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Implantable Radio Frequency Coil for Imaging the Rat Spinal Cord at 11.1 Tesla

Permanent Link: http://ufdc.ufl.edu/UFE0024149/00001

Material Information

Title: Implantable Radio Frequency Coil for Imaging the Rat Spinal Cord at 11.1 Tesla
Physical Description: 1 online resource (62 p.)
Language: english
Creator: Blaskowski, Gary
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: implanted, inductively, mri, rf, spinal
Biomedical Engineering -- Dissertations, Academic -- UF
Genre: Biomedical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We will create an implantable radio frequency coil that is inductively coupled to an outer saddle surface coil during imaging at 11.1 Tesla. The purpose is to increase the signal to noise ratio when imaging the rat spinal cord with MRI compared to the commonly available quadrature saddle surface coil. The coil will be modeled with an electromagnetic field program using the finite difference time domain method. These results will be used to construct a prototype system that shows proof of concept at 11.1 Tesla.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Gary Blaskowski.
Thesis: Thesis (M.S.)--University of Florida, 2009.
Local: Adviser: Mareci, Thomas H.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024149:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024149/00001

Material Information

Title: Implantable Radio Frequency Coil for Imaging the Rat Spinal Cord at 11.1 Tesla
Physical Description: 1 online resource (62 p.)
Language: english
Creator: Blaskowski, Gary
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: implanted, inductively, mri, rf, spinal
Biomedical Engineering -- Dissertations, Academic -- UF
Genre: Biomedical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We will create an implantable radio frequency coil that is inductively coupled to an outer saddle surface coil during imaging at 11.1 Tesla. The purpose is to increase the signal to noise ratio when imaging the rat spinal cord with MRI compared to the commonly available quadrature saddle surface coil. The coil will be modeled with an electromagnetic field program using the finite difference time domain method. These results will be used to construct a prototype system that shows proof of concept at 11.1 Tesla.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Gary Blaskowski.
Thesis: Thesis (M.S.)--University of Florida, 2009.
Local: Adviser: Mareci, Thomas H.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024149:00001


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IMPLANTABLE RADIO FREQUENCY COIL FOR IMAGING THE RAT SPINAL CORD AT 11.1 TESLA By GARY BLASKOWSKI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2009 1

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2009 Gary Blaskowski 2

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To my mom and dad. 3

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ACKNOWLEDGMENTS I would like to thank all my friends past and present at the McKnight Brain Institute for their help. I especially thank Alan Elfstrom for working with me on many early simulations. I thank David Peterson and Barbara Beck for their assistance in building the phantom and prototype coil. I thank Rick Goldberg for his help with the Remcom software when it seemed like nothing would work right. I thank my committee for their guidance. Finally I thank my family (Luann, Blake, and Zoe) for their love and support throughout this work. 4

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TABLE OF CONTENTS page ACKNOWLEDGMENTS ...............................................................................................................4 LIST OF TABLES ...........................................................................................................................6 ABSTRACT ...................................................................................................................................10 CHAPTER 1 INTRODUCTION......................................................................................................................11 2 MAGNETIC RESONANCE......................................................................................................13 3 IMAGING...................................................................................................................................18 4 FINITE DIFFERENCE TIME DOMAIN METHOD................................................................26 5 COIL DESIGN............................................................................................................................33 6 SIMULATED COILS.................................................................................................................37 Internal Coil............................................................................................................................37 External Coil...........................................................................................................................38 Both Coils...............................................................................................................................38 7 COIL BUILDING.......................................................................................................................51 8 PHANTOM IMAGING..............................................................................................................52 9 CONCLUSIONS.........................................................................................................................58 LIST OF REFERENCES...............................................................................................................60 BIOGRAPHICAL SKETCH.........................................................................................................62 5

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LIST OF TABLES Table page 8-1. Rat abdomen tissue equivalent phantom...............................................................................53 8-2. Comparison of signal to noise ratio for different coil arrangements and scan types............53 6

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LIST OF FIGURES Figure page 2-1. Rotating magnet in an external magnetic field. From Higgins DM, 2003-2008, Basic MRI physics Q&A.............................................................................................................17 3-1. Slice selection gradient applied along with an rf pulse with a matching bandwidth excites only protons in that area. From Higgins DM, 2003-2008, Basic MRI physics Q&A...................................................................................................................................23 3-2. Frequency gradient applied during signal acquisition limits the area of interest in the sample by collecting precession frequencies in a specific bandwidth. From Higgins DM, 2003-2008, Basic MRI physics Q&A.......................................................................23 3-3. Timing diagram showing the rf transmission and reception and the slice and frequency gradients. From Higgins DM, 2003-2008, Basic MRI physics Q&A...............................24 3-4. Timing diagram showing the changes in Figure 3-3 needed because the slice and frequency gradients affect the all ready evolving spins. From Higgins DM, 2003-2008, Basic MRI physics Q&A.........................................................................................24 3-5. Timing diagram showing the addition of the phase gradient. From Higgins DM, 2003-2008, Basic MRI physics Q&A.........................................................................................25 4-1. Yee cell..................................................................................................................................30 4-2. View of the internal coil showing Yee cells in the adaptive mesh region.............................30 4-3. Typical simulation input parameters.....................................................................................31 4-4. Total simulation space showing size of Yee cells and space to border region around the sample................................................................................................................................32 5-1. Simple tuned circuit. From Mareci TH, 2005, University of Florida, Gainesville, FL..........35 5-2. Tuned circuit with matching ability. From Mareci TH, 2005, University of Florida, Gainesville, FL...................................................................................................................35 5-3. Resonant conditions for two inductively coupled coils..........................................................36 6-1. View of internal coil with Teflon coating around the outside...............................................39 6-2. View of internal coil in the rat model being driving by a generic surface coil.....................40 6-3. Simulation result showing the voltage response of a discrete broadband source driving an un-tuned coil over the tuned implanted coil loaded with the rat model........................40 7

