(BQ) Part 1 book Physics of thermal therapy - Fundamentals and clinical applications has contents: Physics of electromagnetic energy sources, the physics of ultrasound energy sources, numerical modeling for simulation and treatment planning of thermal therapy,.... and other contents.
Physics of Thermal Therapy Fundamentals and Clinical Applications Edited by Eduardo G Moros Physics of Thermal Therapy ImagIng In medIcal dIagnosIs and Therapy William R Hendee, Series Editor Quality and safety in radiotherapy Targeted molecular Imaging Todd Pawlicki, Peter B Dunscombe, Arno J Mundt, and Pierre Scalliet, Editors ISBN: 978-1-4398-0436-0 Michael J Welch and William C Eckelman, Editors ISBN: 978-1-4398-4195-0 adaptive radiation Therapy C.-M Charlie Ma and Tony Lomax, Editors ISBN: 978-1-4398-1607-3 X Allen Li, Editor ISBN: 978-1-4398-1634-9 proton and carbon Ion Therapy Quantitative mrI in cancer comprehensive Brachytherapy: physical and clinical aspects Thomas E Yankeelov, David R Pickens, and Ronald R Price, Editors ISBN: 978-1-4398-2057-5 Jack Venselaar, Dimos Baltas, Peter J Hoskin, and Ali Soleimani-Meigooni, Editors ISBN: 978-1-4398-4498-4 Informatics in medical Imaging physics of mammographic Imaging George C Kagadis and Steve G Langer, Editors ISBN: 978-1-4398-3124-3 Mia K Markey, Editor ISBN: 978-1-4398-7544-5 adaptive motion compensation in radiotherapy physics of Thermal Therapy: Fundamentals and clinical applications Martin J Murphy, Editor ISBN: 978-1-4398-2193-0 Eduardo Moros, Editor ISBN: 978-1-4398-4890-6 Image-guided radiation Therapy emerging Imaging Technologies in medicine Daniel J Bourland, Editor ISBN: 978-1-4398-0273-1 Mark A Anastasio and Patrick La Riviere, Editors ISBN: 978-1-4398-8041-8 Forthcoming titles in the series Informatics in radiation oncology Image processing in radiation Therapy Bruce H Curran and George Starkschall, Editors ISBN: 978-1-4398-2582-2 Kristy Kay Brock, Editor ISBN: 978-1-4398-3017-8 cancer nanotechnology: principles and applications in radiation oncology stereotactic radiosurgery and radiotherapy Sang Hyun Cho and Sunil Krishnan, Editors ISBN: 978-1-4398-7875-0 Stanley H Benedict, Brian D Kavanagh, and David J Schlesinger, Editors ISBN: 978-1-4398-4197-6 monte carlo Techniques in radiation Therapy cone Beam computed Tomography Joao Seco and Frank Verhaegen, Editors ISBN: 978-1-4398-1875-6 Chris C Shaw, Editor ISBN: 978-1-4398-4626-1 Physics of Thermal Therapy Fundamentals and Clinical Applications Edited by Eduardo G Moros MATLAB® is a trademark of The MathWorks, Inc and is used with permission The MathWorks does not warrant the accuracy of the text or exercises in this book This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2013 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Version Date: 2012928 International Standard Book Number-13: 978-1-4398-4892-0 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com To Kimberly, your love and noble character strengthen me, and to our wonderful sons, Jonas and Ezra Contents Series Preface ix Preface xi Editor xiii Contributors xv part I: Foundations of Thermal Therapy Physics 1 Fundamentals of Bioheat Transfer Kenneth R Diller 2 Thermal Dose Models: Irreversible Alterations in Tissues 23 John A Pearce 3 Practical Clinical Thermometry 41 R Jason Stafford and Brian A Taylor 4 Physics of Electromagnetic Energy Sources 57 Jeffrey W Hand 5 The Physics of Ultrasound Energy Sources 75 Victoria Bull and Gail R ter Haar 6 Numerical Modeling for Simulation and Treatment Planning of Thermal Therapy: Ultrasound 95 Robert J McGough 7 Numerical Modeling for Simulation and Treatment Planning of Thermal Therapy 119 Esra Neufeld, Maarten M Paulides, Gerard C van Rhoon, and Niels Kuster part II: Clinical Thermal Therapy Systems 8 External Electromagnetic Methods and Devices 139 Gerard C van Rhoon 9 Interstitial Electromagnetic Devices for Thermal Ablation 159 Dieter Haemmerich and Chris Brace 10 Clinical External Ultrasonic Treatment Devices 177 11 Endocavity and Catheter-Based Ultrasound Devices 189 Lili Chen, Faqi Li, Feng Wu, and Eduardo G Moros Chris J Diederich vii viii Contents part