Citation
Development of a Multiple Frequency, Inductively Coupled, Implantable, NMR Coil System at 11.1 T

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Title:
Development of a Multiple Frequency, Inductively Coupled, Implantable, NMR Coil System at 11.1 T
Creator:
Harden, Bradley
Publication Date:
Language:
English

Subjects

Subjects / Keywords:
Capacitors ( jstor )
Circuit diagrams ( jstor )
Electric potential ( jstor )
Inductors ( jstor )
LC circuits ( jstor )
Magnetic resonance ( jstor )
Nuclear magnetic resonance ( jstor )
Reactance ( jstor )
Resonant frequencies ( jstor )
Transmission lines ( jstor )
Artificial organs
Diabetes
Pancreas
Genre:
Undergraduate Honors Thesis

Notes

Abstract:
Diabetes is a potentially life threatening disease affecting over 23 million Americans. A new therapeutic approach to treating Type I diabetes involves the implantation of a tissue engineered macro construct. Direct, non-invasive monitoring of this “bioartificial pancreas” can be accomplished with an inductively coupled, implantable, NMR coil system. Development of a multiple frequency system is presented as part of ongoing efforts to characterize the construct’s cells in vivo. Dual frequency implantable coils are shown to provide sufficient Q when resonating within the acceptable range, but require more advanced manufacturing techniques in order to be built reliably. A flaw in one of the design theories used to construct dual frequency surface coils is examined, along with its correction and the subsequent circuit simulations. The long traces required when using variable capacitors are shown to create additional, unexpected resonances, which are corrected with a fixed component design. Further development of the system shows promise, but requires professional construction of both the implantable coil and the surface coil’s printed circuit board to better implement the theoretical designs presented. ( en )
General Note:
Awarded Bachelor of Science in Electrical Engineering; Graduated June 22, 2010 summa cum laude. Major: Electrical Engineering
General Note:
College/School: College of Engineering
General Note:
Advisor: William Eisenstadt

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Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright Bradley Harden. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.

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DEVELOPMENT OF A MULTIPLE FREQUENCY, INDUCTIVELY COUPLED, IMPLANTABLE, NMR COIL SYSTEM AT 11.1 T By BRADLEY J. HARDEN PRESENTED TO THE COLLEGE OF ENGINEERING OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR SUMMA CUM LAUDE HONORS DESIGNATION

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ii 2010 Bradley J. Harden

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iii For my mother I can only hope she would be as proud to be my mother as I was to be her son.

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iv ACKNOWLEDGEMENTS The author would first like to thank Dr. Nicholas Simpson for his advice and support. His enthusiasm for his work spreads to those around him. Thanks are also due to Dr. Thomas Mareci and t he students of his lab. The lab space assistance and advice provided have been invaluable. Special thanks go to Barbara Beck and Sien Wu for the innumerable questions they have endured regarding RF engineering. Finally, the author would like to thank the Howard Hughes Medical

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v TABLE OF CONTENTS CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 2 1.1 Treatment of diabetes ................................ ................................ ................................ .......... 2 1.2 NMR methods ................................ ................................ ................................ ..................... 3 1.2.1 Nuclei of interest ................................ ................................ ................................ ........ 3 1.2.2 Implantable Coils ................................ ................................ ................................ ....... 3 1.2.3 Detection at multiple frequencies ................................ ................................ .............. 4 1.3 Objectives ................................ ................................ ................................ ........................... 4 CHAPTER 2 BACKGROUND ................................ ................................ ................................ ...... 6 2.1 Series Resonant Circuits ................................ ................................ ................................ ..... 6 2.2 Quality Factor ................................ ................................ ................................ ..................... 8 2.3 Transmission Lines ................................ ................................ ................................ ........... 1 1 2.4 In ductive Coupling ................................ ................................ ................................ ............ 13 CHAPTER 3 DEVELOPMENT OF A MULTIPLE FREQUENCY IMPLANTABLE COIL ... 17 3.1 Circuit Theory ................................ ................................ ................................ ................... 17 3.2 Design Constraints ................................ ................................ ................................ ............ 19 3.3 Materials and Methods ................................ ................................ ................................ ...... 19 3.4 Results ................................ ................................ ................................ ............................... 20 3.5 Discussion ................................ ................................ ................................ ......................... 21 CHAPTER 4 DEVELOPMENT OF A MULTIPLE FREQUENCY SURFACE COIL .............. 22 4.1 Circuit Theory ................................ ................................ ................................ ................... 22 4.1.1 Background ................................ ................................ ................................ .............. 22 4.1.2 Previous Work ................................ ................................ ................................ ......... 23 4.1.3 New matching circuit ................................ ................................ ............................... 25 4.1.4 Effects of inductor resistance ................................ ................................ ................... 28 4.1.5 Simulation ................................ ................................ ................................ ................ 30 4.1.6 An alternate design ................................ ................................ ................................ .. 31 4.2 Design Constraints ................................ ................................ ................................ ............ 31 4.3 Materials and Methods ................................ ................................ ................................ ...... 32 4. 4 Results ................................ ................................ ................................ ............................... 33 4.4.1 Three component matching network ................................ ................................ ....... 33 4.4.2 Four component matching network ................................ ................................ ......... 35 4.5 Discussion ................................ ................................ ................................ ......................... 36 4.5.1 Three component matching network ................................ ................................ ....... 36 4.5.2 Four component matching network ................................ ................................ ......... 36

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vi CHAPTER 5 CONCLUSIONS AND FUTURE WORK ................................ ............................. 38 5.1 Conclusions ................................ ................................ ................................ ....................... 38 5.2 Future Work ................................ ................................ ................................ ...................... 38 CHAPTER 6 LIST OF REFERENCES ................................ ................................ ........................ 39