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6-4. Frequency response of an un-tuned coil above the implanted tuned coil in the rat model..................................................................................................................................41 6-5. Simulation result showing the voltage response of a discrete broadband source driving an un-tuned coil over the tuned implanted coil loaded in free space.................................41 6-6. Frequency response of an un-tuned coil above the implanted tuned coil in free space........42 6-7. Frequency response of both the loaded and unloaded situations in figures 6-4 and 6-6 on the same graph..............................................................................................................42 6-8. Simulation result showing the voltage response of a discrete broadband source driving tuned outer coil loaded with the rat model.........................................................................43 6-9. Frequency response of tuned coil above the rat model.........................................................43 6-10. Simulation result showing the voltage response of a discrete broadband source driving tuned outer coil in free space.............................................................................................44 6-11. Frequency response of tuned coil in free space...................................................................44 6-12. Frequency response of both the loaded and unloaded situations in Figures 6-9 and 6-11 on the same graph. Notice how the simulation breaks down at 1.7 GHz due to the shape of the input waveform..............................................................................................45 6-13. View of both coils................................................................................................................45 6-14. Simulation result showing the voltage response of a discrete broadband source driving tuned outer coil over the tuned inner coil loaded with the rat model................................46 6-15. Frequency response of both coils with the rat model..........................................................46 6-16. Zoom of Figure 6-15............................................................................................................47 6-17. Voltage vs. time graph for both coils in air.........................................................................47 6-18. Impedence vs. frequency graph for both coils in air...........................................................48 6-19. Figure 6-15 and 6-18 plotted on the same graph.................................................................48 6-20. Zoom of Figure 6-19............................................................................................................49 6-21. Magnitude of the magnetic field vector when driven at 460 MHz shown through the mid-sagittal plane...............................................................................................................49 6-22. Magnitude of the magnetic field vector when driven at 920 MHz shown through the mid sagittal plane...............................................................................................................50 8

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7-1. Internal coil built on 3 cc syringe container and placed in wax stand-offs. Syringe is filled with rat phantom in Table 7-1..................................................................................51 8-1. Implantable coil inside its phantom body..............................................................................54 8-2. Sagittal image taken of rat spinal cord phantom with FLASH technique.............................54 8-3. Sagittal image of rat spinal cord phantom with RARE technique.........................................55 8-4. Sagittal image of rat spinal cord phantom in receive only mode. FLASH sequence............55 8-5. Sagittal slice in receive only mode, MSME sequence...........................................................56 8-6. Image acquired with a quadrature surface coil, FLASH sequence........................................56 8-7. Image acquired with a quadrature surface coil, RARE sequence. Transmission power was increased to penetrate into the second chamber.........................................................57 9

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Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Master of Science IMPLANTABLE RADIO FREQUENCY COIL FOR IMAGING THE RAT SPINAL CORD AT 11.1 TESLA By Gary Blaskowski May 2009 Chair: Thomas Mareci Major: Biomedical Engineering We have created an implantable radio frequency coil that is inductively coupled to an outer saddle surface coil during imaging at 11.1 Tesla. The system has increased signal to noise ratio when imaging the rat spinal cord with MRI compared to the commonly available quadrature saddle surface coil. The coil has been modeled with an electromagnetic field program using the finite difference time domain method. These results have been used to construct a prototype system that shows proof of concept at 11.1 Tesla. 10

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CHAPTER 1 INTRODUCTION Magnetic resonance imaging (MRI) offers excellent soft tissue contrast and is a promising modality for following pathological changes in rat spinal cord. Rat spinal cord is very small and requires high signal-to-noise ratio (SNR) for high-resolution imaging. Diffusion tensor imaging (DTI) is an applied MRI technique that can be used to evaluate white matter fiber integrity and to tract the fibers in the rat spinal cord. High resolution DTI requires high SNR from the radio frequency (RF) coils that are used to receive RF signals in the MRI experiments. For high SNR, implantable coils have been used for rodent spinal cord imaging at field strengths up to 9.4 T. 1 Phased-array coils have also been used. 2,3 Phased-array coils are a non-invasive way to increase SNR but an implantable coil can still be greater. Motivated by designing an optimized implantable coil for DTI our study explores the precise tuning of the implanted coil at 470 MHz under loading conditions. Our aim was to create in implantable coil that is inductively coupled to an outer saddle surface coil during imaging. Our goal was to increase the SNR versus the commonly available quadrature saddle surface coil. The implanted coil could be inserted during injury phase and used for time course studies of treatment and repair. The coil will be modeled with an electromagnetic field program using the finite difference time domain method. These results will be used to construct a prototype system that can show proof of concept at 473 MHz. Others have noted the problem of imaging the rat spinal cord. The area of interest is small and situated below the surface of the body. This leads to poor signal to noise ratios using regular imaging techniques. 4 Chronically implanted coils, (non-inductive) have been suggested but dismissed because of the problem of infection related to leads breaking the skin. The use of an inductively-coupled implanted coil has been shown as a way to improve imaging in this 11

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area. 1,5,6,7 Silver et al showed the increased SNR with an implanted, inductively-coupled system at 4.7T. 8 Bilgen discussed an implantable coil with a volume excitation that showed increased SNR. 9 What has been lacking has been a systematic way to approach tuning the implanted coil. What is considered here is a complete simulation of the implanted coil conditions. This will aid in building a coil more suited to tune and match with an external coil than those not simulated in this manner. 12

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CHAPTER 2 MAGNETIC RESONANCE According to classical physics a magnet placed in a magnetic field feels a torque to align it with the field (Equation 2-1). B (2-1) Where is torque, is magnetic moment, and B is magnetic field strength. Also, rotating objects conserve their angular momentum unless acted upon by an external torque (Equation 2-2). dtJd/ (2-2) prJ (2-3) Whereis angular momentum, J r is position vector, and p is momentum vector, mass times velocity. Thus a rotating magnet in an external field feels a torque to align it with the field. That torque causes a change in angular momentum which creates precession of the rotating magnet. This is similar to the motion of a gyroscope under the force of a gravitational torque (Figure 2-1). Quantum mechanically, elementary particles like protons and electrons have an intrinsic property that gives them angular momentum. This property is called nuclear spin, and is quantified by the spin quantum number. Protons have a quantum spin number of one half. The nucleus of a hydrogen atom is a single proton. Particles with charge and angular momentum create a situation where the particle interacts with a magnetic field. This interaction is described by the particles magnetic moment. The magnetic moment of a proton is proportional to its angular momentum (Equation 1-4). 13