III: Physical Aspects of Emerging Technology for Thermal Therapy 12 13 Evolving Tools for Navigated Image-Guided Thermal Cancer Therapy 203 Kevin Cleary, Emmanuel Wilson, and Filip Banovac Temperature Imaging Using Ultrasound 219 R Martin Arthur 14 Focused Ultrasound Applications for Brain Cancer 241 15 Extracorporeal Ultrasound-Guided High-Intensity Focused Ultrasound Ablation for Cancer Patients 255 Meaghan A O’Reilly and Kullervo Hynynen Feng Wu 16 Using Hyperthermia to Augment Drug Delivery 279 17 Magnetic Nanoparticles for Cancer Therapy 293 18 Application of Gold Nanoparticles (GNP) in Laser Thermal Therapy 319 19 Thermochemical Ablation 339 Mark W Dewhirst Michael L Etheridge, John C Bischof, and Andreas Jordan Zhenpeng Qin and John C Bischof Erik N K Cressman 162 Physics of Thermal Therapy TABLE 9.1 Electrical Tissue Conductivity at a Frequency of 500 kHz (= RF ablation frequency) Tissue Type Normal Liver (rat, in vivo) Liver Tumor (rat, in vivo) Myocardium (porcine, in vivo) Lung inflated (porcine, ex vivo, 37°C) Fat (porcine, ex vivo, 22°C) Bone (porcine, ex vivo, 20°C) Blood (rabbit, ex vivo, 20°C) Vaporized Tissue Electrical Conductivity σ (S/m) Reference 0.36 (Haemmerich 2003b) 0.45 (Haemmerich 2003b) 0.54 (Tsai 2002) 0.1 (Gabriel 1996b) 0.02 (Pop 2003) 0.03 (Gabriel 1996b) 0.7 (Gabriel 1996a) ~1e-15 Assumed same as air discussion we will always assume this frequency range when referring to electrical tissue conductivity There is a large amount of literature where electrical conductivity has been measured experimentally in various tissues, typically from animals (Duck 1990, Foster 1989, Gabriel 1996a, Gabriel 1996b) It is important to note that most of these studies were performed in extracted tissue, since electrical conductivity changes rapidly after tissue removal from the body (Haemmerich 2002, Tsai 2002) In Table 9.1 we list the electrical conductivity of various tissues from measurements in live animals whenever such data is available Since RF ablation is performed in soft tissue organs such as liver, as well as in bone and lung, the differences in electrical conductivity between these tissues are of relevance In particular, the low value reported for lung tissue (probably due to the presence of electrically insulating air inside lung alveoli) likely is one of the reasons why RF ablation creates smaller ablation zones in lung compared to other tissues (Brace 2009a) Also notable is that multiple studies have reported differences between conductivity of normal and cancer tissue (Esrick 1994, Lu 1992, Surowiec 1988, Swarup 1991, Haemmerich 2003b), where tumor tissue has about 1.3 times higher electrical conductivity, likely due to loss of cell membrane integrity associated with cell necrosis (i.e., cell death) often present in tumors (Haemmerich 2003b) Electrical tissue conductivity varies considerably with temperature as already briefly discussed An increased ion mobility with rising temperature results in a reversible increase in electrical conductivity with a temperature coefficient of approximately 1.5%/°C (Duck 1990) Once tissue temperatures exceed approximately 50°C, irreversible changes are observed In one study electrical conductivity of kidney tissue was measured at temperatures up to 80°C, where permanent increase in conductivity was observed above 50°C (Pop 2003) Similar changes were observed in a recent study performed on surgically removed liver and tumor tissue from liver cancer patients, 0s 30 s Electrical current density 100% 50 FIGURE 9.5 Electrical RF current density for cooled needle electrode at beginning of ablation (left) and after 30 s (right) Due to tissue vaporization around 100°C and associated drop in local electrical conductivity, current density drops after 30 s around the electrode tip (arrow), since at this location temperatures above 100°C are obtained first Current density is shown as percentage of maximum (Reproduced with permission from Haemmerich, D., and Wood, B.J., Int J Hyperthermia 22, 2006b Note also relationship between current density and SAR [Equation 9.1].) where electrical conductivity was measured before