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vii LIST OF FIGURES Figure 2 1. Schematic of a series resonant circuit ................................ ................................ ......... 6 Figure 2 2. Capacitive reactance (red) and the negative of the inductive reactance (blue) vs. angular frequency ................................ ................................ ................................ .. 7 Figure 2 3. Current vs. frequency of a series RLC circuit at various values of Q ......................... 9 Figure 2 4. Lossless model of a transmission line ................................ ................................ ....... 11 Figure 2 5. Schematic of an inductively coupled circuit containing a primary coil (left) and a secondary coil (right) ................................ ................................ ....................... 13 Figure 2 6. Circuit diagram for the primary circuit when considering the effects of the secondary ................................ ................................ ................................ .................. 14 Figure 2 7. Effect of increased coupling on secondary current ................................ ................... 15 Figure 3 1. Three schematics and their corresponding reactance functions representing potential choices for X 2 ................................ ................................ ............................. 18 Figure 3 2. An implementation of a simple inductor as X 1 and the circuit shown in Figure 3 1 (c) as X 2 ................................ ................................ ................................ ... 18 Figure 3 3. Construction of the circuit found in Figure 3 2 ................................ ........................ 20 Figure 4 1. Two commonly used circuit arrangements for tuning and matching ........................ 22 Figure 4 2. Schnall circuit design f or multiple resonance probes ................................ ............... 24 Figure 4 3. Graphical confirmation that the Schnall tuning circuit (blue) can reach the required impedance (red) at the desired frequencies (green) ................................ .... 24 Figure 4 impedance (red) at both frequencies of interest (gr een) simultaneously .................. 25 Figure 4 5. The three component matching circuit cannot match at both frequencies with an inductor great er than ~1.5 nH (5 nH inductance shown) ................................ ..... 26

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viii Figure 4 6. Four component matching circuit ................................ ................................ ............. 27 Figure 4 7. Matching achieved with 39 and 22 nH inductors along with 9.2 pF capacitors ................................ ................................ ................................ .................. 27 Figure 4 8. Tank circuit model when inductor resistance is included ................................ ......... 28 Figure 4 9. Graphical approach to tuning and matching with the inclusion of inductor resistances ................................ ................................ ................................ ................. 29 Figure 4 10. Simulated circuit for a double tuned surface coil ................................ ...................... 30 Figure 4 11. Simulation results showing the predic ted nature of the S parameter ........................ 31 Figure 4 12. Circuit schematic of the first prototype surface coil ................................ ................. 33 Figure 4 13. First surface coil prototype. Variable capacitors and inductors are shown on top. Fixed capacitors are not visible on the bottom. ................................ ................. 34 Figure 4 14. Schematic of the second surface coil prototype ................................ ........................ 35 Figure 4 15. Second surface coil prototype. The coaxial cable pads are visible, but no cable is connected ................................ ................................ ................................ ..... 35

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ix LIST OF TABLES Table 3 1. Resonance frequencies and Qs of coils meeting the design constraints ..................... 21 Table 3 2. Construction details of the coils in Table 3 1 ................................ ............................. 21 Table 4 1. Resonance frequencies and match for the second surface coil prototype .................. 36

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1 ABSTRACT Diabetes is a potentially life threatening disease affecting over 23 million Americans. A new therapeutic approach to treating Type I diabetes involves the implantation of a tissue engineered macro construct. Direct, non invasive monitoring of this be accomplished with an inductively coupled, implantable, NMR coil system. Development of a cells in vivo Dual frequency implantable co ils are shown to provide sufficient Q when resonating within the acceptable range, but require more advanced manufacturing techniques in order to be built reliably. A flaw in one of the design theor ies used to construct dual frequency surface coils is exam ined along with its correction and the subsequent circuit simulations. The long traces required when using variable capacitors are shown to create additional, unexpected resonances which are corrected with a fixed component design. Further development of the system shows promise, but requires professional construction of both the implantable coil and to better implement the theoretical designs presented.

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2 CHAPTER 1 INTRODUCTION 1.1 Treatment of diabetes Diabetes is an extremely serious disease that invokes permanent lifestyle change upon those afflicted. Complications of this disease are numerous, and include cardiovascular disease, blindness, renal disease and limb loss. It is estimated that over 23 million people suffer from diabetes in the US alone [ 1 ]. Five to ten percent of these cases are considered Type I diabetes, a condition in which naturally occurring insulin cells) are damaged or lost. Common treatments for this condition involve in sulin injections whether by manual injection or via insulin pump. These treatments can help patient s manage their blood glucose levels, but not without a severe change in lifestyle. Moreover, despite regulatory efforts, blood sugar changes result in long term damage and a shortened life span. A new therapeutic approach for the treatment of Type I diabetes involves the implantation of a bioartificial pancreatic construc t as a substitute for the cells These constructs consist of replacement cells entrapped within semi permeable alginate beads, which are then contained and implanted within a polymer shell. Typically, therapeutic efficacy is determined by the treatmen maintain normoglycemia. It is important that the construct be monitored closely, to prevent the unexpected onset of end point diabetic effects. give insight into construct function, b ut it cannot, however, provide enough information to fully characterize the health and status of the cells within. In in vitro scenarios t his task would normally be accomplished with histological staining, but in vivo this method is both destructive and in vasive. A n implanted pancreatic construct require s a direct and non invasive technique for cell characterization in vivo in order to predict construct failure Nuclear magnetic resonance

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3 methods are well suited to perform these tasks because of their abili ty to non invasively provide both structural and metabolic information t hrough imaging and spectroscopy of multiple nuclei. 1.2 NMR methods 1.2.1 Nuclei of interest NMR offers a number of methods to monitor cells by the detection of biological nuclei, such as 1 H, 13 C, 19 F, and 31 P. Hydrogen is most often used for imaging, as it is the most sensitive of the nuclei and the most abundant, but it can also be used for spectroscopy. In this application the most relevant molecule of interest for spectroscopic analysis is ch oline, a proportional indicator of oxygen concentration and cell viability [ 2,3 ]. 13 C has been used to observe link s between insulin secretion and metabolic pathways [ 17 ] 19 F NMR can detect oxygen tension within alginate structures using perfluorocarbons to measure pO 2 31 P spectroscopy yield s direct observations of bioenergetic markers, such as ATP, which can be used to monitor cell metabolism [ 4 ] Information from all, or at least some combination, of these nuclei affords the much greater knowledge necessary to predict construct failure once implanted. 1.2.2 Implantable Coils NMR spectroscopy studies correlating spectral intensity to cell structure and f unction typically use external volume or surface coils [ 2,5 ] for observation of their respective nuclei. This is sufficient when a large number of cells are present in vitro but can be a limiting factor in signal to noise ratio (SNR) when dealing with t h e lower cell densities used in vivo It is necessary, then, to maximize the SNR using a number of techniques. To this end an implantable coil can be included within the construct bringing the detection probe closer to the signals of interest while simult aneously eliminating noise from interfering skin, fat and muscle tissues.