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J (2-4) The proportionality constant is called the gyromagnetic ratio 11 and is 2.67 10 8 radians per second per Tesla for protons. Putting equations 2-1, 2-2, and 2-4 together we get the equation of motion (Equation 2-5). )(/Bdtd (2-5) The solution of this equation is known to be a circle as the magnetic moment precesses about the magnetic field vector. If we assign the direction of the magnetic field vector to be along the z axis, the cross product can be simplified (Equation 2-6) Bdtdyx / Bdtdxy / 0/ dtdz (2-6) The frequency of precession is proportional to the external magnetic field and gives equation (2-7), the Larmor equation. 2/Bf (2-7) Where f is frequency and B is magnetic field strength. A large ensemble of hydrogen atoms in an external magnetic field produces a measurable interaction which acts like the classical rotating magnet in an external field. The ensemble average of the magnetic property is called the net magnetization vector, ( M ). Though this is a quantum property it does follow the Bohr Correspondence Principle of large numbers and distances corresponding to a classical description. If there are no interactions between the protons Equation (2-7) becomes, 0/dtdMz zttBMdtMd / (2-8) where z is the direction of the external magnetic field and t is transverse direction to external field. 14

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Placing a macroscopic quantity of water in an external magnetic field of 11.1 Tesla's will cause precession of the net magnetization vector around the direction of the magnetic field at a rate of 473 MHz. This is in the radio frequency range. The time that it takes for a system of magnetic moments to come to equilibrium after a change in external magnetic field is exponential in nature. The time constant of this mechanism is called spin-lattice or T1 relaxation. For an ensemble of precessing magnetic moments, )(/1/01zzMMTdtdM (2-9) where M 0 is the equilibrium value of the M z vector. The solution of equation (2-9) is, 0))(/1()0,()(/),(11MrTterzMrTtetrzM (2-10) Also, an ensemble of freely precessing protons, once put into synchronization with one another can become out of sync with each other due to interactions between the protons. This interaction occurs simultaneously with the previous one, and will cause a shrinking of the net magnetization vector. This process is also exponential in nature and its time constant is called T2 relaxation. Inhomogenaities in the magnetic environment of the individual magnetic moments may increase the loss of synchronicity decreasing the coherence. This overall decay is T2* decay. )0,()(2/),(*rtMtierTtetrtM (2-11) Where z is direction of the static field, r is position vector, t is time and is angular frequency of precession. These equations of motion for the net magnetization vector are called the Bloch equations. 15

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0M is the initial net magnetization vector and has to do with the thermal equilibrium of the proton spins. Using an argument based on the energy states and Boltzmann probability distribution, Haacke et al produces an equation for as: 0M 0220)4/(BkTM (2-12) 12 where is the density of proton spins, and is Boltzmanns constant divided by 2. The theory of superposition applies, and adding a second magnetic field perpendicular to the first will also cause precession. If this magnetic field rotates at the Larmor frequency, it will appear as a static field in the rotating reference frame of the precessing protons. By varying the strength and time that this second magnetic field is applied, the angle through which the net magnetization vector moves can be controlled. When the second magnetic field is turned off, the net magnetization vector will again begin to precess about the single static field. Practically, a magnetic field can be made out of a single loop of wire with a current running through it. A time varying field can be made by varying the current in time. By using a sinusoidally varying current, one can produce two counter rotating magnetic fields rotating at the frequency of the current. This is one way to produce the second magnetic field that can tip the direction of the precessing protons. Since this magnetic field varies at the radio frequency band it is called a radio frequency, (rf) pulse. Faradays law states that this magnetization can be detected through the induction of a voltage, or electromotive force (EMF), in a loop of wire perpendicular to the main magnetic field (Equation 2-13). 16

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dtdEMF/ (2-13) Where magnetic flux, is defined as, coilSdB (2-14) This loop of wire can be the same one that was used to tip the spins earlier. This type of EMF is called a free induction decay (FID). Haacke again shows that with our Block equations of motion and Faradays law we can get an equation for the signal as a function of time (Equation 2-15). ))()(sin()()0,(00)(/30*2rrtrBrMredEMFBttrTt (2-15) 12 Figure 2-1. Rotating magnet in an external magnetic field. From Higgins DM, 2003-2008, Basic MRI physics Q&A 17

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CHAPTER 3 IMAGING Using magnetic resonance to create an image one needs spatially specific information. The way this is done is with another magnetic field applied on top of the large magnetic field needed for precession. This magnetic field is much smaller in magnitude and varies linearly in space along the direction of the large field. This field is called a gradient field. The gradient field, or simply gradient, makes a slightly different magnetic field for protons along the direction of the gradient. It is linear across the sample, either adding a little more, or a little less, to the main magnetic field. The protons in the sample will then precess at a slightly different frequency depending upon where they are. First lets consider a gradient field applied during an exciting rf pulse which adds energy to the system (Figure 3-1). The gradient and the rf pulse bandwidth can be combined to excite only a small section of the sample. This gradient direction specifying where in the sample we are exciting is called the slice direction, and its gradient is called the slice gradient. Similarly, if one acquires an FID while a gradient is applied it will be made up of multiple frequencies occurring at the same time (Figure 3-2). A Fourier transform can take a signal of multiple frequencies and break it down into amplitudes of frequencies of which the original signal contained. This is exactly what will be done to convert the FID into intensities of an image. This direction will be called the frequency or read out direction, and its gradient the frequency gradient. A timing diagram is a representation of the gradients and the rf transmission and reception that aids in the understanding of the MRI image formation. Figure 3-3 shows a timing diagram for the slice and frequency selection gradients. An unavoidable result of applying a gradient is that it will dephase the spinning protons that we want to acquire for our image. One way to remedy this problem is to plan for the dephasing 18

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during acquisition and pre-dephase the spins. Figure 3-4 shows these dephasing lobes in the slice and frequency gradients. But to make an image we need two dimensions. It would be nice to apply this technique again, but when two magnetic fields are applied at the same time, the theory of superposition will give a net gradient magnetization that is the combination of the two, and not give a second direction but a single direction between the two. To remedy this problem, lets look at what is acquired during the FID. It is a frequency encoded signal that has evolved in time. The same thing can be accomplished by applying a gradient field not during rf excitation or acquisition of the FID, but some time in between. By keeping track of the strength of the gradient and the time that the spins evolve during it, the effect of the gradient as a phase offset of the FID as compared to when no extra gradient was applied can be deduced. By repeating this experiment multiple times with varying gradient strengths each time the same result as the simple frequency encoded experiment can be achieved. The direction of this gradient field is called the phase encode direction and the gradient the phase encode gradient (Figure 3-5). In a typical imaging experiment, gradients are applied during RF excitation which will only excite protons along a small slice of the sample. This is the slice selection gradient. If the FID is acquired during a time when the gradients are on, this will have the frequency encoding of the gradients and is the readout gradient. And somewhere in between the spins are allowed to precess for a time under a third gradient field, this is the phase gradient. The experiment is repeated multiple times so multiple phase encoding gradient strengths can used which creates a complete set of points to Fourier transform. Acquiring an MR image requires these three orthogonal gradient directions. 19

