified schematic of the power stage. The rest of the resistor values are chosen based on simulations performed in Cadence PSPICE by trial and error and considering tradeoffs between the power rating of each component, supply voltage and the maximum current passed through the load. The power stage is driven by the driver stage which is illustrated in Figure 56. The driver stage also consists of a class AB amplifier similar to the power stage and is driven by the current amplifier. The complementary transistors (BD675 and BD676) are also Darlington pairs with a high gain (hfe> 750), current capabilities up to 4 A and a max voltage of 45V. The biasing is also performed using a VBE multiplier. The bandwidth of this stage can be extended (up to 30%) by introducing a bootstrapping capacitor (C1) to increase the open loop gain.  128  Vcc Vcc  2N3904  R1  BD675  R3  R3  C1  To Power Stage  R4  R2  BD676  From Current Controller  RL,test  R4 Vee Vee  Figure 56. Simplified schematic of the driver stage. The components of the driver stage were chosen based on simulations assuming a dummy load that has the same the input impedance as the power stage. To control the dynamic response of the current passed through the load and also to correct for linearity and offset errors in the power stage, a current controller is implemented using an OPA453 amplifier as shown in Figure 57. The feedback is the voltage sensed by the current sensing resistor. The op amp is the weakest link in the chain in terms of bandwidth performance. Thus, it is important to select an op amp with high bandwidth, high slew rate and wide output voltage swing (requires high voltage supply). The gain-bandwidth product of the selected op amp is 7.5 MHz and has a slew rate of 28 V/µs and maximum voltage supply of ±40 V (voltage swing is Vsupply – 4), which is more than sufficient for the purposes of this work.  129  Vcc Input To Driver OPA453  Offset Adjustment  Vcc  Vee  R3 R4  R5  R1  C1  C2  R2 P- I Controller  Vee From sense resistor  Figure 57. Simplified schematic of the current controller. The proportional-integral controller is implemented with the combination of resistors (R1 and R2) and capacitors (C1 and C2). The transfer function in the s-domain is  s2 (C1⋅ C2 ⋅ R1⋅ R2) + s(C1⋅ R2 + C2 ⋅ R2 + C1⋅ R1) + 1 TF = s2 (C1⋅ C2 ⋅ R1) + s(C1 + C2)  (76)  with poles at  Poles = 0, −  (C1 + C2 ) . R1C1C2  (77)  By selecting proper values for the R1, C1, and C2, the rise time can be minimized while reducing the oscillations in the output are reduced or eliminated. To remove any residual offset that might appear in the final implementation, a offset adjustment circuit using a voltage divider is used as shown in Figure 57. In order to monitor the current of the load (shown in Figure 58), another op amp is used to drive  130  the oscilloscope probe with the gain of the OP amp being 5 V/V (corresponds to an overall gain 4 A/V). The current monitor can also be used in the feedback loop to increase the voltage from the sense resistor. This feature is not utilized for the purposes of this thesis, but it would be useful for future gradient coils that might have a different load impedance. The current monitor is implemented using the OP37 with a gain bandwidth of 40 MHz and a slew rate of 10 V/µs with a maximum supply voltage of 22V. The fast response of this OP amp ensures that no delay is observed in the monitoring signal.  R2 +18 V R1 OP37  To current monitor  R3 -18 V From sense resistor  R4  Figure 58. Schematic of the current monitor. The gain of this OP amp is found as  Gain =  1 + R2 1 + R4  (78)  R1 , R3  and by selecting the resistors (R1 = 2 kΩ, R2 = 10 kΩ, R3 = 10 kΩ, R4 = 2 kΩ) gain of 5 is obtained; this values was chosen because this op amp is not stable for gains of 5.  131  Each of the described stages was simulated separately and its performance was verified. Then, all the stages were connected to each other and simulated again. The final component values are shown in the screenshot of the simulation shown in Figure 59. The power, driver and current controller stages are marked as shown. The input voltage is adjusted to the specified rage of the data acquisition (DAQ) board to ±10 V using a voltage divider. The circuitry for the current monitor is not shown.  132  133  Figure 59. Screenshot of the final values obtained by simulation.  Vcc  Power Stage  LOAD  Current Controller Gain Adjustment Input  Driver Rsense  Offset Adjustment  Vee  The steady-state output current versus the input voltage of the amplifier is shown in Figure 60. The input voltage is shown on the horizontal axis and ranges between ±10V corresponding to an output current (vertical axis) saturating at ±20 A. Therefore, the total gain of this amplifier (the slope) is 2 A/V, which can be adjusted using the potentiometer at the input of this amplifier (to 1A/V). The saturation of the amplifier will be lower if the supply voltage is decreased. The  Output Current [A]  amplifier is designed to operate with a supply voltage between ±20 - ±36 volts.  RL = 0.5 Ohm Vsupply = ± 36 V  Input Voltage [V]  Figure 60. Simulated steady-state output current versus the input voltage. The input voltage range is set to the maximum range of the DAQ (= ±10 V). The maximum output current is ±20 A. A sample sinusoidal response of a 10 kHz sine wave with a ±10 A amplitude of simulations is shown in Figure 61. The output of resembles a perfect sine wave and there are no visible indication of any offset or cross over distortion. The phase shift at this frequency is less than 3 degrees.  134  Input Voltage [V]  ±5V  RL = 0.5 Ohm LL = 3 µH  Output Current [A]  Vsupply = ± 25 V  ± 10 A  Time [s]  Figure 61. Simulation of a 10 kHz sinusoidal input with ±5 V amplitude (top) resulting in a sinusoidal output of amplitude ±10 A (bottom). A sample pulse response with +20 A amplitude and pulse duration of 20 µs of simulations is shown in Figure 62. The output current closely follows the input with minimal overshoot, rise time of 4 µs and fall time of 3 µs. The settling time is less than 10 µs. The performance characteristics exceed the required switching and current levels needed to drive the gradient coils.  135  Output Current [A]  Input Voltage [V]  RL = 0.4 Ohm LL = 3 µH Vsupply = ± 36 V +10 V  4 µs  3 µs  16 A  20 A  Time [s]  Figure 62. Simulation of a +20 A pulse (bottom) for +10 V input (top). The rise and fall times indicates that this amplifier exceeds the performance requirements necessary for very fast gradient switching. The amplifier that drives the B0 compensation coil has similar characteristics to the other amplifiers. The maximum current capability is ±5 A with a bandwidth of more than 50 kHz. Similar to most linear amplifiers the limiting factor in terms of duty cycle and current rating is the power dissipation of the power stage. The maximum mean dissipation power of a linear amplifier in this case can be found using [87]  Pd =  2 2 Imax Vs = ⋅ 20 ⋅ 36 ≃ 460W , π π  (79)  where Vs is the supply voltage (= 36 V). The maximum mean dissipation power considering a  136  duty cycle of 20% is about 90 W for each amplifier. Therefore, based on theory only, the total power dissipation of four amplifiers would be 360 W which does not take into account the any DC current or non-idealities in the biasing of the power stage. The power dissipation of each component of the power stage (for one amplifier) in steady-state conditions (DC current) was simulated and the result is shown in Figure 63. The top plot shows the power dissipation versus the input voltage while the bottom plot shows the output current versus the input voltages. The resistors (R3 and R4 in Figure 55) dissipated power proportional to the square of the output current and the BJTs have a peak dissipation of about 210 W in the of  the  input  (±6  Resistor (PNP) BJT (PNP)  BJT (NPN)  Output Current [A]  Power Dissipation [W]  midrange  Resistor (NPN)  RL = 0.5 Ohm Vsupply = ± 36 V  Input Voltage [V]  Figure 63. Simulation of the steady-state power dissipation (top) of the most critical components of the power stage for as full range ±10 V input (bottom).  137  V).  The simulated average power dissipation for a 50 kHz sine wave with a ±10 A amplitude is shown in Figure 64. The output of the amplifier is shown in the bottom plot while the power dissipation of the main components of the power stage is shown in the top plot. The total average power dissipation of a single amplifier is shown as a bold line in the top plot that saturates at about 195 W. Therefore, the heat sink of this amplifier has to able to dissipate at least 200 W to ensure its proper operation. The total power dissipation of four amplifiers is about 800 W based on simulations. It is noted that the power dissipation is considerably less for lower supply voltages and higher load impedances. For example if the voltage of the power supply is reduced from ±36 V to ±30 V, the power dissipation for ±10 A sinusoid with output would be 160 W.  BJT (PNP)  Average power dissipated by power BJTs and resistors  PRMS, Total = 195 Watts Resistor (NPN) Resistor (PNP)  Output Current [A]  Power Dissipation [W]  BJT (NPN)  RL = 0.5 Ohm Vsupply = ± 36 V Frequency = 50 kHz  Time [s]  Figure 64. Simulation of the power dissipation (top) of the most critical components of the power stage for a 50 kHz ±10 A sine wave (bottom). The average dissipated power of the combined resistors and BJTs of the power stage is shown as the bold line saturating at about 195 W for each amplifier.  138  With a heat sink capable of dissipating 200 W of thermal power, ideally, a duty cycle of 40% can be achieved for 20 A pulses. This values is calculated by adding the power dissipation of the NPN BJT (100 W) and resistor (400W) at 20 A (input of 10 V) from Figure 63. The duty cycle is then 200/(100+400) = 40%. The schematic layout of the thermal design is shown in Figure 65. The power BJTs (TO-3 package) are attached on the top and resistors (TO-246 package) to the bottom of a ½” thick aluminum plate. The collector connection of the BJTs is connected to its case which has to be electrically isolated (but thermally conducting) from the aluminum plate. As shown in crosssectional view in the top-left in Figure 65, a Mica layer is typically used for this purpose. The pins of the BJTs are not long enough to pass through the aluminum plate; therefore, pin connectors were used to make electrical contact. The connections to the collector (case of TO-3) were transferred to the bottom using a #6 bolt using a semiconductor mounting kit that contains all the necessary fittings, as shown in the cross-sectional view in Figure 65. The power resistors are bolted to the aluminum plate and aligned in such a way that one of the pins of each opposing resistor can be bent downwards to make a direct wire connection to the load (see cross-sectional view in Figure 65 bottom-left). The aluminum plate serves as the mechanical support and the heat-conducting medium between the PC cooling fan and power stage. A copper (with thermal conductivity at least twice as large as aluminum) plate can be used for better performance. The Freezer 7 Pro PC6 cooling fan was chosen for its superior heat removing performance (up to 200W) with a 0.17 °C/W resistance and its relatively low cost.  6  Zalamn Cooling Innovations (www.zalman.com)  139  Figure 65. The top view shows the schematic layout of the BJTs and the power resistors on the heat sink. The three cross-sectional views show the details of the mounting of the BJTs and resistors. When sizing the cooling fans and the aluminum plate, the thermal model shown in Figure 66 was used. Each BJT generates an average of about 65 W of heat with each resistor generating about 35 W. The maximum allowable junction temperature is 200 °C and 175 °C for the BJT and resistor for a case temperature of 25 °C respectively. The shown values for the thermal resistance are mostly found in the data sheets of the selected components. The Thermal resistance of the each power BJT is a reported 0.58 °C/W and 1.50 °C/W for the power resistors. The heat resistance of the aluminum plate is estimated using  140  R thermal =  ∆T d = Pd k AL A  (80)  where d is mean distance between the components, A is the average area the heat flux passes through, and kAl is the thermal conductivity for aluminum (~ 205 W/m/°C). The thermal resistance of aluminum for the resistors is about 0.15 °C/W and 0.2 °C/W, respectively. The Mica insulator has a thermal resistance of about 0.1 °C/W and the thermal paste (applied between the aluminum plate and power components) has a thermal resistance of 0.05 °C/W.  