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4 Early examples of implanted coils required that wires pass through the skin [ 6 ] a design which would not be practical for use in human patients and would leave them susceptible to i nfection Inductive coupling between an implanted coil and an external surface coil solves this problem. Inductive coupling uses the changing magnetic field of the surface coil to wirelessly ind uce a current in the implanted coil. This technique has been s hown to increase SNR in a number of studies [7 10 18 ], including the application described above [ 11 ]. 1.2.3 Detection at multiple frequencies If simultaneous d etection of more than one element is desired additional SNR losses are also incurred. Nuclear magnetic resonance causes all nuclei of a specific element to precess a t a specific frequency. This requires that measurement coils be tuned to all frequencies of interest in a given experiment. Tuning radio frequency (RF) coils to multiple frequencies can be accomplished by the addition of tank circuits [ 12 ] or by the creation of multiple resonance modes [ 13 ]. In both cases the losses in the circuit are increased for each additional frequency add ed, greatly lowering the SNR. For this reason most applications of multiple frequency coils have been limited to two frequencies. Until recently there had been no reports of any attempts to detect nuclei besides 1 H using an implantable coil nor had there been attempts to construct a multiple frequency coil for use at such high magnetic field Develop ment is currently underway on a multiple frequency automatic impedance matching system implemented with a surface and implantable coil However, t his approach maintains a single resonance methodology for circuit tuning by wirelessly adjusting capacitance values using a microcontroller [ 14 ]. Recent attempts to design a multipl e resonant inductively coupled coil system produced suboptimal results [ 11 ]. 1.3 Objectives The aim of this project is to develop a highly sensitive multiple frequency NMR surface coil an d accompanying implantable coil, used to monitor a tissue engineered pancreatic construct

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5 in vivo As mentioned previously, practical designs of mult iple frequency coils are limited to two frequencies b ecause of the losses inherent to them For this reason the scope of the project was limited to detection of 1 H and 31 P, resonating at 470 MHz and 190 MHz respectively in a magnetic field of 11.1 T The se nuclei were chosen because of both their ability to quantify cell viability and metabolism and because of their large separation in resonance frequency (which makes tuning and matching easier).

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6 CHAPTER 2 BACKGROUND 2.1 Series Resonant Circuits In order to produce the highest strength magnetic field po ssible for a given circuit loop loop is maximum when its series impedance is minimum. In circuit analysis, a n impedance is ma de of a real part, called the resistance, and an imaginary part, called the reactance, Given that the resistance in a c ircuit typically varies little with frequency, it is most often the case that the minimum impedance corresponds to t he frequency at which the reactance is zero. A series resonant circuit is one in which the reactance of the circuit goes to zero at a particular frequency called the resonant frequency This can be created by connecting two elements whose reactances vary with frequency as shown in Figure 2 1 Figure 2 1 Schematic of a series resonant circuit The impedance of this series combination is:

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7 where R represents the small but not negligible resistance contributed by the non ideal nature of reactive components X 1 and X 2 It is easy to see that the impedance is minimized when : leaving on ly the resistive component of the impedance. If we let X 1 be an inductor and X 2 a capacitor then w e can graphically visualize this simple example by examining the capacit ive reactance function overlaid on top of the negative of the inductive reactance function as seen in Figure 2 2 Figure 2 2 Capacitive reactance (red) and the negative of the inductive reactance (blue) vs. angular frequency The point of intersection of these two functions is the point at which the total reactance is zero and the total impedance is minimized. For a given induct ance, L, and capacitance, C, this occurs at the resonant frequency:

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8 In the previous example the circuit was found to be resonant at one frequency because the two reactance functions X 1 and X 2 intersected at only one point. The goal, then, of d esigning a circuit with multiple points of resonance is to craft reactance functions which intersect at more than one point. We will revisit this graphical method in future sections when a pplying it to various circumstances relating directly to our applica tion. 2.2 Q uality Factor A series resonant circuit stores energy in its reactive elements and dissipates energy in its resistive elements. It is often convenient when describing a series RLC circuit, to define the ratio between the maximum energy stored at th e resonant frequency to the p ower dissipated per cycle at that same frequency. The maximum energy stored in an inductor is and the maximum energy stored in a capacitor is where I p and V p represent the peak current and peak voltage respectively It can be shown that in a series RLC circuit, these values are equal at resonance: The energy dissipated through the resistor in on e period of time T 0 ( at the resonant frequency ) is equal to:

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9 The ratio between the maximum energy stored and the energy dissipated per cycle is then : Typically this quantit y is multiplied by a factor of 2 and is named the Quality Factor (Q): The reader should note that these equations for Q apply only to a series RLC circuit, whereas the concept of a quality factor extends well beyond th is example. The quality factor of a series RLC circuit also serves as a m easure of peak sharpness on a plot of current vs. frequency. The peak becomes noti ceably taller and sharper a s the Q of the circuit increases ( see Figure 2 3 ). Figure 2 3 Current vs. frequency of a series RLC circuit at various values of Q

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10 The current through the RLC circuit I s induced by and emf voltage V emf can be found to be : In terms of the quality factor this is equal to: The current is maximum at the resonance frequency when The current drops to its RMS value This occurs at frequencies: The separation between these two frequencies is This result allows us to define t he quality factor in an alternate manner, the resonant frequency divided by the 3dB bandwidth:

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11 It is usually desirable to achieve as high a Q as possible when designing resonant circuits. In our application a higher Q results in a higher absolute value of current, improvin g both excitation and detection. There are many factors which limit the Q of a magnetic resonance circuit. It is necessary to coat the implantable coil with a biocompatible polymer to prevent the coil from being ex posed to the fluid and blood within the host Coil coatings often create additional capacitive losses and can shift the resonant frequency of the system and lower its Q. Loading the circuit by placing it next to a lossy tissue sample usually increases the total resistive losses of the circuit, which also lowers its Q. It is important to characterize and account for the effects of coating and loading, however these topics are beyond the scope of this project. It is always important to maximize the Q of a cir cuit, regardless of whether its application will reduce the Q during use 2.3 Transmission Lines At low frequencies (high wavelengths) it can be assumed that wires are at a constant voltage across their entire length. This cannot be assumed, however, at the hi gh frequencies required for magnetic resonance imaging. Voltages and currents must instead be thought of as si nusoidal waves trav elling in both directions along a transmission line. A simple, lossless model for a transmission line is shown in Figure 2 4 Figure 2 4 Lossless model of a transmission line

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12 The conductors on which the waves propagate can be thought of in terms of a distributed inductance, while the separation between the signal path and the return path creates a distributed capacitance. A transmission line, then, does not have zero impedance (as would a perfect conductor) but rather some finite value derived from phy sical parameters. This value is termed the characteristic impedance Z 0 and is often purely real. When dealing with a transmission line it is important to consider the bounda ry conditions which exist at its termination. In the case where the line is of infinite length the impedance seen by the propagating wave is always constant and is equal to the characteristic impedance. This result can be duplicated by terminating the line with an impedance equal to the characteristic impedance, producing the same boundary effects as if the line was, in fact, of infinite length. In this situation the load is said to be matched to the line. If, however, the load impedance Z L deviates from the characteristic impedance t he discontinuity at the boundary will cause a portion of the wave to be reflected and a portion to be transmitted This phenomenon can be described by a reflection coefficient The reflection coefficient is always between 1 and 1 and gives the ratio of the reflected wave to the incident wave, along with the sign of reflection (whether positive or negative). In the case where Z L is equal to Z 0 the reflection coefficient is equal to zero and the entire wave is transmitted to the load. In magnetic resonance applications it is necessary to deliver as high a voltage to the probe as possible during transmission, as well as receive as high a voltage as possible during reception. This requires that the total impedance of the probe circuit be ma tched to the characteristic impedance of the transmission line.

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13 2.4 Inductive Coupling Inductive coupling off ers a distinct advantage in our application by allowing the wireless transmission of power to a coil which was implanted as part of bioartificial organ In ductive coupling uses two coils: one called the primary which receives power from a wired source; another, called the secondary, which receives power from the primary. The circuit in Figure 2 5 shows a typical inductively coupled circuit. The sinusoidal current through the primary circuit creates a varying magnetic field in the primary inductor, L p Some portion, k of the magnetic flux lines creat ed by L p pass through the secondary inductor L s inducing a secondary current The current in the secondary then creates its own magnetic field which induces a current in the primary. Both circuits have effects on each other simultaneously and their circui t equations must be solved as such. Figure 2 5 Schematic of an inductively coupled circuit containing a primary coil (left) and a secondary coil (right) Considered alone the voltage applied to the primary is equal to but if the effects of the secondary current are taken into account, the equation must be rewritten as

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14 Where M is the mutual inductance between the two circuits and is eq ual to k times the maximum possible mutual inductance between the coils: The secondary voltage considered alone would be zero, but is instead These equations can be rearranged to give a clearer picture of the effects of the inductive coupling The primary voltage can be rewritten as Thus the effect of the secondary on the primary can be understood as the addition of an impedance placed in series with the normal impedance Z p (see Figure 2 6 ). Figure 2 6 Circuit diagram for the primary circuit when considering the effects of the secondary It is possible to solve for the primary and secondary currents us ing these equations, producing lengthy and complicated formula s Qualitatively, the effects of increased coupling are as follows [ 15 ] : Bringing the coils closer together increases the overlap of fl ux lines between the m raising k The peak of the primary current which has the typical single peak resonant shape about the common resonant frequency (see Figure 2 3 ) begins to decrease in height

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15 The peak of the secondary current increases in height and exhibits a sharper peak than it would in the absence of the primary (blue curve in Figure 2 7 ) At a certain value of coupling, called the critical coupling point ( k c ) the secondary current peak reaches its maximum possible value (red curve in Figure 2 7 ) Beyond critical coupling the system is said to be over coupled and the peaks of both the primary and secondary current begin to split, creating two peaks which are separated in frequency b y a coupling factor dependant value. The separation increases with coupling. The height of the two peaks in primary current decrease as coupling increases beyond the critical value, whereas the hei ght of the secondary peaks stay substantially the same no matter the coupling (yellow and gree n curves of Figure 2 7 ) Figure 2 7 Effect of increased coupling on secondary current

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16 The above case considered co upling between two circuits which were tuned to the same frequency. In the case where the circuits are tuned to a slightly different fr equency, the results are similar but not identical. In this case t he secondary current exhibits a shape almost identical to that which is obtained by circuits tuned to the same frequency, but with a greater coupling factor. The effect on the primary current peak is to shift its resonant frequency and deform the curve in the vicinity of the resonant frequency of the secondary The primary may or may not develop two peaks, depending on the separation between the resonant frequencies of the two coils. There is also one additional point to consider when designing an over coupled coil system. Although the magnitude of current thro ugh each coil may be identical at both its higher and lower frequency peaks the direction of current flow through the coils is parallel at the lower frequency and anti parallel at the higher frequency. In the higher frequency, counter rotating mode the ma gnetic fields produced by each coil oppose each other and lower the absolute magnitude of the field at all points surrounding the coil in comparison to the lower frequency. co rotating mode. It is often required, then, that both the primary and secondary c oil resonate slightly above the desired resonant frequency in preparation for the eventual split of the single peak into two peaks and the subsequent movement of the lower peak to the desired frequency.