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The acquisition of an MR image requires the transmitting and receiving of energy from the sample in a very precise way. The transmission and reception of energy is done through a tuned resonant circuit. Chapter 5 will discuss the details of circuit design. Electromagnetic energy is transferred to the sample at the resonant frequency of the precessing protons. As mentioned above, if this magnetic field rotates at the Larmor frequency, it will appear as a static field in the rotating reference frame of the precessing protons. By varying the strength and time that this second magnetic field is applied, the angle through which the net magnetization vector moves can be controlled. This is called the tip angle. It is common in many MR experiments to tip the net magnetization vector by 90 or an angle of 2/ radians. It is important to remember that one can only detect the transverse component of the net magnetization vector. A 90 pulse will move the magnetization into the transverse plane. The simplest MR experiment is a 90 pulse followed by an immediate acquisition of the signal. This by no means gives an image, but is an illustrative example of energy transfer. There will be no way to decipher any spatial information from this signal. All protons in the sample will be essentially the same. This experiment is limited by the speed at which the system electronics can switch from a transmitting mode to a receiving mode. It also is limited by the amount of signal one can acquire as seen by the following discussion. It is possible to create a situation where spins that have dephased over time due to T 2 effects can be brought back into coherence. The first method is called a gradient echo. By applying a gradient field in the slice direction, after the original 90 pulse has been given, the spins in the system will progress with differing frequencies as long as the gradient is applied. By exactly reversing the strength and direction of the gradient field for the same amount of time the spins will be brought back into phase. It is similar to runners on a track, each running at a steady 20

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rate, but each running at a different speed. The start of the race is analogous to when the first gradient field is applied. The runners spread out from the starting line. When the gradient is reversed, the runners are told to turn around in their lanes and run back to the starting line. If they do so at the same speed that they went out at, they will again be in a line when they return to the start/finish line. Another way to produce the same result is by a spin-echo. In a spin-echo, the original 90 pulse is followed at some later time by a 180 pulse. Similar to a 90 pulse, one can continue to add energy to the spin system pushing the net magnetization vector through 180 or radians. The 180 pulse flips the spins in the transverse plane giving any spins that had acquired a dephasing angle relative to the net magnetization vector the exact opposite sign. For example, if one spin had gotten a little behind the net magnetization vector by 2, the 180 pulse would flip it across the horizontal plane so that it then is ahead of the net magnetization vector by 2. Waiting the same amount of time after the 180 pulse as between the 90 pulse and the 180 pulse, the spin system will once again become coherent. Using the race analogy, the runners start their race and run at their own pace from the start of the race, where the 90 pulse is the starting gun. The runners progress at different constant speeds for some time. The effect of the 180 pulse would rearrange the runners on the track to a spot exactly opposite the start/finish line to where they are on the track. Then let them continue until they finish the race. One will see that the slow runners who only made it a little way from the start line will end up the closest to the finish line after the 180 pulse. They will continue running at their slow pace to meet up at the finish line with everyone else. When using an echo pulse sequence one can acquire signal for a longer period of time. The signal will grow to a maximum value as the spins return to coherence and then recede as the 21

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spins dephase again. Many pulse sequences take advantage of echoes by repeating the echo sequence multiple times before reapplying a 90 pulse, thereby decreasing the total imaging time. Echoes remove T 2 effects, but are still limited by T 2 dephasing thereby limiting the number of echoes one can acquire before needing another 90 pulse. The rapid acquisition of refocused echoes, or RARE sequence uses multiple spin echoes to speed up acquisition. The number of echoes per excitation pulse is called the RARE factor and can be determined by the user. Because it is used to speed up the process of phase encoding, it must be a factor of the number of phase encoding steps. The cost of using this sequence over a single echo sequence is the loss of signal that occurs during the successive echoes. Under many situations this is a fair compromise. The fast low angle shot, or FLASH sequence is a gradient echo sequence that uses excitation pulses that are much smaller than 90. Because these excitations are small, one can pulse faster than a full 90 pulse and repeat each phase encoding step before complete relaxation occurs. The cost is again loss of signal which may be acceptable to the imager. 22

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Figure 3-1. Slice selection gradient applied along with an rf pulse with a matching bandwidth excites only protons in that area. From Higgins DM, 2003-2008, Basic MRI physics Q&A. Figure 3-2. Frequency gradient applied during signal acquisition limits the area of interest in the sample by collecting precession frequencies in a specific bandwidth. From Higgins DM, 2003-2008, Basic MRI physics Q&A. 23

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Figure 3-3. Timing diagram showing the rf transmission and reception and the slice and frequency gradients. From Higgins DM, 2003-2008, Basic MRI physics Q&A. Figure 3-4. Timing diagram showing the changes in Figure 3-3 needed because the slice and frequency gradients affect the all ready evolving spins. From Higgins DM, 2003-2008, Basic MRI physics Q&A. 24

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Figure 3-5. Timing diagram showing the addition of the phase gradient. From Higgins DM, 2003-2008, Basic MRI physics Q&A. 25

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CHAPTER 4 FINITE DIFFERENCE TIME DOMAIN METHOD Electromagnetic field calculations can be difficult due the large amount of calculations needed. It has been with the advent of the high speed computer that advances have been made in this field. Early software packages focused on the problem of radio antenna design in free space. Only more recently have developers added the ability to model electric and magnetic fields through user defined space. This is needed for MR coil design because the coil is placed very close to the sample. The loading effects of the sample greatly affect the tuning and matching of the coil. The Remcom XFDTD program, State College PA, allows the user to specify different conductivities and permitivities throughout its calculation space. Maxwells laws describe the propagation of electromagnetic waves as a series of differential equations. By stating these differential equations in a Leibniz form of discrete time and position steps one has the basis for computing small discrete changes in the electromagnetic field from its initial conditions. SVVAD (4-1) SAB0 (4-2) SCAtBlE)/(1 (4-3) AtDJlHScC )/( (4-4) where, D = electric displacement E = electric field strength A = area = charge density V = volume B = magnetic field l = linear element t = time H = magnetic field strength J = current density S = integrand over closed surface = integrand over volume V C = integrand over closed curve 26