Power BJTs: TJunction, PBJT=65 W  PBJT=65 W  RJ-C,BJT RMica,TP  RAl Plate 1  BJT (PNP)  116 °C  0.58  TJunction,  RJ-C,BJT RMica,TP  0.1  0.2  TAl Plate 59 °C  RAl Plate 1  BJT (NPN)  (< 200 °C)  RHS TAmbient  PRes=35 W  TJunction,  RTP  RAl Plate 2  ( 25 °C)  Resistor 1  114 °C  PRes=35 W  RJ-C,R  0.17  TJunction,  1.5 RJ-C,R  0.05 RTP  0.15 RAl Plate 2  Resistor 2  (< 175 °C)  Power Resistors:  Figure 66. Lumped model of the power dissipation of the power stage.  The model suggests that the junction temperature of the BJTs reaches 116 °C which is lower than the 200 °C maximum allowable temperature. Similarly for the resistor, the junction  141  temperature of 114 °C is obtained which is lower than the 175 °C maximum allowable temperature. It is important to note that a de-rating of up to 20% is applied to the maximum power rating of the resistors which have a maximum power rating of 100 W. The design heat sink is therefore more than sufficient to dissipate all the heat generated by the power stage of a single amplifier. In order to maximize the performance in terms of heat dissipation of the amplifier, the heat sink was designed to accommodate six fans rather than the required four. In future, it is possible to increase the duty cycle of two of the amplifiers from 40% to approximately 80% by connecting another power stage in parallel to the existing amplifier.  2.3.3 Gradient Amplifier Construction The CAD drawing of the assembled power stage on the heat sink and the printed circuit boards containing the rest of the electronics is shown in Figure 67. Each of the two printed circuit boards (PCBs) that is mounted on the aluminum plate contains the necessary electronics for driving the power stage of two amplifiers. The PCBs are connected to the power stage using separated wires. The cooling fans are power by a switching voltage regulator on the PCBs.  142  12” (305 mm)  14” (356 mm) Figure 67. CAD drawing of the assembled power stage and the PCBs of all four amplifiers. The overall dimensions are also indicated.  143  For a detailed circuit schematic and PCB design for the complete amplifier refer to Appendix B. A photo of the heat sink with the power BJTs attached is shown in Figure 68a and the resistors on the back side in Figure 68b. The cooling fans are mounted on the same side as the power BJTs as shown in Figure 68c. The heat sink has an open design for maximum air flow.  Resistors  BJTs  Location of PC Heat Sinks  Bottom Side  Top Side  (a)  (b) PC Heat Sink  BJTs  (c) Figure 68. Photos of the heat sink with the BJTs mounted on top (a) and resistors mounted on the bottom (b). The PC cooling fans are attached on the top side (c). Photos of the assembled PCBs for all four amplifiers (as marked) are shown in Figure 69. Two 760W 19A 40V Agilent N5746A power supplies are used to generate the needed supply between  144  ±20 to ±36V. The maximum voltage of the amplifiers is limited by the three voltage regulators ±18V and the cooling fan power supply).  Amp B (Gy)  Amp C (Gz)  Amp A (Gx)  Amp D (B0)  Figure 69. Photo of the assembled gradient amplifiers and the B0 compensation amplifier (top). An angled view of the amplifier and the power supply is shown in the bottom photo. 145  2.3.4 Characterization and Testing of the Gradient Amplifiers The assembled amplifiers were connected to a 1.0 Ohm low-inductive load for testing purposes. The current controller is then tuned by adjusting a variable capacitor; and the cross-over distortion was minimized by adjusting the potentiometers at the VBE multipliers. A 30 kHz sine wave with ±0.5 V amplitude is connected to the input of the amplifiers. As expected a sinusoidal output current of ±1 A is observed on the oscilloscope as seen in Figure 70-a. The output closely follows the input with minimal cross-over distortion and offset. In Figure 70-b, a 10 kHz sine  Vin [V]  ± 0.5 V  (a) Iout [A]  ±1A f = 30 kHz 20 µs  RL = 1.0 Ohm  Vin [V]  ±5 V  Iout [A]  ± 10 A  (b)  f = 10 kHz RL = 1.0 Ohm  50 µs  Figure 70. Measured response of the amplifiers for sinusoidal input (a) 1 A at ±30 kHz and (b) 10 A at ±10 kHz. 146  wave with ±5 V is input to the amplifier which is amplified to ±10 A, again with no offset and minimal distortion. The measured frequency response is shown in Figure 71 for a 2 A peak-to-peak (±1 A) and 20 App (±10 A) sine waves. The output of the amplifier was a sine wave throughout the frequency sweep for 2 App, however for the 20 App signal significant cross-over distortion were observed above 40 kHz. The cross-over distortion can be minimized by adjusting the potentiometers of the biasing circuit, doing so however increases the power dissipation of the power stage which could cause permanent damage to the components in the power stage. Another reason for this distortion is that the amplifier was optimized for a 0.5 Ohm load (which was not available at the time of writing this thesis) and it is expected that the distortion is less for the designed load. Because this amplifier is mainly designed to be used mainly as a pulsed current amplifier, further adjustments to the biasing circuitry were not tested. The normalized magnitude (top) and the phase suggest that the amplifier has a -3 dB bandwidth exceeding 180 kHz which significantly exceeds the performance requirements (>50 kHz) described in the beginning of this chapter.  147  Normalized Gain [dB]  0  X: 1.834e+005 Y: -2.978  -5 2 App 20 App  -10  -15 1 10  10  2  3  10 10 Frequency [Hz]  4  10  5  10  6  Phase [Degrees]  0  RL = 1.0 Ohm -90  BW-3dB = 183 kHz  -180 10  1  10  2  3  10 10 Frequency [Hz]  4  10  5  10  6  Figure 71. Measured frequency response of the amplifier into a 1.0 Ohm load is shown for current levels of ±1 A (= 2 App) and ±10 A (= 20 App). The -3 dB bandwidth (at 90° phase shift) is approximately 180 kHz which exceeds the performance requirements. After the frequency response was measured, the amplifiers were connected to the actual gradient coils. The pulse response of the Gz amplifier for +20A and -20A pulses are shown in Figure 72-a and b, respectively. The input voltage is on the top of each plot and the output from the current monitor on the bottom. The maximum duty cycle for which this amplifier was tested for was 5% for the positive pulse and 20% for the negative pulse. Higher duty cycles are possible by increasing the fan speed, but were not tested. The rise and fall times are marked on the plots in Figure 72 and are generally less than 7 µs. The settling time is about 10-15 µs and can be  148  improved by further fine tuning the current controller.  Amp C - Gz +10 V  Vin [V]  (a) +20 A Positive Pulse Trise = 4.6 µs Tfall = 6.3 µs D = 5%  Iout [A] 1 ms  Amp C - Gz  Vin [V] -10 V  (b)  Iout [A] -20 A  10 ms  Negative Pulse Trise = 4.4 µs Tfall = 2.5 µs D = 20%  Figure 72. Measured response of the gradient amplifiers to a step input of (a) +10 A and (b) -10 A with the Gz coil attached. The rise and fall times are below 10 µs which exceed the performance requirements. A close up of the rising edge of a +15 A is shown in Figure 73. The first channel on the top is the input, the second is the voltage at the load, and the third channel is the output from the current 149  monitor. The current on the load closely follows the expected shape predicted by simulation with minimal over shoot and oscillations. The small oscillations at the beginning of the pulse can be further minimized by fine tuning of the current controller (changing resistor values and adjusting variable capacitor)  Amp C – Gz 7.5 V  Vin [V]  3V  Vout [V] +15 A  Iout [A]  10 µs  Positive Pulse Trise = 1.9 µs  Figure 73. Measured transient response of the gradient amplifier to the rising edge of a +15 A pulse with the Gz coil attached attached. The Voltage at the load and its current are shown. The overshoot and the small oscillations can be further minimized by tuning the P-I current controller. The other two gradient amplifiers (i.e., Gx and Gy) have similar performance characteristics and therefore not presented here. The rise and fall times exceed the required performance requirements (< 20 µs ) described in the beginning of this chapter. The response of the current amplifier for the B0 compensation coil is also tested with the coils attached. The results are shown in Figure 74 for a +5 A and -5 A pulse with rise and fall times less than 10 µs. The bandwidth of this channel does not need to be higher than about 100 Hz as  150  its input is varied for each repetition of the imaging sequence; therefore, the bandwidth far exceed the requirement. Constant (DC) currents of ±3 A were tested with no problems.  +2.5 V  Amp D - B0 Compensation  Vin [V]  (a)  +5 A Positive Pulse  Iout [A]  Trise = 7 µs Tfall = 6.5 µs D = 20%  50 µs  Amp D - B0 Compensation  Vin [V] -2.5 V  (b)  Iout [A] -5 A  Negative Pulse Trise = 6 µs Tfall = 6 µs D = 20%  50 µs  Figure 74. Measured response of the amplifiers to a step input of (a) +5 A and (b) -5 A with the B0 compensation coil. The rise and fall times are below 10 µs which by far exceed the performance requirements of this amplifier. 2.3.5 Summary of the Gradient Amplifiers In summary, three gradient amplifiers were designed and constructed mainly using discrete components which can drive bipolar currents up to ±20 A (duty cycles less than 5%) with rise  151  and fall times less than 10 µs into the gradient coils (0.4 Ohm and 3 µH). Also, the amplifiers are capable of continuous sine waves outputs of ±10A into a 1.0 ohm non-inductive load with a bandwidth exceeding 180 kHz. The supply range is ±25 V to ±36 V and the overall gain of the all the amplifiers is adjusted to 2 A/V. The measured results along with the required design specifications are given in  Table 12. Required and measured specification of the gradient amplifiers and the compensation coil amplifier Gradient Amplifier  Compensation Coil Amplifier  Requirement  Measured  Requirement  Measured  Current [A]  ±20  ±20  ±3  ±5  Max duty cycle [%]  2  5  100  100  Rise/Fall time [µs]  < 20  <7  <10000  <10  Settling time [µs]  <100  <20  <1000  <50  152  2.4 RF Electronics The design and characterization of the radio frequency (RF) electronics for detection and excitation of the MR signal are described next. The design of RF electronics and some practical tips are extracted from [32, 38, 53, 56, 59, 88-101].  2.4.1 RF Electronics Requirements There are three components to the RF electronics: the probe, transmitter and receiver. The probe is a coil that generates and/or receives the time-varying magnetic field. In this work, both functions are performed by a single probe. The magnetic field generated by the probe must be orientated perpendicular to the main static magnetic field in order to flip the magnetization. Also, the generated magnetic field must be homogenous over the sample to excite all the spins in the sample by the same angle. Another requirement is that the impedance of the probe has to be matched to that of the transmitter and receiver, in order to ensure maximum power transfer from the transmitter to the probe for excitation, and from the probe the receiver for detection. The transmitter supplies the time varying excitation signal to the probe with the desired shape and amplitude at the Larmor frequency. The shape of the RF excitation signal is determined by the imaging sequence. Refer to Chapter 1.4 for more information. The receiver must detect and amplify the minute MR signals from the background noise. As a result, the receiver must have a high signal-to-noise-ratio (SNR). In addition the amplified voltage swing must be matched to the input voltage range of the data acquisition board. The receiver must also detect a wide range of amplitudes of the MR signal. For example, the receiver must detect a FID signal that has a larger amplitude at the start of the pulse sequence but much  153  smaller amplitudes a few T2* time constants later. Therefore, the requirements of the receiver can be summarized as: high SNR (i.e., low noise floor), high gain and high dynamic ratio. Another important requirement is that the receiver must handle the large amplitudes of the transmitter pulse (= +30 dBm) while detecting a signal on the order of -100 dBm; a voltage difference of 3·106. Therefore, a protection scheme has to be implemented for the receiver.  2.4.2 RF Design 2.4.2.1 The Probe The most commonly employed RF coils for similar applications are the solenoidal and Helmholtz coils [30]. The shape of the two coil types and direction of their magnetic field is depicted in Figure 75. A solenoidal RF coil is simply constructed by winding conductive material, such as copper, onto a cylindrical object. The optimum coil size and wire spacing is discussed in [88, 102, 103].The solenoidal configuration is also well suited for the compact integration of the gradient coils described in section 2.2. The Helmholtz RF coil consist of two equal rings of wires separated by a half the diameter of the rings.  