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17 CHAPTER 3 DEVELOPMENT OF A MULTIPLE FREQUENCY IMPLANTABLE COI L 3.1 Circuit Theory An implantable coil receives its power from an external source. It contains no wire leads ; voltages are induced solely through inductive coupling. The circuit of Figure 2 1 is perfect for describing the behavior of a stand alone loop containing reactive elements. NMR applications require that current flow through a wire which is molded into some well defined shape. I n the case of our implantable coil a single, circular loop gap resonator was found to be the best choice [ 11 ] The loop of wire can be modeled very accurately by an ideal inductor with inductance L s X 1 is then fixed to and X 2 must be chosen to intersect that line at two points, 190 MHz and 470 MHz for our purposes. The resistive losses found in inductors are typically of a much higher magnitude than those found in capacitors. Minimizing the n umber of inductors serves to reduce the total res istance and increase the Q of a circuit Several potential combinations of reactive elements which follow this design philosophy are found in Figure 3 1 : a single capacitor a tank circuit (parallel combination of an inductor and a capacitor) and a capacitor in series with a tank circuit

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18 Figure 3 1 Three schematics and their corresponding reactance functions represent ing potential choices for X 2 It can be seen that only one of these candidates can possibly intersect X 1 ( ) at two points choice (c) The schematic shown in Figure 3 1 (c) can be implemented fairly easily by slightly modifying the circuit design of Figure 2 1 The sing le inductor representing X 1 can be split into an equivalent series combination of two inductors and be used to separate the capacitor from the parallel combination (see Figure 3 2 ). Figure 3 2 An implementation of a simple inductor as X 1 and the circuit shown in Figure 3 1 (c) as X 2

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19 3.2 Design Constraints The construction of Figure 3 2 faces a number of difficulties. The sparse spacing of values found in non magnetic RF chip capacit or s component tolerances, and the inevitable variation in hand made coils makes achieving an exact resonance frequency for a coil impossible. Since we cannot include variable capacitors within the implant, our only option is to use over coupling between the surface and implantable coils to bring the resonance frequency of the system as close as p ossible to the Larmor frequency. In spite of these facts, it is still important that the coil resonate within some minimum acceptable distance from the desired frequency. Should the coil resonate outside of this range the over coupling required to correct the resonance frequency will be too great to achieve It is also important that the system not over couple more than that which is necessary. Over coupling has little effect on the current through the implantable coil, but it has a great effect on the cur rent through the surface coil. Any over coupling reduces the surface coil current (and thereby its magnetic field contribution) and lowers the sensitivity of the system. For this design it is acceptable for the implantable coil to resonate within a 20 MHz range above the Larmor frequencies for each nucleus, 190 210 MHz for 31 P and 470 490 MHz for 1 H. It is also important that the Q at each of the resonance points be sufficient to insure a high SNR in magnet experiments. A Q of 50 will typically be high enou gh for this purpose. 3.3 Materials and Methods Multiple frequency implantable coils were constructed using hand cut 2 mm wide, 202 m thick, copper foil along with ATC 100 B series porcelain multilayer capacitors and hand wound inductors of various sizes. The inductors were made using AWG 25 wire wrapped around wires of other gauges, such as AWG 14, 16, 18, and 20.

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20 Figure 3 3 Construction of the circuit found in Figure 3 2 L oop gap resonator circuits of one centimeter diameter were built according to the schematic of Figure 3 2 as shown in Figure 3 3 The circuit components were all hand cut and soldered. Circuit Q was determined by measuring the resonance frequency and 3 dB bandwidth of each peak These measurements were carried out on the S 21 channel of a n HP 8753E network analyzer. The impedance of hang wound inductors was measured using an HP 4396B impedance analyzer. 3.4 Results The copper foil inductors making up the outside ring of t he coil typically resulted in an inductance on the order of 20 30 nH, while the tank inductor was usually found within the range of 7 13 nH. The capacitors used were often 18 or 20 pF. Although the circuit theory and design constraints would suggest a choi ce of approximately 25 nH for the coil 9 nH for the tank inductor, and two 18 pF capacitors (giving resonance frequencies of 195 MHz and 480 MHz), hand crafting of inductors with nano henry precision is virtually impossible. Upwards of fifteen to twenty coils were constructed, yielding only two which resonated within the acceptable frequency bands at both points These results are found in Table 3 1 The construction details for the two successful coils are detailed in Table 3 2

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21 31 P 1 H Coil f 3dB f 0 f 3dB+ Q f 3dB f 0 f 3dB+ Q 1 207.6 208.9 210.1 83.6 486.4 489.1 491.3 99.8 2 200.3 202.0 203.5 63.1 468.6 472.1 475.3 70.5 Table 3 1 Resonance frequencies and Qs of coils meeting the design constraints Coil L t C t C tt 1 1.5 turns of AWG 25 around AWG 14 18 pF 18 pF 2 2.5 turns of AWG 25 around AWG 16 18 pF 18 pF Table 3 2 Construction details of the coils in Table 3 1 3.5 Discussion The low yield of coils which meet the design constraints is an inevitable result of hand construction and makes a statistical analysis of the variation in coil resonance frequency almost exclusively dependent on their assembly Construction of seemingly identical coils often resulted in resonance frequencies which were different by as much as 40 MHz. A statistical analysis of the results of all coils built would only serve as a mea sure of the expected yield from hand manufacturing. More consistent methods of coil fabrication must be developed if there is any hope to reliably manufacture coils within a reasonable tolerance. For our purposes, however, the coils which were created meet the design criteria and were sufficient for the continued development of a multiple frequency, inductively coupled system.

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22 CHAPTER 4 DEVELOPMENT OF A MULTIPLE FREQUENCY SURFACE COIL 4.1 Circuit Theory 4.1.1 Background When injecting current into a resonant circuit directly it is only required that that reactance at the desired frequencies be equal to zero; the resistance is not constrained to any particular value (it is usually reduced as much as possible to increase Q) The use of a transmission line, however, complicates the circuit. In order to receive maximum current, the resonant circuit must appear to have an impedance equal to that of the characteristic impedance of the transmission line, typically 50 Ohms. The terminating circuit is then matched to the line. Tuning a resonant circuit to a specific set of frequencies and matching it to 50 Ohms at those frequencies requires a more complicated circuit arrangement than that which was previously specified for a simple series resonant circuit. Figure 4 1 shows two arrangements which are in common use for designing both single and multiple resonant circuits. They each combine two impedances that may be specified in any manner desired, along with a resistance, R p and reactance, X p ( p ), representing the fixed geometry of the coil used to transmit and receive the magnetic fields. Figure 4 1 Two commonly used circuit arrangements for t uning and matching