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This is the basis of the finite difference time domain method, FDTD 14 The growing power of computers has allowed this brute force method of solving complex field problems by dividing space into small blocks or cells, and calculating the changes in small amounts of time through those cells. By looking at equation 4-3, the electric field depends upon the rate of change in the magnetic field. So to move through space in the electric field one needs to know its previous value and the time rate of change in the magnetic field. 15 The symmetry of Maxwells equations allows for a similar determination of the magnetic field. The basic spatial elements then become a small cube where the electric fields are defined along the edges and the magnetic fields are defined through the faces. This is called a Yee cell 16 (Figure 4-1). It is important to define what it means to be small. A general rule of thumb is that small spatial steps are taken to be 1/10 th the smallest wavelength of interest. 17 For our work the frequency of interest is 475 MHz. In free space this converts to a wavelength of 638 mm. One tenth of the wavelength is the maximum cell size 64 mm. We have chosen a cell size less that this maximum to accommodate the geometry of our simulation. Our cell is 2 mm on a side. As is the case with many FDTD calculations, the geometry of the physical elements has determined that we need a smaller cell size to accurately model our design, (for the curvature of our implanted coil). The XFDTD software allows the creation of an adaptive mesh area where the mesh gets smaller in a limited space to speed up the calculation overall compared to creating a mesh filled with the smaller cells thoughout. We have created a region around the coil that shrinks the cell size to 1/5 of the outer cells which then become 0.4 mm. This is the best use if the adaptive mesh that can be made by the software program (Figure 4-2). 27

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A time step is then taken to be the time for an electromagnetic wave to transverse a cell. For our 2 mm cell this time in free space is 770.3 fs. The FDTD method defined by Yee 16 starts with the initial conditions defined by the user. The calculation proceeds from the initial electric field at a cell and then calculating the magnetic field at the next point in time for the same cell. This leads to the only assumption needed for the calculation. One must take sufficient time steps to assure the fields have dissipated to a low level. 16 Each simulation will then be run with some number of time steps, up to 512000. Figure 4-3 shows typical input parameters. A discrete source is placed along a cell edge in the outer or driving coil. A 1-volt pulse is inserted over 1000 time steps to produce a frequency profile of the pulse of the desired composition. Simulations that are centered at 500 MHz and have a generally consistent input value for all frequencies of interest are modified Gaussians. Solutions in time can be Fourier transformed to show frequency response. More time steps in the simulation will result in higher resolution in the frequency response. The first result for all simulations will be a voltage vs time graph showing that the initial voltage applied to the exciting port has dissipated to less than -30 db of its initial value of 1 Volt and the simulation is most likely valid. One always must look at the time result for this type of convergence, and critically evaluate any of the other results. If they dont make physical sense, then the solution is to run the simulation for more time steps until the results fit physical parameters. Another consideration when making a simulation is boundary conditions. Because the simulation space is limited by computing power it is best to simulate a limited volume. The simulation cell size is 2mm on a side and a total space of 200 x 200 x 300 cells. That creates a simulation volume 40 cm x 60 cm (Figure 3-4). At the border of this limited space, the simulation will need to make some assumptions about what to do with the fields. There are two 28

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boundary conditions that one can use, Liao and Perfectly Matched Layer (PML). The Liao method uses an estimation of what the fields would be outside the simulation area using the values at the border and previous time information. It uses these estimated values to calculate the fields inside the space. 17 The PML boundary is an absorbing layer that dissipates the energy moving through it. 18 One can define how many cells to use for the absorbing layer. The Liao method works well when there are 10 to 15 empty cells at the border. It takes less memory to use a Liao border than a PML border. The PML border can do a perfect job of absorbing the wave energy, but at the cost of simulation time 18 For my simulation I use the Liao border because I have created a space with at least 15 cells between the border and the geometry. 29

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Figure 4-1. Yee cell Figure 4-2. View of the internal coil showing Yee cells in the adaptive mesh region. 30

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Figure 4-3. Typical simulation input parameters 31

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Figure 4-4. Total simulation space showing size of Yee cells and space to border region around the sample. 32

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CHAPTER 5 COIL DESIGN A RF coil for use in magnetic resonance imaging is at its simplest a resonant AC circuit. Inductive elements balance capacitive elements when driven at the resonant frequency creating a surge of absorbed energy. A single loop of wire with current has an inductance equal to the flux of magnetic field produced per unit of current that creates the magnetic field. By placing a capacitor in series with an AC driving source and wire will create a resonance when the inductive reactance balances the capacitive reactance (Figure 5-1). The equations governing the resonance are: LjXL (5-1) )/(1CjXC (5-2) )/(12LC (5-3) where X stands for reactance; j, the square root of negative 1; L, inductance; C, capacitance; and ,) (2yACfrequenc Alone this would be fine, but power absorbed or outputs limited if the system isnt impedance matched. To allow for impedance matching, a slightly more complicated parallel and series capacitance network is necessary (Figure 5-2). 19 The reactance of this network can be simplified to its equivalent network. Using equivalent reactance principles we can get values for the simplified network: ) /(2ssCjX (5-4) )]/1([)/(22CpLRCRRPeq (5-5) ])/1()[/(]/)([2222PPPeqCLRCCLLRjX (5-6) As one can see, the network will resonate when the reactance adds to zero and the equivalent resistance can be manipulated through C P to match 50 ohms. When any object is placed within the electromagnetic field created by the coil it will disturb the field by creating a change in reactance. This effect is commonly called loading of the coil. 33

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Because MR coils are placed to transmit and receive signal from a sample it is important to take this into consideration when building a coil. The XFDTD program allows for this by letting the user define the permeability and conductivity of any Yee cells. When the goal of a resonant circuit is to produce the highest quality resonance, sources of power loss must be avoided. If the circuit in Figure 5-2 were built there would be a source of energy loss at the capacitors through a conservative electric field. Each capacitor creates an alternating electric field that extends into space. When this field interacts with a biological sample the result is heating of the tissue.20 The energy loss is proportional to the strength of the electric field. Though this mode of energy transfer can not be eliminated, it can be minimized by distributing the capacitance throughout the circuit. An added benefit of distributing the capacitance is improvement in homogeneity of the resonant magnetic field. 20 Inductively coupled coils will have a mutual inductance that depends on spatial geometry. This added inductance makes extra terms in the matching and tuning equations, but can be dealt with. 21 These equations can be cumbersome to solve. By creating the computer simulation one can estimate the effect of this mutual inductance through the fields produced. One way to check the simulations is through the resonances produced. When two resonant coils tuned to the same frequency are placed near each other in space, the effects of driving one with an AC current will create an induced current in the other. Faradays law describes how the changing magnetic flux produced by the first coil will induce the current in the second. Because of the mutual inductance term in the resonance equations, there will now be two possibilities for resonance. There will be one resonance occurring where the two currents are in phase with each other and one when they are out of phase with each other (Figure 5-3). In the counter rotating 34