154  Magnet Pole B0  Capillary Tube B1  B1 Solenoidal Coil  Helmholtz Coil  Magnet Pole  Figure 75. Orientation of the varying magnetic field (B1) in Solenoidal and Helmholtz coils inside a static magnetic field B0. The Helmholtz coil has a larger access compared to the solenoidal coil. However, The SNR of the solenoidal coil is roughly three times greater when compared to the Helmholtz [25, 88, 102, 103]. Both coil types have a relatively homogenous region of excitation as long as the sample volume is sufficiently smaller than the diameter of the coil. A solenoidal coil is chosen for this work due to the simplicity of construction and better SNR performance. It is important to match the impedance of the RF coil to that of the receiver and transmitter for optimum signal transfer. This is achieved by the adding two capacitors to the RF coil as shown in Figure 76. The tuning capacitor, Ctune, is connected parallel to the RF coil while the matching capacitor, Cmatch, is connected in series. By adjusting the capacitors the impedance can be matched (typically to 50 Ohms) to that of the transmitter and receiver. Other impedance matching techniques are described in [89].  155  Capillary Tube Fluid Flow  Detection Coil  To Transmitter and Receiver Cmatch Ctune Figure 76. Impedance matching of the RF probe is achieved by adjusting the two capacitors. There are SNR advantages when using micro coils [99], which will be necessary for high resolution imaging of micro capillaries.  Coil winding: To reliably wind coils with high repeatability with diameters less than 2 mm, a coil-winding machine was constructed by undergraduate students as a side project for this work. This apparatus is depicted in Figure 77. The capillary tube (flexible Teflon tube shown) is tensioned between two drive chucks which are rotated manually. The arm holds multiple wires which are wound on the capillary tube. All the wires are back-tensioned by passing a constant current through the motors attached to the arm. The spacing between the wires is controlled by selecting discrete sizes of fishing lines. The final coil is then glued in place using superglue.  156  Coil Winding Arm  Coil Tensioner  Tube Tensioner  Rotational Motion of Tube Wound Coil Capillary Tube Linear Movement of Arm  175  mm  Figure 77. Coil winding machine capable of winding 20 µm wires onto 100 µm micro capillary tubes.  Integrated RF enclosure: An enclosure is machined out of polypropylene to hold the RF coil, the impedance matching capacitors, the capillary tube and also the gradient coils (described in section 2.2). The dimensions of the probe enclosure are 65 mm (H) x 80 mm (L) x 8.7 mm (T). It consists of two parts with milled slots that hold the capillary tube and the coaxial cable carrying the RF signals to and from the probe. The coaxial cable (RG-223) is of double shielded type and is used for its excellent noise shielding [98]. The directions of the time-varying magnetic field and the static magnetic field are marked in Figure 78-A. The RF coil is located in the middle of the enclosure and surrounds the capillary tube. The impedance matching capacitors, which are of non-magnetic type, are located as close as possible to the RF coil and are accessible for adjustment form the side of the enclosure. The two part of the enclosure are fastened to each other using plastic screws which also define the alignment of the two mirrored polypropylene components. The machined enclosure is shown in Figure 78-B. The final probe enclosure with its elements identified is shown in Figure 78-C. For prototyping purposes a tube with an outer 157  diameter of 5 mm was chosen. In the future, micro-capillary tubes will be used for visualization of flow at microscale.  (a)  (b)  (c)  Figure 78. The CAD drawing of the enclosure for the RF probe along with the orientation of the magnetic fields is shown in (a). The machined enclosure and the components of the probe are shown in (b) and (c). 158  2.4.2.2 The Transmitter To generate the excitation signal, a carrier frequency (pure sine wave) at the Larmor frequency is synthesized by an external source (a function generator specified later in this section). The carrier signal is modulated with the desired pulse shape and is then is then amplified. This is illustrated in Figure 17. The pulse shape is depicted in the top plot and the carrier signal is in the  Sinc Pulse  Rectangular Pulse  Modulated Signal [A.U.]  Carrier Signal [A.U.]  Shaping Pulse [A.U.]  middle plot. The bottom plot shows the modulated carrier.  Time [s]  Figure 79. Pulse shaping of the excitation signal (bottom plot) is achieved by modulating a carrier sine wave (middle plot) with a pulse shaping signal (top plot). The architecture of the transmitter is illustrated in Figure 80. The external sources are connected from the left side and not shown in the figure. The pulse shaping signal is digitally synthesized by a PC and output as an analog signal using a data acquisition board (NI-6259, National Instruments). The analog pulse shaping signal is then amplified by and SRS 560 (Stanford 159  Research Systems Inc., 1290-D Reamwood Avenue Sunnyvale, CA 94089, USA) amplifier. A 25 MHz carrier signal is synthesized by an Agilent 33250A function generator and is split between the transmitter and the receiver. The amplitude of the carrier signal is adjusted by an attenuator and is mixed (MXR-15, RF Bay Inc., 13111 Piney Knoll Lane., Suite B, Potomac, MD 20854, USA) by the pulse shaping signal. The modulated signal is passed through the RF switch (Mini-Circuits model ZYSWA-2-50DR, P.O. Box 350166, Brooklyn, NY 11235 USA) to the power amplifier (MPA-12-30, RF Bay Inc.) that amplifies the signal to +30 dBm (1 Watt). The output of the pulse shaping can be enabled or disabled using the “Pulse” signal. After the end of the excitation phase, the RF switch is disabled to minimize the noise sent to the power amplifier. The set of crossed diodes, placed after the power amplifier, are used to block the noise of the amplifier after the detection phase starts. The noise blocker effectively creates a threshold limit below which the noise (lower than 200 mV) is suppressed. The excitation signal is the sent to the probe via a splitter. The receiver is also connected to the probe via this splitter after passing through the protection electronics. The quarter wave line in addition to the crossed grounded diodes effectively creates high impedance which blocks the receiver form high powered pulses. For more information regarding the properties of the quarter wavelength refer to the book by Fukushima [89], and for other protection techniques refer to [59, 95, 97, 104107].The grounded crossed diodes clip the incoming signal to the reverse biased voltage of the diodes (0.65 V). Thus, only low amplitude signal (such as MR signals) pass through the protection electronics once the high power excitation signal is turned off.  160  Activate Pulse Mixer  Pulse Shape  Noise Blocker  Switch  IF Amplifier  RF Power Amplifier  Attenuator Splitter Splitter  λ/4 Coax  To Probe  25MHz  To Receiver  To LNA  LNA Protection Diodes  Figure 80. Architecture of the transmitter with pulse shaping capability and LNA protection. 2.4.2.3 The Receiver To extract the small MR signals from all the background noise, a receiver with narrow bandwidth is used such that only the signal and the background noise of desired frequencies are amplified to maximize SNR. Various types of receiver architectures exist that are suitable for NMR and MRI applications and are discussed in [88, 89, 92, 93, 97, 100, 108]. To avoid aliasing of positive and negative frequencies, two quadrature channels (90° phase shift) are necessary. In this work, a direct phase-sensitive detection in quadrature architecture is used for its simplicity in design and construction. The schematic of the receiver is shown in Figure 81. In the detection phase, the MR signal from the probe is first amplified using a low noise amplifier (LNA) with a gain of 41 dB and noise figure of 1.4 dB (LNA-605, RF Bay Inc.) and filtered using a homemade band-pass filter with a centered at 25MHz with a bandwidth of 1 MHz. The signal can be further amplified by another 161  LNA (LNA-1050, RF Bay Inc) with a gain of 51 dB and noise figure of 2.0 dB. The gain in this stage is adjusted by selecting appropriate attenuators (Mini-Circuits K1-VAT+). The amplified signal is then mixed down to audio frequencies using two level 7 mixers (MXR-15, RF Bay Inc.). The local oscillator input to each of the mixers is provided by a quadrature hybrid (ZMSCQ-2-50, Mini-Circuits) which generates and in-phase and quadrature reference signals. The audio frequency signals are then amplified by an audio amplifier and filtered using a third order Butterworth filter with a cut off of frequency of 70 kHz. The two channels are then sampled at 200 kS/s each and digitally filtered by a bandwidth defined by the pulse sequence.  To Transmitter  25MHz  Splitter  Quadrature Hybrid 90° 0°  I Splitter Low-pass Filter  IFA  Mixer  From Probe  Q  Attenuator Low-pass Filter  IFA  IFA  Band-pass Filter  LNA  Mixer  Figure 81. Architecture of the receiver with direct quadrature phase-sensitive detection.  162  2.4.2.4 Overall RF Design The block diagram of the overall RF system including the probe, transmitter and receiver is displayed in Figure 82.  Activate Pulse Noise Blocker  Mixer  Detection Coil Pulse Shape  RF Power Amplifier  Switch  IF Amplifier  Capillary Tube Fluid Flow  Attenuator  Splitter  Splitter  Cmatch  25MHz  Ctune Quadrature Hybrid 90° 0°  λ/4 Coax I Splitter Low-pass Filter  IFA  Mixer  Attenuator  Q Low-pass Filter  IFA  IFA  Band-pass Filter  LNA  Mixer  Figure 82. Architecture of the overall RF system for excitation and detection of MR signals. Photographs of the implementation of the RF electronics are shown in Figure 83. The quarterwavelength coaxial cable and the cable connecting the probe are not shown. The transmitter is marked on the right-hand side of Figure 83-A while the receiver is on the left side. The front panel of the homemade box is shown Figure 83-B. The RF components are interconnected using RG-316 coaxial cables.  163  Receiver  Transmitter  Mixer  Mixer  r  LP Fi lter  Switch Powe  Splitter  Splitter Quad Hybrid  Amp  LNA  Mixer  Power Amp  300 mm  Figure 83. The RF electronics box implemented with off-the-shelf components. 2.4.3 Characterization of RF Electronics The probe is made with an AWG 24 laminated copper wire with 7 turns with an inner diameter of 5.5 mm and 7.5mm length. The electrical properties are listed in Table 13.  164  Table 13. Properties of the RF probe. Property  Value  B1 [mT/A]  0.51  DC Resistance [mΩ]  17  Inductance [µH]  0.56  Ctune [pF]  10 - 120  Cmatch [pF]  10 - 120  Q  208  The tuning and matching is achieved by adjusting Ctune and Cmatch until the reflections by the probe are minimized [98, 109] at the Larmor frequency as described in the Appendix C. A snapshot of the screen of the oscilloscope of a matched and tuned probe at 25.2 MHz is shown in Figure 84. Channel one is an indicator of the frequency in sweep mode where the rising edge shows the starting frequency (10 MHz) and the falling edge is the marker frequency (Larmor frequency).  10 MHz  25.2MHz  40 MHz  CH1 - Sync  CH2 - Directional Coupler Output  Resonance Frequency  Figure 84. Scope screenshot of the tuned and matched probe. The probe is tuned to 25.2 MHz and matched to 50 Ohms. 165  Channel two is signal that is reflected from the probe. As visible in Figure 84, the reflections are minimum at the Larmor frequency (25.2 MHz). The input versus output relation of the RF power amplifier in the transmitter is shown in Figure 85. The output power of the RF amplifier saturates at about 30 dBm which is more than sufficient to excite the sample by 90° flip angles with short (< 50 µs) rectangular pulses.  35  30  P1dBm = 29.9 dBm  Output [dBm]  25 Gain = 30.4 dB  20  15  10  5 -15  -10  -5  0  5  10  15  Input [dBm]  Figure 85. Output power of the power amplifier versus the input power. The amplifier has a gain of 30.4 dB and a 1-dB compression point of 29.9 dBm. The results of the RF transmitter are summarized in Table 14.  166  Table 14. Properties of the RF transmitter. Property  Value  P1 dB [dB]  29.9  Gain [dB]  30.4  Frequency Range [MHz]  25 - 1200  Noise Figure [dB]  4.2  Snapshots of rectangular and sinc pulse shaping of the transmitter are shown in Figure 86-A and Figure 86-B respectively.  