and after tissue ablation (Haemmerich 2009) The causes for these irreversible changes are not completely clear, but tissue dehydration and changes in cell membrane properties may be two contributing factors At temperatures above 100°C (boiling temperature of water), tissue begins to vaporize Vapor has a very low electrical conductivity (i.e., it is insulating, see Table 9.1), and therefore has a large impact on RF current pathways As shown in Figure 9.5, current density drops at locations where vapor starts to form, and heating at these locations is effectively eliminated 9.3 Clinical Applications of Radiofrequency Ablation 9.3.1 Cancer Treatment (aka Radiofrequency Tumor Ablation) When thermal ablation using RF current was first clinically used for cancer treatment, it was initially used for liver cancer (both primary and metastatic) since for these patients often surgery is not possible (~80% of patients) (Rhim 1999), and the other two standard therapies—chemo- and radiation therapy—are not effective due to biological reasons Recent clinical studies on patient survival suggest that with proper patient selection similar patient survival rates as with surgical tumor removal (the current gold standard) are attainable (Gillams 2009, Livraghi 2008) In the last few years, tumor ablation has expanded to other cancer types such as lung, kidney, bone, and adrenal gland (Gillams 2008) Especially for lung cancer, tumor ablation has received considerable attention, as there are more than 200,000 new cases of lung cancer reported annually in the United States 163 Interstitial Electromagnetic Devices for Thermal Ablation (a) (b) FIGURE 9.6 Commercial RF electrodes for tumor ablation (a) Multi-prong electrode, with magnified electrode tip shown in insert The prongs are extended once the catheter is placed in the tumor (b) Cooled needle electrode, available as single, or three-needle cluster (see magnified inserts) Active electrode at the tip is cm (single) or 2.5 cm long (cluster), and is internally cooled by circulating water (Reproduced with permission from S Vaezy and V Zderic, eds Image-Guided Therapy Systems, Artech House, Inc., Norwood, MA, 2009 © 2009 by Artech House, Inc.) alone (Jemal 2007), and most patients are not surgical candidates; recent studies suggest that patients with inoperable lung cancer may benefit from RF ablation (Simon 2007, Gillams 2008) A tumor ablation procedure can be performed minimally invasively through a small incision in the skin (by an interventional radiologist), or during laparoscopy or open surgery (by a surgeon) During the procedure, the patient is typically under light general anesthesia or conscious sedation and can leave the hospital the same, or the next, day The treatment goal is to thermally ablate tissue in a zone that encompasses the under imaging visible tumor as well as a ~1 cm margin of normal tissue to ensure destruction of any cancer microsatellites that may surround the tumor (Sasaki 2005) For tumors larger than cm, multiple overlapping ablations created either sequentially (Chen 2004) or simultaneously with multiple electrodes (Laeseke 2007) are typically required Limitations of current RF tumor ablation procedures include: limited performance close to large vasculature, that may result in tumor recurrence due to inadequate temperatures; inadequate intra-procedural imaging feedback on ablation zone growth; the size of the ablation zone of a single ablation is often not adequate to treat large tumors (>3 cm diameter), resulting in prolonged procedural times and higher recurrence rates 9.3.1.1 Radiofrequency Electrodes for Tumor Ablation There are a number of different electrodes commercially available Some electrodes have multiple tines that are extended after the electrode is inserted into the tissue (Figure 9.6a), while another kind uses needle electrodes that are internally cooled (Figure 9.6b) Most of the electrode shaft is insulated such that tissue heating is only produced at the most distal electrode portion (see inlets in Figure 9.6) 9.3.2 Cardiac Arrhythmia Treatment RF ablation (cardiac RF catheter ablation) is