A Dissertation Submitted to the Faculty of the Graduate School of Arts And

STUDIES OF MAGNETIC NANOPARTICLE SHAPE AND SIZE EFFECTS ON T2 RELAXATION IN MAGNETIC RESONANCE IMAGING, AND POWER ABSORPTION IN AN ALTERNATING MAGNETIC FIELD

A Dissertation submitted to the Faculty of the Graduate School of Arts and Sciences of Georgetown University in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in Physics

Joseph Nicholas York, M.S.

ii STUDIES OF MAGNETIC NANOPARTICLE SHAPE AND SIZE EFFECTS ON T2 RELAXATION IN MAGNETIC RESONANCE IMAGING, AND POWER ABSORPTION IN AN ALTERNATING MAGNETIC FIELD Joseph Nicholas York, M.S. Thesis Advisor: Edward Van Keuren, PhD

ABSTRACT

Magnetic nanoparticles have been shown to influence contrast in magnetic resonance imaging (MRI). The magnetic fields of particles vary depending on the shape of the particle. The effects of ferromagnetic nanoparticles with various shapes and sizes on the transverse relaxation rate of water protons were analyzed using T2-weighted MRI. Oblate particles were shown to have a stronger effect on the transverse relaxivity than smaller spherical or prolate particles. A linear relationship between the transverse relaxivity and the ratio of particle surface area to volume was observed. It indicates decreasing the surface area to volume ratio of magnetic nanoparticles enhances the relaxivity. Magnetic particle hyperthermia is the use of ferromagnetic nanoparticles to heat a cancer tumor to destroy it. This phenomenon is possible because these particles absorb power from an alternating magnetic field. Heating trials were performed on aqueous suspensions of ferromagnetic nanoparticles having various shapes and sizes. The results show power absorption of the samples diminishes over time regardless of particle shape or size, and the initial absorption rates of the samples depend on the sample type and preparation conditions. Power absorption was observed in all particles tested, including prolate and oblate particles with anisotropy fields greater than the strength of the applied alternating field. Additionally, large levels of aggregation were observed in all samples tested indicating particle interactions may affect the ability of the samples to absorb energy from the applied field.

iii ACKNOWLEDGEMENTS

This dissertation is the result of many years of preparation and dedication, as well as numerous sources of help, guidance, and influence. I could not have accomplished this without the support of many other people. I thank you all, and would especially like to thank the following individuals. Dr. Ed Van Keuren, you are my thesis advisor, but you are much more than that. You are a true friend and a role model. Your continuous support, guidance, and patience (so much patience) has kept me on track to achieving this goal. I cannot thank you enough. I would like to thank Dr. Jim Freericks, Dr. Mak Paranjape, and Dr. Chris Albanese for being members of my committee. Thank you for your insights, guidance, and support. Dr. Freericks, a special thank you for always playing great music (Bob Dylan) across the hallway while I worked in the lab. Many thanks to all of the faculty and staff of the Physics Department of Georgetown University. You all made the graduate student experience enjoyable. Leon Der, thank you for all your technical support with my laboratory experiments. I would also like to thank all of my fellow graduate students. It has been an honor to go this course with you, and to share all of the bad times (studying) and good times (not studying) with you all. Thank you Dr. Olga Rodriguez and Dr. Yi-Chien Lee for your help and support with the magnetic resonance imaging research. You were a terrific help. Last, and certainly not least, are my friends and family. I would like to thank my parents, John and Becky York, for their endless support and encouragement to be whatever I wanted to be, which turned out to be a physicist. Your unwavering commitment to making

iv your children the best adults they can be has been a source of admiration for me. My three sisters, Anne, Amanda, and Kate, have been sources of inspiration and many laughs. You all make any project feel like a simple task. Thank you. Erika Luth, you have been by my side for nearly all of my graduate school, and you somehow still like me. I could not imagine the world without you in it, and I thank you for all of your support. I will desperately try to repay the kindness, in turn.

v I dedicate this work to the entire York family.

vi May God bless and keep you always, May your wishes all come true, May you always do for others And let others do for you. May you build a ladder to the stars And climb on every rung, May you stay forever young. § May you grow up to be righteous, May you grow up to be true, May you always know the truth And see the lights surrounding you. May you alway be courageous, Stand upright and be strong, May you stay forever young. § May your hands always be busy, May your feet always be swift, May you have a strong foundation When the winds of changes shift. May your heart always be joyful, May your song always be sung, May you stay forever young. - Bob Dylan

vii Table of Contents

1 Statement of Purpose1

2 Cancer 3

I Magnetic Nanoparticle Shape and Size Eﬀects on T2 Relaxation in Magnetic Reso- nance Imaging8

3 Introduction9

4 Magnetic Resonance Imaging 12 4.1 Historical Overview...... 12 4.2 Water Proton MRI...... 14 4.3 Water Proton Magnetic Moment Transverse Relaxation...... 17 4.4 Water Proton Transverse Relaxation Pulse Sequences...... 18 4.5 Particle Eﬀects on Image Contrast ...... 21

5 Materials and Methods 22 5.1 Particle Preparation ...... 22 5.1.1 Synthesis of Particle 1 ...... 24 5.1.2 Synthesis of Particle 3 ...... 25 5.1.3 Synthesis of Particle 4 ...... 25 5.2 Sample Preparation...... 27

viii 5.2.1 Agar Phantoms...... 27 5.2.2 Particle Concentrations Suspended in Agar...... 28 5.3 System Setup...... 31 5.3.1 Machine and Software ...... 31 5.4 Imaging Sequence...... 31 5.4.1 Multiple Slice Multiple Echo...... 31 5.5 Imaging ...... 32 5.6 Imaging Analysis...... 33

6 Results and Discussion 34 6.1 Particle Characterization...... 34 6.2 Inherent Transverse Relaxation of Agar Gels...... 35 6.3 Transverse Relaxation of Agar Containing Contrast Agents...... 41 6.4 Shape and Size Eﬀects on Transverse Relaxivity...... 44

7 Conclusions 48

II Magnetic Nanoparticle Shape and Size Eﬀects on Power Absorption from an Al- ternating Magnetic Field 50

8 Magnetic Particle Hyperthermia 51

9 Historical Overview 56

10 Power Absorption of Ferromagnetic Particles From an Oscillating Applied Field 59 10.1 Theoretical Methods of Calculating The Particle Speciﬁc Absorption Rate . 63 10.1.1 Power Absorption and Particle Orientation Calculations From Particle Energy...... 64

ix 10.1.2 Stoner Wohlfarth Theory...... 69 10.1.3 Linear Response Theory...... 71

11 Calorimetric Measurement of SAR 74

12 Materials and Methods 76 12.1 Theoretical Models...... 76 12.1.1 Particle Rotation in a Viscous Medium...... 76 12.2 Sample Preparations ...... 77 12.2.1 Non-ferromagnetic Sample...... 77 12.2.2 Ferromagnetic Samples...... 78 12.3 Magnetic Particle Hyperthermia Testbed...... 78 12.3.1 Magnetic Field System...... 79 12.3.2 Solenoid Cooling System...... 81 12.3.3 Magnetic Field Probe Calibration...... 82 12.3.4 Testbed Characterization...... 83 12.4 Heating Trials...... 83 12.4.1 Non-ferromagnetic Sample Heating Trials...... 83 12.4.2 Ferromagnetic Sample Heating Trials...... 84