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current mode the magnetic field produced by the coils will be reduced due to the opposing currents of the inner and outer coils. Therefore the co-rotating currents will be used for our coil. Figure 5-1. Simple tuned circuit. From Mareci TH, 2005, University of Florida, Gainesville, FL Figure 5-2. Tuned circuit with matching ability. From Mareci TH, 2005, University of Florida, Gainesville, FL 35

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IIII Figure 5-3. Resonant conditions for two inductively coupled coils. 36

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CHAPTER 6 SIMULATED COILS Internal Coil There are two coils to be modeled; the inner, passive coil and the outer, active coil. The inner coil is a saddle shape constructed of copper tape 0.2 cm wide and 0.1 cm thick (Figure 6-1). The length is 2 cm long and covers an arc length of 1 cm. The radius of the arc is 0.637 cm. The angle subtended is 90 degrees. It has two breaks along its ends 2.2 mm long for tuning capacitors The conductivity of the copper is defined to be 5.8001 x 107 S/m and its relative permeability is 1. Because it will be implanted inside of an electrically conductive rat model, it is coated with a layer of Teflon (Figure 6-2). The Teflon surrounds the coil on all sides with a thickness of 0.8 mm. The conductivity of the Teflon is 1 x 10-19 S/m and a permittivity of 1.94. When the coil is tuned with two 10.9 pf capacitors and driven by a discrete broadband source shown in Figure 6-3 results can be obtained for two conditions; loaded with rat and in free space. First notice the difference between the two voltage vs. time graphs (Figures 6-3 and 6-5). The simulation that was loaded with the rat, shown in Figure 6-3, converges to virtually no signal in 150 ns, whereas the simulation in free space, shown in Figure 6-5, still shows some ringing after 400 ns. Because the simulation is done in free space, there is a longer response in the exciting coil. Looking at the impedance vs frequency graph for the two simulations shows how the results are affected (Figures 6-4 and 6-6). The sample loaded with the rat shows consistently positive resistance and the one in free space dips negative right at the frequency of resonance. Negative resistance is a sign of non-convergence of the simulation. The two resonant peaks occur at 467 MHz and 458 MHz. 37

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External Coil The external coil is similarly simulated both loaded and unloaded. The outer coil has a saddle shape and is constructed of copper tape 0.2 cm wide and 0.1 cm thick. The length is 3 cm long and covers an arc length of 1.5 cm. The radius of the arc is 0.9 cm. The angle subtended is 90 degrees. It has two breaks along its ends 2.2 mm long for the placement of matching capacitors. Similar to the internal coil, when the coil is driven by a discrete broadband source shown in Figure 4-3 results can be obtained for two conditions, both loaded with rat and in free space. What is different is that we will be able to vary the tuning of this coil due to it being accessible outside the rat. Therefore it is constructed with the more complicated series and parallel matching network. It is tuned with two 10.0 pf capacitors in series with the source and a 20 pf and 0.02 pf capacitor parallel to the source. Figures 6-8 through 6-12 show the results of simulation in time and in frequency response. Both Coils The two coils are separated by of 9.3 mm, which includes the 8.3 mm width of standard acrylic which will be used as an animal tray and the 1 mm depth inside the rat where the internal coil will be placed (Figure 6-13). Figure 6-14 shows the voltage response in time. Notice the effect of automatic threshold detection. The simulation was run with the condition of 512,000 time steps, or running until the voltage drop was 30 db. The -30db condition was reached first, and the simulation finished after five days of calculating and zero filled the points past 382500 time steps. Figure 6-15 shows that this condition could still be pushed further because the resistance is still negative. Figure 6-16, which is a zoom of both frequency response curves we see that in air the resonance is higher in frequency by 10 MHz, and of a higher quality factor than the simulation loaded with the rat. From these simulations, the result is that one should create a resonant coil in 38

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air that tunes to a frequency 10 MHz higher, +/2 MHz, than the actual desired resonant frequency to account for the lowering of frequency caused by the load. Figure 6-15 shows two resonant peaks appearing which follows from Chapter 5 on inductively coupled coils. The lower resonance is the co-rotating field resonance and the higher frequency is the counter-rotating resonance. We can see this by viewing the steady state magnitude of the magnetic field vector when the simulation is driven by a single sinusoidal frequency indicated by Figures 6-21 and 6-22. Figure 6-21 shows the fields adding to a more intense region at the implanted coil, and Figure 6-22 shows the fields canceling in the region of the implanted coil. This agrees with the theory of inductively coupled coil resonance stated earlier in Chapter 5. Figure 6-1. View of internal coil with Teflon coating around the outside. 39

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Figure 6-2. View of internal coil in the rat model being driving by a generic surface coil Figure 6-3. Simulation result showing the voltage response of a discrete broadband source driving an un-tuned coil over the tuned implanted coil loaded with the rat model. 40

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Figure 6-4. Frequency response of an un-tuned coil above the implanted tuned coil in the rat model. Figure 6-5. Simulation result showing the voltage response of a discrete broadband source driving an un-tuned coil over the tuned implanted coil loaded in free space. 41

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Figure 6-6. Frequency response of an un-tuned coil above the implanted tuned coil in free space. Figure 6-7. Frequency response of both the loaded and unloaded situations in figures 6-4 and 6-6 on the same graph. 42

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Figure 6-8. Simulation result showing the voltage response of a discrete broadband source driving tuned outer coil loaded with the rat model. Figure 6-9. Frequency response of tuned coil above the rat model. 43

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Figure 6-10. Simulation result showing the voltage response of a discrete broadband source driving tuned outer coil in free space. Figure 6-11. Frequency response of tuned coil in free space. 44

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Figure 6-12. Frequency response of both the loaded and unloaded situations in Figures 6-9 and 6-11 on the same graph. Notice how the simulation breaks down at 1.7 GHz due to the shape of the input waveform. Figure 6-13. View of both coils 45

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Figure 6-14. Simulation result showing the voltage response of a discrete broadband source driving tuned outer coil over the tuned inner coil loaded with the rat model. Figure 6-15. Frequency response of both coils with the rat model. 46