frequently used for treatment of cardiac arrhythmia (i.e., irregular heart beats), e.g., for different types of tachycardia (fast heart rhythm above 150 bpm) and atrial fibrillation (i.e., quivering of the atria) (Huang 2006, Wilber 2007) The spatiotemporal activation pattern of the heart is determined by a specialized conduction system, and abnormalities in electrical conduction in the heart can result in arrhythmia The goal of cardiac RF ablation is to destroy a small region of heart tissue to normalize electrical activation in the heart (e.g., by destroying additional conduction pathways not present in a normal heart) Cardiac ablation is performed in a specifically equipped interventional laboratory by an electrophysiologist A catheter is inserted into a vessel (typically in the groin or neck) and steered into the heart (Figure 9.7) The procedure is guided by x-ray imaging In addition, electrical measurements of local cardiac activity (similar to ECG) are performed via recording electrodes located on the RF catheter (Figure 9.8), as well as by additional specialized recording catheters placed at various locations in the heart Through these local electrical activity measurements, temporal activation patterns of the heart can be determined to diagnose the cause of the arrhythmia and site that needs to be ablated Subsequently, the RF catheter is steered to the target site and RF current is applied to the heart tissue for ~45–120 s, creating an ablation zone of ~5–10 mm in diameter (Figure 9.9) For most commercially available catheters, the applied RF power is adjusted depending on temperature measured by a sensor located within the electrode tip (temperature control) Note, however, that electrode tip temperature is lower than the maximum temperature within the tissue (see Figure 9.9) This is of importance because one of the undesired effects that can occur is tissue perforation (often called “popping” due to the sound associated with this event), which is due to tissue vaporization above 100°C Intracardiac blood flow (i.e., blood velocity at the electrode) considerably affects tissue heating and resulting size of the ablation zone (Figure 9.10) Usually, larger ablation zones are possible at locations with high blood flow (Cao 2001, Tungjitkusolmun 2001, Petersen 1999) To create ablation zones of varying dimensions, catheters of different lengths and diameters are commercially available Some newer catheter designs 164 Physics of Thermal Therapy Reference patch electrode on the dorsal side Handle RF generator Ablation electrode Catheter body FIGURE 9.7 Schematics of cardiac RF ablation system A cardiac RF catheter is inserted through a leg vein and steered to the target site inside the heart (right figure) A reference patch electrode (i.e., ground pad) is placed on the patient’s back The small black region around the RF electrode at the catheter tip depicts the ablation zone (Reproduced with permission from Panescu, D., Whayne, J.G., Fleischman, S.D., Mirotznik, M.S., Swanson, D.K., and Webster, J.G., IEEE Transactions on Biomedical Engineering 42, 1995 © IEEE.) use internal or external cooling to increase ablation zone size (see also Section 9.2) One very common type of cardiac arrhythmia that affects ~2.2 million people in the United States (most prevalent among people older than 60 years) is atrial fibrillation, where the atria quiver instead of beat, thereby reducing the pumping performance of the heart, with a risk of blood clot formation (Wilber 2007) Cardiac ablation is increasingly used for treatment of atrial fibrillation in patients not responsive to medications For treatment of atrial fibrillation, multiple linear, contiguous ablation zones have to be created This is difficult and time consuming with current devices, resulting in long procedural times and high recurrence rates (Wilber 2007) New devices that allow more rapid and reliable creation of linear ablation zones are in development (Burkhardt 2009, Siklody 2010) 9.3.3 Other Applications of RF Ablation whole endometrium (i.e., lining of the uterus) is ablated within typically 3–10 minutes Different methods