167  CH1 – Pulse Shape Envelope Rectangular Pulse  (a) CH2 – Modulated Signal  25 MHz Sine Wave Carrier Signal (not shown)  CH1 – Pulse Shape Envelope Sinc Pulse  (b) CH2 – Modulated Signal  25 MHz Sine Wave Carrier Signal (not shown)  Figure 86. Scope screenshots of the pulse shaping for a rectangular (a) and sinc (b) pulse. The pulse duration of the 90° flip angle for a free induction decay (FID) signal is  π τ= γ  2 =  B1  2  π  2 ≈ 23 µs . (2π) ⋅ (42.6 ⋅10 ) ⋅ (0.5) ⋅ (0.51 ⋅10−3 ) 6  The factor of two in the B1 term takes into account the fact that a solenoidal RF coil generates a field along the coil only. If a coil that could generate a circularly polarized field was used the factor of two is removed. This estimation was verified by measuring the amplitude of the FID signal for various pulse durations of rectangular shape. The result is shown in Figure 87. The amplitude of the MR signal is greatest at duration of about 24 µs. The variation of amplitude  168  with respect to pulse duration closely follows the expected cosine function as seen in Figure 87.  Normalized Signal Amplitude Normalized Signal Amplitude  1.00  0.75  0.50 Normalized Amplitude Theory 0.25  0.00 0  10  20  30  40  50  Pulse Duration (us) Pulse Duration [us]  Figure 87. Amplitude of the FID signal versus the pulse duration of rectangular shape. The 90° flip is achieved for a pulse duration of about 24 µs. The receiver is tested by injecting a -100 dBm sine signal at 25.010 MHz and a carrier frequency of 25.000 MHz. A snapshot of the received signal (averaged 16 times) of both quadrature channels is depicted in Figure 88. The received signal has an expected frequency of 10 kHz. The signal-to-noise ratio of the non-averaged signal is about 18 dB.  169  25.000 MHz Carrier and 25.001 MHz Sine Wave Signal at -100dBm (not shown)  CH1 – In-phase Demodulated Signal (I)  CH2 – Quadrature Demodulated Signal (Q)  Figure 88. 10 kHz sample outputs of the both quadrature channels form the receiver for 100 dBm input signal. The output waveform is averaged 16 times. The SNR of the nonaveraged signal is 18 dB. The dynamic range depends on the expected amplitude of the MR signal, and is set by the correct combination of gains and attenuator in the receiver (see Appendix C for details). The dynamic range is limited by the dynamic range of the mixer (44 dBm). The performance characteristics of the receiver are summarized in Table 15.  Table 15. Properties of the RF receiver. Property  Value  Max Gain [dB]  130  Frequency Range [MHz]  25 - 50  Dynamic Range [dB]  44  Noise Floor [dB]  -120  Dead-time [µs]  100  170  A sample snapshot of an NMR signal acquired by the described RF electronics is shown in Figure 89. The first channel shows a rectangular pulse that is used for triggering of the oscilloscope but is not visible because it has a very short pulse duration (20 µs). The second channel is the I-channel of the receiver detecting an NMR signal. The receiver dead-time is 100 µs as marked. The dead-time of the receiver is partially due to the ringing of the high resonant probe. The recovery time [95]  TRecovery =  Q  Vout  ln   π⋅ f  Vin   (81)  can be calculated for this setup for a quality factor Q of 206, frequency, f (25.2 MHz), voltage of transmitter of +30 dBm (7.1 Vrms) and MR signal voltage down to -120 dBm (0.225 nV) to be 63 µs. This theoretical value is reasonably close to the measured dead-time of the receiver.  CH1 – Trigger Pulse  CH2 – Demodulated NMR Signal  Receiver Dead Time ~100 µs  Figure 89. The dead-time of the receiver for a sample free induction decay MR signal is about 100 µs.  171  2.4.4 Summary of the RF Electronics The design of the RF probe and the architecture of the transmitter and receiver were described. The transmitter and receiver were constructed out of off-the-shelf RF components. The RF probe has a quality factor of above 200 and is tuned and matched at the Larmor frequency (25 MHz). The transmitter has a gain of 30 dB and amplifies signals to + 30dBm that ranging from 25 MHz to 1.2 GHz. The transmitter can modulate the carrier signal with the desired pulse shape. The quadrature phase-sensitive receiver has a maximum gain of 130 dB and a measured noise floor of -120 dBm. The dynamic ratio of the receiver is 44 dB and has a dead time of approximately 100 µs. The RF electronics were successfully used to excite and measure MR signal of water inside a standard 5 mm NMR tube.  172  Chapter 3 - Overall System Design This section describes how the various components are connected to each other and how they are synchronized. The synchronization of two data acquisition boards along with the signal acquisitioning of the raw MR signal is also described.  3.1 Overall Component Layout and Interconnections The design and construction of each component of the MRI microfluidic visualization instrument was described in Chapter 2. A schematic of how the components are interconnected is shown in Figure 53. Depending on the desired imaging parameters (namely, FOV and receiver bandwidth, repetition rate, number of averages, etc) a pulse sequence is synthesized by a computer and saved as a text file. Next, the text file is loaded in a LabVIEW (by National Instruments) program that controls the six analog outputs of the two data acquisition boards (DAQ). Details regarding the LabVIEW program and the DAQ interconnections are described in the following sections. Three of the analog outputs are connected to the three gradient amplifiers that amplify the signal to the desired current level that drive the gradient coils. One analog output is connected to the fourth current amplifier that is used for compensating any drifts of the static magnetic field. The other two analog outputs are directed to the RF electronics; one channel controls the transmitter RF switch and the other channel is first amplified by an SR560 preamplifier (the bandwidth is set to 30 kHz) and then sent to the mixer for pulse shaping.  173  Gx  PC Gy  PCI  B0 Comp. Coil Permanent Magnet Gradient Coils  NI-6259  LabVIEW  Gz  USB  NI-6212  RF Probe Gradient Coils Permanent Magnet  B0  Pulse  B0 Comp. Coil  Sinc  MATLAB USB  MSO6034A  Carrier Signal  RF Electronics  Flow Control  33250A  Figure 90. Schematic of the overall system indicating the connections between system components. The carrier waveform is synthesized by an Agilent 33250A function generator which is connected to the RF electronics. The transmit signal is sent to the RF probe (see Figure 53) via a double shielded coaxial cable. This coaxial cable is also used to send the MR signal picked up by the RF probe to the quadrature phase-sensitive detector in the RF box. The two outputs from the receiver are sent to the Agilent MSO6034A oscilloscope for easy visualization of the raw signals. The oscilloscope is connected to the PC via USB and is configured to amplify and average the detected MR signals which are then sent to the PC for further processing. Details on how the MATLAB (master) and LabVIEW (slave) programs communicate are given in the following sections. A syringe pump (KD Scientific, Model 780210L) is used to control the flow inside the sample (a Teflon tube with an inner diameter of 1.67 mm ) and is marked in Figure 53. A photo of the experimental MRI setup is shown in Figure 91.  174  Figure 91. Experimental setup of the microfluidic visualization instrument with all the components connected.  3.2 Data Acquisition Software The data acquisition software is programmed in LabVIEW 8.5 and controls the analog outputs of two data acquisition boards. The analog inputs are not used in this work, but can be used in future to acquire the MR signals as well. The first DAQ is the NI6259 from National Instruments which has four analog outputs with a maximum sampling rate of 1.25 MS/s (for all four analog output channels together) and is connected to a PCI slot in the PC. It has programmable channels (called PFI) that can be used to  175  access the sampling clock signal and also be used for triggering. The internally generated sampling clock of 500 kHz is routed to the PFI 7. The external trigger is programmed to be on PFI 2. The digital triggering signal is also generated in the NI6259 DAQ using a counter (Cntr 0) which runs at 100 Hz with a 50% duty cycle (i.e., a square shaped waveform). The triggering signal (Cntr 0) must be physically connected to the trigger input of the DAQ (PFI 2) using a BNC cable. The physical connections of the DAQ are shown in Figure 92. The four analog channels of the NI6259 DAQ are connected to the three gradient amplifiers and the RF pulse signal going to the RF electronics box. The second DAQ is the NI6212 also from National Instruments which has two analog outputs with a maximum sampling rate of 250 kS/s (for two analog output channels) and is connected to the PC via USB. This DAQ is programmed (PFI 0) such that the external clock is provided by the first DAQ (NI6259). Using this technique the sampling rate has been tested up to 900 kS/s (which is much higher than the reported specification) with no problems. The sampling rate is the same as the first DAQ which is set to 500 kS/s. The trigger is also programmed to be an external digital signal (PFI 2) which is supplied by the NI6259 DAQ. The physical interconnections between the two DAQs are also shown in Figure 92. The two analog outputs of the NI6212 DAQ are connected to the RF pulse shaping signal (connected to the SR560 preamplifier) and the B0 compensation amplifier.  176  Dev1 = NI6259 (PCI)  Dev2 = NI6212 (USB)  Cntr 0 = Trigger Out PFI 2 = Trigger In  PFI 2 = Trigger In  PFI 7 = Sample Clock  PFI 0 = Sample Clock  AO-0 = RF Pulse  AO-0 = RF Shaping  AO-1 = Gx  AO-1 = B0  AO-2 = Gy AO-3 = Gz  To RF Electronics, Gradient Amplifiers and B0 Compensation Amplifier  Figure 92. The synchronization between the DAQs is achieved by physically interconnecting them as shown. A flow chart of the LabVIEW program controlling the two DAQs is shown in Figure 93. First, the triggering signal that is generated by the counter is initialized. Then, the channel properties (sampling frequency and triggering) of each DAQ is configured as explained before. The pulse sequence that is saved in a data text file is read. The pulse sequence data is then manipulated (i.e., an offset is added or a constant gain is multiplied to each channel) based on the data read from another text file called the MATLAB Link (ML) file. The ML file consists of a series of data, which specify the offsets and gains of the six analog outputs. This file is updated by a MATLAB program called the master. The master program is described in more detail in the following section. After the pulse sequence data is manipulated based on the data in the ML file,  177  the pulse sequence is again manipulated by gains and offsets available on the front panel of the LabVIEW program which are controlled by the user. The user adjustable gains and offsets are for fine tuning of the pulse sequence data. The analog waveforms output by the DAQs are updated every 100 ms. The LabVIEW (slave) program runs continuously until it is manually stopped by the user.  START Dev1 Run counter on Dev1-Cntr 0 for trigger Dev1 Generate sample clock on Dev1-PFI 7  Dev2 Use external clock from Dev2-PFI 7  Dev1 Trigger from Dev1-PFI 2  Dev2 Trigger from Dev2-PFI 0  Read pulse sequence data file Read MATLAB Link (ML) file; apply scaling and offset operations on pulse sequence data Output data No Stop? Yes END  Figure 93. Flow chart of the LabVIEW program that synchronizes the two DAQs to output six analog waveforms read from a data file. The LabVIEW program is in slave mode and controlled by a MATLAB program (master). The front panel with the graphical user interface (GUI) of the LabVIEW program is shown in  178  Figure 94. The location of the ML file and the pulse sequence data file are provided in a text box. The updated analog output waveforms of all six channels are shown on the six graphs on the front panel. The gains read in from the ML file are shown on the bottom left. The user configurable gains are located on the right.  Figure 94. The LabVIEW program with a user friendly GUI in the front panel.  