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23 If it is assumed that the elements Z 1 and Z 2 in each circuit are dominated by their reactive components, we can ignore their resistance and, instead, consider only their respective reactances X 1 and X 2 Solving for a total resistance equal to 50 and a total reactance equal to zero one can obtain the following equations [ 16 ] : for (a) and for (b) Although each circuit play s some role in both tuning and matching, typically X 1 has greater control of the position of the resonance, while X 2 has greater control over the equivalent impedance of the circuit. O ften X 1 is referred to as the tuning circuit, while X 2 is referred to as the matching circuit. 4.1.2 Previous Work Previous attempts to develop a multiple frequency surface coil for use at 11.1 T [ 11 ] were based on designs put forth by Schnall et al. [ 12 ]. Schna ll suggests a tuning circuit consist ing of a capacitor in series with a tank circuit (see equations for Figure 3 1 (c)) and a matching circuit contain ing just a tank circuit (see equations for Figure 3 1 (b) ). The se c ombinations placed in Figure 4 1 (a) can be seen in Figure 4 2

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24 Figure 4 2 Schnall circuit design for multiple resonance probes This design is flawed, however, in that the matching circuit cannot possibly produce the appropriate impedance to match at both frequencies. Given that R L is usually small and tends to vary little with frequency, it is safe, for the time being, to assume that R L is a constant. If we examine the solutions for Figure 4 1 (a), we can see that the equation for X 1 is reduced to X p plus some small constant value. We can visualize this in Figure 4 3 and see that the tuning circuit X 1 can, in fact, intersect at more than one point. Figure 4 3 Graphical confirmation that the Schnall tuning circuit (blue) can reach the required impedance (red) at the desired frequencies (green)

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25 If we apply the same assumptions to the matching circuit, X 2 we realize that this function is simply a small, negative const ant with respect to frequency. The impedance function of a tank circuit, howeve r, cannot possibly intersect a horizontal line at more than a single point, as seen in Figure 4 4 The same analysis can be applied to the circuit of Figure 4 1 (b), producing similar results. Figure 4 4 e (red) at both frequencies of interest (green) simultaneously Previous atte project produced a surface coil which could no mat ch at more than one frequency at a time [ 11 ], which should have been expected given the res ults above In order to match appropriately at more than one frequency, the matching circuit must be changed. 4.1.3 New matching circuit Figure 4 1 (a) was chosen to be implemented because of the simplicity of its equations when R p is assumed to be constant. The tuning circuit proposed by Schnall is sufficient for X 1 in this design, but, as discussed previously, the proposed matching circuit X 2 is not. A circuit must be constructed, then, whose impedance can intersect a negative, horizontal line at more than one point.

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26 4.1.3.1 Three component matching circuit The tuning circuit used above can be used equally as well as a matching circuit. Its impedance function has the ability to intersect the horizontal line at two points and it contains only one additional inductor. At first glance, this circuit appears to solve the problems encountered with the first matching circuit but it does so while creating new issues of its own. The asymptotic nature of the function as it approaches infinite frequency makes it difficult for the function to intersect a horizontal line close to zero. Solutions for th is mat ching circuit (at 190 MHz and 470 MHz) exist only for an induc tance less than roughly 1.5 nH (see Figure 4 5 ). Figure 4 5 The three component matching circuit cannot match at both frequencies with an inductor greater than ~1.5 nH (5 nH inductance shown) An inductance this low is not practical to implement in a circuit. The traces between compo nents can often have inductances of this ma gnitude and a design which relies on an inductance accurate to within such a small margin is simply unsuitable Few chip inductors even exist at this low value, and those that do have large tolerances. The only op tion is to add an additional component to the matching circuit which changes the function to a more convenient shape.

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27 4.1.3.2 Four component matching circuit A second newly proposed matching circuit is shown in Figure 4 6 Its reactance as a function of frequency is Figure 4 6 Four component matching circuit This circuit contains an additional degree of freedom, allowing the designer to choose more practical values than those required by the three component matching circuit. Typically, the inductor values are chosen and fixed, because it is difficult to find h igh quality, non magnetic, variable inductors. Following this design principle, i t is important to note that choosing a high series inductor (L tt ) and low parallel inductor (L t ) makes the control of each intersection point by its corresponding capacitor, more independent. The graph of Figur e 4 7 shows that it is possible to match using components whose values all fall within the practical range for RF circuits. Figur e 4 7 Matching achieved with 39 and 22 nH inductors along with 9.2 pF capacitors

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28 4.1.4 Effects of inductor resistance The circuit theory above can be accurate to a certain extent, but it does not take into accou nt one of primary aspects of real world circuits. Viohl et al. accounted only for the resistance of the probe when solving the equations originally, and this is appropriate for single tuned circuits which require only capacitors. For the double tuned case, however, we must also account for the resistance in the other inductors. First, it is important to discern how the resistance of an inductor is affected by the presence of a parallel capacitance. Figure 4 8 shows the model for a tank circuit when including the resistance. Figure 4 8 Tank circuit model when inductor resistance is included The equivalent impedance of this model is then the parallel combination of and The exact impedance can be written as If the Q is assumed to be high, which is often th e case, the equation can be simplified to

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29 The reactance of this result is identical to that of the tank circuit without a resistor. The resistance, however, varies strongly with frequency in the area of the resonance frequency of the tank circuit. This fact should be accounted for in the graphical process used above. The resistance of the tuning circuit Z 1 can simply be incorporated into the probe resistance R p but the existence of a resistance in the matching circuit requires a bit more effort. If the equations are solved using a pure reactance Z 1 = X 1 and an impedance Z 2 = R 2 + j X 2 the following results are obtained: where R L = R 1 + R p In the even t that R p R 1 and R 2 are all constant with respect to frequency the equations reduce to a form similar to that of the solutions without resistances. These new functions can be graphed and plotted against the reactances as before. Figure 4 9 Graphical approach to tuning and matching with the inclusion of inductor resistances