are employed to thermally ablate the endometrium, including RF ablation devices that employ mesh electrodes with power applied between two electrically insulated meshes (Cooper 2004) RF and laser-based ablation devices are available as treatment modality for varicose veins (Markovic 2009) (endovascular ablation), which are visible, dilated and twisted veins near the skin surface Varicose veins most often affect legs and thighs, due to blood pooling and vein enlargement resulting from insufficiencies in the venous valves The treatment goal is typically closure of the affected veins Methods to obtain vein closure include surgical stripping, injection of a drug that results in vein swelling and closure, and ablation During ablation, a catheter is introduced into the vein, and the vessel wall is heated, resulting in collagen shrinkage and closure of the vein RF ablation is clinically used for treatment of uterine bleeding in women who don’t respond to standard therapies such as medication and scraping of the endometrium During treatment, the 9.4 Biophysics of Microwave Ablation FIGURE 9.8 Cardiac RF ablation catheter (7F = 2.3 mm diameter) Several electrodes are located along the catheter: RF electrode of 4mm length (large arrow), as well as three smaller electrodes used for recording of local electrical activity in the heart (small arrows) Microwaves represent the portion of the electromagnetic spectrum between 300 MHz and 300 GHz The Federal Communications Commission (FCC) or International Telecommunications Union (ITU) permit several unrestricted frequency bands for industrial, scientific, and medical use in several regions of the world, including those most commonly used for microwave ablation procedures today: 915 MHz and 2.45 GHz In addition to these two frequencies, 433 MHz has been explored as an option for microwave hyperthermia in Europe, but not commonly for hightemperature focal microwave ablation Frequencies higher than 2.45 GHz can also be used for some applications but have not been widely explored to date As with other thermal ablation modalities, microwave ablation describes the rapid destruction of tissue by microwave 165 Interstitial Electromagnetic Devices for Thermal Ablation 80°C Ablation zone 50°C Tissue Blood cator Appli Tumor Electrode mm FIGURE 9.9 Tissue temperature around a cardiac RF ablation catheter (2.3 mm diameter) at the end of a 45 s ablation Outermost 50°C boundary estimates boundary of ablation zone heating (Figure 9.11) In this respect, microwave ablation is the more acute and higher-temperature extension of microwave hyperthermia, which has a longer history in the literature and in clinical practice Hyperthermia typically refers to the temperature range of 41°C to 46°C, where most temperatureinduced physiological changes are reversible, while thermal ablation at temperatures over 50°C is associated with irreversible changes such as denaturation of cellular proteins and microvascular coagulation leading to rapid cell death (Dewhirst 2003) Despite the difference in target temperature range, the physics of heating tissues is relatively similar for both therapies A notable exception is when tissue temperatures approach or exceed 100°C, when a series of temperatureinduced changes in tissue properties must be considered (Brace 2010b, Brace 2008) Blood perfus ion Liver FIGURE 9.11 Illustration of a microwave ablation in liver A zone of complete cellular necrosis is created by heat generated from the applicator antenna, which completely encompasses the tumor with a safety margin Electromagnetic energy propagation can be described by solving Maxwell’s equations in a source-free lossless medium: ⋅ E = (9.3) ×E=− ∂B (9.4) ∂t ⋅ B = (9.5) × B = µε ∂E (9.6) ∂t Conduction to myocardium Conduction to electrode Heart Tissue Blood 50°C after s Electrode 50°C after 60 s Catheter body Convective cooling from blood FIGURE 9.10 Thermodynamics of cardiac RF ablation: RF electrode is in contact with myocardium (heart tissue), and RF current results in tissue heating Thermal conduction of heat into the tissue results in growth of the ablation zone (approximated by 50°C isotherm) Heat loss of tissue in proximity of the electrode is due to thermal conduction through the electrode (black arrow); electrode and tissue surface experience convective cooling from blood inside the chamber Surface cooling produces the typical tear drop–shaped ablation zone (Reproduced from Tungjitkusolmun, S., Finite element modeling of radiofrequency cardiac and hepatic ablation PhD thesis University of Wisconsin, Madison, 2000.) 