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Figure 6-16. Zoom of Figure 6-15. Figure 6-17. Voltage vs. time graph for both coils in air. 47

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Figure 6-18. Impedence vs. frequency graph for both coils in air. Figure 6-19. Figure 6-15 and 6-18 plotted on the same graph. 48

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Figure 6-20. Zoom of Figure 6-19 Figure 6-21. Magnitude of the magnetic field vector when driven at 460 MHz shown through the mid-sagittal plane. 49

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Figure 6-22. Magnitude of the magnetic field vector when driven at 920 MHz shown through the mid sagittal plane. 50

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CHAPTER 7 COIL BUILDING A coil was constructed similar to the one suggested by the models. The inner coil was built on a 3 cc syringe container, Kendall Monoject 3 cc, Webster Veterinary Supplies, Sterling, MA. The diameter of the tube tapered slightly from .502 to .508 inches, and the coil was centered at .505 inches Copper tape less than 1 mm thick and 4 mm wide, trimmed by hand to 2 mm, by 3M Corp, St. Paul Minnesota, was used to create the coil body and fixed value capacitors of 12 pf and 15 pf +/10%. The outer coil was similarly constructed on a semi-cylindrical Plexiglas animal tray, Small Parts Inc. Miami Lakes, FL. It was tested with the network analyzer to resonate in free space in the region of 480 MHz. Figure 7-1. Internal coil built on 3 cc syringe container and placed in wax stand-offs. Syringe is filled with rat phantom in Table 7-1. 51

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CHAPTER 8 PHANTOM IMAGING A rat phantom was constructed using a tissue-equivalent substance shown by Beck et al. 22 TX151 is a gelling agent, Oil Center Research, Lafayette, LA and Dowicil is an anti-microbial agent, Dow Chemical (Table 8-1). This material was used to fill the inner coil former, a 3 mL syringe case, and an outer plastic shell 0.781 cm in diameter. The inner coil former was held in place by wax endcaps that were constructed for that purpose (Figure 8-1). Scans were taken with a Magnex 11.1 T 40cm bore magnet with Bruker. Avance console, S116 gradients and software. Scans were taken with two common imaging techniques, a fast gradient echo (flash) and a Hahn spin echo sequence. Spin echoes where recorded with a 1 sec TR, 10.25 ms TE, a 3 mm thick slice, and a 4 cm field of view (Figure 8-3). The flash sequence had a 100 ms TR, 3.36 ms TE, and a 4 cm field of view (Figure 8-2). Because the surface coil was used for excitation, it is highly likely that a region of over-tipping exists in the dark region between the inner coil and outer coil. Therefore a small diode network was created and placed parallel to the coils to effectively turn them off during excitation by an external coil. To effectively cancel the new parallel part of the circuit, they were matched with an inductor to resonate at the driving frequency. The scans were repeated with volume excitation and the two surface coils in a receive only mode. One can see the reduction in the band above the implanted coil (Figures 8-4 and 8-5). As a comparison to standard spinal cord imaging, a quadrature surface coil was placed on the phantom and imaged (Figure 8-6). It may be possible to force the quadrature surface coil to perform better in the region of the spinal cord by putting more power into the coil and overtipping an area near the surface. This would lead to non-uniform image intensity and would 52

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not be used in vivo, but allows us to calculate the greatest signal to noise possible with this type of coil in the region of the spinal cord (Figure 8-7). We can compare the SNR for both methods and coils.23,24 We see in Table 8-2 that the SNR is increased by a factor of six for the implanted coil system over the quadrature surface coil for the spin-echo technique. It is worthwhile to note that the SNR is around 50 for the first two techniques using the quadrature surface coil and three times that using the flash technique. Repeating the same techniques with the implanted coil does not show the difference between spin-echo and flash. It does show an increase in SNR across all techniques done with the quadrature coil, but does not show the increase with flash technique over spin-echo. This indicates that there maybe a mistake in calculation of SNR for the flash quadrature coil. Regardless of technique, the improved SNR will allow for more detailed imaging of the rat spinal cord. Table 8-1. Rat abdomen tissue equivalent phantom. Material Distilled water Sucrose Potasium Chloride TX 151 Dowicil Percentage by weight 64 32 1.55 2.3 0.15 Table 8-2. Comparison of signal to noise ratio for different coil arrangements and scan types. Identifier Area[cm2] Mean Std. Dev SNR Quad_MSME_OV_sig 1.3154 306605 131787 Quad_MSME_OV_nois 0.3623 9918 5253 58.4 Quad_MSME_sig 0.3591 35340 6814 Quad_MSME_noise 0.3379 1525 787 44.9 Quad_flash_Sig 0.6064 245448 40517 Quad_flash_noise 0.3613 3571 1884 130. Induc_MSME_sig 0.4971 2031030 935860 Induc_MSME_noise 0.2705 12664 5984 339 Induc_flash_sig 0.6826 2039470 967388 Induc_flash_noise 0.2158 10354 5254 388 53

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Figure 8-1. Implantable coil inside its phantom body. Figure 8-2. Sagittal image taken of rat spinal cord phantom with FLASH technique. 54

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Figure 8-3. Sagittal image of rat spinal cord phantom with RARE technique. Figure 8-4. Sagittal image of rat spinal cord phantom in receive only mode. FLASH sequence. 55

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Figure 8-5. Sagittal slice in receive only mode, MSME sequence. Figure 8-6. Image acquired with a quadrature surface coil, FLASH sequence. 56

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Figure 8-7. Image acquired with a quadrature surface coil, RARE sequence. Transmission power was increased to penetrate into the second chamber. 57

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CHAPTER 9 CONCLUSIONS We have created an implantable coil prototype that is inductively coupled to an outer coil during imaging. It has shown a factor of six increase in signal to noise ratio versus the commonly available quadrature surface coil. The implanted coil could be inserted during the injury phase of an in vivo experiment and left there for time course studies of treatment and repair. The effects of chronic implantation have not been considered here, but could lead to some foreseeable problems. Changes in tissue surrounding the coil may lead to changes in tuning and quality of the implanted coil. Most notably ischemia and scaring could be expected to occur in vivo and the implanted coil could not react to those changes in loading. The implanted coil method as shown here is difficult to implement on a multiple animal study. The mutual inductance of the coils is difficult to control. It depends heavily upon the geometry of the coils both individually and in relation to one another. If this method were used, I would recommend waiting to coat the coil and checking its frequency at the time of implantation. Small changes in capacitance can be accomplished by deforming the coil slightly. For future research I would recommend creating a way to tune the internal coil in vivo. Perhaps tuning could be accomplished with a variable voltage capacitor controlled by a small circuit board. If this method is implemented I still recommend the 10 MHz offset that we have indicated here. The finite difference time domain method has been shown here as an accurate predictor of coil loading. Variability between individual rats has not been considered here. It would be possible to use a range of rat phenotypes from skinny to fat to give a distribution of possible loading conditions. These ranges loading conditions could be quantified to prepare for 58