3.3 Frequency Shift Compensation and Signal Acquisition A MATLAB program is written to control the analog outputs via the LabVIEW slave program and to perform the signal processing necessary to generate an MRI image. The sequence of program steps is summarized in the flow chart shown in Figure 95. After the user sets the imaging parameters (such and receiver bandwidth, number of averages, etc.) the program first initialized the ML file. Note that the LabVIEW program has to be running before the MATLAB  179  program is started. The oscilloscope is connected via USB to the PC and is configured to acquire the data from the receiver. First, the magnetic flux density of the magnet is verified and controlled. For this purpose, an MR signal (second half of the echo) is acquired by the MSO6034A oscilloscope when the phase and frequency encoding gradients are turned off (their gains are set to zero in the ML file). The frequency shift of the signal is calculated after the offsets of the raw signal are removed and the data is phase corrected. The phase correction is a necessary step to ensure that the real and imaginary part can be both used for calculating the frequency shift. For more information regarding phase correction refer to [88]. The magnetic resonance frequency shift is calculated by taking a fast Fourier transform (FFT) of the phase corrected data. The zero crossing of the imaginary signal of the FFT corresponds to the resonance offset. The function that finds this zero crossing needs a starting point which is set to the maximum of the real part of the FFT. Based on the measured frequency shift, the ML file is updated to adjust the magnetic field in order to compensate for the frequency shift. To acquire an image, the MR signal is acquired with no phase or frequency encoding gradients applied. The drift in the main magnetic field is calculated and a compensation signal is sent to the slave. If the magnet drift is too large (>100 Hz) this iteration is repeated. After the magnetic field is corrected, a single phase encoding step is performed. Again, the oscilloscope is configured to take the echo signal while the proper gradients are applied through the slave. After each phase encoding step, the compensation of the magnetic field is repeated. The averaged raw data (by the oscilloscope) are saved in k-space. After the imaging sequence is finished, the kspace is post processed (filtered and DC offsets removed) and a 2D FFT is performed to form an  180  START  Specify imaging parameters Initialize Write MATLAB Link (ML) file No Setup and acquire FID Signal from MSO6034A (no gradients applied)  Calculate B0 offset and apply compensation; update to ML  Yes  B0 too large? No  Data acquisition finished?  Yes  Construct k-space  Store Data  Perform inverse 2D FFT  Setup and acquire data from MSO6034A  Show Image  Calculate scaling and offset operations and update ML file  END  Figure 95. Flow chart of the MATLAB program (master) that controls the analog outputs via the LabVIEW program (slave). image. A performance of the B0 compensation is shown in Figure 96 as a function of time. A simple integral controller is sufficient to compensate for the phase shift within ±150 Hz peak (average of 40Hz). The first plot shows the measured shift in the resonance frequency. The reference is set to zero Hertz for imaging. The second plot in Figure 96 shows that the error in the frequency shift is within ±150 Hz. The peak at time 6 min occurs because a lab chair (ferromagnetic) was purposely brought within 50 cm of the MRI setup for a minute and then removed.  181  Resonance Shift [kHz]  Figure 96. The performance of the B0 compensation in terms of the shift of the magnetic resonance signal as a function of time is shown in the top figure with the reference is set to zero Hz. The error in the frequency shift is within ±150 Hz. The current inside the compensation coil is also shown as a function of time in the bottom plot. The peak of the error at time 6 min occurred because a lab chair was purposely brought close (50 cm) to the MRI setup. The third plot in Figure 96 shows the current inside the B0 compensation coil as a function of time. Without compensation, the resonance frequency would have shifted by 1.5 kHz over 24 minutes. The response of the compensation system is more clearly shown in Figure 97 where the reference resonance frequency was stepped to 2 kHz for testing purposes. The frequency shift is compensated for within two sampling points. The update rate is dependent on the repetition time of the pulse sequence, the SNR of the signal, the number of averages, and the delay introduced  182  by the data processing and acquisition (combination of the delays by the LabVIEW and the MATLAB programs and the setup time of the oscilloscope). Update rates between 0.01 to 0.5 Hz  Resonance Shift [kHz]  have been used in this work.  Figure 97. The performance of the B0 compensation is shown in terms of the shift of the magnetic resonance signal as a function of time in the top figure with the reference changing by ±2 kHz. The error in the frequency shift is within ±150 Hz. The current inside the compensation coil is also shown as a function of time in the bottom plot. The magnet compensation is applied until the frequency shift is within an acceptable range. After the magnet is compensated, the imaging sequence starts with a single phase encoding step. The phase encoding gain of in the ML file is updated. After a short pause the oscilloscope is configured to acquire data depending on the number of averages and the length of the pulse 183  specified by the user. The recorded data is store in a single line of k-space which is offset from the center depending on the phase encoding step. Then, the cycle for the magnet compensation is repeated before the next phase encoding step is performed. This cycle is repeated until the kspace is filled. After the data acquisition is completed, the raw data in k-space is processed using MATLAB. First, the offsets are removed, and then the data is filtered by a third order Butterworth filter with a cut-off frequency equal to the receiver bandwidth specified by the user. A two-dimensional Fourier transform is then performed to generate the magnetic resonance image.  184  Chapter 4 - Micro MRI Static Imaging and Velocity Visualization The pulse sequences to generate static and velocity encoded images using the described setup is presented next. Information regarding pulse sequence can be found in [15, 29, 30, 32, 88, 89]. A sample static image and a few velocity visualization plots that were obtained by the described instrument are presented.  4.1 Pulse Sequence for Static Imaging The sample in this work is a tube filled with a fluid (water mixed with 40mM CuSO4). For static imaging, the fluid is stationary and for velocity imaging the fluid is flowing at a flow rate set by the syringe pump. The coordinates adopted in this work are shown in Figure 98. The axis of the tube is along the x-axis with the flow moving in positive x-direction. The cross-sectional view of the tube is shown on the right-hand side of Figure 98 where the inner diameter of the tube is marked as Di. The dimensions of the field-of-view (FOV) in the horizontal and vertical directions are marked as FOVy and FOVz, respectively.  185  FOVy  Capillary Tube  y  z  Flow Fluid  z  FOVz  y  x Di  Coordinates of the sample  Cross-sectional view  Figure 98. The coordinate system adopted in this work is shown for the sample. The fieldof-view (FOV) is also defined in the cross-sectional view of the sample. For basic 2D static imaging, the spin echo (SE) pulse sequence described in Chapter 1.4 is chosen for this work as it removes errors due to magnet in-homogeneities. A sample SE pulse sequence is shown in Figure 99 including the amplitude and duration of the each gradient. The echo signal, occurring at time TE, is recorded for imaging purposes. The FID signal immediately after the 90° is not used. To image the cross-section of the tube, a slice along the x-direction is needed. This is accomplished by a combination of a selective (Sinc) RF pulse of a an appropriate bandwidth and a slice selection gradient (Gx in this work). The bandwidth of the RF pulse (BWSinc) and the strength of the slice selection gradient (Gslice) define a slice thickness [m]  SliceTH =  2π⋅ BWSinc 2π⋅ BWSinc = . γ ⋅ Gslice γ ⋅ GS,max  (82)  The 180° RF pulse is non-selective but can be replaced by a selective pulse (in conjunction with a slice selection gradient) if desired.  186  TE 180°  90° RF GS,max GSlice = Gx GP,max  tS GPhase = Gz  GF,max  tP GFrequnecy = Gy  tF  MR Signal Tacq  Figure 99. Basic spin echo (SE) pulse used for 2D imaging. The phase encoding gradient is applied between the 90° and 180° RF pulse. The amplitude and duration of the phase encoding gradient are calculated based on the FOV and the spatial resolution in the phase direction. The gradient used for phase encoding is arbitrarily chosen to be in the z-direction in this work. The Gz gradient is given by  G z,max = G Phase =  2π⋅ NP , FOVP ⋅ γ ⋅ t P  (83)  where Np is the number of phase encoding steps (typically 128 or 256) and tp is the duration for which the phase encoding gradient is applied. The FOV in the phase direction is the same as FOVz defined in Figure 98. To minimize the echo time, TE, the pulse duration of the phase encoding gradient, tp, is chosen to be as small as possible while respecting the restriction posed  187  by equation (83). The de-phasing pulse of the frequency encoding gradient (Gy) is applied between the 90° and 180° RF pulses for a duration tF/2. tF is the pulse duration of the frequency encoding pulse applied during data acquisition of the magnetic resonance signal. This value is chosen based on the sampling frequency, tSampling, and the number of acquired samples NSamples (typically 128 to 512) given by  (84)  t F = NSampling ⋅ tSampling . The gradient strength of the frequency encoding gradient is  G y,max = G Frequency =  2π . FOVF ⋅ γ ⋅ t Sampling  (85)  where FOVF is the field of view in the frequency encoding direction (y-direction) as defined in Figure 98. Equation (85) can be re-written by substituting equation (84) given as  G y,max = G Frequency =  2π⋅ NSampling FOVF ⋅ γ ⋅ t F  .  (86)  To minimize the echo time, TE, the duration of the de-phasing of the frequency encoding pulse is minimized. The frequency encoding gradient applied after the 180° RF pulse is shifted in time such that the rephrasing of the spins coincides with the echo time. The amplitude of this pulse defines the required bandwidth of the receiver given by  188  BWReceiver = G F ⋅ FOVF ⋅ γ .  (87)  It is preferred to have a low bandwidth as the noise increases with bandwidth. A step by step procedure for calculating all the parameters for an imaging pulse sequence is outlined in the Appendix D. The MR signal is recorded for a duration equal to the duration of the frequency encoding gradient with a sampling frequency tSampling. The raw data is then filtered using digital signal processing techniques (described in more detail in the following sections) and stored in k-space. The MR signal from each repetition of the phase encoding cycle occupies a single horizontal line in k-space as shown in Figure 100. The horizontal lines are offset from the center based on the duration and amplitude of the phase encoding gradient (i.e., the area underneath the phase encoding pulse). Each data point is horizontally spaced by  ∆k P = 1 FOVP =  γ ⋅ G P,max ⋅ t P  (88)  2π⋅ NP  in the phase encoding-direction (vertical) and by  ∆k F = 1 FOVF =  γ G F,max t Sampling 2π  in the frequency-encoding (horizontal) direction.  189  (89)  kP = kPhase kP,max kF,max  kF = kFrequency ∆kP ∆kF  Figure 100. A sample k-space showing the points for which the raw data is stored. The recorded MR signal from each phase encoding step is stored in a horizontal line in k-space that is vertically offset from the center depending on the area of the phase encoding pulse.  4.2 Sample Static Image A sample static 2D image of the circular cross-section of the tube (with an inner diameter of 1.67 mm) is shown in Figure 101. The number of averaging, NEX, is 128. The echo time, TE, is 5 ms with a repetition rate, TR, of 102 ms. The field of view for the phase and frequency encoding directions are 15 mm and 58 mm, respectively. A 128 x 1024 image is created by a 2D FFT. The image is cropped to 128 by 128 pixels. The receiver bandwidth is 500 kHz; thus every pixel represents approximately 500 Hz. The nominal resolution is 60 µm in frequency direction and 117 µm in the phase direction. A white/grey disc shaped image was expected to be seen in in the resulting image, which is clearly not the case in Figure 101. Due to time constrains, it was not determined why the image is not a representation of the sample.  190  Figure 101. Sample static image featuring the (cropped to 128x128 pixels) cross-sectional view of the tube (inner diameter = 1.67 mm). TE/TR = 5/102 ms; NEX = 128; FOVP = 15 mm; FOVF = 58 mm. Further effort was put on implementing the velocity encoded imaging because the primary purpose of this instrument is the visualization of velocity, rather than static imaging.  