166 Physics of Thermal Therapy where E is the electric field vector (V/m), B is the magnetic flux density vector (Tesla), and μ and ε are the magnetic permeability (H/m) and dielectric permittivity (F/m) of the ambient medium, respectively Equations 9.4 and 9.6 may then be combined using a vector identity to form the wave equations of electromagnetic propagation: ∂E2 = ν2p ∂t ∂ B2 = ν2p ∂t 2 E (9.7) B (9.8) where vp is the wave propagation velocity (m/s), equal to (με)−1/2 Equations 9.3 through 9.8 imply two important physical processes: (1) time-varying electric or magnetic fields give rise to spatial variations in magnetic and electric fields, respectively, producing a self-propagating electromagnetic field; and (2) wave propagation characteristics are determined by the magnetic permeability and relative permittivity of the wave medium The first process forms the basis for understanding how microwave propagation is created from an antenna, and the second process illustrates how biological tissues respond to applied microwave energy, including both the wavelength of propagation inside tissues, and the absorption of microwave energy 9.4.1 Dielectric Properties of Biological Tissues If the propagation medium is not lossless and has a finite conductivity, σ (S/m), then Equation 9.6 takes the form: ∇ × B = µε factors in biological tissues, the term relative permittivity will be used here The effective conductivity of a material, σ, is defined from the imaginary part of the complex permittivity and is used to describe how well a material absorbs microwave energy Materials with high effective conductivities such as tissue are said to be lossy It is important to note that effective conductivity describes contributions from both moving charges (electrical currents) and time-varying electric fields (displacement currents; Balanis 1989) Displacement currents dominate for most biological tissues in the microwave spectrum and are produced by the rotation of polar molecules such as water in tissue The dielectric properties of tissues are most closely related to their water content (Schwan 1955, Schepps 1980) Tissues with high volumetric water content are characterized by higher relative permittivities and conductivities Most organs and soft tissues are considered to be high water content (Gabriel, C 1996, Gabriel, S 1996a) As a result, the wavelength of an electromagnetic field is shorter and power loss (i.e., heating) is more rapid in these tissues Conversely, tissues with lower water content such as adipose tissue are slower to generate heat Note that there is a distinction between water content by mass, which is more commonly reported in the tissue property literature, and water content by volume For example, consider aerated lung tissue Lung is generally considered to have a relatively high water content by mass but low water content by volume due to its very low density (Duck 1990) Therefore, heat generated in lung tissue is primarily produced in the blood-filled spaces between alveoli, but not in the air-filled alveoli themselves There are no universal models to describe the dependence of tissue permittivity on water content, but mixture models have been proposed for some tissues (Schwan 1977, Ng 2008, Steel 1986) ∂E + J (9.9) ∂t 100 MHz Tissue where J is the electrical current density (A/m2) equal to σE Casting Equation 9.9 into time harmonic form (∂/∂t → jω, where ω = 2πf ) with the current density replaced by σE gives: σ × B = jωµεE + σE = jωµ ε − j E (9.10) ω Equating Equations 9.6 and 9.10 reveals that the dielectric permittivity in a medium with finite conductivity is complex-valued: Adipose Bone (cortical) Breast Kidney