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Each original is also photo g rap h ed in o ne exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. H igher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UM I directly to order. U niversity M icrofilm s International A Bell & H owell Inform ation C o m p a n y 3 0 0 N orth Z e e b R oad . A nn Arbor. Ml 4 8 1 0 6 - 1 3 4 6 U SA 3 1 3 /7 6 1 - 4 7 0 0 8 0 0 /5 2 1 - 0 6 0 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. O r d e r N u m b e r 9308629 C atheter ablation of the heart using microwave energy Mirotznik, Mark Steven, Ph.D. University of Pennsylvania, 1992 UMI 300 N. ZeebRd. Ann Arbor, MI 48106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Catheter Ablation of the Heart Using Microwave Energy Mark S. Mirotznik in B ioengineering Presented to the Faculties o f the U niversity o f P ennsylvania in Partial F ulfillm ent o f the R equirem ents for the D egree o f D octor o f Philosophy 1992 \£cxJ—<,h /''VH (Signature) Supervisor of Dissertation (Signature) Co-Supervisor oi^Dissertation 1 / , 1 Graduate Grot p Chairperson . _ (Signature) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ii Dedication This thesis is dedicated to my parents, Alvin and Charlotte Mirotznik. Their love and support over the last 27 years have allowed me to achieve more than I ever believed possible. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgement I extend ray most sincere appreciation to ray advisor, Dr. Kenneth R. Foster, for his continual support and encouragement throughout the completion of this work. Ken has been a caring advisor and a genuine friend. I would also like to thank my co-advisor, Dr. Nader Engheta. Dr. Engheta's positive outlook and sincere enthusiasm for electromagnetics are a source of endless inspiration. I also extend sincere thanks to my labmates; Dr. Jonathan Leonard and Dr. Amanda Osborne and to the bioengineering staff, Gail, Nancy, Lisa, Monica, Kate and Bill for providing an environment of warmth and friendship from which creativity blossoms naturally. To my roommates Tom, Yale, Casey and Mookie, I am truly thankful. Their good humor and friendship have made living in West Philadelphia a genuinely enjoyable experience. I am particularly indebted to my girlfriend, Rukki, for the countless hours spent proof reading and editing this thesis, while simultaneously being a constant source of encouragement and love. For this I am truly grateful. I would also like to thank Microwave Medical Systems and Arrow International for loan of equipment and financial support. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. iv Abstract Catheter Ablation of the Heart Using Microwave Energy Author: Advisor: Mark S. Mirotznik Kenneth R. Foster, Ph.D Co-Advisor: Nader Engheta, Ph.D The heating properties of helically coiled antennas, for use in a catheter ablation system, are studied both experimentally and theoretically. A theoretical model, based on the sheath helix approximation, is presented and used to predict the antenna's specific absorption rate (SAR) pattern as a function of the geometry of the antenna and the electrical properties of the surrounding tissue. This model is then extended to include the case of an insulating layer. In addition, a thermistor based SAR mapping apparatus was constructed and used to perform experimental studies on helical antennas immersed in aqueous electrolytes of various conductivities. Analytical results agree well with the experimental data, demonstrating the validity of the model. For these antennas, the SAR distribution strongly reflects the presence of standing waves along the antenna. These patterns are found to be particularly sensitive to the helical pitch angle and loss in the external medium. It is shown that, by adding a thin Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. layer of insulation to the outside of the helical antenna, one can produce a more uniform heating pattern which is insensitive to loss. This configuration appears to be suitable for catheter ablation applications. The microwave results are then compared to analytical and experimental results from a radio frequency (RF) ablation device. It is shown that the helical antenna offers the possibility of relatively uniform heating, whereas the RF device heats predominantly at its tip. In vitro experiments are performed in excised sheep hearts. Lesion sizes measured in actual tissue samples agree favorably with calculated responses. These results graphically illustrate that microwave ablation is able to produce larger lesion sizes than presently available techniques. This ability may prove useful in the catheter treatment of a variety of cardiac arrhythmias. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents Page Chapter 1: Introduction 1.1 Scope of this work 1.2 Overview of Thesis 1 2 3 Chapter 2: Background in Catheter Ablation 2.1 Cardiac Arrhythmias Wolff-Parkinson-White Syndrome Supraventricular Tachycardia Ventricular Tachycardia 2.2 Catheter Ablation Techniques DC- Electric Shock Ablation Radio Frequency Ablation Microwave Ablation 2.3 Helical Coil Antennas 5 5 6 7 8 9 10 11 14 15 Chapter 3: 3.1 3.2 3.3 3.4 20 20 23 28 29 Theoretical Background Physical Principles of Microwave Heating Microwave Dielectric Properties of Tissue Microwave Dielectric Properties of Saline Water Solutions Temperature Dependance of Complex Permittivity Chapter 4: Analytical Methods 4.1 Analytical Antenna Modeling 4.1.1 Uninsulated Helical Antenna Helical Sheath Model Formulation Application of Boundary Conditions Electric and Magnetic Field Solutions Determinantal Equation Effect of Feedpoint and Termination Total Electric and Magnetic Fields Summary 4.1.2 Insulated Helical Antenna Antenna Model Formulation Application of Boundary Conditions Electric and Magnetic Field Solutions Determinantal Equation 32 32 32 34 37 42 43 47 52 60 61 62 64 64 67 68 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. vii 4.2 4.3 Chapter 5: 5.1 5.2 5.3 Chapter 6: Effect of Feedpoint and Termination Summary Simple Analytical Model of RF Ablation Summary Analytical Results Results from Uninsulated Helical Sheath Model 5.1.1 Dispersion Characteristics Numerical Method General Properties of the Solutions Numerical Results Summary 5.1.2 Electric Field Distributions 5.1.3 SAR Distributions Results from Insulated Helical Sheath Model 5.2.1 Dispersion Characteristics Numerical Results 5.2.2 Electric Field Distributions 5.2.3 SAR Distributions Summary Experimental Methods 6.1 Measurement of Specific AbsorptionRate (SAR) 6.1.1 SAR Mapping Apparatus 6.1.2 Data Analysis 6.2 Measurement of Input Impedance 6.3 Measurement of Microwave Dielectric Properties Chapter 7: 7.1 7.2 7.3 7.4 72 72 73 78 Experimental Results Helical Antennas 7.1.1 Uninsulated Helical Sheath Antenna SAR Distributions Input Impedance 7.1.2 Insulated Helical Sheath Antenna SAR Distributions RF Ablation Measurements Dielectric Measurements In Vitro Measurements 80 80 81 81 83 85 89 91 97 104 104 105 108 110 117 118 118 118 124 130 133 138 138 139 139 148 151 151 158 162 162 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 8: Conclusion Summary of Results Significance of Results Future Studies 171 171 172 173 Appendix A: Heat Transfer Analysis of Thermistor Based SAR Mapping 174 Apparatus Appendix B; Expansion Coefficients in Open Ended CoaxialProbe Model 180 References Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 181 ix List of Tables Table 7.1: Table 7.2: Helical antennas used for experimental measurements Complex permittivity of NaCl solutions Page 139 140 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Figures Figure 2.1: Figure Figure Figure Figure 2.2: 2.3: 2.4: 2.5: Figure 3.1: Figure 3.2: Figure 3.3: Figure 3.4: Figure 4.1: Figure 4.2: Figure 4.3: Figure 4.4: Figure 4.5: Figure 4.6: Figure 4.7: Figure 5.1: Figure 5.2: Figure 5.3: Figure 5.4: Figure Figure Figure Figure 5.5: 5.6: 5.7: 5.8: Typical reentrant tract formation which leads to ventricular tachycardia Typical catheter used for DC electric shock ablation Typical radiofrequency (RF) ablation system Typical microwave catheter ablation device Illustration of a typical helical antenna studied in this work Page 7 10 12 14 16 The real part of the complex permittivity and conductivity 24 versus frequency Dielectric properties of muscle and blood as a function of 26 frequency Dielectric constant and conductivity of aqueous NaCl 27 solution, muscle and blood as a function of frequency [Foster and Schwan, 1986] Conductivity of 0.8% NaCl as a function of frequency for 31 temperatures ranging from 25 to 55°C. Helical antenna immersed in an external lossy medium Helical sheath model of antenna immersed in an external lossy medium Illustration of forward and backward traveling waves along sheath helix antenna model Insulated helical antenna immersed in an external lossy medium Insulated helical sheath model of antenna immersed in an external lossy medium Model of an RF ablation device An approximate model of Figure 4.6 Dispersion characteristics of slow and fast mode lqb vs.(3b Dispersion characteristics of slow and fast mode propagation constant vs. pitch angle Dispersion characteristics of slow and fast mode propagation constant vs. sheath radius Illustration of the antenna geometry used for the electric field calculations Magnitude of electric fields at various frequencies Magnitude of electric fields at various pitch angles Magnitude of electric fields for various loss tangents Illustration of the antenna geometry used for the SAR calculations with permission of the copyright owner. Further reproduction prohibited without permission. 33 36 52 63 65 74 76 86 88 90 92 93 95 96 97 Figure 5.9: Normalized SAR distribution at various pitch angles Figure 5.10: Normalized SAR distribution for various loss tangents Figure 5.11: An illustration of the insulated helical sheath model used for the dispersion calculations Figure 5.12: Dispersion characteristics of slow and fast mode of insulated antenna vs. insulation thickness Figure 5.13: An illustration of the antenna geometry used for the electric field calculations of the insulated antenna Figure 5.14: Magnitude of electric field for insulated antenna for various insulation thicknesses Figure 5.15: Illustration of geometry used in the SAR calculations for the insulated antennas Figure 5.16: Normalized SAR distribution of insulated antenna for various insulation thicknesses Figure 5.17: Comparison of normalized SAR distributions for insulated and uninsulated helical antennas in various lossy media 100 103 105 Figure Figure Figure Figure Figure Thermistor based SAR mapping apparatus Positioning table Thermistor probe used for SAR measurements Electronic circuit used to record thermistor temperature Heat transfer model used to subtract the thermal artifact out of the thermistor temperature measurement Illustration of how the measured thermistor temperature will differ from the actual temperature of the external medium Measured temperature vs. time curves for several different input power levels The initial and second slopes identified in Figure 6.7 vs. input power Reference plane for input impedance measurements Experimental setup used to measure input impedance Experimental setup used to measure complex dielectric constant of tissue at microwave frequencies. The measurement is based on the open-ended coaxial probe technique 119 121 122 123 125 Measured and calculated SAR distributions for antennas of various pitch angles in distilled water Measured and calculated SAR distributions for Antenna #3 (in Table 7.1) in distilled water and various saline solutions Measured and calculated SAR distributions for Antenna #4 (in Table 7.1) in distilled water and various saline solutions Measured resistive and reactive components of the input impedance vs. frequency for antennas with various pitch angles immersed in 0.8% saline 142 6.1: 6.2: 6.3: 6.4: 6.5: Figure 6.6: Figure 6.7: Figure 6.8: Figure 6.9: Figure 6.10: Figure 6.11: Figure 7.1: Figure 7.2: Figure 7.3: Figure 7.4a: 107 108 109 110 113 116 127 128 129 130 132 134 145 147 149 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 7.4b: Figure 7.5: Figure 7.6: Figure 7.7: Figure 7.8: Figure 7.9: Figure 7.10: Figure 7.11: Figure 7.12: Figure 7.13: Figure 7.14: Figure A.1: Measured resistive and reactive components of the input 150 impedance vs. frequency for antennas with various pitch angles immersed in 0.4% saline 154 Measured and calculated SAR distributions of uninsulated and insulated antennas in distilled water for Antennas #1,#3 and #4 (in Table 7.1) Measured and calculated SAR distribution of uninsulated and 157 insulated antennas in 0.8% NaCl solution for Antennas #3 and #4 Illustration of the RF catheter geometry used for the SAR 158 measurements Measured and calculated SAR distributions of the RF catheter 159 in 0.4% and 0.8% NaCl solution Measured and calculated SAR distributions for the RF catheter 161 and the insulated helical antenna (Antenna #3) in 0.8% NaCl solution The experimental setups used for microwave and RF ablation 163 studies Illustration of lesion geometry used in lesion size measurements 165 167 Measured lesion de"th and SAR pattern for Antenna #1 with and without insul*. Measured lesion dt and SAR pattern for Antenna #3 with 168 and without insulaL i Measured lesion depth and SAR pattern for the insulated 170 Antenna #3 and the RF catheter Spherical thermistor immersed in a homogenous lossy medium and exposed to a uniform electric field Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 174 List of Symbols C h ap ter 3: 1. 2. 3. 4. 5. 6. a e = effective electrical conductivity of lossy media p = tissue density e* = complex permittivity to = radian frequency E = electric field vector SAR = specific absorption rate C h ap ter 4: Noninsulating Antenna 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14 15. e, = permittivity of interior region of helical antenna ee = permittivity of lossy exterior region ofhelical antenna a = radius of inner core of helical antenna b = outer radius of helical antenna a = helical pitch angle p = helical pitch L = length of helical antenna ne= electric Hertz vector 11”= magnetic Hertz vector I,, = modified Bessel function of the first kindof order n K„ = modified Bessel function of the second kindof order n I, = total axial current flow (5 = propagation constant lq = wavenumber of interior region kj = wavenumber of exterior region 16. u = V(P2 - k,2) 17. v = V(P2 - kg2) Insulating Antenna 18. 19. 20. 21. 22 23. 24. d = thickness of insulation £t= permittivity of interior region 62= permittivity of insulating region £3= permittivity of exterior lossy region lq = wavenumber of interior region kj = wavenumber of insulating region k3 = wavenumber of exterior lossy region Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xiv 25. 26. 27. u = V(p2 - kj2) x = V(p2 - k22) v = V(p2 - k32) C hapter 5: 1. tan(y) = loss tangent of external lossy medium C hapter 6: 1. 2. 3. 4. 5. 6. 7. 8. 9. 0,,, = electrical conductivity of thermistor k*,, = thermal conductivity of thermistor otu, = thermal diffusivity of thermistor c m = electrical conductivity of outside medium ]£„, = thermal conductivity of outside medium pm = density of outside medium Z0 = characteristicimpedance of transmission line Zfc = input impedance tohelical antenna = power reflection coefficient Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 Chapter 1: Introduction In the past, pharmacological and surgical treatments were the only alternatives for diseases such as cardiovascular disease, colorectal cancer and endobronchial cancer. Surgery for some of these patients, depending on the severity of their illnesses, posed a considerable risk. Over the last 50 years, attempts to treat high risk patients without subjecting them to major surgical intervention have led to a range of nonsurgical techniques using catheters. One promising area of investigation is the treatment of certain cardiovascular diseases resulting from arrhythmias. The idea is to use a catheter to ablate (destroy by heating) the diseased cardiac tissue near the catheter tip. Recent advances in the techniques of electrode catheter mapping have allowed the catheter to be localized near arrythmogenic cardiac tissue. This tissue can then be ablated to produce the desired therapeutic effect. A range of catheter ablation techniques have been explored, including the use of direct current (DC) electric shocks, cryotherapy, radiofrequency energy (RF) and lasers. The common goal of all these techniques is to destroy abnormal tissue sites while sparing neighboring healthy tissue. The effectiveness of any of these techniques depends on the type and location of the arrythmogenic site The aim of this thesis is to complete a careful study into the design and application of a relatively new catheter ablation technique which uses microwave Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 energy. The goal is to design a microwave antenna that is able to ablate larger volumes of tissue than is currently possible with other catheter ablation techniques. This ability may allow catheter treatment of a class of arrythmias (ventricular tachycardia) which is untreatable with current catheter ablation techniques. 1.1 Scope of this work The ability of a microwave ablation device to produce a desired heating pattern depends on a variety of factors. These include the geometry and operating frequency of the antenna and the electrical properties of the tissue. Presently, however, microwave ablation is still in its infancy and most of these factors have not been carefully studied. Moreover, a careful comparison between microwave and radiofrequency ablation is needed to address the relative merits of the former. To better understand the significance and limitations of microwave ablation, it is necessary to have a clearer understanding of the heating characteristics of the microwave ablation antenna. This study focuses on a particular antenna design, the helically coiled antenna, whose heating properties have not been previously understood. This work answers the basic question of how the geometry of the antenna, the electrical properties of the tissue and the frequency of operation effect the heating characteristics of the antenna. Additionally, I compare the heating patterns of microwave antennas with those of commercially available RF devices. This research follows the sequence: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1) Present an analytical model for insulated and uninsulated helical antennas immersed in a general lossy medium. 2) Experimentally study the heating characteristics of helical antennas to validate the analytical models. 3) Perform in vitro studies on excised sheep hearts, to qualitatively estimate the lesion sizes microwave ablation is capable of producing. 4) Compare the analytical and experimental data from the microwave ablation results to those from commercially available RF catheters. This comparison allowed me to evaluate the relative merits of the different techniques. 1.2 Overview of Thesis The remainder of this thesis presents the results obtained from the above study sequence. Chapter 2 presents the physiological and technical background underlying catheter ablation. It includes a survey of previously investigated catheter ablation techniques including studies in microwave ablation. Also included is a detailed description of the helical antennas studied in this work. The next chapter addresses the theoretical background of microwave ablation. A presentation of the physical principles underlying microwave heating is followed by a discussion of the electrical properties of cardiac tissue at microwave frequencies. Chapter 4 presents the analytical models used during this study. It begins with analytical models for the helical antenna with and without insulation followed by a simple analytical model for commercially available RF catheters. In Chapter 5, the theoretical results which follow Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the analytical antenna models is presented. These results consist of the dispersion characteristics, the electric Held distributions and the SAR patterns for the uninsulated and insulated helical antenna. Chapter 6 describes the experimental techniques used to validate the analytical models. After the analytical and experimental methods have been explained, Chapter 7 presents and discusses the experimental results obtained. From these results, Chapter 8 draws pertinent conclusions regarding the utility of microwave ablation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 Chapter 2: Background in Catheter Ablation This chapter addresses some of the medical and technical aspects of catheter ablation. The first section describes a few of the most common cardiac arrhythmias for which catheter ablation techniques have been used. The next section describes some of the most commonly investigated catheter ablation techniques. The last section discusses the microwave antenna studied in this work, the helical coiled antenna. 2.1 Cardiac Arrhythmias Cardiac arrhythmias are, by definition, any variation from normal cardiac rhythm. This definition encompasses a wide variety of cardiac diseases, some of which have been successfully treated using catheter ablation techniques. In this section, a few of the most common cardiac arrhythmias which are treatable using catheter ablation are described. In addition, there is a description of a class of arrhythmias, termed ventricular tachycardia, which are presently not treatable using catheters. Wolff - Parkinson - White Syndrome The Wolff-Parkinson-White syndrome (WPW) is a functional disorder affecting many young people in the United States1. This disorder, it is believed, stems from an Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. accessory muscle bundle (bundle of Kent) connecting the atria to the ventricles. Rapid depolarization over this bundle results in ventricular preexcitation. Consequently, WPW patients have frequent attacks of supraventricular or even ventricular tachycardia. Conventional treatments, such as open heart surgery or drug therapy, are found to have either too high a risk or undesirable side effects. This has motivated the use of catheter ablation. With this technique, the ablation catheter first locates the accessory pathway and then deposits enough energy to destroy it. The accessory pathways are typically thin muscle bundles lying anywhere within the myocardium. Consequently, the ablation catheter needs only to produce a small discrete lesion large enough to reach this pathway. Presently, RF catheters are in wide clinical use in the treatment of WPW syndrome. They have an extremely high success rate ( > 90 %) and do not subject the patient to unnecessary risk2,3,4. Supraventricular Tachycardia This class of arrhythmias includes all tachyarrhythmias which originate above the ventricle. While the mechanism behind the tachyarrhythmia varies depending on the exact nature of the disease, the treatments have certain similarities. For many cases of supraventricular tachycardia, the solution is to ablate the AV node. By doing, this the physician effectively separates the conducting system between the atria and the ventricle. Hence ensuring that the supraventricular tachycardia does not interfere with the otherwise normal contraction of the ventricle. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A variety of catheter ablation techniques have been introduced to ablate the AV node in patients with supraventricular tachycardia5,6,7. These techniques, like those used in the treatment of WPW, need only produce a small discrete lesion at the AV node to successfully disrupt conduction. Techniques using DC-electric shocks and RF energy have been used and are discussed in more detail below. Ventricular Tachycardia Of all the arrhythmias mentioned here this is the most life threatening and the most difficult to treat. Each year, several hundred thousand people in the United States die of ventricular fibrillation (sudden death), often the result of ventricular tachycardia (VT). This condition is primarily due to rapid impulse formation in a ventricle. Ventricular contractions during VT are generally greater than 100 beats per Infarct Normal Pathway Unidirectional Block Delayed Conduction Proximal Reexcitation Figure 2.1 Typical reentrant tract formation which leads to ventricular tachycardia. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. minute and may be as high as 150 to 175. If sustained, VT can degenerate to ventricular fibrillation and death. This tachycardia often results from an impulse repetitively circling a reentry loop. Figure 2.1 shows a typical reentrant tract formation which can lead to VT. It is presently believed that abnormal tissues which border transmural infarcts are a major source of this formation. Electrophysiological studies have shown that reentrant circuits require the formation of separate conduction paths around a transmural infarct8, with a unidirectional block and a slowed impulse conduction velocity in one pathway. This combination can lead to reexcitation of the proximal node and, consequently, an impulse which repetitively circles around the reentrant circuit9. Long-term management of patients with recurrent ventricular tachycardia is a challenging therapeutic problem. Usually, antiarrhythmic drugs are initially prescribed, but many patients either fail to respond or cannot tolerate the drug therapy. Invasive therapies include open-chest surgical intervention or the placement of automatic implantable cardioverter defibrillator (AICD). The surgical procedure is designed to excise or ablate the reentrant pathways whereas the implantable defibrillator is used to shock the heart into a normal sinus rhythm when the onset of tachycardia is detected. The success rate of these procedures is approximately 15% with a relatively high operative mortality rate of 7-9%10. Moreover, due to additional medical problems, a significant subset of VT patients are not ideal candidates for either procedure. Consequently, a relatively noninvasive therapy such as catheter ablation is highly desirable. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A number of groups have reported varying success in their attempts at catheter ablation of VT foci111213. For this procedure, multipolar electrode catheters are inserted into the right or left ventricle. VT is induced by using standard stimulation protocols and the catheters are manipulated within the ventricles to determine the exact foci of the VT. Once this foci has been located, the ablation catheter is manipulated to that location and used to apply large amounts of energy directly at the tachycardia foci. Previous studies using low frequency (DC) electric shocks or RF energy have shown modest success rates (< 50%) with several additional complications11,12,13. Because of these drawbacks the procedure has not gained clinical acceptance. It is believed, among other things, that presently available ablation devices are incapable of producing lesion sizes necessary to successfully treat this disorder. Unlike the situation with WPW treatment or AV nodal ablation, ventricular tachycardia foci can be quite large, and can necessitate an ablation area which is larger than currently available devices are able to produce. This is the primary motivation for investigating microwave ablation devices. 2.2 Catheter Ablation Techniques A variety of catheter ablation techniques have been introduced for the management of patients with cardiac arrhythmias. These techniques include the use of large direct current (DC) electric shocks, radiofrequency (RF) energy and microwaves. In this section, each of these techniques is examined in detail. I also include a review of previously reported studies which use these techniques. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. DC - Electric Shock Ablation The first catheter ablation procedures used low frequency electric shocks14. Figure 2.2 shows a typical catheter used in these initial experiments. As shown in the figure, the catheter is placed along the endocardium and large DC - electric shocks, ranging from 30-200 J of energy, are applied via two catheter electrodes. In most cases, two additional EKG mapping electrodes are included. These electrodes are used to help the physician locate the arrhythmogenic site. Tissue V Mapping Electrodes Figure 2.2 Typical catheter used for DC - electric shock ablation. Initial approaches used high-energy DC shocks, with good results, to ablate the AV node in patients with supraventricular tachycardia. Data accumulated from a world-wide voluntary catheter ablation registry reports over 500 attempted catheter ablations of the AV node using this method15. Overall, the results show that arrhythmia control was achieved in 85% of the patients. High-energy shocks were also applied, with moderate success, to ablate accessory pathways in patients with Wolff-Parkinson-White Syndrome. Fisher et al. delivered shocks, ranging from 40 to 150 J, to 20 patients with WPW16. Complete arrhythmia control, without need for antiarrhythmic drugs, was achieved in 75% of the patients. However, the use of DC shock has been associated with significant complications. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. These include induction of arrhythmias17, thromboembolism18, transmural necrosis and cardiac perforation16. The DC energy is also difficult to control and is associated with a high incidence of catheter damage19. Attempts at treating ventricular tachycardia using DC-electric shock ablation have been met with a limited success rate. The most comprehensive study reporting the results of this comes from a voluntary international registry20. In 164 patients with VT, the overall incidence of complete tachycardia cure (without the need for supplemental antiarrhythmic drugs) was 18%. An additional 41% improved with a supplement of antiarrhythmic drug therapy. Radio Frequency Ablation The moderate success of electric shock ablation encouraged investigators to search out alternative energy sources that could be incorporated into a catheter ablation system. The use of radiofrequency (RF) energy was the next major advance in catheter ablation technology. RF ablation was introduced because the energy delivered can be more accurately controlled and the lesions produced can be well circumscribed. Several groups, employing a variety of techniques, have described catheter ablation with radiofrequency. For example, Huang et al. used 750 kHz RF energy from an electrosurgical generator in the bipolar mode (exactly analogous to the DC-shock ablation catheter except operating at a much higher frequency, see Figure 2.2) to ablate the AV node in dogs21. Complete heart block occurred in most of the dogs used in these studies. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. More recent studies of RF ablation in humans have used standard electrophysiological catheters operating in a unipolar mode22,23,24. Figure 2.3 shows a typical RF catheter ablation system operating in the unipolar mode. T ic c i ip Ground Plane Mapping Eledrodi Figure 2.3 Typical radiofrequency (RF) ablation system. The RF currents (typically between 350 kHz - 1.0 MHz) are passed between one electrode placed at the distal end of the ablation catheter and a ground electrode fixed on the patient's back. Jackman et al. “ used this arrangement in the treatment of 166 patients with Wolff-Parkinson-White syndrome. In 164 of these, accessory-pathway conduction was eliminated resulting in a successful treatment. Complications from the application of RF energy occurred in only three patients. There were no associated mortalities. It became apparent that RF ablation had many distinct advantages over DCelectric shock: 1) RF currents (unlike the low-frequency currents) are unable to excite cells and thus cannot produce painful tetanic muscle stimulation ; 2) RF currents cannot directly induce fibrillation; 3) The RF lesion’s location and size are more Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. controllable and, therefore, the risk of perforating the ventricular wall is remote. Presently, RF ablation is in wide clinical use. However, due to the nature of the RF lesions (small discrete lesions concentrated near the tip of the catheter) the procedure has been limited to bypass tract ablations. These ablations include accessory pathway ablation in Wolff-Parkinson-White Syndrome, ablation of the His bundle and ablation of the AV node. Experience in the treatment of ventricular tachycardia with RF energy has been limited to a small number of patients. Borggefe et a l reported successful radiofrequency ablation in two out of five patients. Morady et al.n recently reported favorable outcomes in only 45% of 31 patients with VT using RF ablation treatments. One of the main problems indicated by Morady and Borggefe is the exact localization of the arrhythmogenic focus, i.e., the reentry circuit of the tachycardia. Since the RF lesions are small discrete lesions, if the catheter is not positioned exactly over the arrhythmogenic focus then the procedure is not successful. Moreover, the patients reported by Morady and Borggefe represent a highly selective group with a single type of monoraorphic tachycardia. This type of tachycardia stems from a single focus. In constrast, in many cases of VT, there may be several foci located near an infarcted region. In these situations, the small lesions produced by an RF catheter make it impossible for a single ablation procedure to eliminate all of the necessary foci. Consequently, several investigators have turned to energy sources which are capable of ablating larger regions of tissue. One promising source is the use of microwave energy. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 Microwave Ablation The idea of using microwave energy to facilitate tissue heating is by no means new. In the hyperthermia treatment of cancer, some invasive (interstitial) radiofrequency and microwave devices have been developed in order to heat deepseated tumors. Several investigators have compared devices using RF energy to those using microwaves26,27,28. In particular, Stauffer et al. showed that microwave devices are capable of heating much larger volumes of tissue than the RF devices. From these earlier studies evolved the idea of incorporating microwave energy into a catheter ablation device. A typical microwave catheter is shown in Figure 2.4. The microwave energy, typically 915 MHz or 2.45 GHz, is released via a microwave antenna placed at the catheter tip. The objective is to produce a uniform heating pattern along the entire 1 in n itrs Volumetric Heating Figure 2.4 Typical microwave catheter ablation device. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. length of the antenna, such that a large region of tissue can be ablated. Previous work in microwave catheter ablation is limited to only a few studies29,30. In particular, Landberg et a l recently reported the use of a microwave catheter for AV nodal ablation in dogs. Landberg used a modified helical coiled antenna, operating at 2.45 GHz, to successfully block AV nodal conduction in 6 out of 6 dogs used in this study. To date, the use of microwave catheters in the treatment of VT has not been studied. Studies using microwave catheters have been hampered by a lack of good theoretical models to predict the interaction between the antenna and lossy tissue. This work provides a theoretical model for a particular antenna design, a helically coiled antenna. This antenna (similar to the one used by Landberg) is described in the next section. 2.3 Helical Coiled Antennas Helical antennas are well known for their applications in communications, where they have been studied for over 50 years. Quite a different application involves their use for heating. Little work has been done to characterize such helical antennas in lossy media, and most previous reports are experimental in nature. Satoh et al?1 along with Wu and Carr32 described the heating patterns of several helical antennas used as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 short circuit termination Helically wound wire inner core dielectric outer shield coaxial transmission line Figure 2.5 Illustration o f a typical helical antenna studied in this work. The antenna is fabricated by wrapping a thin wire around the exposed dielectric of a coaxial transmission line in a helical fashion. The wire is then soldered to the inner core of the coaxial line at the distal tip of the antenna (short circuit termination) and fixed to the outer shield at its base. The pitch angle a is used to define the angle between the helical wire and the plane normal to the antenna axis. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. interstitial hyperthermia applicators. Figure 2.5 shows a typical helical antenna design used by Satoh and Wu. The details of this antenna are presented in the Analytical Methods chapter. The results from these experimental studies show a relatively uniform heating pattern along the length of the antenna with a shallow depth of heating. For their application of hyperthermia, where the goal is to achieve the greatest depth of heating, other antenna designs are preferable. For catheter ablation, where great depth of heating is often neither required nor desirable, helical antennas may be preferable. However, Satoh and Wu did not present a comprehensive theoretical or numerical analysis of the antenna characteristics as functions of helical pitch or other design parameters. There is a need for a more general analysis of such antennas, in order to describe their heating characteristics as functions of antenna geometry and dielectric properties of the lossy medium. There is a considerable amount of literature relevant to the theoretical problem. However, due to the complicated boundary conditions imposed by the actual wire wound helix, all studies have relied on approximate models. One such model, originally studied by Sensiper33, approximated the actual wire wound helix by an anisotropically conducting sheath. The sheath is assumed to be perfectly conducting at some angle a parallel to the actual helical wire and perfectly insulating normal to that direction. Sensiper obtained the field distributions and dispersion characteristics assuming the antennas were radiating into free space. The antennas studied by Sensiper did not include an inner core and were assumed to be infinite in length. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 Neureuther et al?* extended Sensiper’s model to include a conducting core (the case of interest here). He also studied a more physically realistic, but mathematically difficult, model consisting of a perfectly conducting spiral tape, and showed that the sheath model was an excellent approximation. Neureuther et al., like Sensiper, assumed the antennas were radiating into free space and were infinite in length. Perini35 carried out an extensive theoretical and experimental study of helical antennas. He used a spiral tape model with conducting core to compare calculated results to experimental measurements. He was primarily concerned with the antenna's radiation pattern and, consequently, only considered antennas radiating into free space. He found that the results obtained under the assumption of an infinite length helix did not agree satisfactorily with those determined experimentally. Perini suggested that more accurate calculations could be obtained by taking into account end reflections. However, he did not include these reflections in his model nor, to the best of my knowledge, has any other investigator. More recently, Hill and Wait36, carried out an elegant theoretical study on wave propagation along coaxial cables with helical shields. Their model consisted of a dielectric coated conductor which was shielded by a finite number of helices. Utilizing a modal expansion technique, they solved for the propagation characteristics of waves traveling along their antennas. They were particularly interested in the leaky feeder technique used to provide radio communication in mine tunnels. Although that is quite different from the application of interest here, the method used by Hill and Wait is utilized and extended to the case of lossy medium in this work. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 Only a few investigators have studied the helical antenna immersed in a lossy medium and these studies have been numerical in nature. Chen37 developed an integral equation for an infinite length helical sheath antenna radiating into a lossy medium. His equations could then be solved numerically for the antenna's current distribution. Casey and Bansal38 extended Chen's work to the case of a finite length helical sheath antenna in a general lossy medium. In addition, they developed a numerical method based on the moment method for solving their integral equation. The antennas considered by these investigators did not include a dielectrically coated inner conductor or the effects of a coaxial feed point and short circuit termination, as is the case here. However, the solutions reported by Casey and Bansal are used here to check limiting cases for the model presented in this work. The models presented in this thesis extends the analytical work of Hill and Wait, Neureuther et al., Sensiper, and others to include the effects of a coaxial feed point, a short circuit termination, an insulating layer and an external lossy medium. These considerations, which are necessary for the present application of microwave catheter ablation, have not, to the best of my knowledge, been studied by any previous investigator of helical antennas. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 Chapter 3: Theoretical Background The mechanisms responsible for myocardial tissue damage which occurs during ablation, depend on the ablation technique used. In RF and microwave ablation the primary phenomenon responsible for cellular damage is excessive heating of tissue. The first section of this chapter describes the physical principles underlying the microwave heating of tissue. The discussion continues with an overview of the dielectric properties of cardiac tissues at microwave frequencies. These properties are then compared to those of saline solutions used to evaluate the antenna. 3.1 Physical Principles of Microwave Heating In heating applications, the quantity of interest is the power, Pd, dissipated in the medium. If we assume a nonmagnetic medium occupies some volume V, then the total dissipated power is given by Pd ‘ } J J c , \ E \ 2d v Warn [3.1] where e is the instantaneous electric field vector in units of V/cm and a e is the electrical conductivity of the medium in units of S/cm. The quantity being integrated, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Pd = o j ^ l 2’ *s 1116 dissipated power density in units of Watts/cm3. If the electromagnetic energy has a sinusoidal time variation (time-harmonic), then the instantaneous electric field vector can be related to its complex phasor equivalent in a very simple manner by E { x M ) = R e [ E ( x ^ at] [32] where e represents the complex phasor form of the electric field, to is the angular frequency ( to = 2 n f) and j= V -l. Throughout this thesis, the time-harmonic case will be assumed and the term e**0*will be omitted for convenience. The dissipated power density can also be related to the complex electric field by P, - k * |£ |2 ~ s cm3 M In Equations 3.1- 3.3, cre represents the total or effective conductivity of a general lossy material. This effective conductivity is actually composed of a static term, cts, due to the ability of free charges to migrate under the influence of an electric field, and an alternating part, a a, caused by the rotation of dipoles as they attempt to align with an alternating applied electric field. The static conductivity contributes to the ohmic or resistivelosses of a material, while the alternating conductivity contributes to dielectric losses. The ratio of resistive to dielectric losses depends onthe material and the frequency of the applied electric field. For most biological tissues, resistive heating dominates up to several MHz after which dielectric effects begin to contribute. At microwave frequencies (f>300 MHz), dielectric losses play a more significant role Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 in the generation of heat. When energy is deposited in tissue faster than it can be dissipated by thermal conduction or convection, the inevitable result is an elevation of temperature. In the early transient regime (where heat conduction and convection effects are negligible) the temperature increase can be related to the applied power density distribution and the thermal properties of the tissue by q(xy& t) = pc at [3.6] where T(x,y,z,t) = Time dependent temperature distribution ( °C) q(x,y,z,t) = Applied power density distribution(Watts/cm3 ) p = Density of tissue (gm/cm3 ) c = Specific heat capacity of tissue (Joule/(gm °C)) Substituting [3.3] into [3.6], the rate of tissue temperature increase is found in terms of the applied electric field and the effective conductivity of the tissue by, SAR = f l ! ! ! ! - Ito v , c <L Z . 2p M) dt gm [3.7] The left side of [3.7] is termed the specific absorption rate (SAR) and is used extensively in biomedical applications to determine the amount of power deposited per unit mass in biological tissue. The SAR distribution specifies the power deposition patterns of antennas used in hyperthermia or ablation applications and is used extensively in this work. At longer times, the temperature distribution will be different Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 from the results shown in [3.7] due to the affects of thermal conduction and convection. Thus, the SAR distribution can be experimentally measured by determining the initial rate of temperature increase directly after the power is applied. 