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30 Figure 4 9 demonstrates that although the necessary reactance has changed significantly in the vicinity of the resonance frequencies of both the tuning and matching circuits, the separation between the desired detection frequencies (190 MHz and 470 MHz) is sufficient to allow us to disregard most of the resistor effects. 4.1.5 Simulation Before construction of a circuit it is often useful to simulate the design in order to test for additional effec ts arising in real world circuits. The four component matching circuit was simulated, along with the estimated resistance of each inductor based on Q values provided by the manufacturer. The S parameter of interest was S 11 or, equivalently, the reflection coefficient. Figure 4 10 Simulated circuit for a double tuned surface coil

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31 Figure 4 11 Simulation results showing the predicted nature of the S parameter The simulation produced excellent results using only two significant figures for the specification of all components. Although not apparent from Figure 4 11 due to t he limitation in frequency resolution, this circuit achieved a 38 dB match at 190 MHz and a 33 dB match at 470 MHz. 4.1.6 An alternate design The work shown has concentrated on the circuit configuration of Figure 4 1 (a). The method described, however, can apply equally well to configuration (b). Although the analysis may be a bit more complicated and may rely more heavily on graphing tools and simulations rather than hand calculations, the results may be more favorable, as this circuit allows for a balanced match configuration. A balanced match reduces shield currents in the coaxial cable and, combined with cable traps, can greatly reduce the noise received by the RF ampli fier. 4.2 Design Constraints There are two types of design constraints for the surface coil: those that are inherent to all MR coils, and those that are specific to this project. Like all coils, t he circuit must be able to match at each frequency to at least 20 dB, transferring 90% of the power to the coil. It must also, of course, consist exclusively of non magnetic components, a factor which can be quite limiting in the choice of inductors and capacitors.

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32 For this specific project, however, the coil must be able to tune and match over a range of frequencies for each nucleus, instead of just a single frequency. This requirement arises from the fact that the surface coil is part of an inductively coupled coil system. The system will always be in the over coupl ed state, meaning that there will be a split in the resonance at each detection frequency. The large tolerances of the implantable coil increase this requirement. This is made up for by the ability of the surface coil to over couple to a varying extent, mo ving the split resonance frequency to the desired point. It is estimated that the surface coil should be able to tune and match over a range of approximately 20 MHz above the Larmor frequency, in accordance with the design constraints of the implantable co il. 4.3 Materials and Methods The circuits described above were designed for and constructed on prototype printed was used to lay out the boards, which were then milled onto one ounce copper, FR 4 prototype PCB s Board designs for the three component matching circuit used ATC 100 B series porcelain multilayer chip capacitors, ATC WL series chip inductors, Voltronics NMA_HV series variable capacitors, and AWG 20 copper wire in a two centimeter loop for the probe. The four component matching circuit used ATC 100 B series chip capacitors, Coilcraft 1812SMS series air core inductors, and AWG 12 copper wire in a two centi meter loop for the probe. All components were hand soldered to each board along with a 50 coaxial cabl e Measurements of resonance frequency were obtained on the reflection (S 11 ) channel of a n HP 8753E network analyzer

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33 4.4 Results 4.4.1 Three component matchi ng network Alth ough it is clearer now that the first prototype design is impractical to implement, at the time it was thought that thi s was possible. The schematic in Figure 4 12 shows the component values which were chosen for construction The circuit was built on a double sided prototype PCB which was raised and mounted to a Plexiglas surface. The coil probe ran through a hole from the other side of the Plexiglas to solder to the board (see Figure 4 13 ). The results for this probe were greatly different than predicted. The surface coil resonated at upwards of 4 or 5 frequencies over the 150 600 MHz band. Changing each variable capacitor had an effect on the circuit, but to a widely varying extent. Figure 4 12 Circuit schematic of the first prototype surface coil

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34 Figure 4 13 First surface coi l prototype. Variable capacitors and inductors are shown on top. Fixed capacitors are not visible on the bottom. The matching tank capacitor (C m ) had a small effect on the matching of all resonances present over the frequency band of interest, but changed the circuit little otherwise. The matching series capacitor (C mm ) had the ability to match two of the resonance points present in the 200 270 MHz range. It also was the sole affecter of a higher frequency resonance which was able to tune from approximately 440 to over 600 MHz. The tuning capacitors had effects which were more expected. The tuning tank capacitor changed the frequency of several resonances over the 200 270 MHz range, while the tuning series capacitor had an effect on only one of these peaks. Combining changes in the two tuning capacitors and the matching capacitor C m allowed for one of the peaks to tune and match adequately ( 20 dB) over quite a larger range (210 265 MHz). There were other peaks present in the circuit as well, but they were no t controlled in any well defined manner by any of the variable capacitors present. It was found that, if a probe was attached to the transmission port of the network analyzer, the transmission at various areas of the board had tremendously different streng ths of resonance. The vicinity of the coaxial cable

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35 connections was found to have a relatively flat response. In the areas between the variable capacitors, however, there were strong responses at the resonance frequencies for their respectively controlled peaks. 4.4.2 Four component matching network The second surface coil prototype was constructed with only fixed components using the four component matching circut Figure 4 14 shows the selected component values. Figure 4 14 Schematic of the second surface coil prototype Figure 4 15 shows the constructed prototype based on the new design. It can be seen that there are no variable capacitors, such as those found in Figure 4 13 This circuit resulted in only two resonances. The results of these tests are summarized in Table 4 1 Figure 4 15 Second surface coil prototype. The coaxial cable pads are visible, but no cable is connected

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36 Lower Upper Resonance Frequency 176.1 381.8 Match 3.15 dB 9.99 dB Table 4 1 Resonance frequencies and match for the second surface coil prototype 4.5 Discussion 4.5.1 Three component matching network Althou gh not necessarily bad by themselves the additional resonances produced by this circuit indicate that there are large effects whi ch were not accounted for in the theory presented above. Considering the results of the transmission probe test it was determined that these additional resonances were likely due to the large loop area and extra inductances seen by the circuit as a result of the long traces. Long traces and large loop areas are somewhat inherent to the use of variable capacitors because of the ir size. This effect is exacerba ted by the fact that at least four degrees of freedom are needed to be able to properly tune at match and two different frequencies. The presence of four variable capacitors makes for a prototype board with large parasitics. This was the motivation behind the decision to test a fixed component design when implementing the new four component matching circu it. 4.5.2 Four component matching network The results of the second prototype were promising. The size of the PCB for this design was greatly reduced in comparison to its predecessor. The lack of any additional resonances beyond the expected two indicates that t he problems associated with the previous iteration were likely due to the board size and trace length.