13 Experimental Testbed Characterization 86 13.1 Magnetic Field Probe Calibration...... 86 13.2 Operating Range ...... 86 13.3 Solenoid Chamber Temperature Stability...... 88

14 Results and Discussion 90 14.1 Theoretical Results for Laboratory Particles...... 90 14.2 Experimental Results...... 95

x 14.3 Theoretical Optimization of SAR and MPH ...... 107

15 Conclusions 109

A LCOrientation.m 111

B Temperature and Field Dependent SWT Code 114

References 118

xi List of Figures

4.1 CPMG spin echo sequence shown with transverse signal response. The axis is time...... 20

5.1 Schematics of the five different types of single domain ferromagnetic nanoparticles used in this research. Their magnetization orientation is shown on the left, and their dimensions on the left. In the figure c is one half of the magnetic easy axis length, and a is one half of the magnetic hard axis length of the particle...... 23 5.2 Contrast agent sample: multiple layers of agar containing varying concentrations of magnetic particles...... 29 5.3 Image slice shown from the (a) front, (b) side, and (c) top...... 33

6.1 SEM image taken of particle type 1. It is made of magnetite, and has a spherical shape (mean radius ≈ 6 nm). This particle type was synthesized using the Massart Method [1]...... 36 6.2 SEM image taken of particle type 2. It is made of magnetite, and has a spherical shape (mean radius ≈ 16 nm). This particle type was purchased (Sigma Aldrich)...... 37

xii 6.3 SEM image taken of particle type 3. It is made of magnetite, and has a prolate shape with half axes lengths a = 7 nm (magnetic hard axis) and c = 24 nm (magnetic easy axis). This particle type was synthesized using a method similar to that of Kumar and Koltypin et al. [2]...... 38 6.4 SEM image taken of particle type 4. It is made of magnetite, and has a prolate shape with half axes lengths a = 15 nm (magnetic hard axis) and c = 225 nm (magnetic easy axis). This particle type was synthesized using a method proposed by Matsui [3]...... 39 6.5 SEM image taken of particle type 5. It is made of barium ferrite, and has an oblate shape with half axes lengths a = 100 nm (magnetic hard axis) and c = 10 nm (magnetic easy axis). This particle type was purchased (Sigma Aldrich)...... 40 6.6 Transverse relaxation rate of agar phantoms A-1 and A-2 versus the published

results of Davies et al. for agar R2 versus concentration [4]...... 42 6.7 Transverse relaxation rates as a function of particle concentration. The slopes

r2 are given in table 6.4...... 43

6.8 Particle parameter eﬀects on relaxivity r2...... 45

10.1 Schematic of a prolate particle with with half axes lengths c and a and aspect ratio α = c/a suspended in a viscous medium with viscosity η whose magnetization M makes an angle θ with respect to the particle easy axis (a), and particle easy axis makes an angle φ with respect to the applied ﬁeld direction. 60

xiii 10.2 Schematic of a typical hysteresis loop for coherent magnetization reversal in a single domain ferromagnetic particle whose easy axis makes some small angle φ with respect to the applied ﬁeld direction. The loop is characterized

by the particle remanent magnetization MR (the aligned component of the particle material saturation magnetization at zero applied ﬁeld strength), and

the coercive ﬁeld µ0HC (the applied ﬁeld strength required to remove the remanent magnetization of the particle). The enclosed area of the loop is

A = 4µ0MRHC ...... 62

12.1 High power magnetic field system. Subfigure (a) is the system wire circuit. Subfigure (b) is a schematic of the system...... 79 12.2 High power magnetic field system solenoid and water-jacket...... 81 12.3 Cooling system for testbed solenoid...... 82

13.1 Calculated solenoid magnetic ﬁeld strength H0 from voltage drop measurements across a resistor as a function of magnetic ﬁeld probe (MC162) voltage. 87 13.2 Nonmagnetic sample heating trial...... 89

14.1 Plot of κ, equation 10.17, as a function of eﬀective particle anisotropy Keff

2 1/3 and volume, denoted as the radius of an equivalent sphere RES = (a c) .

An applied ﬁeld with H0 = 45 (kA/m) and frequency f = 30 (kHz) was used. Temperature is taken to be T = 37 (◦C). Equations 10.16- 10.20 are valid for κ < 0.7...... 92

xiv 14.2 Particle easy axis orientation φ versus time for particles a, b, c, and d suspended in water, η = 0.001 (Pa·s), and subjected to a static ﬁeld in the minus

x direction with H0 = 10 (kA/m). The method and equations of Lissberger and Comstock [5] were used for the results. Subfigure (a) depicts orientation time for spherical particles, which is applicable for particles a and b. Subfig- ure (b) and (c) are for prolates having the aspect ratio of particles c and d, respectively...... 94 14.3 Time dependent temperature difference (a) and calorimetric heating (b) of

particles a, b, c, d, and e using an applied ﬁeld strength H0 = 45 (kA/m) and frequency f = 30 (kHz). Each individual heat trial was initiated after a

time tFS ≈ 180 (s) from when the sample being tested was removed from the sonicator...... 97 14.4 Time dependent temperature diﬀerence (a) and calorimetric heating (b) of

particles a, b, c, d, and e using an applied ﬁeld strength H0 = 45 (kA/m) and frequency f = 30 (kHz). Each individual heat trial was initiated after a

time tFS ≈ 90 (s) from when the sample being tested was removed from the sonicator...... 97 14.5 Time dependent temperature diﬀerence (a) and calorimetric heating (b) of

particles a, b, c, d, and e using an applied ﬁeld strength H0 = 45 (kA/m) and frequency f = 30 (kHz). Each individual heat trial was initiated after a

time tFS ≈ 180 (s) from when the sample being tested was removed from the sonicator...... 98 14.6 Time dependent temperature diﬀerence (a) and calorimetric heating (b) of

samples containing particle a using an applied ﬁeld strength H0 = 45 (kA/m) and frequency f = 30 (kHz)...... 98

xv 14.7 Time dependent temperature diﬀerence (a) and calorimetric heating (b) of

samples containing particle b using an applied ﬁeld strength H0 = 45 (kA/m) and frequency f = 30 (kHz)...... 99 14.8 Time dependent temperature diﬀerence (a) and calorimetric heating (b) of

samples containing particle c using an applied ﬁeld strength H0 = 45 (kA/m) and frequency f = 30 (kHz)...... 99 14.9 Time dependent temperature diﬀerence (a) and calorimetric heating (b) of

samples containing particle d using an applied ﬁeld strength H0 = 45 (kA/m) and frequency f = 30 (kHz)...... 100 14.10Time dependent temperature diﬀerence (a) and calorimetric heating (b) of

samples containing particle e using an applied ﬁeld strength H0 = 45 (kA/m) and frequency f = 30 (kHz)...... 100