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implantation. This method of coil simulation is limited by computer speed and storage capacity. As these computer characteristics are improved the method will become even more widely used. MRI is the modality of choice for imaging the primary nervous system at this time. It is possible that in the future another imaging modality may be created that will surpass this method for high resolution imaging in this area. Heating of the tissue could possibly occur using the MRI method leading to necrosis and other related problems. Care must be taken by the experimenter not to use too much transmitted power. 59

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LIST OF REFERENCES 1. Bilgen M. Magnetic Resonance Microscopy of Spinal Cord Injury in Mouse Using a Miniaturized Implantable RF Coil. J. Neuro Methods 2007; 159:93-97. 2. Mogatadakala KV, Bankson JA, Narayana PA. Three-element phased-array coil for imaging of rat spinal cord at 7T. Magn Reson Med. 2008; 60:1498-505. 3. Yung A, Kozlowski P. Signal-to-noise ratio comparison of phased-array vs. implantable coil for rat spinal cord MRI. Magn Reson Imaging 2007; 25:1215-1221 4. Bilgen M, Abbe R, Narayana P. Dynamic contrast-enhanced MRI of experimental spinal cord injury: In vivo serial studies. Magn Reson Med 2001; 45:4 614-622. 5. Flack FC, James E, Schlapp D. Mutual inductance of air-cored coils: effect on design of radio-frequency coupled implants. Med & biol Engng 1971; 9:79-85. 6. Schnall M, Barlow C, Subramanian V, Leigh L Jr. Wireless implanted magnetic resonance probes for in vivo NMR. J Magn Reson 1986; 68:161-167. 7. Hoult DI, Tomanek B. Use of mutually inductive coupling in probe design. Concepts in Magnetic Resonance 2002; 15:4 262-285. 8. Silver X, Xu W, Mercer EV, Beck BL, Bossart EL, Inglis B, Mareci TH. In vivo 1H magnetic resonance imaging and spectroscopy of the rat spinal cord using an inductively-coupled chronically implanted RF coil. Magn Reson Med 2001; 46:1216-1222. 9. Bilgen M. Inductively-overcoupled coil design for high resolution magnetic resonance imaging. Biomed Eng Online 2006; 5:3 doi:10.1186/1475-925X-5-3 10. Hwang F, Hoult DI. Automatic probe tuning and matching. Magn Reson Med 1998; 39:214. 11. Higgins DM, 2003-2008, What is the Larmor equation? Basic MRI physics Q&A, reviseMRI, Reigate, UK. http://www.revisemri.com/questions/basicphysics/larmor_eqn 12. Haacke EM, Brown RW, Thompson MR, Venkatesan R. Magnetic Resonance Imaging Physical Principals and Sequence Design. New York: Wiley and Sons; 1999. 13. Higgins DM, 2003-2008, Creating a MRI image Q&A, Basic MRI physics Q&A, reviseMRI, Reigate, UK. http://www.revisemri.com/questions/creating_an_image 14. Yee KS. Numerical Solution of Initial Boundary Value Problems Involving Maxwells Equations In Isotropic Media. IEEE AP-S Transactions; May 1966. 15. Kunz KS, Luebbers RJ. The finite difference time domain method for electromagnetics. Boca Raton, FL: CRC Press LLC, 1993. 60

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16. Users Manual for XFDTD the Finite Difference Time Domain Graphical User Interface for Electromagnetic Calculations, Version 5.04, Remcom, Inc., February 1999. 17. Liao ZP, Wong HL, Yang GP, Yuan YF. A transmitting boundary for transient wave analysis. Scientia Sinica 1984; 28:1063-1076. 18. Brenger JP. A perfectly matched layer for the absorption of electromagnetic waves. J. Comput Phys 1994; 114:185. 19. Mareci TH, 2005, Basic Design Principles of Magnetic Resonance (MR) Coils, University of Florida, Gainesville, FL http://faraday.mbi.ufl.edu/marecilab/research_results/MR_coils/basic_design_MR_coils.pdf 20. Chen C, Holt D. Biomedical Magnetic Resonance Technology. Adam Hilger Publishing, New York, NY, 1989, pp 142-143. 21. Grist TM, Jesmanowics A, Kneelan JB, Froncisz W, Hyde JS. Doubly tuned local coils for MRI and MRS at 1.5T. Magn Reson Med 1988; 6:253-264. 22. Beck B, Jenkins K, Rocca J, Fitzsimmons J. Tissue-equivalent phantoms for high frequencies. Concepts in Magn Reson Part B 2004; 20:30-33. 23. Kaufman L, Kramer DM, Crooks LE, Ortendahl DA. Measuring signal-to noise ratios in MR imaging. Radiology 1989; 173:265-267. 24. MdGibney G, Smith MR. An unbiased signal-to-noise ratio measure for magnetic resonance images. Med Phys 1993; 20:1077-1079. 61

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BIOGRAPHICAL SKETCH Gary Blaskowski was born in Wausau, Wisconsin. He attended the University of Wisconsin Marathon Campus and graduated with a Bachelor of Science degree in physics from the University of Wisconsin Madison in 1989. As an undergraduate, he worked as a research assistant at Tracor Northern (Middleton, Wisconsin). His work there included improvements to production and pre-production optical equipment. After graduation he received an internship with the Department of Energy and spent 6 months with the Smith research group in Hanford Washington. The focus of his work was on a novel method of ion injection for mass spectrometry. In 1994 he graduated from the University of Florida with a Master of Science degree in physics. He then spent seven years teaching AP Physics at Eastside High School in Gainesville, Florida. In 2001 he began work as a research assistant at the McKnight Brain Institute of the University of Florida. His work there was in support of the advanced magnetic resonance imaging and spectroscopy facility. He now works as a system engineer for General Electric Healthcare working in support of magnetic resonance imaging hardware. 62