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37 Although the resonances are not at the predicted frequencies and do not match appropriately, the results shown suggest that it may be possible to constru ct a multiple frequency surface coil at such high frequencies if variable capacitors could be reintroduced to this circuit. It is possible that more advanced RF PCB engineering could help solve this problem, such as the use of microstrips and ground planes It appears as though the major contributing factor to the issues plaguing this design stem from RF parasitics inherent to prototype PCBs.

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38 CHAPTER 5 CONCLUSION S AND FUTURE WORK 5.1 Conclusions The work presented has shown that there is strong potential for the development of a multiple frequency, inductively coupled coil system. The design of the implantable coils, although somewhat difficult to manufacture reliably, is sufficient to provide a high Q circuit for coupling to a surface coil. Although a functioning surface coil which could tune and match at both frequencies of interest was not achieved, results indicate that there exists a strong theoretical foundation for its continued development. 5.2 Future Work Future work on the project should revolve around advanc ing both theory and practical application relating to the surface coil. It would be helpful to derive a theoretical estimation of the magnetic field produced by the surface coil, while in the over coupled state, at the point of interest, namely the center of the implantable coil. This theory could be simulated using CST Microwave Studio and eventually experimentally tested in the lab. Further RF board development would also be of great benefit to the project. It is likely that the primary issues arising in the design of the surface coil are derived from high frequency RF PCB effects. This can be mitigated, as was shown, by reducing the size of the board, but it is a requirement of the project that the design include variable capacitors. Reintroducing them to the design will inevitably increase the board size and trace lengths, likely to a level unacceptable for our application. More advanced RF design techniques should be investigated for mitigating the effects of RF parasitics.

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39 CHAPTER 6 LIST OF REFERENCES 1. Centers for Disease Control and Prevention. National diabetes fact sheet: general information and national estimates on diabetes in the United States, 2007. Atlanta, GA: U.S. Department of Health and Human Services, Centers for Disease Control and Preventi on, 2008. 2. Stabler CL, Long RCJ, Constantinidis I, Sambanis A. Noninvasive measurement of viable cell number in tissue engineered constructs in vitro using 1H NMR spectroscopy. Tissue Engineering 2005;11(3 4):404 414. 3. Long RCJ, Papas KK, Sambanis A, Const antinidis I. In vitro monitoring of total choline levels in a bioartificial pancreas: 1H NMR spectroscopic studies of the effects of oxygen level. Journal of Magnetic Resonance 2000;146(1):49 57. 4. Papas KK, R. C. Long J, Constantinidis I, Sambanis A. Role of ATP and Pi in the mechanism of insulin secretion in the mouse insulinoma TC3 cell line. Biochemistry Journal 1997;326:807 814. 5. Constantinidis I, Long RCJ, Weber C, Safley S, Sambanis A. Noninvasive monitoring of a bioartificial pancreas in vitro and i n vivo. Annals of the New York Academy of Sciences 2001;944:83 95. 6. Koretsky AP, Wang S, Murphy Boesch J, Klein MP, James TL, Weiner MW. 31P NMR spectroscopy of rat organs, in situ using chronically implanted radiofrequency coils. Proceedings of the National Academy of Sciences USA: Biochemistry 1983;80:7491 7495. 7. Schnall MD, Barlow C, Subramanian VH, Leigh JSJ. Wireless implanted magnetic resonance probes for in vivo NMR. Journal of Magnetic Resonance 1986;68:161 167. 8. Wirth EDI, Mareci TH, Beck BL, Fitzsimmons JR, Reier PJ. A comparison of an inductively coupled implanted coil with optimized surface coils for in vivo NMR imaging of the spinal cord. Magnetic Resonance in Medicine 1993;30:626 633. 9. Silver X, Ni WX, Mercer EV, Beck BL, Bos sart EL, Inglis B, Mareci TH. In vivo 1H magnetic resonance imaging and spectroscopy of the rat spinal cord using an inductivelycoupled chronically implanted RF coil. Magnetic Resonance in Medicine 2001;46:1216 1222. 10. Bilgen M, Elshafiey I, Narayana PA. In vivo magnetic resonance microscopy of rat spinal cord at 7 T using implantable RF coils. Magnetic Resonance in Medicine 2001;46(6):1250 1253. 11. Volland N. Sensitivity Improvement of a Nuclear Magnetic Resonance Method to Monitor a Bioartificial Pancreas. P h.D. dissertation, University of Florida; 2009.

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40 12. Schnall MD, V. HS, J. S. Leigh J, B. C. A new double tuned probe for concurrent 1H and 31P NMR. Journal of Magnetic Resonance 1985;65:122 129. 13. Fitzsimmons JR, Brooker HR, Beck BL. A comparison of double tun ed surface coil. Magnetic Resonance in Medicine 1989;10:302 309. 14. Lentzen B. Development of an Implantable Multiple Frequency Antenna System for MRI/S of a Bioartificial Pancreas. Honors Thesis, University of Florida; 2008. 15. Terman F. Radio Engineering Second Edition. McGraw Hill; 1937. p 79 16. Viohl I, Gullberg G. Tuning and Matching Networks for MR Imaging and Spectroscopy. Journal of Magnetic Resonance Imaging 1994;4:627 630 17. Simpson N, Khokhlova N, Oca Cossio J, Constantinidis I. Insights into the role of anaplerosis in insulin secretion: a 13 C NMR study Diabetologia 2006;49:1338 1348. 18. Volland N A Mareci T H, Constantinidis I, Simpson NE. Development of an inductively coupled MR coil system for imaging and spectroscopic analysi s of an implantable bioartificial construct at 11.1 T Magnetic Resonance in Medicine 2010;63:998 1006.