191  4.3 Pulse Sequence for Velocity Imaging As discussed in Chapter 1.4, the two pulse sequence implementations for flow encoded imaging are time-of-flight (TOF) and phase contrast (PC) techniques. For PC flow encoded imaging, again a spin echo pulse sequence as shown in Figure 102 can be used. The pulse sequence is very similar to the static imaging case. The first notable difference is that the role of the gradients has changed. For example the slice selection gradient is now Gz which excites slice parallel to the fluid flow. The next difference is that the phase-encoding pulse is applied twice with a delay of T and duration tP: once before and once after the 180° RF pulse.  TE 180°  90° RF GS,max GSlice = Gz T  GP,max  tS GPhase = Gx  GF,max  tP GFrequnecy = Gy  tF  MR Signal Tacq  Figure 102. The basic modified spin echo (SE) pulse for phase sensitive velocity encoding. The effect of this type on phase-encoding is that stationary spins do not experience any net phase shift; however, spins traveling with a speed V in the x-direction, experience a net phase shift  192  ∆φP = γG P,max t P ⋅ T ⋅ V .  (90)  The pulse duration tP and the time delay T are chosen based on the maximum expected velocity. To avoid any uncertainties due to phase warps greater than 2π, the parameters are chosen such that the maximum velocity is lower than  V<  2π . γG P,max t P ⋅ T  (91)  The k-space is filled similar to the static imaging case after the raw data is digitally processed. Due to time constraints PC velocity imaging was not implemented. However, a TOF based velocity encoded pulse sequence was implemented for fast flows as a proof-of-concept that the experimental instrument is functional. As described in Chapter 1.4, a one-dimensional plot of the velocity can be made using a SE pulse sequence shown in Figure 13a. A selective 90° RF pulse of with a bandwidth of 6 kHz was applied along with a slice selection gradient (Gx) with an amplitude of 0.06 T/m for a duration of 2 ms to excite a 2.3 mm slice. The rephrasing lobe of the slice selection gradient is applied for 1 ms with an amplitude of -0.06 T/m. The de-phasing pulse of the frequency encoding direction (Gy) can be applied anywhere between the 90° and 180° RF pulses. To minimize TE (5 ms) however, this pulse is applied when the re-phasing of the slice selection gradient starts. The amplitude of this gradient was 0.3 T/m and was applied for a duration of 1 ms. The frequency encoding gradient is applied again for a duration of 2 ms and sample amplitude after the 180° pulse, and is centered at the TE = 5ms. The phase encoding was not used in this experiment as only a 1D plot of bulk velocity in a slice was measured. The resulting echo was sampled during  193  the second pulse of the frequency encoding gradient at 500 kHz. A repetition rate of TR = 102 ms was used. The actual pulse sequence used in the instrument is shown in Figure 13b.  TE 180°  90° RF GS,max GSlice = Gx  (a)  tS GPhase = Gz GF,max GFrequnecy = Gy  tF  MR Signal Tacq  +3 V  PW180 = 20 µs  TE = 5 ms TR =102 ms  2 ms BWSinc = 6 kHz  0.06 T/m  1 ms  2 ms  0.3 T/m  (b) 2 ms  1 ms  Figure 103. Simplified SE for 1D imaging of fast flow (a) and the sequencing diagram used for the instrument (b). The pulse sequence shown in Figure 13 is repeated twice, once when the fluid in the tube is  194  stationary and once when the fluid is flowing at a certain flow rate (set by the syringe pump). The idea is that stationary spins produce and echo signal at time TE because the experience both RF pulses; however, moving spins (if sufficiently fast) do no experience the 180° pulse thus produce a reduced echo signal as they have moved out of the detection area (approximately 3 mm). A difference between the magnitudes of their Fourier transform is an indication of the velocity profile inside the tube. Sample 1D plots that show the flow field inside a tube are shown in the flowing section.  4.4 Sample Velocity Visualization Measurements The normalized reduction of the echo amplitude as a function of the flow rate inside the tube is shown in Figure 104. As expected the echo amplitude is reduced as a function of flow rate for fast flows. The reduction in echo amplitude is observed for flow rates above 100 ml/hr, and it is semi-linear for flows up to 2,300 ml/hr. The echo amplitude further decreases (non-linearly) as a function of flow rate for flows above 2300 ml/hr. The average velocity is related to the flow rate was calculated to be 0.127 (mm/s) / (ml/hr). For example, a flow rate of 393 ml/hr corresponds to an average velocity of 50 mm/s inside the tube. The flow is still laminar for these experiments as the Reynolds number is approximately 85 which is below the critical value of 2300. This is true even for the maximum flow rate of 4000 ml/hr (0.5 m/s) where RE = 840.  195  Flow Rate Inside Tube [ml/hr]  Figure 104. A reduction in the amplitude of the MR echo is observed as the flow rate is increased in the tube. The reduction in echo amplitude is observed for flow rates above 100 ml/hr, and it is semi-linear for flows up to 2,300 ml/hr. The pulse sequence that was described in the previous section is used to extract one-dimensional bulk velocity information inside the 1.67 mm diameter tube, as shown in Figure 105. The flow rate was set to zero and the magnitude of the 1024-point Fourier transform was plotted as shown in the top plot of Figure 105a. The horizontal axis is the frequency where the bandwidth (22 kHz) of the signal is related to the gradient amplitude (0.3 T/m) and the diameter of the tube (1.67 mm). This corresponds to a FOVF = 40 mm and a spatial resolution of approximately 40 µm. The center of the tube is offset from the center of the gradient which results in the offset observed in the spectrum shown in Figure 105. Also, at the zero-frequency a sudden dip in the magnitude is observed with a bandwidth of about 500 Hz which is due to the in-homogeneity of the magnet (20 ppm). After the initial experiment, the flow rate was set to 393 ml/hr (Vmean = 50 mm/s) and the experiment was repeated. The result is shown in the middle plot shown in Figure  196  105a. The difference between the first two plots is an indication of the velocity in the tube. This is shown in the bottom plot in Figure 105a where the vertical axis is in arbitrary units. As expected the maximum signal occurs in the center of the tube while the signal is zero at the walls of the tube. The experiment was repeated for a flow rate of 786 ml/hr (Vmean = 100 mm/s) as shown in Figure 105b. The residual signal shown in the bottom plot of Figure 105b has the largest signal (i.e., velocity), again, in the middle of the tube. The overall amplitude of the residual plot of this case (Vmean = 100 mm/s) is also larger than the one for the previous case (Vmean = 50 mm/s). The signal is zero at the walls of the tube as expected. From laminar flow theory, the velocity profile in expected to be a parabolic with a shaped defined as [5]    r 2  V(r) = Vmax 1 −     R    where the velocity at each point r can be found if the maximum velocity Vmax is known.  197  (92)  Q = 0 ml/hr  Q = 393 ml/hr  Residual = Static - Velocity  (a) Q = 0 ml/hr  Q = 786 ml/hr  Residual = Static - Velocity  (b) Figure 105. The 1D flow imaging experiment for a flow rate of 393 ml/hr (Vmean = 50 mm/s) in (a) and a flow rate of 786 ml/hr (Vmean = 100 mm/s) in (b). The residual plots in the bottom plot of (a) and (b) are an indication of the velocity profile in the tube.  198  The residual signals are presented in one plot and are fitted with a parabolic curve expected from laminar flow, as shown in Figure 106. The measured velocity profiles fit well with the parabolic curves expected by theory.  V2 = 10 cm/s  V1 = 5 cm/s  Di = 1.67 mm Figure 106. The velocity profile inside a 1.67 mm tube at flow rates of 393 ml/hr (Vmean = 50 mm/s) and 786 ml/hr (Vmean = 100 mm/s). The velocity profiles fit well with an expected parabolic profile of laminar flow. As shown in Figure 106, this instrument can be used, at least for relatively fast flows, to visualize flow in one dimension. This result also suggest that the main components of this MRI imaging instrument are functional and can be used to set up more elaborate pulse sequences to accurately measure flow field of slower flows.  199  Chapter 5 - Conclusion and Future Work 5.1 Conclusion In conclusion, the development of the world’s smallest magnetic resonance imaging (MRI) instrument specifically designed for flow imaging is described. Static images can also be generated by this instrument. The hardware of this instrument has been developed for imaging resolutions approximately (30 µm)3 and velocity resolutions of 50 µm/s in all three spatial directions in a 2 mm diameter-spherical –volume (DSV). The work done in this thesis includes: 1. Analysis of requirements for high resolution magnetic resonance static and velocity encoded imaging (Chap 1.4) 2. Design, simulation and characterization of two permanent magnets (Chap 2.1) one of which was used for imaging at 0.6 T with a 20 ppm homogeneity over a 2 mm DSV. 3. Design, simulation and characterization of a triaxial gradient module capable of gradient fields up to 2.8 T/m with rise times below 50 µs, with integrated RF probe (Chap 2.2). A new fabrication process for MRI gradient coils was developed. 4. Design, simulation and characterization of four high-current gradient amplifiers capable of currents up 20 A with a bandwidth greater than 150 kHz at a combined power of 0.8 kW (Chap 2.3) 5. Design and characterization of a quadrature phase-sensitive detector capable of detecting magnetic resonance down to -120 dBm and RF transmitter with pulse shaping capability (Chap 2.4).  200  6. Implementation of the synchronization and timing of the overall magnetic resonance system (Chap 3). 7. Due to time constraints only a time-of-flight velocity encoded images were taken as a proof of concept. Visualization of flow inside a 1.67 mm diameter tube with average velocities of 50 mm/s and 100 mm/s were demonstrated in one-dimensional plots with a spatial resolution of approximately 40 µm. The images fit well with the parabolic curve expected from theory for laminar flows.  5.2 Future Work The micro MRI for flow visualization has been built, the functionality of which was demonstrated. However, there are several improvements that can be made to this instrument for future which are discussed next.  5.2.1 Elimination of Noise in the Experimental Setup The current setup is limited by to major noise sources: (1) the noise from the gradient amplifier and its power supplies which are picked up by the RF amplifier, and (2) noise from ground loop which deteriorates the signals received by the data acquisition system (the oscilloscope). A quantification of this noise was not made due to time constraints; qualitatively however, the voltage of the noise is about 30% of that of the MR signal which was partially reduced by averaging the signal over 128 repetitions. The switching noise of the power supply is directly picked up by the RF electronics as they are very close to each other and are connected to the same power bar. This noise is observed when the power supply is turned ON while its outputs are disabled. The second way that switching  201  noise affects this system is when the noise from the outputs of the power supply travels through the current amplifier to the gradient coils. As the gradient coils are very close the RF probe, the noise is picked up from the RF probe and further amplified by the sensitive detector. The noise that is internally generated by the current amplifier also appears on the outputs which again couples through the gradient coils to the RF probe which again deteriorates the received signal The noise is at 60 Hz and was partially reduced by using common mode powder core filters connected to each output of the current amplifier. Another major source of noise is ground loops in the current setup. It is very difficult to avoid ground loops as almost all RF components are one way or the other connected to ground. The ground of the power supplies (three for the RF electronics and two for the gradient amplifiers) are connected to a single point. The ground connections of the power supplies are also connected to the earth connection of the power bar. The ground of the audio amplifier, which is nonisolated, was connected to the system ground but is also connected to the ground of the power bar (which makes a ground loop by itself). The data acquisition boards are also non-isolated and therefore the grounds of all the connections to the DAQs are interconnected. In future this problem needs to be addressed by carefully interconnecting the components while keeping the ground loops to a minimum. Ideally, the noise observed in the digitized signal on the PC is only the noise from the RF probe and the RF electronics.  