Liver Lung (inflated) Muscle ε = ε′ − j σ = ε ′ − jε ′′ (9.11) ω Tissues are characterized by substantial complex-valued dielectric permittivities but magnetic permeabilities approximately equal to that of vacuum Relative permittivity, εr, is the real part of the complex permittivity and quantifies the ability of a material to store electrical energy relative to a vacuum It is frequently referred to as dielectric constant; however, since it is variable depending on frequency, temperature, and other GHz εr σ εr σ 6.07 15.3 5.69 98.1 69 31.6 65.9 0.04 0.64 0.3 0.81 0.49 0.31 0.71 5.45 12.4 5.41 57.9 46.4 21.8 54.8 0.05 0.16 0.05 1.45 0.90 0.47 0.98 10 GHz 100 GHz εr εr σ 3.56 3.3 2.59 8.04 6.87 8.63 3.56 8.66 1.84 57.1 42.9 21.4 62.5 σ 4.6 0.59 8.12 2.14 3.88 0.74 40.3 11.6 32.5 9.39 16.2 4.21 42.8 10.6 The dielectric properties of tissue are also dependent on frequency, temperature, and structure The frequency dependence of each tissue can be described analytically using a multipole ColeCole model Each pole, n, of the model describes the complex permittivity dispersion within a frequency range, with the summation of multiple dispersions describing a larger frequency spectrum: εˆ(ω ) = ε ∞ + ∑ n ε s ,n − ε ∞,n (1 − α n ) + ( jωτn ) + σi (F//m), (9.12) jωε where ε∞ is the relative permittivity at a frequency where ωτ » 1, εs is the relative permittivity at a frequency where ωτ « 1, τ is Interstitial Electromagnetic Devices for Thermal Ablation 167 the relaxation time constant (s), α is an empirical distribution parameter that broadens the dispersion in each frequency range, σi is the static ionic conductivity, and ε0 is the permittivity of free space Relaxation, permittivity, and distribution parameters have been described in the literature for most common biological tissues (Gabriel 1996b) The dielectric properties of tissue are generally thought to be relatively isotropic, but this assumption may not be valid in some situations in tissue with anisotropic structure such as muscle (Epstein 1983) The temperature dependencies of relative permittivity and conductivity are not yet as fully characterized as frequency dependence Most of the reported data are clustered around the ambient room temperature (~20°C) or normal body temperature (~37°C) In the range of 10–60°C, temperature dependence for both relative permittivity and conductivity have been assumed to take a linear form (Duck 1990, Lazebnick 2006, Pop 2003, Bircan 2002, Chin 2001, Stauffer 2003) However, near temperatures of water phase change (0°C or 100°C), substantial deviations from the linear model have been reported (Brace 2008) At temperatures in excess of approximately 60°C, protein and cellular structure changes can also affect tissue permittivity Such effects have not been well characterized in the available literature on bulk tissue, but some data suggest that protein denaturation may be detectable by a sudden slight change in the temperature curve of relative permittivity or conductivity in some tissues (Bircan 2002, Wall 1999) Penetration of a microwave field into a tissue medium is also dependent on the dielectric properties of the tissue For a plane wave in a homogenous isotropic medium, the penetration depth, δ (m), of an electromagnetic field is defined as the distance required for the electric field to attenuate to 1/e (~37%) of its initial value: One final point of consideration: much of the energy radiated by microwave antennas in lossy media such as biological tissues is absorbed in the near or Fresnel zones of the antenna and may not propagate as a plane wave In this case, the aforementioned calculations of attenuation and penetration depth should be viewed as approximations Electromagnetic simulation and, in particular, simulation of electromagnetic-thermal interactions can provide more accurate estimates of temporal heating for comparing devices and frequencies for microwave ablation δ= 1/2 1 σ ω µε + − ωε (m) (9.13) In the case of most tissues and microwave ablation frequencies, the penetration depth from Equation 9.13 can be approximated by assuming the tissue is a good dielectric ([σ/ωε]2