14.11Coercive ﬁeld strength HC (A/m) as a function of anisotropy and particle

2 1/3 volume, denoted as the radius of an equivalent sphere RES = (a c) , for aligned (equation 10.16) and randomly oriented particles (equation 10.18) using the temperature and frequency dependent SWT model. An applied ﬁeld

with H0 = 45 (kA/m) and f = 30 (kHz) was used for this data. Temperature is taken to be T = 37 (◦C). The white circles encapsulate the areas about the values taken to be those of particles a, b, c, and d. Particle e has an anisotropy over three times that of particle d. It is not shown in the ﬁgure. 101

xvi 14.12Energy absorption density (hysteresis losses due to irreversible magnetization rotation along the particle easy axis) A (J/m3) as a function of eﬀective

particle anisotropy Keff and particle volume, denoted as the radius of an

2 1/3 equivalent sphere RES = (a c) , for aligned (equation 10.19) and randomly oriented particles (equation 10.20) using the temperature and frequency de-

pendent SWT model. An applied ﬁeld with H0 = 45 (kA/m) and f = 30 (kHz) was used for this data. Temperature is taken to be T = 37 (◦C). The white circles encapsulate the areas about the values taken to be those of particles a, b, c, and d. Particle e has an anisotropy over three times that of particle d. It is not shown in the ﬁgure...... 102 14.13Heating rates predicted by the linear response theory proposed by Rosensweig

[6] for magnetic particles subjected to an applied field with H0 = 45 (kA/m) and f = 30 (kHz). A temperature of T = 37 (◦C) was used in this data. Viscosity values of η = 0.001 and 0.01 (Pa·s) are used to determine LRT sensitivity to the particle environment. Sample concentration used is 20 (mg/mL), unless otherwise specified on a graph. In the figure (a) is the result of spherical SPM particles, and (b) and (c) are for prolate SPM particles in a fluid viscosity η = 0.001 and η = 0.01 (Pa·s), respectively...... 105

xvii List of Tables

5.1 Magnetic nanoparticles used in this research along with their compositions, dimensions, and research identiﬁers. In the table, c refers to one half of the magnetic easy axis length and a refers to one half of the magnetic hard axis length...... 26 5.2 Agar phantoms without contrast prepared to calculate agar transverse relax-

ation time T2 as a function of concentration, and the stability of 3% agar gel

T2 under various conditions...... 28 5.3 Magnetic material and concentration used in the phantoms containing contrast agents...... 30 5.4 Main parameters of the MSME imaging sequence...... 32

6.1 Physical properties of sample particles: c is on half of the magnetic easy axis length, a is one half of the magnetic hard axis length, α is the aspect ratio,

ρ is particle density, MS is the saturation magnetization of the material, and

Keff is the effective particle anisotropy. Values of ρ and MS for magnetite are those reported by Rosensweig [6], those of barium ferrite are taken from Coey [7]. The effective particle anisotropy is the sum of the magnetocrystalline anisotropy and shape anisotropy taken from Coey [7]. Any particle dimension that could not be confirmed via SEM is denoted with an asterisk...... 35 6.2 Observed relaxation rate of agar under various conditions...... 41

xviii 6.3 The transverse relaxation time T2 of samples containing particles A, B, C, and D in varying concentrations, that are separated into layers. Two surfactants were used to coat the particles: SDS and AOT...... 43 6.4 Transverse relaxavities of particles...... 44

13.1 Magnetic ﬁeld strength conversion chart for peak to peak voltage readings of the magnetic ﬁeld probe...... 87

13.2 Largest obtainable magnetic ﬁeld strength H0 as a function of frequency f for our laboratory system...... 88

14.1 Speciﬁc absorption rates SARP and coercive ﬁeld strengths HC of particles

a, b, c, d, and e for an applied ﬁeld with H0 = 45 (kA/m) and f = 30 (kHz). Temperature is T = 37 (◦C). For particles where κ < 0.7, equations 10.19 and 10.20 (using b = 1 and n = 0.8 for the random case) were used in equation 10.2

to calculate SARP for aligned and random distributions of particles, respectively. For these particles, equations 10.16 and 10.18 were used to calculate

HC for aligned and random distributions of particles, respectively. For particles where κ > 0.7, equations 10.14 and 10.15 were used in equation 10.2 to

calculate SARP for aligned and random distributions of particles, respectively.

For these particles the aligned orientation coercive ﬁeld is HC = 2KEﬀ/µ0MS,

and the random orientation coercive ﬁeld is HC = 0.48(2KEﬀ/µ0MS). The

SARP and HC values for particles having κ > 0.7 are denoted with asterisks. Values followed by a star imply that the predicted coercivity is greater than that of the laboratory applied ﬁeld strength...... 91

xix 14.2 Predicted speciﬁc power absorption rates using LRT, where power losses are given by equation 10.21, for particles a, b, c, d, and e. An applied ﬁeld with

H0 = 45 (kA/m) and f = 30 (kHz) was used. Temperature was taken to be T = 37 (◦C), and the viscosity of water η = 0.001 (Pa·s). Initial heating rates were calculated by setting equations 10.21 and 11.1 (without the fraction

mS/mP ) equal and solving for the heating rate...... 92 14.3 Relevant times of the laboratory experiment: the time the applied ﬁeld is in

a single direction τH , the Brownian relaxation time τB of each particle, and

the N´eelrelaxation time τN for each particle. For calculations of τB and τN it is assumed ﬂuid viscosity is η = 0.001 (Pa·s) and temperature is T = 37 (◦C). 93

xx Chapter 1

Statement of Purpose

“There’s a formality in academia that can’t be ignored, even if a man is busy with other things, like trying not to die.” - Randy Pausch, The Last Lecture

Advances in electromagnetics and nanotechnology are starting to make early cancer tumor detection and cell-selective therapy a reality. Magnetic resonance imaging (MRI) is now a common practice, which allows us to render two and three dimensional images of internal tissues in the body, without invasive surgical procedures, by monitoring the relaxation behavior of hydrogen nuclei in water molecules (water protons) in the patient [8,9, 10]. Research has shown that contrast agents, magnetic ions or particles, disrupt the relaxation of water protons, which changes the contrast of magnetic resonance images [11, 12]. Localizing these agents around a given structure, in our case a tumor, allows smaller regions of interest to become more clearly visible in rendered MRI images [9, 10]. Contrast agents can see further use since nanoparticles can now be made at sizes that can cross cell membranes, and localize around and within the desired cell structure [10]. Today,

1 we are finding ways to make magnetic nanoparticles cell or region selective [10, 13], and thus generating powerful contrast agents that will aid in tumor detection. Magnetic nanoparticles can also be used for tumor therapy [6, 10, 11, 12, 14, 15]. When these particles are placed in an oscillating magnetic field they absorb power (energy per time) from the field. To minimize their own energy, the particles will give the acquired energy off as heat. Thus, nanoparticle power absorption from an applied field can be utilized as a form of local heating therapy for tumors when the particles are confined to the tumor region because tumor cells have been shown to be more susceptible to temperature changes than healthy cells [14]. While there is an abundance of research on spherical magnetic particle size effects on MRI contrast and particle power absorption, I hypothesize that particle shape is a control- lable parameter to be utilized in engineering better MRI contrast agents and local heating agents. In this dissertation I analyze the effects of varying magnetic nanoparticle size and shape on water proton relaxation within the particle vicinity in MRI, and power absorption of the particle material from an alternating applied field. Each topic, detection and therapy, is discussed as a separate body of work. In the following work, cancer is briefly explained, as well as why tumor cells are more susceptible to heat than healthy cells. The use of spherical and non-spherical magnetic nanoparticles to alter the contrast of MRI is then discussed. The effects of shape and size on magnetic nanoparticle power absorption as a cancer therapy is discussed last.