5.2.2 Design of Optimized Pulse Sequences In order to achieve the resolutions for which the components were designed for, the pulse sequence needs to be optimized. Due to time constraints only simple pulse sequences were  202  tested. For high resolution microscopy, typically phase encoding in all three spatial dimensions is used. Other elaborate pulse sequences exist for multi-echo acquisition which will speed up the imaging process significantly.  5.2.3 Improvements of the Magnet The magnetic field and homogeneity of the 0.6 T permanent magnet can be improved by increasing the thickness of the iron plates of the efflux return path. According to simulations, the magnetic field will increase to 0.7 T for the same gap because the regions of saturation are reduced. This also has the effect of slightly improving the homogeneity. The homogeneity can be improved by another factor of 2-3 by using 2 mm thick high permeability material on the pole faces, according to simulations. Increasing the thickness of the Iron plates has the effect of reducing the leakage around the magnet which would make the magnet less susceptible to ferromagnetic objects around the magnet. In terms of temperature stability, it is probably best if a NMR lock was used to continuously control the magnetic field as opposed to the current implementation that compensates for the magnetic field every 10-20 seconds during which the magnetic field drifts. If an NMR lock is used, the imaging time would be approximately halved because the B0 compensation software loop can be removed. The magnet also needs to be enclosed in a well insulated (thermal) and shielded (magnetic) area. This significantly reduces the image artifacts due to drifts and disturbances. All these improvements will be necessary to achieve high resolution images. The main limitation for achieving higher resolutions is the magnet. With a higher magnetic field (SNR increases) it is possible to achieve higher resolutions. The magnetic field of the existing magnet can be improved such that saturation effects are minimized (by increasing the thickness  203  of the return path) and the area of the gap (i.e., the area of pole the pole faces) is reduced.  5.2.4 Improvements for Gradient Coils and Amplifiers The existing triaxial gradient module and the gradient amplifiers should be sufficient for imaging resolutions of approximately (30 µm)3. If higher resolutions are desired the gradients coils can be redesigned to have a greater number of turns and/or have reduced dimensions. Reducing the dimensions also has the effect of reducing the volume over which the gradients are linear. Alternatively, the driving current supplied by the gradient amplifiers can be increased by redesigning the gradient amplifiers to allow for larger supply voltages. A potentially problematic issue that was not fully investigated for the existing setup is the eddy currents that are generated in the conductive pole faces of the magnet by the rapidly switching gradient fields. This is especially true of large gradient strengths are used in the pulse sequence. A simple and effective fix for this to insert a few slotted conducting discs between the poles of the magnet and the gradients (discussed in Chapter 2.2). This will limit the eddy current in a smaller region where they are quickly diminished. Active shielding can also be used if additional sets of coils are added to the gradient module to cancel the effect of the gradient fields close to the magnet pole faces.  5.2.5 Improvements of the RF Electronics The main performance limitations of the receiver are the dynamic range (i.e., SNR) and the rather large offsets on the outputs. The problem with large offsets at the output must be addressed as it introduces significant saturation in the RF components that leads to image artefacts. An effective solution is to use phase cycling [88, 89] to eliminate the offsets and other  204  common mode errors which would require a transmitter whose phase can be changed dynamically. The home audio amplifier used as the final stage of amplification is also quite noisy with a limited bandwidth which must to be replaced by a suitable amplifier in near future. The aforementioned problems can be avoided by replacing the existing analog receiver with a digital receiver. In a digital receiver, the signal picked up by the RF probe is first amplified by a low noise amplifier and then directly digitized by a fast analog-to-digital converter. The signal, now being in the digital domain, can be further processed using digital signal processing techniques with no additional loss or degradation of the signal. 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Miller, "A broadband nuclear magnetic resonance spectrometer: Digital phase shifting and flexible pulse programmer," Review of Scientific Instruments, vol. 64, pp. 127-131, 1993. [109] J. C. H. Tso, Private Communication: Practical Information on Impedance Matching of the RF Probe, 2008. [110] O. Cakir, "Copper etching with cupric chloride and regeneration of waste etchant," Journal of Materials Processing Tech., vol. 175, pp. 63-68, 2006.  213  Appendix A – Gradients The fabrication process of the gradient coils is described next.  A.1 Gradient Fabrication Process The designed assembly of the gradient coils and the mask is shown in Figure 107. The mask (positive) is printed on the shiny side of a toner transfer paper (TTS-5 from Pulsar) using a laser printer. The ink of the laser printer temporarily adheres to this special paper and is released when the printed side is pressured on the desired substrate and heated.  Figure 107. The assembly of the designed gradient coils (left) and its mask (right). The fabrication process is schematically illustrated in Figure 108. In Step 1, a 0.01” (254 µm) thick copper sheet is cut (see Figure 109-1), flattened, cleaned using a Scotch-Brite® sponge and later cleaned again (see Figure 109-2) using a solvent (MG Chemicals Cat. # 827B-425G). In Step 2, the shiny side of the toner transfer paper which has the mask printed on it is laid on top of  214  the copper sheet. A completely black mask that is printed on a separate toner transfer paper is laid on the bottom of the copper sheet (see Figure 109-3). The masks are fastened by two strips of masking tape on each side; this is then sandwiched between to ¼” thick Aluminum plates that are used for applying pressure on this structure (see Figure 109-4). To ensure that a uniform pressure is applied on the mask and copper sheet, a few layers (~10) of paper are added (see Figure 108 Step 2) on each side. The two plates have are fastened to each other using five screws and baked for 30-45 minutes at 175 °C. After the backing process is finished, the plates are taken out of the oven (using pliers) and left in air to be cooled. The taped TTS-Copper-TTS is removed in Step 3 and placed in a bowl with warm water for 10 to 20 minutes (see Figure 109-5). The toner transfer paper is carefully pealed off. The mask is now transferred form the TTS paper to the copper sheet (see Figure 109-6). The texture of the transferred mask is relatively porous and will peal off if the etching is performed with no other intermediate step. To ensure that the mask adheres to the copper sheet, a green film (called Green TRF) is applied to both sides (see Figure 109-7) of the copper sheet using pressure and heat in Step 4 (same as Step 2). The only exception is that the film can be pealed off with no wetting step involved. The green TRF adheres to the mask only and it is repeatedly (usually 2 to 3 times) applied until the whole mask is covered (see Figure 109-9). In Step 5, the copper is etched in an etching solution typically containing 200 mL HCl, 100 mL H2O2 and 200 mL H20. The etching solution is commonly referred to as Cupric Chloride (CuCl2) and has an etching rate of up to 50 µm/min at 50° C and very high dissolved copper capacity. The main chemical reactions are [110]: CuCl2 + Cu → 2CuCl  (93)  CuCl2 + 2HCl + Cu → 2CuCl + 2HCl .  (94)  215  The etching process also produces bubbles which ensure uniform and rapid etching. For easy handling of the sample a tape is attached to its back side. After the etching process is complete (see Figure 109-10), the sample is flushed with water for several minutes to clean it from any residual etchant (Step 6). To remove the mask in Step 7, either a degreaser or Acetone can be used (see Figure 109-11).  Step 1: Clean and flatten Copper Sheet. Heat  Pressure  Step 2: Apply Mask (printed on TTS using Laser printer). Bake for 30 - 45 min at 175° while pressed by aluminum plates.  Step 3: Wet paper for 10 - 20 min and peel off TTS. Heat  Cu TTS Paper Paper Aluminum Plate  Pressure  Step 4: Apply Green TRF and bake for 2 minutes at 175 while pressed by aluminum plates (same as Step 2) X2  Printer Ink Green TRF Tape Parylene Polypropylene Plate  Step 5: Etch in HCl (200 mL)+ H2O2 (100 mL) + H2O (200 mL), takes 10-30 min.  Epoxy  Step 6: Soak in water to remove any residual etching solution. Step 7: Clean (wet and wipe) using Acetone and degreaser Step 8: Apply 15 µm Parylene Coat X2  Step 9: Assembles coils and putty in epoxy to get final product  Figure 108. Fabrication process of the gradient coils is represented schematically from a cross-sectional point of view. For more information refer to the text. In Step 8, each coil is coated with a 15 µm layer of Parylene for electrical insulation purposes  216  (see Figure 109-12 and Figure 109-13). After that, the coils are assembled and spaced using strips of cut transparency slides and another 15 µm Parylene coating is applied (see Figure 10914).  Figure 109. Photos of the fabrication steps. For more info refer to the text. 217  After the insulation is tested using a multimeter, all the coils are assembled using the four alignment pins and glued in place using super glue (see Figure 109-16). Unfortunately, the super glue reacts with the Parylene coating and the Polypropylene insert that produce a white coloured dust. The coils remain intact, however. In future other fast drying adhesives need to be explored to fix this (aesthetics) problem. In the final step (Step 8) the coils are puttied in a clear epoxy. Teflon strips are used to restrict the epoxy on each side (see Figure 109-17). The finished triaxial gradient insert is shown in Figure 109-18.  218  Appendix B – Gradient Amplifiers B.1 Circuit Schematics of the Gradient Amplifiers  219  220  221  222  223  224  225  226  B.2 Printed Circuit Board (PCB) Layout of the Gradient Amplifiers The driver and the current controller are implemented in a four-layered PCB that was fabricated by an outside vendor (Sierra Proto Express) and its components are soldered in-house. Each PCB contains the electronics for two of the amplifiers, power for cooling fans, and LEDs which can be connected to thermostats to indicate temperature spikes in the power stage (see Figure 110Figure). Each amplifier is powered separately such that in the event of a major failure of one amplifier, the rest of the amplifiers can be salvaged.  227  FAN POWER -18V  Offset  +18V  Gain  OP37 Input  Driver  P–I Controller  OPA453  Current Monitor  LEDs  Amp B Amp A  LEDs  Figure 110. Final CAD drawing of the four-layered PCB design of two of the gradient amplifiers.  228  All 4 Layers  229  Top Layer  230  Bottom Layer  231  Power Plane  +36V Line +36V Fuse Protected  -36V Line +36V Line  -36V Line  -36V +36V -36V Fuse Protected Fuse Protected Fuse Protected  +18V  232  +18V  -18V  -18V  Ground Plane  233  Analog  Analog  Digital  PCB Construction Summary  Board Construction Summary  234  B.3 Gradient Amplifier Bill of Material (BOM)  ALLIED ORDER  DIGIKEY ORDER  235  236  237  Appendix C - RF C.1 Impedance Matching Setup The electronics schematic for the RF probe is shown in Figure 111. The probe is modeled as an inductor (Lcoil) with a given quality factor, a tuning capacitor (Ctune) and a matching capacitor (Cmatch).  