3.2 Microwave Dielectric Properties of Tissue Propagation and absorption of electromagnetic energy within biological tissue depends upon the electrical properties of the tissue. Biological tissue is essentially nonmagnetic, so its permeability p is essentially the same as the permeability of free space, p0 (= 4k x 10 '7 Henrys/meter). However, the complex dielectric permittivity, e*, of tissue is a complicated quantity depending on the type of tissue and the frequency of the applied fields. The complex permittivity is expressed as e* = e0( e'- j e#) where e0 is the permittivity of free space (=8.85 x 10'12 Farads/meter) and j= V -l. The quantity z' is the relative dielectric constant and the imaginary part, z ‘ = oe /coe0, accounts for tissue losses. A large amount of literature exists on the electrical properties of tissue. In particular Foster and Schwan39 gave a thorough review of measured data and discussed some of the mechanisms responsible for changes found in e*. The dielectric constant, e', and conductivity, o e, of most soft tissues are qualitatively similar in there frequency dependence. In Figure 3.1 (from [Foster & Schwan, 1986]) the frequency dependence found in typical tissues is shown. The frequency of interest here (915 MHz) corresponds, for muscle tissue, to the region just before the beginning of the y dispersion. At this frequency, the cell membranes Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 10.0 915 M H z a S/m FREQUENCY. Hz Figure 3.1 Hie real part of the complex permittivity and conductivity versus frequency showing a, p and v dispersions [Schwan & Foster, 1986]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 are shorted out and no longer contribute to the bulk permittivity. Hence, the bulk permittivity at these frequencies, may appropriately be treated as a suspension of proteins and other macromolecules in electrolytic solution. Dielectric properties of tissue depend also on the particular tissue type. For microwave ablation, the tissues of direct interest are cardiac muscle and blood. In Figure 3.2, the electrical properties of canine muscle tissue are compared with the those of whole blood as a function of frequency (data taken from Foster & Schwan 1986). The dielectric constant of blood and muscle show a broad dispersion with a plateau region between 100 MHz and 1.0 GHz. The conductivity shows a slow increase from a low frequency value of 1.0 S/m to a high frequency value at 10.0 GHz, of 10 S/m . Figure 3.2 shows that, at microwave frequencies, blood and muscle have similar electrical properties. This fact justifies the assumption that muscle tissue and blood form an electrically homogenous medium over the frequencies of interest here. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 i° j • Muscle o — Whole Blood b 0.1 — 0.01 1 0.1 10 frequency, GHz 1000 Muscle - Whole Blood s 1 100 a tr 0.01 1 0.1 frequency, GHz Figure 3.2 Dielectric properties of muscle and blood as a function of frequency, [data taken from Foster and Schwan 1986] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 27 • - o -1 CA b frequency, GHz O K OOf ond 1.QX Noa OMftaee frequency, GHz Figure 3.3 Dielectric constant and conductivity of aqueous NaCl solution, muscle and blood as a function of frequency. [ data taken from Foster and Schwan 1986] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 3.3 Microwave Dielectric Properties of Saline Water Solutions To facilitate the placement of temperature and electric field probes, sodium chloride solutions were used as an electrical tissue phantom. In this section the electrical properties, at microwave frequencies, of different concentrations of saline solutions are compared with those of tissue. The complex permittivity as a function of frequency, water temperature and salinity, can be calculated from the equations published by Stogryn40. These equations assume the complex permittivity of saline solutions can be represented by an equation of the Debye form given by € * = € „ + (6* ~ € J - y A . 1 + you coe^ where es and permittivity, [3.8] are the low and high frequency limit, respectively, of the cs is the low frequency conductivity and t is the relaxation time constant. Stogryn obtained equations for the parameters in the Debye expression by interpolating measured data over a broad range of frequency, temperature, and salinity ranges. In Figure 3.3, the dielectric permittivity e' and conductivity cre, of various concentrations of sodium chlorine solutions (at 25°) are compared with those of blood and muscle tissue, as a function of frequency. The figure shows that, between 1.0 and 10.0 GHz, the conductivity of 0.8% NaCl agrees well with the conductivity of blood and muscle tissue. In the same frequency range the dielectric permittivity of the NaCl Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 solutions is slightly higher than that of tissue (by 5 to 10 dielectric units), but varies only slightly with NaCl concentration. Subsequent analysis will show that this small difference does not significantly change the heating patterns for the microwave antennas studied here. 3.4 Temperature Dependance of Complex Permittivity The dielectric properties of tissue is known to vary as a function of temperature. In the frequency range of 0.5 - 5.0 GHz, the conductivity of high water content tissue has a temperature coefficient which varies from 0.5-2%/°C. The dielectric permittivity reflects that of water, which has a very small temperature coefficient. Since the SAR calculations are dependent on conductivity, it is obvious from [3.3] that the SAR will also be a function of temperature. However, if a temperature dependent conductivity is included in Maxwell's equations, then the equations become non-linear and extremely difficult to solve. I will estimate the maximum change in conductivity that is likely to occur during the ablation process (during the ablation process tissue temperatures may increase as much as 20-30°). Since, at microwave frequencies, the electrical properties of high water content tissue reflect that of electrolytic NaCl solutions, the equations of Stogryn were used to calculated the conductivity as a function of frequency and temperature. In Figure 3.4, the calculated conductivity of 0.8% NaCl solution as a function of frequency is shown for a range of temperatures. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The results show a slight temperature coefficient around 1.0 GHz. This results from the cancellation of two different temperature dependent mechanisms. As discussed earlier, the total conductivity arises from two different mechanisms: the movement of ions under the influence of an externally applied electric field and the rotation of electric dipoles (of water) with the applied electric field. Both mechanisms are dependent on temperature, but show quite different dependencies. The conductivity due to the movement of ions has a positive temperature coefficient in this frequency range, whereas the conductivity due to the alignment of dipoles has a negative temperature coefficient in this frequency range. In the neighborhood of 1.0 GHz, the two temperature coefficients approximately cancel each other out, resulting in only a slight total temperature coefficient. It is fortunate that the microwave antennas used in this study are being driven at 915 MHz, near the region of small temperature dependence. Consequently, it is hypothesized that temperature elevation will cause only a small change in the electrical properties of tissue, which will not significantly alter the antenna's SAR pattern. In the experimental section, care was taken to monitor the reflection coefficient of the microwave antennas during the time course of ablation, to confirm the conclusions of this section. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 3.0 T em p eratu re C o - 25° 35 45' 2.5 55 \ E ■ > 2.0 o 3 XJ c o o 1.5 1.0 0.5 1.5 1.0 F requency 2.0 GHz Figure 3.4 Conductivity of 0.8% NaCl as a function o f frequency for temperatures ranging from 25°C to 55°. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 Chapter 4: Analytical Methods This chapter describes the analytical methods used in the study. The chapter consists of two sections. The first section describes the analytical antenna modeling for the insulated and uninsulated helical antenna. The second section presents a simple analytical model of presently available RF ablation catheters. 4.1 Analytical Antenna Modeling To understand the heating characteristics of helical antennas, the underlying theory will be presented. I present an analytical model which, given the various parameters of helical antennas and electrical properties of the tissue, can approximately predict the SAR distribution in the lossy tissue medium. I then extend the model to the case where a layer of insulation is added to the outside of the antenna. 4.1.1 Uninsulated Helical Antenna An uninsulated helical antenna for catheter ablation applications is shown in Figure 4.1 This antenna is fabricated from a coaxial transmission line with inner and outer conductors of radii a and b, respectively, which are separated by a (loss-free) dielectric medium of permittivity e, and permeability p0. At the end of this transmission line, the outer conductor is stripped back a distance L and the helical antenna placed over the line. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 — H 2a 2b Figure 4.1 Helical antenna o f length L, outer radius b, inner radius a, helical pitch angle a and pitch p, immersed in an external lossy media. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 The antenna is connected at the distal end to the inner conductor of the coaxial line, and at the proximal end to the outer conductor of the transmission line. Thus, this antenna has length L, and in this arrangement it can be regarded as being terminated at one end by the coaxial feedpoint, and at the other by a short circuit. The medium outside the helical coil is considered homogeneous, isotropic, linear, nonmagnetic and lossy, characterized by complex permittivity ee* = e/+ j e / and real permeability p0 To describe the helix, as is usually done, I define the pitch, p, as the distance between adjacent coils, and the pitch angle, a , as the angle the helix makes with the plane normal to the helical antenna axis (shown in Figure 4.1). The pitch angle is related to the pitch by . - i / 2 rc b , a = cot (----- ) P where b is the radius of the helix. If Maxwell’s equations could be formulated in an appropriate coordinate system in which the surface of the wire is described by keeping one of the coordinates constant, then an exact solution could be obtained using techniques similar to those used in waveguide problems. While it is possible to formulate the problem in a proper coordinate system, it appears the resulting equations cannot be easily solved exactly. Consequently, investigators must rely on approximate models when analyzing helical antennas. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 35 Helical Sheath Model One approach has been to replace the physical helix with an anistropically conducting cylindrical sheath33,34. The sheath model shown in Figure 4.2, is assumed to be perfectly conducting at an angle a which is parallel to the helix wire ( denoted by unit vector ) and perfectly insulating normal to that direction (denoted by unit vector a j . The short circuit termination is modeled approximately by a perfectly conducting cap at the end of the antenna (shaded region). In this section, I describe an analytical model for the helical sheath antenna immersed in lossy media, and then extend the analysis to include a coaxial feed point and a short circuit termination. A modal expansion technique is utilized. This method is based on the techniques used by Hill and Wait, Casey41, Delogne42 and others in treating loosely braided coaxial cables with helical shields. The procedure is outlined in the following steps. 1. Solve Maxwell's equations in the source-free region with the appropriate coordinate system. 2. Apply the necessary boundary conditions. Specifically, the tangential electric field must vanish on the surface of all perfect conductors. For the sheath helix, in particular, the tangential electric field component in the direction of a„ must be zero. 3. Formulate the determinantal equation. Solving this determinantal equation yields which wave guide modes can exist along the helical structure. 4. Expand the fields at the coaxial feed point into a sum of the allowed guided modes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 5. Introduce a set of reflected modes such that the net tangential electric field vanishes at the short circuit termination. 6. Add all the forward and reflected modes together to calculate the total electric and magnetic fields present in the lossy exterior media. The SAR can then be easily calculated using Equation 3.7. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 z=L Z=0 h— H 2a 2b Figure 4.2 Helical sheath model of length L, outer radius b, inner radius a and helical pitch angle a, immersed in an external lossy media. The sheath helix model defined as an anisotropically conducting cylinder, which is perfectly conducting at an angle a w.r.t. to the plane normal to the antenna axis and perfectly insulating normal to that direction. A perfectly conducting cap models the short circuit termination. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 Formulation In a source-free homogenous region, the time harmonic form of Maxwell's equations are written as VxE = -ju\iH Vxtf =jae*E I4-2) V*£ = 0 [4-3^ V-tf = 0 t4-4! where e* and p are the complex permittivity and permeability of the region. It is well known that due to the divergenceless properties of the source free Maxwell's equations ( Equations 4.3 and 4.4) the electric and magnetic fields may be expressed in terms of electric and magnetic Hertz vector potentials IP and IT”, respectively, by43.44.45 e h =VxVxir - j<onVxnm ^ =y we* Vxiie + Vxyxir* ^ Substituting [4.5] and [4.6] into [4.1] and [4.2], one obtains the following set of decoupled equations in terms of either IP or IT” given by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 VxVxne - vcv-ep) - k2ne =o [4.7] VxVxIF - V(V-IP) - *2D* = 0 [4.8] Here, jfc = o>/jxe* represents the complex wavenumber of the medium. Solutions to [4.7] and [4.8] when inserted into [4.5] and [4.6] provide general solutions to Maxwell's equations in a source free region. Thus the next step is to solve these equations in the appropriate coordinate system. I use a circular cylindrical coordinate system, defined in the usual manner, by coordinates (p,<)>,z), shown in Figure 4.2. The unit vectors a/7 and ax are related to the unit vectors in the cylindrical coordinate system ap, a*, and a* by &u = az sin(a) + cos(a) [4.9] ax = az cos(a) - sin(a) [4.10] where a is the helical pitch angle defined earlier. For an antenna, or waveguide, of infinite length whose cross section is uniform along the z direction, it can be shown that only the z component of the Hertz vectors IT and IF 1are necessary to obtain a general solution to Maxwell's equations3,4. Thus, if the Hertz vectors are assumed to have only a z component, the electric and magnetic field components in cylindrical coordinates are given by [4.5] and [4.6] as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 X lii t i S p *8p r H .ll] p a* 1 j 2i p ozdy E H dp , . ^ + dz2 = p X, - * p j 3<j) ^ ap k% 14.13] + ¥H L dzdp [4-14] ♦ i f £ [4.15] p a z ft Hz = ^ - + k 2I% dz2 [4-16] In addition, Equations [4.7] and [4.8] simplify to the scalar Helmholtz equation given by (V2 + k 2)% - i - i ( p — ) * — — * — p 3 p K dp p> a* 2 *2 < * ♦ * * . + *2I ? - 0 I f p f C , ♦ p 3p 3p . o p2 3<j,2 [4-17] 14181 az2 where the vector identity VxVxA = V(Vvl) - V2/! has been utilized. As in [Hill and Wait], the solutions to [4.17] and [4.18] are obtained using a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 standard separation of variables approach which assumes solutions of the form HZ = P(p)-3>(<J))-Z(z) [4.19] Substituting [4.19] into [4.17] or [4.18] and separating the equation yields three ordinary differential equations in terms of <(>, z and p . — + n 2® = 0 [4.20] — + p2Z = 0 d z2 [4.21] P y K p f ) ~ [(P2- * V + » 2]P = 0 dp dp [4.22] Here, n is an arbitrary positive or negative integer and P is the unknown propagation constant. The solutions to [4.20] and [4.21] can be expressed in exponential form: ®(4>) = e ~J " * and Z(z) = e J p z. The solutions of [4.22] are the modified Bessel functions of order n and argument J $ 2- k 2- The general solutions to [4.17] and [4.18] are thus given by l£(P><M = [An / „ ( / F P p ) + Bn Kn( ^ P p)] e 'j P * e-J" ♦ [4.23] n zw( p , ^ ) = [Cn U < / f T 2p) + Dn * n( / F F p ) ] e-J e-J where I,, and K„ are the modified Bessel functions of the first and second kind respectively, and A„ , Bn ,Cn and Dn are unknown coefficients determined by the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 boundary conditions and source distribution [ Hill and Wait]. The general form of [4.23] is valid in both the interior (a< p <b), and exterior (b< p < oo) regions. However, due to differences in electrical properties and boundary conditions in the two regions, the actual solutions can be quite different. Consequently as in [Hill and Wait], the fundamental solutions in the two regions are given by a< p <b I?(p,4«) = ]An P) * K [4.24] +d„c.f/F^p)] t ->' ! rCfp.fe) =[c. ♦ b< p < <» K M A =F n * n( / F * ? P ) e-J P1 e -j" * Y [4.25] K M * ) = Gn Kn( J W ^ P ) e - ^ z e - j n * where, the subscripts i and e refers to the interior and exterior regions, respectively. In forming the solutions in the exterior region, the modified Bessel functions I„, is excluded to insure that the electric and magnetic fields remain finite as p approaches infinity. The electric and magnetic fields can now be calculated by substituting [4.24] and [4.25] into [4.11] through [4.16]. The unknown coefficients Bn, Cn, D„, F„ and Gn are then related by applying the appropriate boundary conditions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 Application of Boundary Conditions Along a boundary between two different media the electric and magnetic fields must obey certain boundary conditions. If both media have a finite conductivity, the tangential electric and tangential magnetic fields must be continuous at the boundary. If one medium has an infinite conductivity (a perfect conductor), then the tangential electric field will vanish and the tangential magnetic field will be discontinuous by an amount equal to the induced surface current density. These conditions are written mathematically as n x (E2 - Et) = 0 for finite Oj and c 2 [4.26] n x (ff2 - £ ,) = 0 n x E2 = 0 CTj infinite and ct2 finite [4.27] n x H2 = J S where subscripts 1 or 2 denotes media 1 or 2 respectively, n is a unit vector normal to the perfectly conducting surface, and Jt is the surface current density. Applying the above boundary conditions to the helical sheath results in the following six conditions: i. At the surface of the inner conductor p = a [4.28] [4.29] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 ii. At the surface of the helical sheath p = b, with current I, flowing in the direction a;/ Ez(b)t = Ez( b \ [4.30] = E Jb\ [4.31] P , H M t = H A b \ - ^ - Sm-(g) ♦ * * e 2*6 [4.33] where the subscripts i and e again refer to the interior and exterior regions, respectively, and a is the helical pitch angle defined earlier. It must be remembered that since the sheath model is perfectly conducting along the a„ direction, the tangential electric field along a„ should also vanish, (i.e. E„ = 0 ). This boundary condition is used later in formulating the determinantal equation. Electric and Magnetic Field Solutions Applying the boundary conditions above to the fields calculated by [4.11] through [4.16], one obtains a set of six simultaneous equations for the unknown coefficients An, Bn, Cn, Dn, Fn and Gn. In principle, this set of six equations can be solved directly to obtain all the coefficients in terms of It. However, this leads to a horrible algebraic mess. A simpler approach, as done in Hill and Wait35, involves satisfying the boundary conditions at p=a, Equations [4.28] and [4.29], separately. This leads to the simple relationships Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where the variables u = \j$2 - k f and v = - k] are defined for notational convenience, and the prime refers to the first derivative with respect to the entire argument. Following the method used in Hill and W ait36 and using a similar notation, it is convenient to define the variables IJu a) Zn(u p) = /„(« p) - f - j - Kn(u p) 1 Z'n(u p) = /„(« p) - [4.35] Kn(u p) < (« a) Substituting [4.34] and [4.35] into [4.24], one obtains a much simpler expression for the Hertz vectors in the interior region, given by n^C p,^) = A Z (u p) e ~J *z e~j n * [4.36] n*(ps^ ) = C „ Z > p) e~j Q2 e~J n * Now, applying the boundary conditions at the helical sheath, Equations [4.30] [4.33], one obtains a set of four simultaneous equations for the unknown coefficients Cn, Fn and Gn, instead of six. This set of equations was solved using a symbolic manipulating program (Maple) and the following electric and magnetic field Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. components were obtained b< p < oo Ep - - ( ^ A, < (v P » e-Jt ' e-J' * [4.37] C„ < (v p) - i ! / l „ AT„(v p)) e-J * ‘e - l " ♦ [4.38] C. K„(v p ) + j £„ = U P<o V p V [4.39] £ ; = -v 2 ^„ X„(v p) ff„ - € (OR JC„(v p) - J H„ - -( j coe.v X. < (v = -V 2 P) , p v C„ AT„'(v p)) + KJy p)) « -J U e -J . ♦ C„ tf„(v p) e 'n ze~J n * [4.40] [4.41] [4.42] a< p <b Eo = ^ P) ^ P h F „ Z > p ) ) ^ P V J M [4.43] = ( 7 p u u Gn Z*'(u p) - ^ F n Zn(u p)) e~J " e l * * [4.44] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 [4.45] Ez = -u 2 Fn Zn(u €•(07* „ # p - ( - ^ - ^ Zn(u p ) - j p « G „ Z > p)) ^ . . [4.46] H = - ( j <o€j« F„ Z„'(« p) h- £ J L Gn Z*(u p)) [4.47] = -k 2 G„ Z > p) e -j *ze-j n * [4.48] where .4,, = y |iw /, « 2 Z„ [ sin(a)v « (« Z„* - v Z„*' *„) [4.49] + cos(a) n P (v K ' Zn* - « Kn Z„')] / 2ttD* Cn = /, [ p n Z„ Zn* Kn(v2- u 2) (i«2 * sin(<|>)- p « cos(4») [4.50] + <o2p «2 z ; ' cos(4>) v (€,. Kn z'n v - €e K'n z n «)] I 2%b D* F = A " v2 Kn" [4.51] «2Z„ [4.52] and D* = p2 «2 z; z; Z * 2 («2 - v2)2 v 2 (« < V z;' ( « 6e k' Z„ - v e, z ' ) and for notational convenience 7^ = Z„ (u,, b), Z^’= ^ ’(t^b), 2,,*= Zp’ ^ b ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [4.53] 48 K,, = K„(vnb) and K j = K„ (vn b). The solutions derived above for the electric and magnetic fields depend on the arbitrary integer n. Each value of n corresponds to a different independent solution and, in waveguide terminology, is referred to as a guided mode . The guided modes with different n values will have different azimuthal variation due to the e j n* term. The propagation constant, P, is still undetermined and requires an additional boundary condition to completely specify it. The needed relation is termed the determinantal equation or the dispersion relation. Determinantal Equation The determinantal equation can be derived from the boundary condition that on the surface of the helical sheath the electric field component in the direction %, must vanish. This condition is written as Ez sin(a) + cos(a) = 0 at p = b [4.54] Substituting [4.38] and [4.39] into [4.54] results in the determinantal equation - y2 \ KJy b) sin(a) + [ j pw v C„ k '„(v p) - ^ - A n Kn(v p)] cos(cc) = 0 P [4.55] where A„ and C„ are given by Equations [4.49] and [4.50] respectively. The propagation constant, P, which is complex in general, is calculated by determining the roots of [4.55]. For any particular integer value for n (guided mode with azimuthal variation e'jn*), Equation [4.55] may have several roots. Each of these roots Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 corresponds to a guided mode with a different radial variation. The set of guided modes described above provide only a partial solution to the helical sheath problem. In addition to the guided modes there exists a set of radiation modes needed to account for radiation phenomena. The complete solution is then formed by summing all of the individual guided modes along with the radiation modes. This is written as «° N Ktai = £ E £ n,(P) e~JPvZ e ' J n * + radiation modes »—-/>-i « Htotai = £ N £ e~J [4.561 L 1 e~j n * + radiation modes n~-oop<i where n and p represent guided modes with azimuthal variation n and radial variation p. Also N represents the number of roots of Equation 4.55 given any particular value for n. The radiation modes become important when the antenna's far field ( observation points several wavelengths away from the antenna) is of interest, whereas the guided modes play are crucial role in the distribution of the antenna's near field. Since we are interested in the heating characteristics seen near the surface of the helical antennas, the radiation modes are not included in this analysis. In principle, all of the guided modes along with the radiation modes must be summed to exactly model the characteristics of the helical sheath. However, in practice only a few of the guided modes may be needed to adequately approximate the heating characteristics of the helical sheath. For the helical antennas considered here, the dominant guided modes are those modes where n=0 (modes with no azimuthal Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 variation). These dominant modes are examined in more detail in the next section. Dominant Modes n=0 There are two reasons for believing the n=0 modes will be of importance here: 1) The electric and magnetic fields of the coaxial line source are independent of azimuthal angle, <J>. It is easy to show that if the source fields are independent of <j> then the modes excited by that source will also be independent of <|>. 