2 Chapter 2

Cancer

“So it is said that if you know your enemies and know yourself, you can win a hundred battles without a single loss. If you only know yourself, but not your opponent, you may win or may lose. If you know neither yourself nor your enemy, you will always endanger yourself.” - Sun Tzu, The Art of War

Cancer is a group of diseases that are broadly defined by the main characteristics of excessive or uncontrollable cell proliferation (increased affected cell population) and under certain circumstances, the ability of cells to leave the primary tumor site, spreading to different areas of the body (termed metastasis). Disruptions in the normal regulatory mechanisms that exist in healthy cells, allows cancerous cells to both proliferate indefinitely and to survive under conditions that normal cells cannot. One of the main, but not the only, inducers of cancer is the occurrence of mutations in important regulatory genes. Because cells repli- cate their deoxyribonucleic acid (DNA), before dividing, gene mutations of the cancer cell are passed to its offspring. Today, we know cancer progresses as a time dependent accrual of mutations on a cellular level, passed along via cellular division, which gives rise to unique traits that make it easier for affected cells to increase their population [16].

3 There are over 100 known types of human cancer [16]. Cancer can afflict any type of cell capable of division. Many forms of cancer become life-threatening because the excessive population of the affected cells form invasive primary tumors and metastases, which negatively impact healthy components of the body. It is becoming clear that while there are numerous available pathways to the formation of malignant cancer tumors there are only a limited number of attributes, being defined as hallmarks of cancer, unique to most and questionably all forms of these cancers [17, 18]. Six of these hallmarks are known today: sustained proliferative signaling, evading growth suppressors, evading apoptosis (a natural induced cell death), enabling replicative immortality, sustained angiogenesis (the formation of new blood vessels), and activating tissue invasion and metastasis [17, 18]. Two potential hallmarks being researched are the deregulation of cellular metabolism and the avoidance of immune surveillance [18]. The first four hallmarks of cancer are acquired at the cellular level, and are found in all forms of cancer [18]. When these afflictions occur, they result in the formation of micro-tumors, small abnormal masses of cells. As a micro-tumor grows it eventually cannot sustain itself without its own source of nutrients and waste removal. To obtain them a tumor will induce angiogenesis, the formation of new blood vessels, to transfer oxygen and other nutrients to the tumor and remove metabolic waste and carbon dioxide [18]. Angiogenesis is typically used by adult bodies as a temporary mechanism to heal wounds, or support female reproductive cycles. However, angiogenesis is also induced during tumor progression [19]. This is done by inducing pro-angiogenic signaling pathways in endothelial cells [18]. The blood vessels created by tumor-induced angiogenesis are immature [20, 21]. They have poor circulation, are easily broken, and are often not closed systems because of perpetual building. Tissue invasion and metastasis can result from growing tumors. These processes are now understood to follow a multistep process [22, 23]. The tumor begins by growing and

4 invading neighboring space in its tissue of origin. At some point it becomes connected to the lymph and blood systems where it can place affected cells into those systems. These cells are dispersed to other parts of the body where they can attach and form new micro- tumors. If these new tumors take hold in the new tissue they can fully colonize and form macroscopic metastases. This process requires a tremendous amount of adaptation and cooperation between cancer cells and normal cells. Even if a primary tumor can distribute micro-metastases there is no guarantee they will successfully form macroscopic tumors [22, 23]. Two more possible hallmarks of cancer are being identified in recent research: the deregulation of cellular metabolism, and immune system avoidance [18]. It is clear that numerous cancerous tissues change their cellular metabolism to support the strain of continuous cell proliferation [24, 25, 26]. They may change what they use for nutrients and excrete, or may change their energy consumption rates. It is also clear that the existence of tumors is made possible by them evading normally efficient portions of the immune system designed to eradicate foreign entities and damaged cells [18]. These mechanisms of cancer are still not fully understood. They are being labeled as emerging hallmarks of cancer. In addition to hallmarks, which are responsible for cancers abilities to survive, proliferate, and spread, cancer enablers are being identified. Enablers provide cancer cells the means to collect hallmarks. Obviously, genetic mutation is one of the known enablers, but an additional mechanism is immune evasion. The immune system has a double standard when it comes to cancer. It is largely responsible for damaging and limiting many forms of cancer cells, but also appears to be helping others [18]. One natural defense of the immune system is inflammation of the affected area. Inflammation induces angiogenesis, the presence of cellular growth factors, and other enabling factors that can actually cause cancer to grow rather than destroy it [18]. The onset of inflammation around small tumors may even provide genome instability and enhance the capacity of cancer cells to acquire more

5 mutations or traits. This selective nature of the immune system is being researched, but is not yet fully understood. It is being treated as a second form of enabling cancer in addition to genetic mutation [18]. These six hallmarks and two potential hallmarks are being found in a variety of cancers. There are also two recognized enablers enhancing the ability of cancer cells to acquire these traits. For a cancer to establish itself at the cellular level, survive, proliferate into a micro-tumor, invade (become macroscopic) and form other macroscopic metastasis it must obtain all of these traits. This process is time dependent and can take years to accomplish. The accumulation of these cells as tumors in the body places life-threatening stresses on the body that disrupt its ability to sustain life. Since the blood vessels created by tumor-induced angiogenesis are erratic and inefficient [20, 21], tumors are more susceptible to heat because increasing the tissue temperature will tax the tumor cells, making their inefficient circulatory system inadequate to sustain cell life. A local temperature of T = 44 (◦C), maintained for a duration of one hour, has been shown to successfully cause large scale tumor cell death [14]. In order to not adversely affect other surrounding cellular structures, which are also susceptible to changes in temperature, the heating must be confined to the tumor region. Heat dissipation of localized magnetic nanoparticles due to an alternating magnetic field is a promising method of local heat therapy. It can be aided, as well as other therapy techniques such as radiation or chemotherapy, by the use of the magnetic particles as contrast agents in MRI in order to detect smaller primary tumors and metastases. Ad- ditionally, local magnetic particle heating therapy does not have the same constraints on patient treatment intervals as other therapies (chemotherapy and radiation), which harm both healthy and cancerous cells on large scales. Well controlled heating therapy treatments can be administered independently and more frequently than these other therapies, or they can be used in combination with these other therapies to enhance their effects. These two

6 potential beneﬁts of magnetic particles, local heating therapy and tumor detection, make them promising agents to be used in cancer healthcare.