Lcoil  To Transmitter and Receiver Cmatch Ctune  Figure 111. Lumped model of the RF probe. The resonant frequency ω is tuned (to 25.2 MHz) when the resonance condition (95)  ω2 LcoilCtune = 1 , is satisfied and the input impedance (50 Ohms) is matched when the matching condition,  QωLC2tune = 50 , Ctune + Cmatch  (96)  is satisfied [90, 99]. The quality factor of the coil can be estimated by approximation values found in [88]. An approximate value for the capacitors can be determined using (95) and (96). The matching capacitor is usually larger than calculated due to the parasitic capacitances of the  238  probe. To accurately tune and match the probe the test setup shown in Figure 108a is used. The function generator (Agilent 33250A) is used to generate a swept sine wave from 10 MHz to 40 MHz at 2 Vpp with a sweep rate of 0.2 seconds. The frequency marker (falling edge of Sync pulse) is set to the frequency (25.2 MHz) where to probe is tuned and matched. The Sync pulse is connected to CH1 of the oscilloscope (Agilent MSO6034A). The output of the function generator is connected to “RF Output” port of the directional couples (i.e., the “Magic T”). The “Coupled” is connected to CH2 of the oscilloscope and the “RF Input” port is connected to the probe as shown in Figure 108a. This setup measures the signal reflected by the probe. At the resonance frequency of the probe, most of the signal does not reflect from the probe provided that the probe is matched as well. The time axis of the oscilloscope is effectively related to frequency as shown in Figure 108b. The envelope of the signal seen on CH2 of the oscilloscope is an indication of the reflected signal from the probe at the range of selected frequencies. The tuning and matching capacitors are adjusted until the reflected signal form the probe is minimized at the Larmor frequency. The expected envelope of the reflected signal from a tuned and matched probe is shown in Figure 108b.  239  Function Generator (Agilent 33250A)  Sync  CH 1  Swept Sine  CH 2  Oscilloscope (Agilent MSO6034A)  “Magic T” (Mini-Circuits ZDC-10-1) Lcoil  (a)  Cmatch Ctune  Start Frequency Amplitude  Stop Frequency  Resonance Frequency  CH 1  (b) CH 2  Time Frequency  Envelope of Reflected Signal  Figure 112. Test setup to measure reflections of the probe (a) and the resulting signals on the oscilloscope (b).  C.2 Selection of Gains in the Receiver A simplified version of the components of the receiver are shown in Figure 113a.The signal path of the MR signal is shown in Figure 113b as it passes though various components. The signal 240  power of each stare is represented by the letter P and the gains and attenuators are represented by the letter G.  Splitter  LO  MR Signal  (a)  I  DAQ  RF  IF Low-pass Filter  IFA  (b)  IFA  Band-pass Filter  LNA  Mixer  Pout,LNA-2 GAtten-2, Splitter  Attenuator  PMRSignal  Pout, LNA-1  Pin-max, LNA-2 GFilter-1, Atten-1  GLNA-2  GLNA-1  Pout, Mixer Pin, Mixer  GConv-Loss, Mixer Pin, IF  PLO, Mixer  GLO-IF, Mixer  Pout, I-Q  Pout, IF GIF  GFilter-2  DAQ  Pnon-linearity, Mixer  Figure 113. The components of the receiver are shown in (a), and the signal path of the MR signal is shown in (b).  The MR signal passes through the first LNA, the band pass filter, a second LNA, an attenuator and splitter, the down conversion mixer, the audio amplifier and filter, and finally the data acquisition board (DAQ). The dominant noise source of this particular receiver come from the noise picked up by the probe which is further amplified by th first LNA depending on its noise  241  figure. A signal distortion is also introduced at the mixer which comes from the leakage of the signal from the LO to IF ports of the mixer. The leaked signal of LO is at Larmor frequency and when mixed with the MR signal (also at Larmor frequency) results in a DC offset which could saturate the mixer and resulting in non-linearities. The proper selection of gains ensures that the noise level stays at the optimal value and no component of the receiver is overloaded and/or saturated. The combination is of gains and attenuator has to satisfy the following equations:  242  Pout,LNA−1 = PMRSignal + G LNA−1 < P1−dB,LNA−1  (97)  Pin,LNA−2 = Pout,LNA−1 − G Filter −1,Atten −1 < Pin −max,LNA−2  (98)  Pout,LNA−2 = Pin,LNA−2 + G LNA−2 < P1−dB,LNA−2  (99)  Pin,Mixer = Pout,LNA−2 − G Filter −2,Atten −2 < Pin −max,Mixer  (100)  Pout,Mixer = Pin,Mixer − GConv−Loss,Mixer  (101)  Pnon −linearity,Mixer = PLO,Mixer − G LO−IF,Mixer  (102)  Pnon −linearity,Mixer < Pout,Mixer  (103)  Pout,Mixer < Pin −max,IF  (104)  Pout,IF = Pin,IF + G IF < P1−dB,IF  (105)  Pout,I−Q = Pout,IF − G Filter −3  (106)  Pin −min,DAQ < Pout,I−Q < Pin −max,DAQ  (107)  G Receiver = PMRSignal + G LNA−1 − G Filter −1,Atten −1 + G LNA−2  (108)  − G Filter −2,Atten −2 − GConv−Loss,Mixer + G IF − G Filter −3 (Pin −min,DAQ + PNoise,Receiver ) < G receiver < (Pin −max,DAQ + PNoise,Receiver − DR Receiver )  (109)  DR Receiver = 1 − (PConv−Loss,Mixer − PLO−IF,Mixer )  (110)  The constants of these equations can be found from the data sheets which have been summarized below in Table 13.  243  Table 16. Specification of the RF receiver. GLNA-1 [dB]  51  P1-dB, LNA-1 [dBm]  17  GLNA-2 [dB]  42  Pin-max, LNA-2 [dBm]  13  P1-dB, LNA-2 [dBm]  14  Pin-max, Mixer [dBm]  1  GConv-Loss, Mixer [dB]  7.5  GLO-IF, Mixer [dB]  50  P1-dB, IF [dBm]  30  Pin-min,DAQ - Pin-max,DAQ [dBm]  (-58) – (30)  Each component has to be checked for maximum input and output saturation which are given by (97)-(107). The total gain of the receiver can be calculated by (108) the information given in Table 13. The maximum gain has to be selected such that it satisfies (109), where the noise of the receiver is measured experimentally. Equation (109) ensures that the output swing of the receiver is match to the input voltage range of the DAQ. The dynamic range can be calculated as given by (110) and is limited to around 44 dB.  244  Appendix D - Pulse Sequence Design The procedure to calculate the parameters of the static and velocity imaging pulse sequences is described in this section.  D1. Static Imaging The SE pulse that is used for static imaging is shown in Figure 99. The SE pulse sequence is described in the text in Chapter 1 and 3 therefore not repeated here. However, the calculations of the pulse duration and the gradient amplitudes are detailed here. It is important to note that the  TE 180°  90° RF GS,max GSlice = Gx GP,max  tS GPhase = Gz  GF,max  tP GFrequnecy = Gy  tF  MR Signal Tacq  Figure 114. Basic spin echo (SE) pulse used for 2D imaging. calculations presented here are valid for the pulse sequence shown in Figure 99 only and would  245  change if a different pulse sequence was used. The bandwidth of the selective RF pulse (BWSinc) and the strength of the slice selection gradient (Gslice) define a slice thickness [m]  SliceTH =  2π⋅ BWSinc 2π⋅ BWSinc = . γ ⋅ Gslice γ ⋅ GS,max  (111)  To make the echo time as small as possible, the duration of the selective RF pulse is increased. This is achieved by increasing the bandwidth of the receiver (total length of the selective pulse is approximately 4/BWSinc). The duration of the slice selection gradient, tS, is equal to the duration of the selective pulse (approximately 4/BWSinc). The rephrasing lobe of the slice selection gradient is tS/2 if an equal but opposite (-GS,max) is used. To reduce problem with eddy currents, the maximum gradient strength of the slice selection gradient is limited to about 0.3 T/m. In this work, the bandwidth of the slice selection was 6 kHz, which defines a 2 mm slice with a gradient amplitude of 0.07 T/m. The bandwidth of the receiver, BWReceiver, is a parameter that needs to be specified by the user. Ideally the bandwidth over a single pixel has to be greater than the errors due to the magnet inhomogeneities and the drifts. The average error of compensation of the magnetic resonance shift is approximately 100 Hz; therefore, for a 128 by 128 pixel image the received bandwidth has to be at least 12.8 kHz. The homogeneity of the magnet in this work produces a bandwidth of at least 350 Hz. This requires that the bandwidth of the receiver is better than 44.8 kHz. This bandwidth is too large for the RF electronics as the bandwidth is limited by the audio amplifier to about 30 kHz. Another limitation is that for larger bandwidth the SNR increases. Therefore, a bandwidth of 15 kHz was chosen for testing purposes.  246  The field of view of the phase encoding  (112)  FOVP = δP NP  Depends on the resolution in the phase encoding direction, δP, and the number of phase encoding steps NP are specified by the user. A FOVP of 5.12 mm was used for testing purposes for 128 phase encoding steps and a nominal resolution of 40 µm. The amplitude of the phase encoding gradient is given by  G z,max = G Phase =  2π⋅ NP , FOVP ⋅ γ ⋅ t P  (113)  where tP is the duration for which this gradient is applied. To minimized TE, the pulse duration,  tP, needs to be minimized. To avoid eddy current problems which would distort the echo signal, the gradient was limited to about 0.3 T/m. For testing purposes the gradient was applied for 4 ms (tP) with a maximum amplitude (GP,max) of 0.148 T/m. The FOV in the frequency encoding direction is given by  FOVF = δF NSampling ,  (114)  where the resolution in the frequency encoding direction, δF, and the number of acquired samples  NSampling are specified by the user. A FOVF of 5.12 mm was used for testing purposes for a 128 samples in the frequency encoding direction and a nominal resolution of 40 µm. The amplitude of the frequency encoding gradient is given by  247  G F,max =  2π⋅ BWReceiver . γ ⋅ FOVF  (115)  The amplitude of the frequency gradients is found to be 0.069 T/m for a receiver bandwidth of 15 kHz and a FOVF of 5.12mm. The pulse duration of the frequency encoding gradient after the 180° RF pulse is equal to the total acquisition time, tF, that is given by  tF =  2π⋅ NSamplig FOVF ⋅ γ ⋅ G y,max  =  NSamplig BWReceiver  .  (116)  For a total of 128 samples and a receiver bandwidth of 15 kHz, the acquisition time (equal to the duration of the frequency encoding gradient after the non-selective RF pulse) is approximately 8.5 ms. The echo time can now be calculated based on the pulse durations found above. The echo time is approximately    t  2 2 + max  TE ≈ 2  , tP , F   .  BW 2   Receiver  BWReceiver   (117)  The first term is the half the duration of the selective RF pulse. The second term specifies the largest time among: the rephrasing time of the slice select gradient, duration of the phase encoding gradient, and half the duration of the frequency encoding gradient. The pulse duration of the non-selective RF pulse is very short (< 50 µs) compared to the duration of the gradients and therefore ignored. The echo time, TE, is set to 10 ms; this value is mainly defined by the duration of the frequency encoding gradient. The final pulse sequence is shown in where all the pulse durations and gradient amplitudes are marked.  248  +3 V  PW180 = 20 µs  TE = 11 ms TR =102 ms FOVY = FOVZ = 5.1 mm  2 ms  BWReceiver = 15 kHz  BWSinc = 6 kHz  128 x 128  2 ms GX,max = 0.056 T/m 2 ms GY,max = 0.069 T/m 8.53 ms  4.27 ms GZ,max = 0.138 T/m 4.27 ms  Figure 115. A sample spin echo (SE) pulse used for 2D imaging in this work.  D2. Velocity Imaging For flow encoded imaging, again a spin echo pulse sequence as shown in Figure 102 can be used. The SE pulse sequence for flow imaging is described in the text in Chapter 1 and Chapter 4 therefore not repeated here.  249  TE 180°  90° RF GS,max GSlice = Gz T  GP,max  tS GPhase = Gx  GF,max  tP GFrequnecy = Gy  tF  MR Signal Tacq  Figure 116. The basic modified spin echo (SE) pulse for phase sensitive velocity encoding. All the parameters calculated for the static imaging case are valid for flow imaging except for the phase encoding direction and also the echo time. The effect of this type on phase-encoding is that stationary spins do not experience any net phase shift whereas spins traveling with a speed V in the x-direction, experience a net phase shift  ∆φP = γG P,max t P ⋅ T ⋅ V .  (118)  The pulse duration tP and the time delay T are chosen based on the maximum expected velocity. To avoid any uncertainties due to phase warps greater than 2π, the parameters are chosen such that the maximum velocity is lower than  250  V<  2π . γG P,max t P ⋅ T  (119)  For testing purposes the maximum velocity was set to 13 mm/s. The maximum phase encoding (velocity encoding) gradient is set to about 0.3 T/m with a duration, tP, of 2 ms and a delay, T, of 3 ms. The nominal velocity step resolution  δv =  V NP  (120)  depends on the number of phase encoding steps Np. For 128 phase-encoding steps the velocity resolution for this case is approximately 100 µm/s. The echo time can now be approximated as  t   T − tP TE ≈ 2  + tP + F  2  2  (121)  The first term is the delay after the non-selective RF pulse needed for the re-phasing lobe of the phase encoding gradient. The second term is the pulse duration of the phase encoding gradient (needs to be applied before the frequency encoding gradient). The last term is half the duration for which the frequency encoding gradient is applied. An echo time, TE, of 13 ms is used for testing purposes.  251  

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