2) Experimental measurements revealed no changes in field intensity as the helical antennas were rotated in the <j>direction. Consequently, I considered only those modes with n=0. Equations [4.37] - [4.55] simplify for those modes to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 Ep - 'J Pp v A . <<v P> J * * £« c„ < (v p) «•*»■> [4.57] £; = -v 2 i i . * .(v p) p * M - - / P„ v C„ < (v p) e 'n - 2 ff, - - J o « ,v ,4. J t f v p ) e 2" - ' [4.58] H, = -v2 C„ K J y p) p2" * 2 a<p<b £p - - i P . « F. z > p) ^ £t ■ J P„“ K 0„ z ;\u p) e~J i f [4.59] Ez = -» 2 F0 Z„(« p ) e ‘, s -‘ ffp “ -J Pp « G. z;'(» p) p2" * 2 H4 - - j . eI v F . z 4 * p ) « 2' , *‘ [4.60] H ,= -u2 G „ Z > p) e ‘-'M where the subscript 0 indicates the n=0 modes. The constants A0, C0, F0, G0 reduce to Equation [4.61], where again It is the total surface current along the a„ direction of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 j It u Z0(u ft) sin(a) Ao = 27tft CO v [ ee k 0' (v b) Z0(u b) u - €, K Jy b) z'0(u b) v ] It Z*'(u b) cos(a) 2*ft v [ < ( v b) Z*'(u b) v - < ( v ft) Z > ft) « ] [4.61] j It v K0(v ft) sin(a) 2%b (0 [ ee < ( v ft) Z0(« ft) « - e,. ^ 0(v ft) Z'(u ft) v ] /{ < ( v ft) cos(a) 2nft«[ <(v ft) z; («ft) v - <(v ft) z;(« ft)«] the n=0 mode. The determinantal Equation [4.55] reduces to the following for n=0 [4.62] w2 \i0 cos(a)2 K Jy ft) 2 Z > ft) —2 v - ------------ « v <(v ft) z; («ft) u sin(a)2 e < ( v ft) e » =Q z'0(u ft) u - e . ---------- ' zo(« *) The propagation constants, P0, of the n=0 modes are now calculated by determining the roots of [4.62]. It should be noted that, although only the n=0 mode is being considered, there may still be several roots to [4.62], each corresponding to a mode with different radial variation. The approximate solution is thus the sum of all modes with n=0 with the appropriate weighting function. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 - E fy p ) ci,r ‘ [4.63] .p-1 where the subscript n=0 has been omitted, for simplicity. Effect of Feedpoint and Termination The above results describe the characteristics of waves that can propagate along an infinitely long helical sheath structure. However, it is necessary to account for the terminations of the helical antenna which, for present purposes, consist of a coaxial feed point at one end and a short circuit at the other. The effect of these terminations can be modeled by expanding the given source backw ard traveling m odes En »Hn 77 source field E„ H, forw ard traveling m odes E n\ H n+ Figure 4.3 Illustration of forward and backward traveling waves along sheath helix antenna model, fields on an orthogonal basis using orthogonality relations that can be obtained for Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 modes in the helical sheath. In the next section, I present the such relations. This specific analysis follows from a more general analysis given by Mclsaac46. Orthogonality Relations Consider any two different modes, viz., m4 and n* modes, propagating in the helical sheath shown in Figure 4.3. The Lorentz reciprocity theorem, which is derived in Appendix B, is used to relate them by3,4 [4.64] (P. ♦ P„) / / [£> .< « » 4(P .40 - 4(P .+ ) x HnM ) ]• 4, p <#> dp = 0 P ” fi <jp=0 For those modes in which Pn * - Pm, the Lorentz reciprocity theorem implies that the integral must evaluate to zero. That is “ 2n / / [£„(p,40 x JyM(p,<j>) - £m(p,4)) X Hn(p,4)) ]• az p d$ dp = 0 p » j <{>=0 [4.65] For those modes in which P„ = - Pm the above integral may obtain a non-zero value. In general, for a bidirectional reciprocal waveguide, the electric and magnetic fields of an arbitrary source (represented by superscript s) can be expanded into an infinite sum of forward and backward traveling modes (represented by superscripts +,- respectively ): Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 N E °(p M = E E ( % n --« p -l ^ ( p . 4 # e 'J ^ +K e ~J ^ + Radiation modes [4.66] iT M d = ± f > v STV(p,<J)) e '; p ^ n--~ p-1 + Radiation modes + bv e'J^ ) [4.67] For our helical sheath case where we take n=0 and ignore the radiation modes, Equations [4.66] and [4.67] reduce to « £ > , # W ) * E (S # > ) e p‘ i 'J M + h £;<p> [4-681 + bp » > ) e ~J f4-69] where the subscript n=0 has been omitted for convenience, and the <{>dependance removed to illustrate that only those modes independent of azimuthal variation are being considered. The unknown coefficients \ and bp are obtained by applying the Lorentz reciprocity theorem, Equation [4.64], to Equations [4.68] and [4.69]. This results in the following expressions for a,, and bp as given in Maclsaac46. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56 /[ E~x S’ - £-'* «;] pip ar ‘ ~ h / [ £ ,x a ; - e ; x [470] h; \ Pi ? J [ e ; * S ’ - e ’ x h 'p] p ip h - —a _____________________________________ -----------------------------------P 00 [4.71] /[ £;x s; - e;* s;i ?dP Equations [4.70] and [4.71] are general orthoganality relations valid for any reciprical waveguide. If possible, it is convenient to express these relations only in terms of the forward traveling modes and source field. For the case of the sheath helix this can be accomplished, by noting the relationship between the propagation constant and field components of the forward and backward traveling waves given as |3 0- = - P0+ , Ep‘= - Ep+, E,'= E*+, E- = Ez+, Hp'= - Hp+, H, = H,+ and H ^ H / . Substituting these equalities into Equations [4.70] and [4.71] results in the following expansion coefficients a,, and bp for the sheath helix; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 [4.72] 7> 2/ ( £ ; , ♦ e ; , h > ^ a /( e ; , e *‘ - e^ e; - e; ^ ♦ e; e (> « / p [4.73] a where Es and Hs refer to the known source fields, E+pp, E \ p, H+pp, H \ p and E'p E ^ p, H p p and H*^p are the transverse field components of the forward and backward traveling waves. Introduction of the Feedpoint from the Coaxial Source I assume that the helical antennas are driven by a coaxial line source. In considering the source field distribution, I ignore higher order modes in the coaxial cable, and I also use the normalized electric and magnetic fields. At the feedpoint of the antenna z = 0 (i.e. the junction of the antenna with the inner conductor of the coaxial line), the normalized electric and magnetic field distributions are assumed to be Assuming that the waveguide is infinitely long, contribution of the coaxial source Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 Hi = e; =o — — P HI = 0 a<p<b [4.74] p>i> to the forward traveling modes can be expressed as ^-1 [4.75] />»i To obtain the unknown coefficients, a,,, I substitute the electric and magnetic fields from the coaxial source, Equation [4.74], into the orthoganality relation, Equation [4.72]. This results in -/(K , [4.76] ap ‘ 2/ ( e „ ‘ h ;„ * e ;„ h ; , > P</P Introduction of a Short Circuit Termination The antennas are shorted at the proximal end (z=L) where the helix is connected to the inner line of the coaxial cable. The reflections from this short circuit termination can also be included in the model. For simplicity, I assume that only a single reflection occurs at the end of the antenna and that the reflected wave does not Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 significantly disturb the source fields at z=0. In a lossy external medium, this is justified due to the rapid attenuation of both forward and reflected waves along the antenna. Since, I model the short circuit termination as a perfectly conducting cap at z=L, the sum of theforward and reflected waves must then satisfy the condition that the total tangential electric field vanishes at this termination. To satisfy this condition, the tangential component of electric field for the reflected wave must be related to the incident wave at z=L by e ] = e ;p + e;4> = -£ p+p - <4> f4-77] where superscripts - and + refer again to the reflected and incident traveling waves, respectively, and subscript t denotes the tangential component of electric field. Since this effectively means that the reflection coefficient at z=L is essentially unity, the tangential component of magnetic field, of the reflected wave, is related to those of the forward traveling wave as h; = h ;$ + h $ = h ; p + h$ r4-78J The reflected wave, defined in Equations [4.77] and [4.78], can now be interpreted as a secondary source located at the antenna's termination, z=L. This secondary source, denoted by superscript ss, can also be expanded into a sum of n=0 propagating modes. However, in this case I choose the expansion in terms of modes propagating in the - z direction. Hence, The total incident electric and magnetic fields at z=L are obtained by substituting Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 £ ”(P) - I 0, £,(P) « '* p-1 [4.79] />-l [4.76] into [4.75] and replacing z by L. Thus [4.80] e. N EP +(p) e 'n > -E p -l 2/< 2? ;, h ; p + ^ N - / ( K* ) prfp e.— + " if ) if C (P ) e '1 p;i 2/ ( h ; , * e ; , H; f ) p</p [4.81] The unknown coefficients, bp, for the reflected wave can now be obtained by applying [4.77] through [4.81] to the orthogonality relation [4.73] and are given as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 ~ f(P p ,p tot + E+,p Hp, tot “ Ep, tot H $ ,p ~ E $,tot H p , p )P^P a______________________________________________________________ | 4 .oZ ] 2! K , K p * K .p K p M p where the terras which contain subscript tot refers to the field components defined in Equations [4.80] and [4.81]. The total reflected wave is now obtained by substituting Equation [4.82] into [4.79]. Total Electric and Magnetic fields After including the coaxial line source and the short circuit termination, the total electric and magnetic fields are obtained as the sum of forward plus reflected modes as 4 , - **„ * - X > , % *~n > .+ b, 4 e ' ' 1* ) I4-83! p -1 where the coefficients a,, and bp are given by Equations [4.76] and [4.82] respectively. The quantity of interest here is the specific absorption rate (SAR) which is now given Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 N * bp h; ) [4.84] />"1 as & 4i? = — o | £ j 2p 1 *fl 2 — [4.85] Kg where c and p are the conductivity and density of the outside medium respectively. Summary In summary the procedure for calculated the SAR pattern of the uninsulated helical antenna is outlined in the following steps: 1) The propagation constants for the dominant are calculated by determining the roots of the determinantal Equation [4.62]. 2) are 3) The electric and magnetic fields associated with each of these modes then formulated using Equations [4.57] - [4.61]. The coaxial line source is then expanded into each of the dominant modes by Equations [4.74] - [4.76]. 4) A set of reflected modes due to the short circuit termination are then derived by Equations [4.79]-[4.82]. 5) All of the forward and reflected modes are then summed to get the total electric and magnetic fields, Equations [4.83]-[4.84] . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 6) Finally, the specific absorption rate (SAR) in the exterior lossy media is determined by Equation [4.85]. 4.1.2 Insulated Helical Antenna This section investigates the electrical properties of insulated helical antennas which are immersed in a lossy dielectric medium. I extend the previous analytical model based on the uninsulated helical sheath approximation, to include a thin layer of insulation. In comparison to an uninsulated antenna, the presence of a thin insulating layer can significantly alter the field distribution of the antennas, the mode structure of the antennas, and the SAR pattern in the outside medium. I will illustrate that the insulated helical antenna may offer the possibility of short-range and relatively uniform heating in the outer media. In subsequent chapters, it will be shown that these properties make it well-suited for catheter based applications in biomedicine. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 Insulating layer - - e 3>^3 Figure 4.4 Insulated helical antenna of length L immersed in an external lossy media, insulation is assumed to be lossless with a permittivity of e* and thickness (c-b). The layer Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 Antenna Model The antenna to be considered is shown in Figure 4.4. Extending the previous analysis for uninsulated antennas, the antenna is considered to be covered by an insulating layer of thickness d with lossless dielectric medium of permittivity e^. The outside medium is infinite in extent, lossy and nonmagnetic, with permittivity £3’ = Ej' - j £3" and permeability p0 The inside dielectric medium is assumed lossless of permittivity ey Following the previous analysis, I model the helical antenna in cylindrical coordinates (p,<t>,z) by a sheath helix shown in Figure 4.5. Formulation I utilize the same modal expansion technique used before to solve for the total electric and magnetic fields present in now three regions; inner dielectric (a<p<b), insulating layer (b<p<c) and lossy outside medium (c<p<°°). These regions are denoted as 1,2 and 3, respectively. The electric and magnetic Hertz vectors must satisfy the scalar Helmholtz equation in each of the three regions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 z=L- £ 3»G 3 2=0 Figure 4.5 Insulated helical sheath model. This figure is identical to the uninsulated helical sheath model shown in Figure 4.2, with the addition of a layer of lossless insulation of permittivity e* and thickness (c-b). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 (V* + k 2)l£ = ^ P aP p aP (V2 + fc2) l £ = i l ( p i ) ^ p dp H dP p2ad>2 ^ as2 + k 2^ =0 t4-863 + -L .?? 5 l + . ^ E l +k2! ^ = 0 t4-8?] p2 a<|,2 dz2 The solutions to [4.86] and [4.87] in each region are given by: a<p<b n'(p.4v) - [ D , K J x p) ♦ F, I„(x p) ] e ' J " * « ' » « n -(p ,(M - [ d ; K,(x [4 gg] p) ♦ f ; 1„ (x p) ] e '1 "* e b<p<c n*(p,^) = [ fl„ K fa p) + C„ /„(* p) ] «-•'!>« n"(p,<t>^) » [ b ; k ,( x P) ♦ c; /„(* P) ] « >"♦ [4.89] c<p< IT(p,(j,^) = 4 J n( v p ) e ^ ^ ^ J [4.90] n m( p , 4 ) ^ ) = A ; K „ \ v p ) * / * ♦ e V M where u = y p 2-*,2, * = /p M £ v = ^ p 2-*32 and K„ and I„ are the modified Bessel functions of the first and second kind, respectively, and n is an arbitrary positive or negative integer. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [4.91] 68 Application of Boundary Conditions \ \ Bn, Bn’, Cn, Cn*. D„, Dn’, Fn and Fn* are obtained The unknown constants, in terras of the total current I, flowing in the direction a,, by applying the appropriate boundary conditions to the electric and magnetic fields at the interfaces p = a, p = b and p = c. Thus: i. At the interface p = a the tangential electric fields Ez and E+ vanish. =E Ja\ = 0 [4.92] ii. At the sheath interface p = b, with current I, flowing in the direction a//( the tangential electric fields are continuous and the tangential magnetic fields are discontinuous. E f i ) i = E^(b)2 [4.93] where again a is the helical pitch angle of the insulated helical antenna, iii. At the interface between the insulation and the exterior lossy medium, the tangential electric and tangential magnetic fields are continuous. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 Ez(c)2 = Ez(c)3 = Et (c)3 Hz(c)2 = Hz(c)s H^(c)2 = ff*(c)3 [4.94] Solutions for Electric and Magnetic Fields The insulated helical antennas are also being driven by a coaxial line source. Consequently, since the symmetry considerations discussed for the uninsulated helical antennas are still valid, only modes independent of azimuthal angle <|) will be considered (i.e. n=0). The solutions presented below for n=0 are valid for all modes with radial variation p, but for convenience I omit the subscript p. Applying the boundary conditions found in Equations [4.92] to [4.94] leads to the following electric and magnetic field solutions for the modes where n=0: a<p<b E, - -JK%DozXkp*** Et -•/>„<■> K D 'o t i W * * Ez - - u l D„ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [4.95] 70 B , - -J K u0 d ; z ; ( u oP)e-J^ = "7«€i uQ Do Z'(u0p)e hz [4.96] = - u l d ; z ;( u09)e-J>* b<p <c Et - -J K \ K lw W . E„= jp.w *„ [s ; < < *.p )-c ; 4'(i0p)]e * [4.97] B, - -*? [fl, % p ) » C 0 ;„(j,p)]e * - -7P„ * . [ « ; < k p ) < »* / X p )]< - -j< * V . IB, K '( V ) * C , ^ . p ) ] e ^ [4.98] ff; - - J 2 [B; « „ ( « „ p ) < / . ( j y > ) ] « * c<p Ep " -7'Pp v0 * - > oP) ^ E , - jp .w v„ /]„■ K^r,p)e'JP'‘ E, = -v2 4 , Kp(v„p)e* Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I4-" ! 71 H , - -]» . v. a ; K f a j e * * ' Ht ■ -J“ e3v. j4. * :»(voP)<! ^ hz [4.100] - -v„2 a ; A y v „ p ) e * The constants A0, A0*, B0, B0‘, C0, CG*, Dc and D0*are given as: A = j €^ f sin((x)Xo“0 Zo(u0b)U0(cx^K iicx^ ° 2it£> <x>v0 Ql , _ /, cos(a)*0 Z0* ( « ^ ) [ ^ 5 ( « ^ ( a o ) - / 0( « X ( « » ) ] 2rcfc v0 (?2 B = j Jt sin(«)«0 ~ 2ltf> (jiXQ Ql ° B. . = h « » (« ) ^ ( # , ljc x } K 0(cvJ - x 0 IJjcxJK & cvjl 2nb x0 Q2 [4.101 _ j I, sin(a)«0 Z0(u0b)[e2v0K0(cvJK!,(cxJ - ejcJKfcvJKJicxJH c .2tc6 ojc0 <?j c _ A cos(a) z;W > )[* o KjLcvJKJLaJ - v. W < ( « o ) ] 2nb x0 Q2 D _ Jh8fa(«W yw y, - €3x0<(cv0)^] ° 2lti> (|)M0 Qj 2j* = h c o s(a)[j:X (c0 ^ 3 ~ 2itf» q2 where: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission 72 Q{ = + £1e^ X ( < * X ( Wo W - ' W C2 = +V o W ^ W [ 4102] ) ^ c ( “. W 4 - x 0vpK0(cvJZ*'(u0b)W2 - x 0u0K'0{cv^Z *(u0b)W3 and wx =W W " =W < ( « .) " f e w W3 = l'0(bx^K0(cx^ - l 0(cx^K ,0(bxt) [4.103] W4 = jfcxJK 'C cxJ - I'icxJK'CbxJ where I0 and K„ are the zeroth order modified Bessel functions of the first and second kind respectively. For the limiting cases where c = b or ^ = £3, the solutions for the insulated helical sheath reduce to those obtained previously for the uninsulated antenna. Determinantal Equation The determinantal equation for the insulated helical antenna is constructed using the same boundary condition as the uninsulated case, namely that on the surface of the helical sheath the electric field component in the direction a„ must vanish. Thus: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 £> 2sin(oc) + cos(a) = 0 at p=b [4.104] i|x0w D* Z*(u0b) cos(a) - u0 D0 Z0(uob) sin(a) = 0 The propagation constants, p0, of the n=0 modes for the insulated antennas are now calculated by determining the roots of Equation [4.104]. Analogous to the uninsulated results Equation [4.104] may have several roots each having a different radial variation. The total fields are then approximated summing all modes where n=0. Feedpoint and Termination The feedpoint and short circuit termination for the insulated antenna are accounted for by using the same approach as for the uninsulated antenna. To briefly review the procedure; 1) Expand the coaxial line source into a sum of forward traveling modes using Equations [4.73] - [4.75]. 2) Calculate the reflected modes by insuring the net tangential electric field on the short circuit termination vanishes, using Equations [4.80] - [4.82]. 3) The total fields are then formed by summing the forward and reflected waves together. Summary In summary, the procedure for calculated the SAR pattern of the insulated helical antenna is outlined in the following steps; 1) The propagation constants for the dominant radial modes are first Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 calculated by determining the roots of the determinantal Equation [4.104]. 2) The electric and magnetic fields associated with each of these modes are then calculated using Equations [4.95] - [4.100]. 3) Using Equations [4.74] - [4.76] the coaxial line source is expanded into each of the dominant modes. 4) A set of reflected modes due to the short circuit termination are then calculated using Equations [4.79] - [4.82]. 5) All of the forward and reflected modes are now added together to calculate the total electric and magnetic fields using Equations [4.83]-[4.84] . 6) Finally, the specific absorption rate (SAR) in the exterior lossy media is calculated using Equation [4.85]. 4.2 Simple Analytical Model of RF Ablation To better comprehend the significance of the microwave antenna results and to investigate its merits relative to presently available RF techniques, I developed an analytical model describing an RF ablation catheter. Figure 4.6 presents an illustration of a currently available RF ablation catheter operating in a unipolar mode. The geometry of the problem is also shown in Figure 4.6. The tip of the catheter is a hemispherical electrode of radius b. A grounding plane is placed a distanced D away from the electrode tip. The interior region is assumed to be filled with a homogenous tissue medium of permittivity e and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 RF Ablation Model Tissue Figure 4.6 Model of an RF ablation device, consisting of a hemispherical tip electrode of radius b, placed a distance D from a grounding plane. The space between the tip electrode and grounding plane is assumed to be filled with tissue. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 conductivity o. The operating frequency for a typical RF ablation device is 500 kHz. At this frequency, the wavelength, k, is approximately 60 meters in tissue. This wavelength is several orders of magnitude larger than the largest dimension of the devise. Consequently, a quasi-static approximation is well justified. The standard procedure for solving quasi-static problems is outlined in the following steps. 1) Solve Laplace's equation in the tissue medium for the scalar potential O. V2® = 0 2) Express the electric field in terms of the scalar potential O. E = -V® 3) [4.105] [4.106] Use electric field to calculate SAR pattern. SAR = — o |E |2 2p Kg [4.107] The geometry shown if Figure 4.6 does not lend itself to a closed form analytical solution to Laplace's equation. However, if the distance, D, is much larger than the radius of the hemispherical tip electrode, b, the following model is a valid approximation. The approximate model shown in Figure 4.7 consists of the same hemispherical tip electrode surrounded by a larger hemisphere a distance D away. The space between the two electrodes is assumed to be filled by tissue. The inner electrode is held at a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 Tissue SAR « Figure 4.7 An approximate model of Figure 4.6, consisting of the same hemispherical tip electrode of radius b surrounded by a larger hemisphere of radius D. The inner electrode is at potential V0, whereas the outer electrode is fixed at ground potential. The space between the two hemispheres is assumed to be filled with tissue medium. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 constant potential of V0, while the outer electrode is fixed at ground potential. Both electrodes are assumed to be perfect conductors. It is also assumed that current flows only radially between the two electrodes ( i.e. a perfectly insulating boundary along the flat surfaces of the hemispheres ). Consequently, the electric field will have only a radial component. The problem is stated mathematically as [4.108] ^®(M>.6) ‘ - y ( r 2~ ) * — ~ ( s in ( 4 > ) ~ ) + ■ 1 - 0 r 2 dr dr r 2sm(<J)) &1> r 2sin2(<]>) 502 It is clear from Equation [4.106] that if the electric field has only a radial component then the scalar potential O will vary only in the radial direction. Consequently, Equation [4.108] simplifies to V ^ fr) = r 2 dr dr =0 [4.109] with the following boundary conditions <b(b) = V0 0 < <]>< 7t, - n/2 <0< n/2 <b(D) = 0 0 < <p < 7C, - tc/2 <0< n/2 The solution to Equation [4.108] is given by = i Id ° ^ b<r <D [4.110] 1 The electric field in the region between the electrodes is obtained using Equation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 [4.110], and is given by 1 ------ f 72 ~D r b<r <D [4.111] ~b Taking, the outer radius D to be far larger than the radius of the inner electrode b, results in the following approximate form E(r) « - ar b<r <D [4.112] r The SAR pattern of the RF catheter is now obtained using Equation [4.107] as [4.113] where a and p are the conductivity and density of the tissue medium respectively. According to Equation [4.113] the SAR from RF catheters drops off as r4 away from the electrode tip. This rapid attenuation in SAR explains why RF ablation produces small discrete lesions located near the tip of the catheter. 4.3 Summ ary In this chapter, the analytical methods used for this study were derived. Analytical antenna models based on the sheath helix approximation were developed to calculate the SAR distributions of insulated and uninsulated helical antennas. In addition, an Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. analytical model of a RF ablation catheter was presented. In the following chapters the models presented here are presented in comparison with experimental results. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 Chapter 5: Analytical Results This chapter presents the results which follow from the analytical antenna models derived in Chapter 4. It consists of two sections. The first presents the theoretical results obtained from the uninsulated helical sheath model. The second shows the results for the insulated case. Each section calculates the antenna's dispersion characteristics, its electric field distribution and its SAR pattern under a variety of configurations. For the present application, it is of particular importance to investigate how these properties change with respect to loss in the external medium, operating frequency, pitch angle, antenna radius and insulation thickness. The dependence on these parameters will indicate how the actual helical antenna performs with respect to changes in configuration. Chapter 6 will present the experimental methods and Chapter 7 will compare the analytical results presented here to the experimental results. Results from the RF ablation model (derived in Section 4.3) will also be discussed in Chapter 7. 5.1 Results From Uninsulated Helical Sheath Model This section presents the theoretical results of the uninsulated helical sheath model derived in Section 4.1. Using this model, I calculate the dispersion characteristics, the electric field distributions and the SAR patterns of the uninsulated helical sheath. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission 82 5.1.1 Dispersion Characteristics The dispersion characteristics of an antenna or waveguide characterize the existence and the general nature of waves which can propagate along the structure. The propagation constant, p, is used to quantify these characteristics. It can be determined by calculating the roots of the determinantal equation. For the dominant modes (n=0) of the uninsulated helical sheath antenna, the determinantal equation is given by Equation 4.62. The intent here is to investigate how the propagation constant of the uninsulated antenna varies with loss in the external medium, frequency, pitch angle and antenna radius. Numerical Method A general closed form solution to Equation 4.62 appears not to be possible. Consequently, I rely on numerical techniques. I develop a numerical method to solve for the complex propagation constant, p. The method is based on a modified NewtonRaphson procedure47, and is implemented using the mathematical software package Matlab [MathWorks, Inc]. The Newton-Raphson algorithm is an iterative procedure normally used to locate real roots. I modify the procedure to make it applicable for finding roots in the complex plane. To accomplish this, I first write the determinantal equation in the form D(P,p|,p2,p3,..,p„) = 0, where pj though pn are known parameters (such as size, frequency, pitch angle, etc.). The propagation constant, which is complex in general, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 83 can be written as p = pr + j p , [5.1] where pr and ft are the phase constant and attenuation constant, respectively. Likewise, the deterrainantal equation can be separated into real and imaginary parts given by £ (P r> P p P v P v •• » Pn) = D r +j Di When given an initial guess for the solution, the Newton-Raphson computes a corrected value, pr = pf + Apr p r = pf * Ap. where APr and Aps are correction terms. These correction terms are calculated by applying a first order Taylor expansion to both the real and imaginary parts of the determinantal equation. The following set of simultaneous equations results: 3D 3D -D, = 3p’ . 3D. 3D. -D . = — iA Pr + — ^Apf ' ap r 3P; [5.4] where the partial derivatives are calculated by a central difference method. Solving these equations gives the correction terms. This procedure is then iterated until the procedure adequately converges to a root. For present purposes, I assume Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 convergence when both I Pnew - P°ldl < 10 6 and I D(P“ W, Pi,..,pn) - D(p0ld, plf..,pn)l < 10 12. To insure that the result is a true root, the alteration of sign of the real and imaginary parts is checked in the vicinity of the root. Verification of Numerical Method The accuracy of the numerical technique was checked by examining some limiting cases. For the special case where the inner and outer media are identical and lossless, Equation 4.62 reduces to that reported earlier by Neureuther et al.34. For this case, the results of the present technique agree well with Neureuther's et al34. solutions. Additionally, when the inner conductor does not exist (a=0), Equation 4.62 reduces to that reported by Sensiper33. This case was also checked and the results obtained here are identical to those obtained by Sensiper. It was therefore concluded that the numerical method worked properly and hence was utilized in finding solutions for the cases of interest here. General Properties of the Solutions Before proceeding with the numerical results, it is useful to consider some general properties of the solutions. In considering these general properties, I distinguish between antennas or waveguides in lossless and lossy media. Only the former case was discussed by Neureuther et al.34. lossless media In the case of a lossless external medium, where Rel kj I >Rel kj, the propagation constants of the n=0 modes are real and greater than the wave number of the external medium k,.. The corresponding waves propagate more slowly along the antenna than in the medium and are the well known slow Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 modes of helical structures. For guided modes where n>0, the wave numbers can be complex. This case is studied in some detail in34. lossy media For a lossy external medium, where I k, I >1 kj I ( the case of interest here), the propagation constants of the n=0 modes are, in general, complex and their magnitudes are not necessarily greater than the wave number of the external medium. In this case, the following three distinct regions can be distinguished: the very fast mode region, Pr< Re( k,)< Re( k j; the fast mode region, Re( lq)< |3r< Re(k,.); and the slow mode region, Re( kj)< Re( ke)< Pr. The very fast mode region corresponds to waves which travel along the antenna faster than plane wave propagation in both the external lossy medium and the internal dielectric. These waves are found to be improper and do not correspond to physically realistic solutions of interest here. The fast mode region corresponds to waves propagating faster than plane wave propagation in the external medium1, but slower than it in the internal dielectric. It appears that this mode in our problem can only exist if the outside medium has loss. Consequently, it has not been reported in previous studies dealing with helical antennas radiating into free space. The slow mode region corresponds to waves which travel slower than plane wave propagation in both the external medium and the internal dielectric. These waves, which also exist in the lossless case, are the primary contributors to the antenna's 1 The term fast mode generally referrs to modes with phase velocities greater than both the internal and external medium. Here, I use the term fast mode to refer to modes with phase velocities greater than the external medium, but less than the internal medium. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 heating properties. For all the helical sheath antennas investigated here, only two distinct propagating modes are found to exist. Based on the definition given above, one in the fast mode region and the other is in the slow mode region. The total solution is formed by summing the two modes. In the numerical results which follow, the characteristics of each of these modes are investigated as a function of the antenna design parameters and the electrical properties of the external medium. Numerical Results The effects of loss, operating frequency, pitch angle and antenna radius on the propagation characteristics of the uninsulated helical antennas are investigated. The dependence on these parameters will give an indication of the actual antenna heating performance with respect to configuration changes. It should be noted that only the dominant modes, n=0, are considered in the results and discussion which follow. Effect of Varying Frequency and Loss Figure 5.1 illustrates how the two modes of propagation (slow mode and fast mode) are affected by changes in frequency. Here, Pb vs. kjb (ki=toV(|i0ei)) is used to normalize the results with respect to antenna radius, b, and inner medium permittivity, In this figure, the pitch angle is fixed at 20°, the ratio of outer to inner core radii is fixed at b/a=3.0, the real part of the outside medium's permittivity is taken to be 30 times that of the inside insulation, z j /e / = 30, and the loss tangent, tan(y)=z'lz', of the outside medium is varied from 0.1 to 0.7. Several points can be made about this figure. First, the phase constant (Pr) and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 F ast Wave 0.6 tan(y) Ra(flb) ImQib) 0.4 fib 0.2 0.0 - 0.2 0.0 0.1 0.2, . k.b 0.3 0.4 Slow Wave B tanM Refffb) Imfflbl 6 4 fib 2 0 -2 0.0 0.1 0.2, . i 0.3 0.4 Figure 5.1 Slow and fast mode k,(= o /c v ^ b vs Pb diagram for b/a=3, pitch angle a=20°, and ee7e/=30. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 attenuation constant (ft) are nearly linear functions of frequency for both the slow and fast modes. However, the slow mode is much more sensitive to frequency changes as is seen by the steeper slopes. Second, the attenuation constant of the slow mode is much larger than that of the fast mode. Consequently, the slow mode will release its energy to the exterior medium more rapidly. Third, the slow mode shows a higher sensitivity to changes in loss than the fast mode, particularly in its attenuation constant. Consequently, the fast mode will propagate in a very lossy medium with little attenuation whereas the slow mode will release its energy quickly. Effect of Varying Pitch Angle and Loss Figure 5.2 illustrates how the two modes of propagation are affected by changes in the pitch angle, a. Here, b/a is 3.0 with b=0.1 cm, e j /&/ = 30 with e/=2, and tan(Y)=e#/e/ is once again varied from 0.1 to 0.7. In addition, the frequency is fixed at 915 MHz (the frequency used in the experimental measurements) and the pitch angle is varied from 5° to 40°. Clearly, for the slow mode, as a decreases (corresponding to a more tightly coiled helix) the phase constants and attenuation also increase.. This observation is consistent with the results reported earlier by Neureuther et a l . The effect becomes much more pronounced at smaller pitch angles (5° < a < 10°) when the helix is very tightly coiled. The fast mode, however, shows little sensitivity to pitch angle. This suggests that any changes in heating pattern caused by changes in pitch angle are primarily due to the slow mode. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 F a s t Wave 0.6 c 0.4 o -M n c o ° 0.2 lmag(/3) 0.0 a. 0.2 10 30 Pitch Angle a deg. 20 40 Slow Wave tanM R o (ff) Real(0) a> L. Q. Imag(/?) -5 10 20 30 40 Pitch Angle a deg. Figure 5.2 Slow and fast mode propagation constant vs. pitch angle for f=915 MHz, b=0.1 cm, a=0.03 cm, and €c7e/=30 with §'=2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 Effect of Varying Antenna Radius Figure 5.3 illustrates how the two modes of propagation are affected by changes in the helical sheath radius. Here, b/a is 3.0 , e j lz( = 30 with e/=2, and tanCy^'/eMXS. In this figure the frequency is fixed at 915 MHz, the pitch angle is fixed at 20°, but the outer sheath radius, b, is varied from 0.03 to 0.3 cm. For catheter ablation applications, the radius of a realistic antenna should be somewhere within this range. The important point of this figure is that, as the radius changes by a factor of 10, the propagation constants for both the slow and fast modes changes by, at most, 40%. Thus, I conclude that the propagation constants for both the slow and fast modes are rather insensitive to changes in sheath radius (at least for the antenna sizes of interest here). Summary From the determinantal equation for the uninsulated helical sheath antenna, I determine that two dominant modes of propagation exist in lossy media. One mode, termed a slow mode, corresponds to the same slow modes found in lossless media. The other is a newly discovered faster mode. The propagation constant of the slow mode is sensitive to changes in frequency, pitch angle and loss of the external medium, whereas, that of the fast mode is not. Also, both slow and fast modes are insensitive to changes in sheath radius for the antenna diameters of interest here. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 Fast Wave 0.5 0.4 *»- R eal(l) 0.3 CO c o CJ 0.2 c o Vj o o> O a. 0.1 k_ o.o a Q_ - lmag(/S) 0.1 0.05 0.10 0.15 0.20 0.25 0.30 Sheath radius b (cm ) Slow Wave Real(/9) c a 01 c o o c o □ OI a a. oL. o- lmag(/S) -1 -2 0.05 0.10 0.15 0.20 0.25 0.30 Sheath radius b (cm ) Figure 5.3 Slow and fast mode propagation constant vs. sheath radius, for f=915 MHz, b/a=3, e^ /e^ O with e/=2, pitch angle o=20° and the loss tangent tan(y)=0.5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 5.1.2 Electric Field Distributions In addition to the dispersion characteristics presented above, it is useful to consider how the actual electric field distribution varies with respect to changes in the antenna or the external medium. The dependence on the frequency, the pitch angle and the loss in the external medium are investigated in this section. Using these results along with the dispersion diagrams given above, one can gain some physical insight into the characteristics of an actual wire wound helical antenna. These results follow from the analytical model presented in Section 4.1. The electric fields in the external medium are obtained using Equations 4.56 through 4.60 with the unknown propagation constants being calculated from the procedure described above. For these plots, the electric fields are calculated at the feedpoint, z=0, as a function of radial distance, p, from the antenna surface. Also, the antenna is assumed to be semi-infinite in length with a coaxial source of unit voltage placed at z=0. The coaxial source is included so that the electric fields obtained can be normalized with respect to a fixed source voltage using the orthogonality conditions derived in Section 4.1. Figures 5.5 through 5.7 indicate how, for both the slow and fast modes, the magnitude of the electric field varies with changes in frequency, pitch angle and loss. In these figures, the outer and inner radii of the antenna are fixed at 0.1 and 0.03 cm, respectively. Also, eJ /&/ = 30 with e/= 2. Figure 5.4 illustrates the geometry of the antenna used in these calculations (In these figures the arrows are used simply to indicate the envelope of the electric field magnitude, not its direction). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 Effect of Varying Frequency Figure 5.5 illustrates how the magnitude of the electric z-0 ta n ( Y ) - ■ e l-6 0 .0 soiree voltage f V0- 1 .0 V o lt JS s e m i infinite in le n g th Figure 5.4 An illustration of the antenna geometry used for the electric field calculations, field in the external medium, varies with respect to frequency given a source of one volt. Here, the pitch angle is fixed at 20°, the loss tangent of the outside medium is fixed at 0.5 and the frequency ranges from 100 MHz to 10 GHz. It is clear from the figure that the magnitude of the electric field at the surface of the antenna increases with increasing frequency (assuming the voltage source remains at unity). Although this is true for both slow and fast modes, it is somewhat less pronounced in the slow mode. This figure also shows that, with increasing frequency, the electric field attenuation in the radial direction is more rapid. In fact, for the 10.0 GHz slow mode, the electric field is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 Fast Mode 1.5 1.0 10.0 1.0 E S ui 0.5 O.o 0.0 ------------- 1-----r ” 0.00 i i i .................... 0.25 0.50 0.75 Radial Distance (cm) 1.00 Slow Mode Frequency, GHz 1.5 o - 0.1 • - 1.0 * - 10.0 E u Ui 0.5 0.0 — 0.00 0.25 0.50 0.75 1.00 Radial Distance (cm) Figure 5.5 Magnitude of electric field in external medium at various frequencies. Here, b=0.1 cm, a=0.03 cm, eJ/^30 with e/=2, pitch angle a =20° and the loss tangent tan(y)=0.5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 essentially zero at a distance of only 1.0 ram from the antenna surface. Consequently, it is important to choose a frequency range which provides the proper heating depth for the desired application. The antennas in this study were operated at 915 MHz. Figure 5.5 indicates that, at this frequency, significant heating occurs in the first few millimeters from the antenna. This depth of heating is believed to be adequate for most of the cardiac arrhythmias treatable by catheters. Effect of Varying Pitch Angle Figure 5.6 indicates how the magnitude of the electric field varies with changes in pitch angle. Here, the loss tangent is fixed at 0.5, the frequency is fixed at 915 MHz and the pitch angle is varied from 10° to 30°. This figure clearly indicates that, as the pitch angle decreases, the magnitude of the electric field distribution increases. This is true for both slow and fast modes, however, much more pronounced for the former. Consequently, for catheter ablation applications, greater heating depths can be achieved by decreasing the helical antenna's pitch angle. Figure 5.6 also shows that, for the pitch angles studied here, the magnitude of the electric field of the slow mode is greater than that of the fast mode. This suggests that heating is primarily due to the slow mode. Effect of Varying Loss Figure 5.7 illustrates the effect of varying the loss of the external medium. In this figure, the pitch angle is fixed at 20°, the frequency is fixed at 915 MHz and the loss tangent is varied from 0.1 to 0.7. The plot shows that the magnitude of the electric field distribution in the transverse plane for both slow and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 Fast Mode 0.5 Pitch Angle a o - 30° • - 20 ° 0.4 E 0.3 § HL 0.2 0.1 0.0 - >— 0.00 0.25 0.50 0.75 Radial Distance (cm) Slow Mode 4.0 Pitch Angle a o - 30° • - 20 ° 3.0 E o 1.00 2.0 uj 1.0 0 .0 I— 0.00 0.25 0.50 0.75 Radial Distance (cm) 1.00 Figure 5.6 Magnitude of electric field in external medium at various pitch angles. Here, b=0.1 cm, a=0.03 cm, with e/=2, f=915 MHz and the loss tangent tan(y)=0.5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 Fast Mode 0.4 E o loss o • • • tangent tan(y) - 0.1 - 0.3 - 0.5 - 0.7 0.2 0.0 0.00 0.25 0.50 0.75 Radial Distance (cm) 1.00 Slow Mode 1.5 loss tangent tan(?) o - 0.1 1.0 * - 0.3 * - 0.5 * - 0.7 0.5 0.0 ■— 0.00 0.25 0.75 0.50 Radial Distance (cm) 1.00 Figure S.7 Magnitude c f electric field in external medium for various loss tangents. Here, b=0.1 cm, a=0.03 cm, ej/e'=30 with e/=2, pitch angle a=20°, f=915 MHz and the loss tangent ranges from 0.1 to 0.7. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 fast modes is insensitive to changes in external medium loss. It should be noted however, that while the field distribution in the transverse plane is insensitive to loss, the fields will decay as they propagate in the longitudinal direction. This attenuation is sensitive to external medium loss as indicated by the dispersion diagram in Figure 5.2. 