7 Part I

Magnetic Nanoparticle Shape and Size Eﬀects on T2 Relaxation in Magnetic Resonance Imaging

8 Chapter 3

Introduction

Magnetic resonance imaging (MRI) is a non-invasive form of imaging capable of mapping tissue components of living organisms in vivo. MRI is most commonly done by observing the nuclear magnetic resonance (NMR) behavior of the protons of hydrogen atoms in water molecules (water protons), which are present in large quantities in tissues, organs, and tumors (if they are present). In MRI a large static external magnetic field (usually taken to be in the zˆ direction) is used, causing the z-component of the water proton magnetic moments to align parallel or anti-parallel with the external field, and the individual moments to precess about the external field. Over time, the proton ensemble reaches a thermal equilibrium of protons in an excited state N↑ (z-component anti-parallel) and ground state N↓ (z-component parallel). These discrete spin states are caused by Zeeman splitting due to the external magnetic field. The transverse (ˆx andy ˆ) components of the water proton magnetic moments precess about the external field with random orientations to one another. Due to these effects of the static field, the thermal equilibrium of the water protons has a net magnetization oriented along the static field axis. An electromagnetic pulse of energy applied to the water protons alters the water proton energy states. After the pulse is removed the water proton ensemble returns to thermal equilibrium. While the ensemble relaxes it emits electromagnetic energy. This

9 emitted energy is the observed signal used for magnetic resonance imaging. The contrast of MRI images depends on local water proton densities and their relaxation processes. Various tissues will contain different levels of water protons, and their relaxation rates are influenced by the composition of the local tissue environment: static field strength, field inhomogeneities, neighboring charges or perturbations, and diffusion length scales and times. It is this combination of proton density and unique local environment that provides the inherent contrast of MRI. This natural source of contrast in MRI has proved useful, but there is a growing demand to further enhance the contrast of MRI in order to distinguish smaller features or highlight specific areas of interest either by diminishing the background contrast, or increasing the contrast of a desired focal point. One method of influencing contrast in MRI is the addition of contrast agents to the body. Contrast agents are usually ions or magnetic nanoparticles placed in the patient to affect the relaxation processes of water protons within their vicinity, which provides the desired changes in contrast to the MRI image. Ferromagnetic materials, especially iron oxides such as magnetite, are being developed as particle contrast agents because they have

a large inherent saturation magnetization MS. Single domain particles of this substance (or other ferromagnetic material) create strong field gradients about them, which influence the precession behavior of water protons within the particle gradient. Also, most iron oxides have long been known to be relatively biologically inert [27]. These two attributes make iron oxides a desirable contrast agent to utilize. Particle size effects for spheres and clusters of spheres (of ferromagnetic materials) on image contrast are well referenced in contrast agent literature [9, 11, 28, 29], but there is a lack of discussion on the effects of particle shape, which also change the magnetic properties of a material. Most notably, changing a magnetic sphere to be more flat, disc-shaped, or elongated, needle-shaped, causes the field about the particle to change [30, 31]. Thus, we can control the field gradient of particle contrast agents by controlling their size and shape.

10 Nanofabrication techniques already exist to create magnetic particles of various shapes and sizes [1,2,3, 32, 33]. In this research I examine the effects of ferromagnetic nanoparticle shape and size on the decay time of the transverse emitted signal from a water-based material undergoing MRI. To do this, MRI using a Carr-Purcell-Meiboom-Gill spin echo sequence, defined in the theory section, was carried out on particle-infused agar gel samples, called phantoms, with various concentrations of ferromagnetic particles having various shapes and sizes. The results indicate the effective transverse relaxation time (T2), a measure of time required to lose the majority of the return signal intensity, is influenced by the size and shape of particle contrast agents. A linear relationship between transverse relaxivity and the ratio of particle surface area to volume was observed, which indicates reducing the surface area to volume ratio of a magnetic particle enhances the transverse relaxivity. The data indicates a stronger

T2 contrast enhancement for larger low-aspect ratio barium ferrite platelets compared to smaller spherical and prolate shapes of magnetite nanoparticles.

11 Chapter 4

Magnetic Resonance Imaging

“We often plough so much energy into the big picture, we forget the pixels.” - Silvia Cartwright

4.1 Historical Overview

In 1924 Wolfgang Pauli proposed the presence of nuclear spin to account for inconsistencies between observed atomic spectra and the theory of quantum mechanics. Pauli believed these discrepancies were caused by the interactions of nuclei with their local environment, where they experienced unique magnetic ﬁelds from the moving charges around them. Taking into account the nuclear spin, the atomic spectra matched the quantum mechanical model of the system, proving the existence of quantum mechanic magnetic moments, which occur on the nuclear level. From this, it was distinguished that nuclei with odd mass numbers possess half-integer spin, nuclei with an even mass number and odd charge number exhibit integer spin, and nuclei with both an even mass number and charge number exhibit no spin.

12 Felix Bloch and Edward Purcell in 1946 individually found that nuclei with a non- zero spin angular momentum can absorb and emit electromagnetic radiation when they are located in an external magnetic field [34, 35]. They each found a direct relationship between external magnetic field strength and nuclei angular momentum precession rate. This rate of precession is known as the Larmor frequency. With this knowledge the application of nuclear magnetic resonance (NMR), the ability to identify structures via their components’ Larmor frequencies, became viable. Both Bloch and Purcell received the Nobel Prize in Physics for this discovery in 1952. The chemical shift effect, the change of a nuclei’s (charged particle) Larmor frequency due to molecular bonding, was discovered in 1950 by Proctor and Dickinson [36, 37]. Their groups discovered that the Larmor frequency of a given nucleus can vary depending on the electron distribution surrounding it, thus proving that molecules could be identified using NMR because their molecular bonds would cause known changes in the Larmor frequency of the nuclei being observed. The link between differences in a nuclei’s Larmor frequency and its position within a molecular structure made NMR a power spectral tool. The first commercial NMR spectrometers were released in 1953. Years later, in 1971, Damadian found that cancer tissue has different relaxation properties than healthy tissue [8]. This result was found using ex vivo tissue samples. Later, Damadian developed field focusing NMR, which he labeled FONAR [38]. FONAR was used to measure the relaxation time of tissue in vivo, and create images by physically moving the patient through the machine. Damadian was the first person to show in vivo tissues can be resolved using NMR. While Damadian was perfecting his FONAR technique, in 1972 Paul Lauterbur completed a new imaging technique, which he called zeugmatography. Using a number of NMR measurements associated with a linear magnetic field gradient, Lauterbur created two dimensional images of water filled samples. For this work, he was awarded the Nobel Prize

13 in Medicine and Physiology in 2003, and his article, published in Nature in 1973 [39], is considered the basis of MRI. Hinshaw expanded the capabilities of imaging, via magnetic resonance techniques, by developing what he called the sensitive-point technique. This method allowed image selection planes of a three dimensional object using three magnetic ﬁeld gradients [40]. This version of the multiple sensitive-point technique is the underlying principle for modern MRI imaging and slice selection. Hinshaw’s method provided the small spatial resolution and stationary patient required to make accurate and repeatable image mapping of the interior of a three dimensional structure. MRI has grown to be one of the most powerful tools in medical research and healthcare, where it is used to evaluate and diagnose patients. The ability to discern soft-tissues in patients non-invasively has had a dramatic impact in science and healthcare. MRI can achieve greater potential through the use of contrast agents. Recently published data con- ﬁrms that 30% of all MRI analysis uses some sort of contrast enhancement medium [41].

4.2 Water Proton MRI

MRI is most commonly done by observing the nuclear magnetic resonance (NMR) behavior of the protons of hydrogen atoms in water molecules (water protons), which have a magnetic moment µ and spin S because they are a moving (rotating) body with a net charge, and are present in large quantities in tissues, organs, and tumors (if they are present). The magnetic moment and spin of a hydrogen atom proton are related to one another,

µ = γS, (4.1) where γ is the gyromagnetic ratio of the hydrogen proton [7, 42].

14 When a patient or object undergoes this type of MRI they are placed in a large static external magnetic field B = B0zˆ causing the z-component of the moments to align parallel or anti-parallel with the external field, and the transverse (ˆx andy ˆ) components of the water proton magnetic moments to precess about the external field at a rate

ω = γB0. (4.2)

This rate of precession is known as the proton Larmor frequency. For water protons γ = 267.53 (MHz/T) [7]. The gyromagnetic ratio contains information speciﬁc to the water proton (spin and magnetic moment), making the water proton Larmor frequency unique. The water proton magnetic moments eventually reach a thermal equilibrium of populations whosez ˆ-components are aligned (N↓) and anti-aligned (N↑) because of Zeeman splitting of the hydrogen atom proton energy states due to the external magnetic ﬁeld. The proton energies are given by

U = −µ · B = ±γSzB0, (4.3)

where +γSzB0 is the high energy state for a counter-aligned magnetic momentz ˆ-component, and −γSzB0 is the low energy state for an aligned magnetic momentz ˆ-component. Although water protons are quantum mechanical in nature, classical physics can be used to describe the ratio of energy states for an ensemble of them. The distribution of proton magnetic moments between the ground and excited state for a macroscopic ensemble in an external field is given classically by N ∆U ↓ = exp( ) (4.4) N↑ kBT where ∆U = 2γSzB0 is the energy difference between the two possible energy states, kB is the Boltzmann constant, and T is the absolute temperature [43]. This population difference results in a net magnetization along the axis of the static external field (M = M0zˆ). There are

15 nox ˆ andy ˆ components to the net magnetization because the individual magnetic moments of the water protons are randomly oriented with respect to one another. After the water protons reach a thermal equilibrium an electromagnetic pulse is used to alter the populations of energy states. The electromagnetic pulse is a circularly polarized magnetic field, oscillating at the Larmor frequency ω of the water protons, and is applied for a duration of time [44]. The magnetic field of the pulse appears as a constant field to the water proton magnetic moments, causing them to rotate about the circularly polarized field (as well as the static field). The amount of rotation is specific to the time the rotating field is applied. This electromagnetic pulse causes two things to occur: it changes the populations of protons having either energy state in the static field, and synchronizes the magnetic moments of each water proton (causes them to be coherently phased). Once the electromagnetic pulse is removed, the water protons relax to thermal equilibrium, and the individual water proton magnetic moments fall out of synchronization (they fall out of phase with each other and themselves). This loss of synchronization is known as dephasing [44]. While the water protons relax back to a thermal equilibrium they emit an electromagnetic pulse perpendicular to the static external field. This emitted energy pulse is the observable signal used for MRI. It is observed using a solenoid as a detector. To supply positional information within the emitted signal magnetic field gradients are used to create small areas or volumes of the external applied field, each with a unique static field strength. This causes the areas or volumes to have different Larmor frequencies [44]. There are two methods of analyzing an area or volume return signal due to an excitation electromagnetic pulse in MRI. One is used to calculate the longitudinal net magnetization

growth rate (denoted as T1) related to the water proton energy populations, and the other is used to calculate the transverse net magnetization decay rate (denoted as T2) of the water protons related to the dephasing of their magnetic moments [44]. The images determined

from these two rates are referred to as T1- and T2-weighted images.

16 The contrast of MRI pixels or voxels depends on the densities of the local water protons in the area or volume associated with the pixel or voxel, rotational movements of the water protons, and the water proton translational movements (diﬀusion). The density of water protons determines the initial magnitude of the pixel or voxel signal intensity, and the rotational and translational movements of the water protons determine the decay rate.

4.3 Water Proton Magnetic Moment Transverse Re-

laxation

The effects of ferromagnetic nanoparticles on the transverse relaxation processes of water proton magnetic moments that occur after an electromagnetic pulse disrupts the thermal equilibrium of the water protons is the primary focus for this body of research. There are two physical mechanisms due to the presence of ferromagnetic material that cause the loss of water proton magnetic moment phase coherence [45]. They both stem from the contrast agents creating static field inhomogeneities in the applied field. The first is dephasing due to the ensemble of stationary water protons in the magnetic field gradient of the particle. This gives rise to a distribution of proton precession fields in a given spatial region. The proton spins from different water molecules will experience small differences in the local field, and so will precess at different Larmor frequencies. The total emitted signal from an ensemble of these spins will decay as the magnetic moments from the individual water protons dephase with respect to one another. This occurs at a rate 1/T2SD (SD denotes static dephasing), which is faster than the inherent rate of the medium without the particles present 1/T2 < 1/T2SD. This type of dephasing is known as static dephasing. The second type of dephasing is due to individual water proton diffusion on a large enough scale for the magnetic moments of the water protons to move across the field gradient of the particle. These types of movements cause the water proton magnetic moment to

17 precess at diﬀerent Larmor frequencies as it travels through the ﬁeld gradient of the particle, causing it to dephase from its initial Larmor frequency after the electromagnetic pulse is removed. This results in an irreversible loss of phase coherence and signal intensity [46]. This loss of phase coherence is also faster than the inherent relaxation rate of the water

protons in the absence of the particle gradient 1/T2 < 1/T2DD (DD denotes diffusion based dephasing). These two relaxation mechanisms are complex functions of particle shape and size. In addition to static dephasing and diffusion dephasing near a particle contrast agent there are other sources for the dephasing of the protons. The proton magnetic moments are susceptible to neighbor interactions, inherent field perturbations in the static field itself, and other phenomenon that distort the local magnetic field. Because of this the relaxation rate

∗ of a water proton density T2 is given by

1 1 1 1 ∗ = + + (4.5) T2 T2 T2SD T2DD

where 1/T2 is the natural relaxation rate of the water proton density, 1/T2SD is the static dephasing rate near a magnetic particle, 1/T2DD is the diﬀusion dephasing rate near a magnetic particle [47].