5.1.3 SAR Distributions For catheter ablation, the quantity of direct interest is the specific absorption rate (SAR). This section investigates how the SAR pattern of the uninsulated helical sheath antenna is affected by changes in pitch angle and loss in the external medium. z*=0 0 e0 z=3.0 cm tan(Y)*» — * e'=60.0 * source Held Vo-1.0 Volt p -0 p=0.4 cm Figure 5.8 An illustration of the antenna geometry used for the SAR calculations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The procedure summarized at the end of Section 4.1 is used to calculate these patterns. Since, the experimental results are obtained at one frequency, 915 MHz, the effect of changes in frequency is not considered here. For these calculations, the antenna is assumed to be 3.0 cm in length with outer and inner radii of 0.1 cm and 0.03 cm, respectively. Figure 5.8 illustrates the geometry of this antenna. As before, the real part of the permittivity of the outside medium is taken to be 30 times greater than that of the inside insulation, e j /e/ = 30 with &/= 2. The SAR distributions are calculated as a function of radial distance, p, from the antenna surface and longitudinal distance, z, along the axis of the antenna. It is of particular interest here to compare the pattern of the SAR distribution as a function of pitch angle and of loss. For this reason, I normalize each pattern such that it has a maximum value of one. Effect of Varying Pitch Angle Figure 5.9 indicates how the SAR distribution varies with changes in pitch angle. Here, the loss tangent is fixed at 0.1, the frequency is fixed at 915 MHz and the pitch angle ranges from 10° to 30°. Several observations can be made regarding these figures. First, the results show a distinct standing wave pattern along the length of the antenna. This sinusoidal variation in SAR has been observed experimentally by other investigators48,49 and in this work. Second, the spatial frequency of the standing wave depends strongly on the pitch angle. As the pitch angle decreases, the spatial frequency increases. Physically, this phenomenon corresponds to waves which propagate more slowly as the pitch angle Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 A. Pitch angle a=30° mm jliiiill |||» B. Pitch angle a=20° Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 101 C. Pitch angle a=10° Figure 5.9 Normalized SAR distributions at various pitch angles. Here, b=0.1 cm, a=0.03 cm, L=3.0 cm, eJ/^30 with e/=2, f=915 MHz and the loss tangent is tan(Y)=0.1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 decreases. This observation is consistent with the dispersion characteristics of the slow mode. Third, the SAR pattern attenuation in the radial direction is more rapid for the smaller pitch angles. Thus, to heat tissue at extended depth, one must either increase the pitch angle of the antenna or wait for the effects of thermal conduction to carry energy deep into the tissue. For the catheter ablation treatment of ventricular tachycardia, this does not pose a significant problem. As mentioned in Chapter 2, the reentrant tract formations are believed to lie near the endocardial surface. Consequently, there is no direct need for heating at large depths. Effect of Varying Loss Figure 5.10 indicates how the SAR distribution varies with loss in the external medium. Here, the frequency is 915 MHz, the pitch angle is 10° and the loss tangent ranges from 0.1 to 0.7. Clearly, as the loss tangent increases the SAR peaks shift toward the antenna's distal tip. In fact, when the loss tangent is comparable to muscle tissue, tan(y)=0.5, almost all of the energy is located near the tip. Physically, this phenomenon can be explained by examining the propagation of the dominant modes along the helical structure. At the coaxial feedpoint, z=0, the fast mode is excited strongly by the coaxial source. As shown earlier, this mode has a small attenuation constant which is insensitive to loss in the external medium. Consequently, the fast mode carries most of its energy to the end of the antenna with little loss to the external medium. At the distal end of the antenna, a reflected slow Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 A. Loss tangent, tan(Y)=e#/ e - 0.1 B. Loss tangent, tan(y)=e#/ e - 0.3 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 104 C. Loss tangent, tan(Y)=e#/e'= 0.5 D. Loss tangent, tan(y)=e#/e/= 0.7 Figure 5.10 Normalized SAR distributions for various loss tangents. Here, b=0.1 cm, a=0.03 cm, L=3.0 cm, eZ/ei-30 with e/=2 the pitch angle o=10° and f=915 MHz. Here, the loss tangent is varied from 0.1 to 0.7. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 mode is excited in order to satisfy the short circuit boundary condition there. This mode has a relatively large attenuation constant which becomes larger as the loss in the external medium increases. Hence, this reflected slow mode loses its energy rapidly to the external medium and is responsible for the large hot spot seen at the antenna's tip. The present application of catheter ablation, calls for a more uniform heating distribution than the uninsulated helical antenna exhibits. It will be shown in the next section, that a thin layer of insulation can be used to produce such a heating pattern. 5.2 Results From Insulated Helical Sheath Model This section presents the theoretical results of the insulated helical sheath model derived in Section 4.2. Using this model, I calculate the dispersion characteristics, the electric field distributions and the SAR patterns of the insulated helical sheath. 5.2.1 Dispersion Characteristics The intent of this section is to investigate how the dispersion characteristics of the helical sheath antenna change due to the addition of an insulating layer. These properties are determined by solving the determinantal equation, Equation 4.104, given in Section 4.2. The numerical method described above is used to calculate the propagation constant, |5, as a function of insulation thickness and loss in the external medium. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Numerical Results Figure 5.11 illustrates the geometry of the insulated helical sheath antenna studied here. insulating layer infinite In length +z Figure 5.11 An illustration of the insulated helical sheath model used. The insulated helical sheath model consists of the helical sheath antenna studied above, with an insulating layer surrounding the antenna. The insulating layer is assumed to be lossless with a permittivity equal to that of the internal dielectric, e / ^ / . The external medium is taken to be lossy with complex permittivity e3’=e3/- It is also assumed that the real part of the permittivity of the external medium is greater than the permittivity of the internal dielectric and insulating layer, Under these assumptions, only two distinct propagating modes are found to exist. These modes, as in the uninsulated case, consist of one in the fast mode region and another in the slow mode region. The numerical results which follow investigate the Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. effect of insulation thickness and loss in the external medium on each of these modes. Effect of Insulation Thickness and Loss Figure 5.12 illustrates how the two modes of propagation are affected by changes in the thickness of the insulation and loss in the external medium. Here, the outer radius, b, and inner radius, a, are taken to be 0.1 and 0.03 cm, respectively, ^ /e / = 30 with e/= ^ = 2 , and tan(y)=e3#/e3/ is once again, varied from 0.1 to 0.7. In this figure, the frequency is fixed at 915 MHz and the pitch angle is fixed at 20°, but the insulation thickness, d=c-b, is varied from 0.001 to 0.1 cm. Several important observations can be made regarding Figure 5.12. First, the slow mode is sensitive to the addition of an insulating layer, whereas the fast mode is not. Second, as the insulation thickness increases, the attenuation and phase constant of the slow mode decrease. This results in a slower release of energy and a decrease in wave number for thicker insulations. Third, as the insulation thickness increases, the slow mode becomes less sensitive to loss in the external medium. In fact, as the insulation thickness becomes greater than 0.025 cm, both modes are essentially independent of the external medium. I conclude from these results that, by judiciously choosing a proper insulation thickness, one can adjust the attenuation constant of the slow mode to produce a more uniform heating distribution. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 108 Fast Mode 0.5 0.4 0.3 Lobb ta n g e n t 0.1 0.3 0.5 0.7 0.2 0.1 0.0 - 0.1 — 0.000 0.025 0.075 0.050 insulation thickness, d (cm) 0.100 S lo w M ode 2.5 Lo bb ta n g e n t 0.1 0.3 0.5 0.7 2.0 1.5 1.0 0.5 Ol 0.0 Imag(jJ) -0 .5 - 1.0 — 0.000 0.025 0.050 0.075 0.100 Insulation thickness, d (cm) Figure 5.12 Slow and fast mode propagation constant vs. insulation thickness. Here, f=915 MHz, b=0.1 cm, a=0.3 cm, e3V€/=30 with e/=€2/=2, pitch angle a=20° and the loss tangent is varied from 0.1 to 0.7. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 109 5.2.2 Electric Field Distributions These results follow from the analytical model presented in Section 4.2. The electric fields in the external medium are obtained using Equations 4.99 through 4.103, with the unknown propagation constants being calculated by the procedure described above. Like the previous electric field plots, the values are determined at the feedpoint, z=0, as a function of radial distance, p, from the outer surface of the insulating layer. Also, the antenna is assumed to be semi-infinite in length with a coaxial source of unit voltage placed at z=0. Figure 5.13 illustrates the geometry of the antenna used in these calculations. tan(Y)' 4 - 6 0 .0 ► sem i Infinite !n length source voltage V „ - 1 .0 V o lt In s u la tin g la y e r Figure 5.13 An illustration of the antenna geometry used for the electric field calculations of the insulated antenna.. Effect of Insulation Thickness Figure 5.14 shows how the magnitudes of the electric field distributions of the two dominant modes are affected by insulation thickness. Here, the outer radius, b, and inner radius, a, are taken to be 0.1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. F a s t M ode 0.4 0.3 Insulation Thickness (cm ) E ° 0.2 • - • - 0.01 0.001 Ll I 0.1 - 0.0 — 0.00 0.25 0.50 0.75 Radial Distance (cm) 1.00 Slow Mode 1.00 Insulation Thickness (cm ) 0.75 • - - - 0.01 0.001 >_ 0.50 ui 0.25 0.00 *— 0.00 0.50 0.75 0.25 Radial Distance (cm ) 1.00 Figure 5.14 Magnitude of electric field in external medium for various insulation thicknesses. Here, b=0.1 cm, a=0.03 cm, e3//e/= 30 with e,/=e2/=2, pitch angle a=20°, f=915 MHz and the loss tangent tan(y)=0.5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I ll and 0.03 cm, respectively and z j /e / = 30 with e/= &/=2. In this figure, the frequency is 915 MHz, the pitch angle is 20°, the loss tangent is 0.5 and the insulation thickness is varied from 0.001 to 0.1 cm. Clearly, for the slow mode, as the insulation thickness increases, the magnitude of the electric field at the surface of the insulating layer decreases. It should be noted however, that these plots are the electric field distribution at the antenna's feedpoint, z=0, and will attenuate along the length of the antenna as e^‘z- The electric field distribution of the fast mode, as in the previous results, shows little sensitivity to insulation. 5.2.3 SAR Distributions This section investigates how an insulating layer affects the SAR pattern of the helical sheath antenna. The complete procedure summarized at the end of Section 4.2 is used to calculate the SAR as a function of insulation thickness and loss. p -0 insulating layer Figure 5.15 Illustration of geometry used in the SAR calculation for the insulated antennas. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 112 Figure 5.15 illustrates the geometry of this antenna. For these calculations, the antenna is assumed to be 3.0 cm in length with an outer radius, b, and inner radius, a, taken to be 0.1 and 0.03 cm, respectively, e3/ /e / = 30 with e/= e / =2 and the frequency is fixed at 915 MHz. The SAR distributions are calculated as a function of radial distance, p, from the insulation surface and longitudinal distance, z, along the axis of the antenna. Here, I examine the SAR distribution as a function of insulation thickness and loss. As done in the uninsulated case, the SAR patterns are normalized to a maximum value of unity. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 113 A. Uninsulation / llliP B. Insulation thickness d=0.001 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. >- C. Insulation thickness d=0.01 i M E I I I -P- :: : D. Insulation thickness d=0.1 Figure 5.16 Normalized SAR distributions for various insulation thicknesses. Here, b=0.1 cm, a=0.03 cm, L=3.0 cm, e3//e/= 3 0 with €/=e2'=2 the pitch angle o=20°, f=915 MHz and the loss tangent tan(Y)=0.1. Here, the insulation thickness is varied from 0.001 to 0.1 cm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 115 Effect of Insulation Thickness Figure 5.16 illustrates the effect of insulation thickness on the SAR distribution. Here, the pitch angle is 20° and the insulation thickness is varied from 0.0 to 0.1 cm. As in the uninsulated case, the SAR pattern shows a distinct standing wave pattern whose wavenumber varies with insulation thickness. As the thickness increases, the wavenumber decreases, resulting in less cycles of the standing wave. Effect of Insulation Thickness and Loss Figure 5.17 compares the effects of loss on the SAR distributions of the insulated and uninsulated antennas. Here, for both antennas, the pitch angle is 10° and the loss tangent is varied from 0.1 to 0.7. For the insulated antenna, the insulation thickness is fixed at 0.01 cm. These figures show how adding an insulating layer decreases the sensitivity to loss. For the uninsulated antenna, as shown earlier, increasing loss shifted the SAR distribution towards the antenna's tip. This resulted in an undesirable SAR pattern for catheter ablation applications. However, the SAR pattern of the insulated antenna is insensitive to changes in loss. Thus, by adding a thin layer of insulation, one can achieve a nearly uniform heating pattern along the length of the antenna, even for loss tangents comparable to that of heart tissue. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Uninsulated Insulated A. Loss tangent, tan(Y)=e#/e'= 0.1 B. Loss tangent, tan(y)=e#/ e - 0.3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ^ Uninsulated Insulated C. Loss tangent, tan(y)=erle'= 0.5 D. Loss tangent, tan(Y)=e'/e'= 0.7 Figure 5.17 Comparison of normalized SAR distributions for insulated and uninsulated helical antennas in various lossy media. Here, c=0.11 cm, b=0.1 cm, a=0.03 cm, L=3.0 cm, €3//e/=30 with e /= e 2/=2 and the pitch angle a=10°and f=915 MHz. Here, the loss tangent is varied from 0.1 to 0.7. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 118 Summary This chapter presented some analytical results from the helical sheath antenna model presented in Chapter 4. The results, which included dispersion diagrams, electric field plots and SAR distributions, reveals several interesting observations. First, when the antennas are immersed in an external lossy medium, there exist only two dominant (n=0) modes of propagation. One mode, corresponds to a previously reported slow mode found in lossless medium. The other is a newly discovered faster mode, which appears only to exist when the external medium has loss. The propagation constant and electric field distribution of the slow mode were found to be sensitive to changes in the antenna configuration and the electrical properties of the external medium, whereas those of the fast mode were not. It was also determined that the slow mode contributes primarily to the heating of the external medium. Second, the SAR distributions for the uninsulated antennas show a significant shift in the heating pattern as the external medium became more lossy. This resulted in an undesirable heating pattern for the present application of catheter ablation. Lastly, it was discovered, that by adding a thin layer of insulation to the outside of the helical antenna, one can produce a more uniform heating pattern which is insensitive to loss in the external medium. This configuration appears to be suitable for catheter ablation applications. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 119 Chapter 6: Experimental Methods This chapter describes the experiments that were performed. It consists of three sections. The first section describes the apparatus used to measure SAR patterns, the second describes the input impedance measurements; the third section describes the measurement of the microwave dielectric properties of normal and coagulated heart tissue. 6.1 Measurement of Specific Absorption Rate (SAR) To confirm that the calculated SAR patterns of the helical coil antenna and RF ablation catheter were correct, I designed and built a system to measure the SAR directly. This section consists of two parts. In the first part the hardware of the SAR mapping apparatus is described. The second part explains the methods used to correct for artifacts introduced by the thermistor probe. 6.1.1 SAR Mapping Apparatus A block diagram of the experimental apparatus is shown in Figure 6.1. The antennas were mounted in a tank filled with aqueous electrolyte of varying Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 120 (x,y,z) Positioning Command Vectra Computer Thermistor Signal Thermistor Digital Pulse 915 MHz Microwave Generator Tttt ' Aiteua Microwave Energy Blectroltfe Solatiga Figure 6.1 Thermistor based SAR mapping apparatus. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 121 conductivities. In Chapter 2, this solution was shown to be adequate as a muscle phantom in the frequency range used in this study. The probe consisted of a small thermistor encased in a glass micropipette, whose thermal response time was less than 0.1 seconds. The leads of the thermistor were a twisted pair of shielded wires which were aligned perpendicular to the field of the antenna. This arrangement was found to be sufficiently immune from interference to permit accurate measurements of the antenna SAR patterns. The probe was mounted on a translating platform which was controlled by stepping motors. The entire measurement process, including movement of the probe and the recording and analysis of the thermistor output, was controlled by a laboratory computer (Hewlett-Packard Vectra) running ASYST. To measure the SAR pattern, I recorder the transient temperature increase in the outside medium following a short pulse (0.5 sec) from either a 30 Watt 915 - MHz microwave transmitter or a 30 Watt 500-kHz RF generator. The thermistor probe was then repositioned to record the local SAR at a new position. Using this apparatus, I could map the three-dimensional SAR pattern of an antenna with a 0.5 mm spatial resolution over a total time period of a few hours. The whole process was done under computer control. A more detailed description of the specific hardware follows. Hardware The hardware for the SAR mapping apparatus consisted of the x-y-z positioning table, thermistor probe, power generators, computer and data acquisition system with associated electronics. A few of these components are discussed in detail below. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 122 Positioning Table Accurate measurement of the antenna's SAR distribution requires precise positioning of the thermistor relative to the antenna. The device shown in Figure 6.2 was able to position the thermistor in three dimensions with the necessary accuracy. This device is described in detail by Cheever 198950 and will be discussed here briefly. Figure 6.2 Positioning table The antenna mapping system was constructed using an x-y plotter drive mechanism for transverse motion and a stepper motor mounted on the plotter pen carriage for vertical, z-axis, positioning. The x-y plotter was able to position the thermistor in the x-y plane with an accuracy of 0.1 mm. The stepper motor allowed Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 123 positioning in the z-axis with an accuracy of 0.04 ram. The x-y-z positioning was under computer control. Given the required x, y and z coordinates at which to move the thermistor, the computer would translate these coordinates into the proper plotter and stepper motor commands. The plotter possessed its own language which defined movements. These commands were sent via a serial line from the computer. The z-axis movement was accomplished by calculating the appropriate number of step increments to send to the stepper motor. This sequence of steps was sent to the stepper motor controller via a digital I/O port on the data acquisition system which in turn incremented the stepper motor. Thermistor Probe A thermistor probe was constructed by securing a small thermistor at the tip of a glass micropipette. A schematic of the probe is shown in Figure 6.3. The thermistor had a diameter of 0.25 mm and a measured thermal response time of 0.08 sec. It was Therm istor G lass Micropipette 0.25 mm Twisted Fair Insulated W ires Epoxy Figure 6.3 Thermistor probe used for SAR measurements. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 124 secured and sealed by a layer of epoxy placed at its base. The wire leads were insulated and twisted to reduce electrical interference. Electronics Any temperature changes are reflected as changes in the thermistor resistance. The circuit shown in Figure 6.4 was designed to transform the resistive changes of the thermistor into an appropriate voltage signal which was capable of being sampled by the computer's data acquisition unit. The first stage was a transducer bridge circuit used to obtain an output voltage proportional to temperature variations. In this case, a potentiometer was added to the bridge so that the bridge could be balanced at some reference temperature and then calibrated to read variations above and below that reference. The bridge output was sent though an instrumentation amplifier (variable gain) and then passed through an active two-pole low pass filter with a 1.0 kHz cutoff Vrr = 2.0 V 6.0 KQ f 9.6 KQ — o | 14.0KQ 7.4 KQ 5.0 KQ ''I --------W V - p A A r vw 9.6 KQ out 14.0 KQ 1.0 KQ 1.0 KQ S i— V W i — V W -1 o I Figure 6.4 Electronic circuit used to record thermistor temperature. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 125 frequency. This temperature signal (Vout) was then sampled by a multichannel 12-bit analog-to-digital converter (Data Translations DT2800) at 100 Hz using an HPvectra™ computer with the data acquisition software ASYST™. 6.1.2 Data Analysis Thermistor probes have been used by a variety of investigators to facilitate SAR measurements. In particular, Bowman et a/.51 studied the accuracy of using thermistor probes and reported some potential problems when the probes were inserted into strong electromagnetic fields. Bowman's chief problem was that due to differences in the electrical and thermal properties of the thermistor and the external phantom material the measured temperature does not purely reflect the heating of the external medium (the desired phenomenon), but rather a combination of thermistor and external medium heating. This section examines this thermal artifact in more detail. Thermal Artifact Due to differences in electrical and thermal properties between the thermistor and external medium, the recorded temperature will reflect a combination of thermistor and external medium heating. However, since it is desirable to measure only the external medium temperature uncontaminated by the presence of the thermistor, a heat transfer model was developed to extract the external medium temperature from the recorder temperature signal. The model shown in Figure 6.5 consisted of a spherical thermistor of radius R, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 126 ^ th ®th» Kh» **th \ ®m> ®m» P ► Figure 6.5 Heat transfer model used to subtract the thermal artifact out of thermistor temperature measurement. electrical conductivity coefficient of heat conduction and thermal diffusivity immersed in an uniform electric field of strength E0 and surrounded by an external medium. The external medium was assumed to be infinite in extent with electrical conductivity a m, coefficient of heat conduction k,,,, specific heat cm and density pm. Initially, the thermistor and external medium were assumed to be in thermal equilibrium at temperature, Tj, and at time t=0 the electric field was turned on. The fields internal to the thermistor, E^, will be different than the external fields Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 127 and can easily be calculated by52, Ea 34=— E„ [6.1] 24 + 4 where e** = e*' - j a j (to e0 ) and em* = e j - j o m/ (to e0 ) are the complex permittivity of the thermistor and external media respectively. The goal of this analysis is to calculate the average temperature of a thermistor immersed in a uniform external electric field. A heat transfer analysis, described in detail in Appendix A , finds the average temperature, T,ve,within the thermistor to be a Tt * * £ [ 2 . n4 4 where 5 = - — ][1 - e {« ** ] + S t [6-2] «* ° m is the rate of temperature increase in the external medium 2 PmCm Q = M and a* 2 is the time averaged power density dissipated in the thermistor. Without the presence of the thermistor the external medium temperature, T , ^ , is simply given as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. r — i - T> * s ‘ [M 1 By comparing Equations [6.2] and [6.3], it is clear that the temperature measured by the thermistor differs from the actual temperature of the external medium by an Measure temperature signal using thermistor o. Actual temperature signal time Figure 6.6 Illustration of how the measured thermistor temperature will differ from the actual temperature of the external medium. exponentially decaying transient component and a constant offset. This is illustrated in Figure 6.6. The quantity of interest here is the SAR in the external medium. By definition, this is given by the rate of temperature increase multiplied by the specific heat of the external medium ( SAR = cmS ). Therefore, measuring the slope of the thermistor signal after the transient component has sufficiently died out, results in a quantity Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 129 directly proportional to the SAR in the external medium. To verify this prediction of the model, I performed the following experiment. The thermistor probe was positioned and fixed in place next to a microwave antenna. A 915 MHz microwave generator was then used to pulse microwave energy at several 26.0 Second Slope 25.0 Input Power, W atts 40.0 Watts O 6> <D X J 24.0 30.0 Watts a> a. E 20.0 Watts 23.0 10.0 Watts 22.0 Initial Slope Power On 21.0 0.00 0.25 Power Off 0.50 0.75 1.00 Time, se c o n d s Figure 6.7 Measured temperature versus time curves for several different input power levels. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 130 different power levels for 0.5 seconds each2. The resulting temperature signals were recorded as a function of time (Figure 6.7). From these curves, I calculated the slope of the initial temperature rise and the steady state temperature rise (identified in Figure 6.7 as the second slope). These values were then plotted as a function of the input power. The results shown in Figure 7.0 0.0 O a) o — initial slope • - seco n d slope 5.0 ■a 3.0 2.0 1.0 0.0i 40 Input Pow er Watts Figure 6.8 The initial and second slopes identified in Figure 6.7 verses input power. 2 The microwave generator was custom built by Microwave Medical System, Littleton MA. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 131 6.8 illustrate the utility of the model. The second slope shown in Figure 6.8 is nearly a linear function of input power, as predicted by the heat transfer model. This is a necessary requirement for accurately measuring SAR patterns. If the initial slope was used, as shown in Figure 6.8, the SAR pattern would be highly distorted by the non-linear relationship to input power. 6.2 Measurement of Input Impedance The SAR mapping apparatus described in Section 6.1 was used to measure the normalized distribution of energy released by the ablation device. This, however, does not complete the entire picture. It is also necessary to consider what percentage of the energy supplied by the generator is actually being released by the antenna. Since, a considerable amount of the supplied energy may be reflected back into the generator, the actual energy being released by the antenna may be too little to produce an adequate ablation. It is possible to calculate the amount of reflected energy from Z=0 Z=L Figure 6.9 Reference plane for input impedance measurements. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 132 HP 8720 Network Analyzer GPIBBns Apple Macintosh n Computer Figure 6.10 Experimental setup used to measure input impedance. knowledge of the antenna's input impedance. In this section, I describe the experimental setup used to measure the input impedance of insulated and uninsulated helical antennas radiating into a variety of external media. Using a standard transmission line approach, the input impedance is defined as the impedance looking into the antenna at the plane z=0 ( shown in Figure 6.9 ). The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 133 input impedance which is a complex quantity, z both the input resistance = +j represents and the input reactance X ^ ,. The percentage of reflected power is related to the input impedance through the simple relationship, |j . 5p;p % Reflected Power = r /wer x 100 = ---------- — x 100 [6.4] U + _^|2 Here, Z0 is the characteristic impedance of the transmission line (Z0= 50 £2 typically) P is the power reflection coefficient defined as Tpower = ——• where Pref is and ^inc the reflected power and the incident power. Figure 6.10 illustrates the test fixture used for the input impedance measurements. The antennas were immersed in a basin filled with electrolytic NaCl solution. The basin was made large enough to insure that any reflections off its sides or top did not significantly disturb the antenna's input impedance. A flexible 50 £2 coaxial cable was used to attach the antenna to a Hewlett Packard 8720 Network Analyzer. The network analyzer was calibrated at the reference plane ( shown in Figure 6.9) using three factory standard loads — a short circuit, an open circuit and a 50 £2 termination. An Apple Macintosh II computer running the data acquisition program Labview™ was programmed to control the network analyzer. The network analyzer swept through the frequency range 130.0 MHz to 3.0 GHz recording the input impedance every 50.0 MHz. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.3 Measurement of Microwave Dielectric Properties The SAR calculations depend on the dielectric properties of the medium into which the microwave antenna is radiating. During the time course of ablation, the dielectric properties of heart tissue may change due to tissue coagulation. Therefore, it was necessary to measure these properties at the low microwave frequencies for both normal and coagulated heart tissue. To accomplish this, I used an open ended coaxial probe technique developed by Tanabe et al.. 53 The probe (typically a length of the common 3.6 mm OD semirigid coaxial line) was placed against the unknown sample and its complex reflection coefficient, T, was measured with an automated network analyzer (ANA). This technique relies on the fact that the reflection coefficient of the coaxial probe is strongly dependant on the dielectric properties of the sample. Given the reflection coefficient and a good electrical model for the coaxial probe, one cal calculate the complex dielectric constant of the medium. The instrument used to implement the coaxial probe technique is shown in Figure 6.11. For these measurements the reflection coefficients were measured using a Hewlett- Packard 8410 A Network Analyzer. A computer program was written ( Miranda, 1990) on a Hewlett-Packard 9000 series computer, to calibrate the network analyzer, record the reflection coefficients and calculate the complex dielectric constant. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 135 To Computer To Computer A HP8410A Network Analyzer HP5-I42A Freqtjency Counter HP8620C Sweep Oscillator 6412 A Display 862090A RF Plug-In " HP8743 A Reflection / Frequency Converter Transmission Test Unit Reflection Port Transmission Port r~ i Coaxial Probe Saline 7 Figure 6.11 Experimental setup used to measure the complex dielectric constant o f tissue at microwave frequencies. The measurement is based on the open-ended coaxial probe technique developed by Tanabe3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 136 In order to back calculate the dielectric properties from the measured reflection coefficient, a good electrical model for the coaxial probe is required. While no rigorous closed-form solution exists, several approximate models are available. Marcuvitz54 expressed the admittance of the probe as an integral over its aperture; this approximation was rederived in equivalent forms by Levine and Papas55 and Misra56. Marcuvitz's approximation can be expanded into a series that is convenient for numerical solution, and in this form can be used in experimental studies employing the probe technique. According to Marcuvitz's approximation, the admittance at the plane separating the coaxial probe and the sample (z=0 in Figure 6.11) is Y = G +jB 71 [6.5] a v - Si(2k0y/e a sin(—)) - Si(2k0\fe b sin(—))] dQ 2 2 In these expressions, a and b are the inner and outer radii of the line, respectively; k„ is the propagation constant in free space, k ^ rc f/c where c is the velocity of light in vacuum and f is the frequency; e and ec are the relative permittivities of the material under test and the dielectric in the transmission line, respectively; J0 is the Bessel function of order zero; and Si is the sine integral; Y0 is the characteristic admittance of the line (Y0 = 0.02 S in these experiments). In this model, k0, ec, Y0, a and b are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 137 known parameters, G and B are measured quantities and e is the unknown permittivity. Equation [6.5] was expanded in a power series in e in order to numerically calculated e from the other known and measured quantities. The power series given by Misra et al.56 is valid over the frequency range and probe dimensions used in this study. For these experiments, probes were constructed from a semirigid 50 £2 coaxial line (3.5 mm OD line with a type SMA connector) of length 6.0 cm. The end distal to the connector was machined flat and polished with a fine crocus cloth. The probe was then connected to the test cable of the ANA (automated network analyzer). The reference plane of the measurement was defined by shorting the end of the probe with aluminum foil and adjusting the electrical delay of the ANA until a constant 180° phase angle was observed. The reflection coefficients from three standard load each with a known reflection coefficient were used to correct for imperfections in connectors and the network analyzer. Additional test samples with known dielectric properties were used to access the accuracy of the technique. Over the range of frequencies used in this study, the technique was found to be accurate to within 6.0% of the known values. This was considered acceptable for the present study. The dielectric measurements were performed on freshly excised sheep hearts (0 - 2 hrs old) that were obtained from a local slaughterhouse and put on ice for less than 1.0 hour prior to the experiment. For uncoagulated tissue measurements, the left ventricle was excised and sliced Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. into 3 cm x 3 cm sections with thicknesses ranging form 1 cm - 2.0 cm. The slices were then immersed in a tank filled with 0.9% physiological saline at 25°C. The tissue was left in the tank at least 30 minutes prior to measurements to insure that it reached thermal equilibrium with the temperature of the external medium. After the coaxial probe was properly calibrated, it was pressed firmly against the epicardial surface. The dielectric measurements were then recorder at a variety of frequencies. The procedure was then repeated on the endocardial and myocardial surfaces. For coagulated tissue measurements, the left ventricle was again excised and sliced into 3 cm x 3 cm sections. However, this time, the heart tissue was cooked in a microwave oven at 300 watts for 10 minutes. It was then cooled for 30 minutes at 25°C prior to dielectric measurements. At this point the tissue was visibly coagulated. The tissue was then immersed in the tank of physiological saline and the same measurement process, as that for uncoagulated tissue, was performed. The results for the dielectric properties of coagulated and uncoagulated tissue's dielectric properties are presented in Chapter 6 (the results chapter). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 139 Chapter 7: Experimental Results This chapter presents the experimental results obtained during the course of this study. It consists of four sections. The first begins by presenting the experimental measurements for the uninsulated and insulated helical antennas described in Chapter 6. It then compares these experimental results to the analytical results of Chapter 5. The second section presents experimentally measured SAR patterns for an RF ablation device and compares these to the results calculated by using the analytical model derived in Section 4.3 . The third section briefly summarizes the complex permittivity measurements for normal and coagulated heart muscle at microwave frequencies. Finally, the last section shows the results of the in vitro experiments which were performed. 7.1 Helical Antennas This section presents the measured results for the uninsulated and insulated helical antennas. These experimental results consist of SAR patterns and input impedance measurements. To validate the analytical models developed in Chapter 4 , 1 compare the experimentally obtained SAR patterns with the calculated results in Chapter 5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 140 7.1.1 Uninsulated Helical Sheath Antenna A series of helical antennas were fabricated and used in these experiments. Their physical characteristics are summarized in Table 7.1. The antenna was constructed by placing a helically wound, 0.01 cm diameter, copper wire around the exposed dielectric of a standard semirigid coaxial line. The helix was wound using a precision spring-winding facility at Arrow International (Reading, PA), with pitch angles ranging from 5.25° to 17.1°. All the antennas were of length L=3.0 cm with an inner conductor of radius a=0.03 cm and outer conductor of radius b=0.095 cm. The dielectric core was made of teflon which was assumed to have a dielectric constant £i=2.1. For all these experiments, the frequency was fixed at 915 MHz. Length, Radius of inner Radius of outer Pitch angle, a, L, (cm) conductor,a, (cm) conductor,b, (cm) deg. Antenna #1 3.0 0.03 0.095 17.1 Antenna #2 3.0 0.03 0.095 11.2 Antenna #3 3.0 0.03 0.095 8.6 Antenna #4 3.0 0.03 0.095 5.25 Table 7.1 Geometry o f helical antennas used for experimental measurements SAR Distributions The apparatus described in Section 6.1.1 was used to measure the SAR patterns of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 141 the helical antennas described in Table 7.1. It is of particular interest, for the present application, to investigate how the SAR patterns of these antennas vary with respect to changes in pitch angle and loss in the external medium, and to compare these experimental results to the analytical model of Section 4.1. Table 7.2 summarizes the dielectric properties of the various concentrations of NaCl solutions used in these experiments. These properties were calculated from the equations given by Stogryn40 (which were discussed in Section 3.3 and used for the analytical calculations). The calculated and measured SAR patterns are normalized in the same manner as described in Chapter 5. Concentration of NaCl solutions (wt./wt.) distilled water 0.2% NaCl (wt/wt) 0.4% NaCl (wt/wt) 0.6% NaCl (wt/wt) 0.8% NaCl (wt/wt) 1.0% NaCl (wt/wt) Complex Permittivity E^E'-je' 78.2-j3.41 77.5-J10.3 76.8-jl7.1 76.2-j23.8 75.5-j30.4 74.9-J36.9 Loss Tangent tan(y)=E//E/ 0.044 0.13 0.22 0.31 0.40 0.49 Table 7.2 Complex permittivity and loss tangent of various concentrations of NaCl solutions. For reference, the complex permittivity of muscle tissue at 915 MHz was measured to be approximately E*=56-j27. Effect of Pitch Angle In Figure 7.1, the SAR patterns measured for the antennas in Table 7.1 immersed in distilled water are compared to the calculated SAR patterns. Clearly, the analytical results are in good agreement with experimental data, demonstrating the validity of the sheath helix model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Measured Calculated A. Antenna #1, pitch angle a=17.1 B. Antenna #2, pitch angle a=11.2' Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Measured Calculated mm C. Antenna #3, pitch angle a= 8.6‘ D. Antenna #4, pitch angle a=5.25° Fto* 7.1 Measured and calculated SAR distribudons fo, antennas of various ptteb angles in disdlled Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 143 144 As discussed in Chapter 5, the SAR pattern shows a distinct standing wave pattern along the length of the antenna and its wavenumber increases as the pitch angle decreases. This implies a standing wave with higher spatial frequency for more tightly coiled helices. Figure 7.1 also shows that the majority of the energy is held close to the helical surface. The penetration depth at 915 MHz is shown here to be only a few millimeters, and decreases as the pitch angle decreases. Consequently, if a deep lesion is required, one must rely on the effects of thermal conduction. Effect of Loss Since the conductivity of muscle tissue is much greater than that of distilled water, the SAR plots shown above do not reflect the pattern of the actual antenna when used for ablation. Figures 7.2 and 7.3 illustrate how tissue loss would change the SAR pattern of the uninsulated helical antenna. In Figure 7.2, Antenna #3 (a= 8.6°) is immersed in solutions ranging from distilled water to 0.8% saline. As predicted by the analytical results, as the conductivity of the external medium increases, the heating pattern shifts toward the antenna's tip. In fact, when the conductivity is similar to muscle tissue (0.8%) almost all of the heating takes place at the tip. Figure 7.3, shows the same phenomenon for Antenna #4 (a=5.25°). In this case however, the shift in the heating pattern is ever more pronounced than for Antenna #3. This is consistent with the analytical results which predict larger attenuations with decreasing pitch angle. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Measured Calculated A. Antenna #3 in distilled water B. Antenna #3 in 0.2% (wt/wt) NaCl solution Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 145 Measured Calculated V C. Antenna #3 in 0.4% (wt/wt) NaCl solution D. Antenna #3 in 0.8% (wt/wt) NaCl solution Figure 7.2 Measured and calculated SAR distributions for Antenna #3 in distilled water and various saline solutions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Measured Calculated A. Antenna #4 in distilled water B. Antenna #4 in 0.2% (wt/wt) NaCl solution Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Measured Calculated 148 C. Antenna #4 in 0.4% (wt/wt) NaCl solution D. Antenna #4 in 0.8% (wt/wt) NaCl solution Figure 7.3 Measured and calculated SAR distributions for Antenna #4 in distilled water and various saline solutions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 149 In p u t Impedance To gain a deeper understanding of the heating characteristics of the helical antenna, it is necessary to also consider the input impedance. The experimental setup described in Chapter 6 was used to measure the input impedance of the uninsulated helical antennas described in Table 7.1. Unfortunately, the helical sheath model is unable to adequately predict the measured input impedance. This is not surprising since the input impedance is far more sensitive to current distribution near the feed point and to angular variations in electric field than the SAR distribution. The helical sheath model assumes no azimuthal variation in electric field and current distribution. Moreover, higher order modes in the helical structure and the coaxial transmission line are not considered here and could significantly affect the input impedance. In order to accurately predict the input impedance, the helical sheath model should be replaced by a more realistic (and complicated) model, such as a helical tape model. In Figure 7.4a, the measured resistive and reactive components of the input impedance are plotted as a function of frequency. The antenna pitch angle varies from 5.25° to 17.1° and the outside medium is 0.8% saline. Figure 7.4b shows the same antennas radiating into 0.4% saline. The results show an insensitivity of input impedance to changes in helical pitch angle and loss in the external medium. This most likely reflects the contribution of the fast mode since it is also rather insensitive to variations in pitch angle and loss. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 150 140 Helical Pitch Angle deg. o - 14.B 120 | 100 .c o ua> c 80 a .52 *01 u a: 0.16 0.60 1.04 1.4B 1.92 Frequency GHz 2.36 2.80 60 40 cn |o Helical Pitch Angle deg. o - 14.8 20 0) 2 o a -M o a a -2 0 O ' -4 0 -6 0 -B0 0.16 0.60 1.04 1.4B 1.92 Frequency GHz . 2.36 2.80 Figure 7.4a Measured resistive and reactive components of the input impedance vs. frequency for antennas with various pitch angles immeresed in 0.8% saline. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 151 140 Helical Pitch Angle deg. •2 60 0.16 0.60 1.04 1.48 1.92 2.36 2.80 Frequency GHz Helical » -2 0 1.04 1.48 1.92 Frequency GHz 2.36 2.B0 Figure 7.4b Measured resistive and reactive components of the input impedance vs. frequency for antennas with various pitch angles immeresed in 0.4% saline. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 152 7.1.2 Insulated Helical Sheath Antenna An insulating layer was formed by tightly wrapping the antennas, described in Table 7.1, in a thin layer of Teflon tape (insulation thickness d=0.01 mm, permittivity £2=2.1). Here, I intend to investigate the effect of this insulating layer on the SAR pattern and input impedance of the helical antenna. I also compare the measured results to the analytical model developed in Section 4.2. SAR Distributions The SAR patterns of the insulated antennas were measured using the same apparatus and method used for the uninsulating antennas. Effect of Insulation and Pitch Angle Figure 7.5 shows the effect of insulation on three antennas (a=17.1°, 8.6° and 5.25°) immersed in distilled water. Here, the measured results are compared to the calculated results. First of all, the insulation dramatically alters the antenna's SAR pattern. Secondly, the measured and calculated results are in fairly good agreement, with the exception of an additional peak in the SAR pattern of the calculated results. This discrepancy is most likely due to inaccuracy in measuring the insulation thickness. As shown in Chapter 5, a very small change in this thickness can significantly alter the antenna's SAR pattern. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Calculated Measured 153 A. Antenna #1, uninsulated in distilled water. 1 m m B. Antenna #1, insulated in distilled water Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Measured Calculated C. Antenna #3, uninsulated in distilled water. D. Antenna #3, insulated in distilled water Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 154 Calculated Measured l V s3 .O -cC O 1 E. Antenna #4, uninsulated in distilled water. »*'\\V .v F. Antenna #4, insulated in distilled water. Figure 7.5 Measured and calculated SAR distributions of uninsulated and insulated antennas in distilled water for Antennas #1, #3 and #4. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 156 Effect of Insulation and Loss Figure 7.6 compares the normalized SAR patterns of Antenna #3 and #4 for the insulated and uninsulated cases when immersed in 0.8% NaCl solution. It is evident that a thin layer of insulation greatly reduces the effect of loss on the antenna's SAR pattern. As seen earlier, the SAR peaks of the uninsulated antenna rapidly shifts toward the antenna's tip with increasing conductivity of external medium. In the insulated antennas, however, this effect is greatly reduced. In fact, the measured and calculated results show no significant change in the normalized SAR pattern as the loss of the exterior region approaches that of tissue. This result has significant implications for the prospective biomedical application of catheter ablation. Clearly, due to the reduction in tip heating, the insulated antenna produces a relatively uniform heating pattern even when immersed in a medium as conductive as tissue. Consequently, insulated helical antennas may offer the possibility of treating cardiac arrhythmias (such as ventricular tachycardia) which require relatively large lesion sizes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Measured Calculated A. Antenna #3, uninsulated in 0.8% (wt/wt) NaCl solution. B. Antenna #3, insulated in 0.8% (wt/wt) NaCl solution. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Measured Calculated 158 5&> C. Antenna #4, uninsulated in 0.8% (wt/wt) NaCl solution. iccD. Antenna #4, insulated in 0.8% (wt/wt) 1^£1* solution. Figure 7.6 Measured and calculated SAR distributions of uninsulated and insulated antennas in 0.8% NaCl solution for Antennas #3 and #4. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 159 7.2 RF Ablation Measurements To better comprehend the significance of the microwave antenna results with respect to presently available RF techniques, I measured a series of SAR patterns for the RF device. The SAR mapping apparatus, described in Section 6.1.1, was used to obtain the patterns. The RF catheter used in these experiments was a standard quadrapole catheter (manufactured by the Bard corporation) which is used clinically in RF ablation procedures. The catheter was driven by a 100 Watt, 500 kHz RF source. It was immersed in a basin filled with NaCl solution and oriented perpendicular to a grounding plane which was placed 15.0 cm from the catheter tip. Figure 7.7 illustrates the geometry of the RF catheter used for these experiments. The SAR patterns were measured in both 0.4% and 0.8% NaCl solutions, and were catheter tip electrode b=3.0mm p=0 T 3-0 mm grounding plane p=0.4 cm z=2.'0 cm 2=3.0 cm Figure 7.7 An illustration of the RF catheter geometry used for the SAR measurements, normalized in the same manner as the microwave antenna results. Figure 7.8 compares the measured SAR patterns to the results calculated from Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Measured Calculated 160 sc*- A. RF catheter in 0.4% (wt/wt) NaCl solution. B. RF catheter in 0.8% (wt/wt) NaCl solution. Figure 7.8 Measured and calculated SAR distributions of the RF catheter in 0.4% and 0.8% NaCl solution. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 161 the analytical model derived in Section 4.2. Several observations can be made regarding this figure. First, the measured and calculated results are in excellent agreement. Second, the SAR of the RF device is largest at the tip of the catheter and rapidly attenuates away from this point; it drops off as 1/r4 as predicted by the model. Third, the normalized SAR patterns are independent of the conductivity in the external medium. Consequently, the penetration depth of the RF device in tissue, or any other lossy medium, will only be 2.0 to 3.0 mm. Figure 7.9 compares the normalized SAR pattern of an RF catheter with that of an insulated helical antenna. This figure clearly demonstrates that the microwave antenna is capable of heating a much larger region than the RF device. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Measured Calculated 162 .O A. RF catheter in 0.8% (wt/wt) NaCl solution. B. Insulated helical antenna in 0.8% (wt/wt) NaCl solution. Figure 7.9 Measured and calculated SAR distributions for the RF catheter and the insulated Antenna #3 in 0.8% NaCl solution. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 163 7.3 Dielectric Measurements The open ended coaxial probe technique, discussed in Section 6.3, was used to measure the complex permittivity of cardiac tissue at microwave frequencies. These properties were measured for coagulated and uncoagulated tissue in order to determine if the tissue's electrical properties are likely to change during the time course of ablation. Table 7.3 summarizes the average dielectric data obtained from 5 samples of raw sheep heart muscle and 5 samples of coagulated heart muscle at 915 MHz. Measurements were performed on the endocardial surface. Complex Permittivity * / • f e =e -je Uncoagulated Endocardium Coagulated Endocardium 56.3 - j27.4 57.4-j26.8 Table 7.3 Summary o f dielectric properties for raw and coagulated sheep heart at 913 MHz. The data illustrates that coagulation does not significantly alter the complex permittivity of cardiac tissue at this frequency. Consequently, the antenna's SAR pattern should not change during the time course of ablation. 7.4 I n Vitro Measurements A series of in vitro experiments were performed. Their purpose was to estimate the size of lesions likely to be produced by microwave ablation. These results could then be correlated with measured SAR patterns. Additional experiments using RF Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 164 In Vitro Experiments Epicardial surface P e g s u s e d to hold a n te n n a In p la c e Helical Antenna Heart tissue sam ple a. Microwave ablation setup RF c a th e te r RF tip e le ctro d e P e g s u s e d to hold a n te n n a In p la c e Epicardial surface G rounding p la n e Heart tissue sam ple b. RF ablation setup Figure 7.10 The setups used for microwave and RF ablation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 165 ablation catheters were performed to provide a comparison for the microwave results. The experiments were done on excised sheep hearts obtained from a local slaughterhouse (0 - 2 hrs old). The left ventricles were excised and sliced into 8 cm x 8 cm sections, which ranged in thickness from 1.0 - 2.0 cm. As illustrated in Figure 7.10, the tissue slices were fixed in placed and the ablation device (either a microwave antenna or an RF catheter) was placed along the epicardial surface. To approximate the effect of intraventricular blood, the entire setup was immersed in a tank of 0.9% NaCl solution at 25°C . An HP -vectra computer was used to pulse microwave or RF energy to the ablation device for a set period of time (either 5.0, 10.0 or 30.0 seconds). The power levels of the microwave or RF device was adjusted such that the total power being delivered from either was the same. This power was measured and recorded during ablation to assess any changes due to tissue coagulation. After the ablation was completed, the tissue was removed from the saline bath and the lesions were excised and immediately placed in formalin for tissue fixation. Lesion Size Determination After two weeks, the tissue was removed from the formalin and the lesion was carefully sectioned parallel to its base. All lesions were identified by a sharply demarcated homogenous area of tissue coagulation. After sectioning through the lesion's cross section, the depth, d, was measured every 2.0 mm along the antenna's length. If, at any location, the lesion could not be visually identified, it was assigned a depth of zero. Figure 7.11 illustrates the geometry of a typical lesion. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 166 epicardial su rfa c e lesion lesion d ep th Figure 7.11 Illustration of lesion geometry used in measurements. Results Microwave Antenna Results In Figures 7.12A and 7.13A, lesion depth is plotted verses location along antenna axis for several different heating times. These figures compare the results for Antenna #1 and #3 with and without insulation. The measured normalized SAR pattern for each antenna is given below each plot. The patterns is included as a means of correlating the actual lesion size with the SAR plot. For these experiments, the microwave power was set to 30 Watts and monitored during the ablation process. There are several observations which can be made regarding these experiments. First, the depth of the lesions along the antenna axis correlates well with the SAR Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. plots, particularly for the shorter heating times (5.0 and 10.0 seconds). Second, the effects of thermal conduction become significant after 20 to 30 seconds of heating, tending to produce a larger more uniform lesion. Third, the insulated antennas Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 168 Uninsulated Antenna Insulated Antenna 0.4 E u .—* Heating Time sec. o 0.3 - c o 'm u Heating Time sec. E o — 5.0 10.0 • - • - 30.0 c o 5€) - 3 0 .0 o c c o o E (U c V 0. «c u CL 0.0 0.5 1.0 1.5 2.0 2JS 3.0 Length along antenna axis (cm) I 0.1 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Length along antenna axis (cm) S Figure 7.12 Measured lesion depth and SAR pattern for Antenna #1 with and without insulation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 169 Insulated Antenna 0.4 Uninsulated Antenna 0.4 Heatir.g Tima see. E o E u 0.3 co * m e o c o - 30.0 Heating Time sec. o Hm 0.2 o c o -l-l 10.0 • - v - 30.0 -M e *8 cc 0. 0.3 c o e ec * -4 0.1 e 0.1 CL 0.0 0.0 0.5 1.0 1.5 2.0 2.5 Length along antenna axle (cm) 3.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Length along antenna axis (cm) Figure 7.13 Measured lesion depth and SAR pattern for Antenna #3 with and without insulation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 170 clearly produce a larger, more uniform lesion than the same antenna without insulation. This effect, which is more pronounced in Figure 7.13, is of particular importance for the catheter treatment of VT, which requires a large lesion size. RF Results Figure 7.14 compares lesions produced by the insulated helical antenna to those produced by an RF catheter. The RF source, like the microwave source, was set at 30 Watts and monitored during the ablation process. The main finding from these experiments is that the RF catheter produced a very localized lesion near the tip of the catheter, whereas the microwave antenna produced a far larger lesion which extended over its entire length. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 171 insulated Antenna 0.5 Heating Time eec. E c o m o o co 33 0.2 E e ac i i . _0 o c o s 0.2 - - e 0.1 - o ▼ - 30.0 1----- 1-------1 0.4 . Heating Time eec. I 0 - 5.0 1 . ■ - 10.0 11 0.3 V -3 0 .0 E ci 0.3 RF Catheter c o *2 11 E 0.1 a. a. 0.01i • eii» m * • 0.0 0.5 1.0 1.5 0.0 Length along antenna axis (cm) - • — 2.0 2.5 3.0 Length along antenna axis (cm) Figure 7.14 Measured lesion depth and SAR pattern for the insulated Antenna #3 and the RF catheter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 172 Chapter 8: Conclusion Over the past 10 years, a number of exciting new catheter techniques have been introduced for the ablation of cardiac arrhythmias. Motivated by a need to ablate large regions of tissue, this study has examined a relatively new technique using microwave energy. Using combined theoretical and experimental methods, it has focused on a particular antenna design, the helical coiled antenna. The properties of such antennas immersed in a lossy medium have not been previously understood fully. Analytical models based on a sheath helix approximation are described in Chapters 4 and 5 and experimentally verified in Chapter 7. These analytical and experimental results are then compared to those of an RF ablation catheter as a means of estimating the relative merits of the former. Summary of Results Near field SAR patterns in a homogenous lossy medium were calculated and measured to characterize the antennas used in this study. It was found that, in general, the SAR patterns strongly reflects the presence of standing waves. These patterns depend in a complex way on the geometry of the antenna and the electrical properties of the external medium. . The patterns were found to be strongly sensitive to helical pitch angle, such that a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 173 tighter helix results in a higher propagation constant and a tighter standing wave pattern. Another prominent feature is the high sensitivity of the antenna SAR pattern to the loss of the external medium. This results, for the case of an uninsulated antenna, in an antenna which heats predominantly at its distal tip. It was shown, however, that if a thin layer of insulation is added to the outside of the helical antenna, a more uniform heating pattern can result. This results even when immersed in very lossy medium, such as tissue. Analytical and experimental results using an RF ablation catheter, revealed a SAR pattern which was concentrated at the catheter’s tip. In vitro experiments were performed on excised sheep hearts. These experiments were done to correlate the SAR mappings with measurements of the induced tissue lesions. It was found, that for short heating times (less than 10 seconds) thermal conduction had little effect on tissue heating, consequently, the SAR patterns closely matched the lesion geometry. Prolonged heating, on the other hand, lead to a more diffuse heating distribution. Significance of Results The practical significance of this work is two fold. First, it was found that the insulated helical antenna design offers the ability to ablate large regions of tissue. This ability, which is not found in currently available ablation techniques, should be useful in the treatment of ventricular tachycardia. Second, this study broadened our understanding of helical antenna behavior in lossy matter. This may have practical significance is other applications such as angioplasty or biotelemety. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 174 Future Studies While this work has taken a step in the right direction, there are still many details to be considered before microwave ablation becomes a clinical reality. First, further studies are needed to quantify the effects of ventricular blood flow, heat conduction and myocardial perfusion on the size of the induced lesion. Possibly a combined electromagnetic/heat transfer model can be developed to account for these parameters. Second, the possible dangers associated with this technique, such as the formation of blood clots or induced catheter damage, should be accessed. It may be possible to modify the helical antenna design ( such as shielding blood from the microwaves) to minimize these potential hazards. Third, a series of in vivo experiments should be performed in an animal model of ventricular tachycardia to determine the effectiveness of the technique. These experiments can also be used to estimate the optimal ablation parameters (frequency, pitch angle, antenna length, power level) required to produce a particular lesion. Clearly, microwave catheter ablation has potential in the treatment of certain cardiac arrhythmias which are currently unbeatable using present techniques. However, further engineering and medical studies are needed before this ambitious goal is realized. This work has hopefully increased our understanding of the capabilities of microwave ablation, and provided a foundation for further research. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 175 Appendix A: Heat Transfer Analysis of Thermistor Based SAR Mapping Apparatus As discussed in Chapter 6, a heat transfer analysis is utilized in subtracting out any thermal artifact introduced by the presence of the thermistor. The model shown in Figure A.1 consists of a spherical thermistor of radius, R, electrical conductivity, om, coefficient of heat conduction, k^, and thermal diffusivity, o^, immersed in an uniform electric field of strength, E0, and surrounded by an external medium. The external medium is assumed to be infinite in extent with electrical conductivity, c m, coefficient of heat conduction, k,,,, specific heat, cm, and density, pm. Initially, the thermistor and external medium are assumed to be in thermal equilibrium at temperature, Ti( and at ^ th Kh» ®th \ ^m> Pm ► Figure A .l Spherical thermistor immersed in a homogenous lossy medium and exposed to a uniform electric field. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 176 time t=0 the electric field is turned on. The fields internal to the thermistor, E*, will be different than the external fields and can easily be calculated by, - --3e,< E. [A.1] 2ert + €w (co e0 ) and e*’ = e j - j a j (to e0 ) are the complex where e*’ = e*' - j permittivity of the thermistor and external media respectively. The goal of this analysis is to calculate the average temperature of a thermistor immersed in a uniform external electric field. The temperature distribution inside the thermistor, T(r,8,<l>), is determined by solving the heat conduction equation with the appropriate boundary conditions. The heat conduction equation is written in spherical coordinates as l i f r23r i l ) * -J_-L (sm (0) i l ) + - J — H dr r!sio(0) 00 00 r!sin!(0) 0*1 cta dt [A.2] If the electric field distribution is assumed to be uniform both insideand outside the thermistor (as the case here), then the resultant temperature distribution will vary only in the radial direction. The heat conduction equation under this condition simplifies to 1 * 0 . 7) r dr2 where, q = ♦^ - ± E kA [A.3] aA dt pa js the time averaged power density dissipated in the thermistor. 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 177 The appropriate boundary conditions are 7(0, t) = finite [A.4] 7 (R ,t) = Tt + S t where S = ^ 0|W is the rate of temperature increase in the external medium, 2PmC« and T; is the initial thermistor temperature. Before attempting to solve Equation [A.3] it is useful to first introduce the variable U(r, t) = r [2(r, t) - TJ [A.5] Substituting Equation [A.5] into [A.3] and [A.4] results in &E + r St = —— dr2 [A.6] dt with the boundary conditions U(0, t) = 0 [A.7] U(R, t) = R S t and the initial condition U(r, 0) = 0 To solve Equations [A.6] and [A.7], I utilize the method of eigenfunction Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [A.8] 178 expansion. Using this method the function U(r,t) is expanded into a sum of eigenfunctions, On(r), with time varying coefficients, bn(t), written as OD U(r,t) = £ > „(* ) ®„(r) [A-9] n-1 where the eigenfunctions are obtained by solving the appropriate eigenfunction problem. For the present problem the eigenfunctions are obtained as $>(r) = sin(— r) R [A. 10] Substituting Equation [A. 10] into [A.9] results in oo t'W ) ■ £ *>.«) s i n ( ^ r ) [A.11] & n-1 Substituting this result into Equation [A.6], and collecting like terms, results in the following differential equation for the unknown coefficients bn(t); t * M f J2 *UMQ - ( f r ) - [A.12] Utilizing the orthogonality property of sinusoids this expression can be expressed for any value of n as ^ K «) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [A-13! 179 where, - 2 a A Q R cos(/iit) c, = '* kA n% -2art nn S cos(/ni) C2 = Solving this first order differential equation for the unknown coefficients, bn(t), results in (nn)2, b ^ = -2R3 cos(/m)( ^ (nn)3 _ _1] (1 - e *2 * ) + (n7c) 5 f ) tA-14l kA aA R2 The thermistor temperature distribution can now be calculated by substituting Equation [A. 14] into Equation [A. 11] and substituting that result into Equation [A.5]. This results in the following thermistor temperature distribution o> T(r, t) = - 2 R 3 ( - ^ - — ) Y Kh ath «-1 (nit)3 - e . ,m tr . j “rt ) --------------+ s t + Ti r [A. 15] The measured thermistor signal actually reflects its average temperature. Consequently, it is necessary to determine the average temperature using Equation [A. 15] and [A. 16]. T J t) = - i - it / 4n r2 n r , 0 dr 3 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. [A. 16] 180 This results in [A. 17] *th th "-1 (W7t) By keeping only the first term in Equation [A. 17], I find the approximate average thermistor temperature as it* *» “* It is this expression which is used in Chapter 6 in analyzing the thermal artifact introduced by the thermistor. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 181 Appendix B: Derivation of the Lorentz Reciprocity Theorem In deriving the orthoganality relations for the helical sheath it was necessary to impose the Lorentz reciprocity theorem without derivation. I will now derive the theorem for the particular case of interest here. Consider any two different modes, viz., E„ H, and Ej, H2, propagating in the z direction, shown in Figure 4.3, of a guiding structure (a helical sheath in this case). In a source free region these fields will satisfy Maxwell's equations given as V x Ex = ■H 1 V x UY = j a e • 4 [B.U [B.2] V x E2 = -ja m • H2 [B.3] V x H2 = ja>e ■4 [B.4] Utilizing the vector identity V-(4xB) = B'VxA - A-VxB (where A and B are arbitrary vectors) we can expand the quantity v ( E ^ H J and V- (4> < 4) “ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 182 - Ey-VxH2 [B.5] V* (E2xHy) = Hy'VxE2 - E2-VxHy [B.6] V- (£xx / y = respectively. Subtracting [B.6] from [B.5] results in V* (EyxH2 - E2xHy) = H2 VxEy - Ey-VxH2 - H{-VxE2 + E^VxHy [B.7] Substituting equations [B.l] through [B.4] for the curl of the field vectors given on the right hand side of [B.7] results in V- (EyXH2 - E2xHy) = 0 [B.8] The del operator in Equation [B.8] can be separating into its transverse, denoted by subscript t, and longitudinal parts, given as V,- (Eu xHy - E24xHu ) + ^ (E y x H 2 - E2xHy)-az = 0 [B.9] Applying the divergence theorem to Equation [B.9] results in ~ V ^ i P ’ dl + c ~ az ds = 0 [B.10] s The above expression must be true for any surface, s, with contour, c. If the surface integral is chosen to be over the entire transverse plane, then the contour integral must be evaluated at infinity. Since the fields must vanish at infinity the first integral in Equation [B.9] must also vanish. This results in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 183 *> 2u f / - |( ^ x f f 2 - &z p d* dp = 0 [B.11] p»0 $ -0 For the present case each mode is propagating in the z direction with a propagation constant p. The fields can thus be written as £ ,(p A z) = £ i ( p ,<B « ,( P ,M - A ,(P ,« (B 12] 4(p,<|>d = £ 2( p ,« e * * S 2( p M - J?2(P,4>) e * Substituting Equation [B.12] into [B .ll] and preforming the derivative results in the form of the Lorentz reciprocity theorem used in Chapter 4. oo ( P i + P2 ) / 2n / (frpADxBjp*) - ^(p^)^i(p^))* a* p d<Mp =0 P “0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 184 1. F. H. Netter, The CIBA Collection o f Medical Illustrations, vol. 5, Ciba - Geigy corporation, pp. 60-61, 1978. 2. W. M. Jackson, X. Wang, K. 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Cetas., "Heating pattern of helical microwave intracavitary oesophageal applicator." Int J. Hyperthermia; 7:577586, 1991 50. EA Cheever, "Capabilities of Microwave Radiometry for Detecting Subcutaneous Targets," Ph.D Thesis, University o f Pennsylvania 1989. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 188 51. H.F. Bowman, E.G. Cravalho, and W. Woods, "Theory, Measurement and Application of Thermal Properties of Bioraaterial," Annual Review o f Biophysics and Bioengineering; 4:43-80, 1975 52. J.D. Jackson, Classical Electrodynamics, John Wiley & Sons, New York 1975. 53. E Tanabe and W Joines, "A nondestructive method for measuring the complex permittivity of dielectric materials at microwave frequencies using an open transmission line resonator," IEEE Trans. Instrum, ideas:, 25:222-226,1976 54. N Marcuvitz, Waveguide Handbook, New York: McGraw-Hill 1951, pp. 213-216. 55. H.R. Levine and C.H. Papas, "Theory of the circular diffraction antenna," J. Appl. Phys.; 22:29-43, 1951 56. D. Misra, M. Chabbra, B. Epstein, M. Mirotznik and K.R. Foster, "Noninvasive Electrical Characterization of Materials at Microwave Frequencies Using an Open-Ended Coaxial Line: Test of an Improved Calibration Technique," IEEE Trans. Microwave Theory Tech.; 38:8-14, 1990 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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