4.4 Water Proton Transverse Relaxation Pulse Sequences

Water protons only emit an observable signal after absorbing energy from an electromagnetic pulse. To determine characteristics of the system of water protons from their emitted signal

? (such as T2 and T2 ) requires using sequences of excitation pulses with diﬀerent rotation planes and diﬀerent duration times. The degree of rotation of the water proton magnetic moments due to an electromagnetic pulse is controlled by the time duration of the excitation pulse.

18 MRI commonly uses time durations that rotate the component of the magnetic moments 180 degrees and 90 degrees, and refer to these excitation pulses as π and π/2 pulses, respectively [44]. The π pulse is used to invert the component of the magnetic moments causing the net magnetization about the axis perpendicular to the plane of the excitation pulse. As an example, a π pulse in the (x, y) plane transposes the thermal equilibrium populations of energy states, which flips the thermal equilibrium net magnetization vector along the z axis. The π/2 pulse is used to rotate the magnetic moments causing the net magnetization into the plane of the excitation field. A π/2 pulse in the (x, y) plane rotates the thermal equilibrium z-components of the magnetic moments into the (x, y) plane. This equalizes the populations of energy states in the static field. There are two MRI pulse sequences commonly used to measure the transverse re-

laxation time T2 [44]: the Carr-Purcell-Meiboom-Gill (CPMG) spin echo sequence and the

∗ Gradient Echo (GE) pulse sequence. Whether the observed decay time is T2 or T2 , depends on the imaging sequence used. When there are perturbations and other factors influencing the local magnetic field of the water proton magnetic moments the decay time for the magnetic moments to dephase is reduced, which shortens the length of the water proton return signal. If these influences are static sources that do not vary over time their effects can be reversed using the CPMG sequence. The CPMG spin echo sequence uses a π/2 pulse followed by a series of π pulses at specific echo times τ to re-phase the proton spins experi- encing static fields in the transverse plane at a time 2τ, which is when the signal intensity is observed [48]. By repeating this re-phasing π pulse at odd intervals of τ and measuring the signal intensity at even intervals of τ the transverse relaxation time observed is T2 as shown

∗ in ﬁgure 4.1 and not T2 [44], which is also illustrated in ﬁgure 4.1. The relaxation time T2

∗ is clearly longer than T2 . The equation used to obtain T2 is [44]

−τ/T2 Mx,y(τ) = Mx,y(0)e . (4.6)

19 Each signal intensity observed at the even intervals of τ will be a maximum, but each successive maximum is decreased by an amount proportional the the percentage of the magnetic moment population that has relaxed through natural irreversible processes with out the eﬀects of local perturbations. This loss is shown in ﬁgure 4.1.

Figure 4.1: CPMG spin echo sequence shown with transverse signal response. The axis is time.

Gradient echo imaging sequences do not use the re-phasing π pulses. They are simply

a π/2 pulse followed by observation and a delay time τD (normally 5·T1) to allow the ensemble to fully relax if multiple data sets are required [44]. Because of this the transverse relaxation

∗ time observed is the shorter T2 and has contributions from both the sample material and the perturbations. There is no rephasing and the decay is rapid as shown in 4.1. The equation

∗ to obtain T2 is [44]

−τ/T ∗ Mx,y(τ) = Mx,y(0)e 2 , (4.7)

where τ is time.

20 4.5 Particle Eﬀects on Image Contrast

The eﬀect of contrast agents on the transverse relaxation of water proton magnetic moments is mostly discussed in terms of the observed transverse relaxation rate R2, the inverse of transverse relaxation time T2O (O denotes observed). At low concentrations of contrast agents the relaxation rate is linearly proportional to the concentration [12],

1 1 R2 = = r2 · C + (4.8) T2O T2I where T2O is the observed relaxation time, r2 is the slope known as the relaxivity, C is the concentration of the contrast agent, and T2I is the inherent relaxation of the sample without contrast agents. The goal of particle contrast agents is to cause large known changes to the observed transverse relaxation rate using minimal amounts of a particle contrast agent for the purpose of enhancing a specific tissue in an MR image, which is made possible by the particle contrast agent having a large transverse relaxivity r2. This research is focused on contrast agents that affect transverse relaxation, however, contrast agents also have effects on the longitudinal relaxation rates of water protons.

21 Chapter 5

Materials and Methods

5.1 Particle Preparation

Five different types of magnetic nanoparticles were investigated during my research. The particles are modeled as spheroids. Their shapes and dimensions (confirmed via SEM) are given in figure 5.1. The shapes are drawn such that the magnetization points up, and rotating the schematic about the particle magnetic easy axis (the vertical axis) gives the overall shape. In the figure c is one half of the magnetic easy axis length, and a is one half of the magnetic hard axis length of the particle.

Particles 1 through 4 are made of magnetite (Fe3O4), a ferromagnetic substance with a cubic crystal structure. Particle 5 is made of barium ferrite (BaFe12O19), a ferromagnetic substance with a hexagonal crystal structure. Particle shape and crystal structure both affect the particle magnetization orientation. The magnetization induces an internal field, called the demagnetization field. This field is shape dependent and the energy is minimized when the magnetization lies along the axis of this field (the easy axis), which is along the particle major axis (the longer axis). The crystal structure also affects the magnetization. Energy is also minimized when the magnetization lies along the crystal easy axis. Both

22 Figure 5.1: Schematics of the five different types of single domain ferromagnetic nanoparticles used in this research. Their magnetization orientation is shown on the left, and their dimensions on the left. In the figure c is one half of the magnetic easy axis length, and a is one half of the magnetic hard axis length of the particle.

23 shape and crystal effects determine the direction of magnetization. For particles 3 and 4, shape effects cause the magnetization to lie along the particle major axis. Particle 5 has a strong crystal easy axis, which makes the magnetization lie normal to the disc plane. It was necessary to change the particle material in order to have a particle whose magnetization lies perpendicular to the shape easy axis. Barium ferrite was chosen because its magnetization is on the same scale as magnetite, is affordable, and readily available. Particles 2 and 5 were purchased (Sigma Alrich). Particles 1, 3, and 4 were synthesized using already established techniques shown by Massart [1], Kumar and Koltypin et al. [2], and Matsui [3], respectively. The synthesis method of these particles are given below.

5.1.1 Synthesis of Particle 1

Particle 1 was prepared with a coprecipitation technique similar to Massart’s method [1] by placing a 2:1 molar ratio of ferrous and ferric chloride in a sodium hydroxide solution. For this 400 mL of 0.75M NaOH, made from sodium hydroxide crystals (Fisher Scientiﬁc) and distilled water, were placed in a 2 L beaker and heated to 100 degrees Celsius while being continuously agitated with a stirring rod rotating at 550 rpm. Next, 40 mL of 1M

FeCl3 and 10 mL of 2M FeCl2 were added to the solution. These solutions were prepared

using FeCl3·6H2O and FeCl2·4H2O (both Fisher Scientiﬁc) in distilled water, respectively. Following this, 20 mL of 2M HCl (Sigma Aldrich) was added dropwise. The contents were agitated and heated for a duration of ten minutes. After this time the beaker was placed on a permanent magnet and the contents of the beaker were allowed to cool to room temperature.

The supernatant was then siphoned oﬀ and 200 mL of 1M HNO3 (Acros Organics) was used to wash the particles. The mixture of particles and HNO3 was agitated using a stirring rod rotating at 700 rpm for ten minutes. Afterwards the beaker was placed on the permanent magnet, the contents allowed to settle, and then the supernatant was removed. This washing process was repeated ﬁve times. After this process the particles were dried and placed in

24 storage vials.

5.1.2 Synthesis of Particle 3

Particle 3 was prepared using a sonochemical method published by Kumar and Koltypin et al. [2]. Initially, 100 mL of distilled water was placed in a 250 mL beaker and deoxygenated by bubbling nitrogen gas through the water for ten minutes. Afterwards, 1 g of iron(II) acetate (Strem Chemicals) and 20 mg of β-cyclodextrin (Acros Organics) were added to the beaker. The contents were sonicated with a high-intensity probe (Fisher Scientiﬁc Model 550 Sonic Dismembrator, horn model CL4) set to power level 6 at room temperature and atmosphere for 3 hours. Nitrogen gas was bubbled through the mixture throughout the sonication process. After sonication the beaker was placed on a permanent magnet and the contents were allowed to cool to room temperature. Once cooled, the supernatant was removed and the particles were washed repeatedly by swirling them in 100 mL of distilled/deoxygenated water for ﬁve minutes and then setting the beaker on a permanent magnet. Washing was repeated until the supernatant was clear. After this the particles were dried and placed in storage vials.

5.1.3 Synthesis of Particle 4

Particle 4 was prepared by creating acicular goethite particles and reducing them to magnetite by baking them in a furnace . The goethite particles were prepared in the following manner based on Matsui’s method [3]. In a 2 L beaker 62.6 g of FeSO4·7H2O (MP Biomedi-

cals) was added to 250 mL of distilled water to create a 0.45 M FeSO4 solution. This solution was agitated with a magnetic stirring rod rotating at 500 rpm. In a 500 mL beaker 43.3 g of NaOH crystals (Fisher Scientiﬁc) were placed in 225 mL of distilled water and dissolved by agitating with a magnetic stirring rod at 500 rpm. Once the sodium hydroxide solution

25 was made it was placed in the 2 L beaker with the ferrous sulfate solution. Distilled water was added to the beaker until a total volume of 500 mL was achieved. The 2 L beaker was then heated to 45 degrees Celsius and agitated at 500 rpm. This temperature and agitation were maintained for four hours. Afterwards, the contents of the beaker were allowed to settle overnight. The yellow precipitate was then removed from the clear supernatant and washed using distilled water. The water was siphoned off and the beaker was placed in an oven at 100 degrees Celsius to dry. Once the particles were dry, the furnace temperature was increased to 350 degrees Celsius and the precipitate was baked for three hours at this temperature. The beaker was shaken every half an hour to agitate the powder. After baking, the beaker and now black precipitate was allowed to cool to room temperature. The black powder was then placed in vials and labeled. Unfortunately, particle 4 was not available during my contrast agent research, but was used in my power absorption research. Two separate sets of particle identifiers were used to distinguish the particles used in each research topics. They are given in table 5.1. Additionally, the color-coding used in figure 5.1 is carried out through both bodies of research.

Particle Composition c a Contrast Agent Power Absorption (nm) (nm) Identifier Identifier 1 Fe3O4 6 6 A a 2 Fe3O4 16 16 B b 3 Fe3O4 24 7 C c 4 Fe3O4 225 15.0 - d 5 BaFe12O19 10 100 D e Table 5.1: Magnetic nanoparticles used in this research along with their compositions, dimensions, and research identifiers. In the table, c refers to one half of the magnetic easy axis length and a refers to one half of the magnetic hard axis length.

26 5.2 Sample Preparation

5.2.1 Agar Phantoms

The agar gel samples are called phantoms. This is the term used to define a sample containing a homogeneous medium to test the performance of an MRI machine, or the effects of a contrast agent on a homogeneous medium MR image. Four agar phantoms without contrast agents were prepared using granulated agar (Fisher Brand, molecular genetic granulated): two uniform phantoms with different agar concentrations, and two bilayer phantoms of three percent agar allowed to set in different environments. These phantoms were prepared to determine the effects of agar gel concentration and agar gel setting (solidifying process) environment upon the transverse relaxation time T2 of agar gels. The agar granules were used as delivered for all phantoms. The sample preparation conditions are summarized in table 5.2. Phantom A-1 was a two percent agar (w/w) phantom made by placing 2 g of agar granules in 100 mL of distilled water in a 250 mL beaker. The solution was stirred at 1000 rpm and heated to 90 ◦C until a thick clear viscous solution formed. This solution was poured into a disposable culture tube (Fisher Brand, 12mm x 75 mm) and left to solidify at room temperature. The opening was sealed with parafilm after the agar set into a cloudy white gel. Phantom A-2, a three percent gel, was made using the same method described above. The two bilayer phantoms use three percent agar in a disposable culture tube. Phantom A-3 was filled half-way with liquid agar that was placed in a freezer to set. A second layer was placed on top of the frozen layer and left to set at room temperature. Parafilm was used to seal the culture tube. Phantom A-4 has a bottom layer of agar allowed to solidify at room temperature and a top layer of agar left on a hot plate for an extended time to evaporate also allowed to set at room temperature. Again, parafilm was used to seal the sample.

27 5.2.2 Particle Concentrations Suspended in Agar

Eight contrast agent phantoms were prepared using the four available types of sample material: A, B, C, and D. One set of contrast agent phantoms used a small amount (1% w/w) of sodium dodecyl sulfate (SDS, purchased from Fluka Chemika) as a particle coating for each type of magnetic material. The other set of phantoms used dioctyl sodium sulfosuccinate (AOT, purchased from Fluka Chemika) to coat the particles. The contrast agent phantoms are summarized in table 5.3. The contrast agent phantom samples were made of 500 µL layers of 3% agar (w/w) with a different concentration (mg/mL) of nanoparticles in each layer. Figure 5.2 is a schematic of a phantom containing contrast agents. The bottom layer is pure agar upon which the most concentrated layer is placed with each successive layer having a lower concentration of magnetic particles. All agar layers were allowed to set at room temperature before the next layer was added. The remaining volume of each sample tube was filled with pure agar. Parafilm was used to seal the sample tubes. The multi-layered contrast agent phantoms have nine layers as shown in figure 5.2. The process for constructing a single phantom is given below. First, 100 mL of 3% agar gel was made and kept on a hot plate at 90◦C and agitated with a magnetic stirring rod at 1000 rpm. Second, seven aqueous solutions of magnetic particles were created by initially placing

Phantom Composition Layer Solidifying Environment A-1 2% agar - room temp. A-2 3% agar - room temp. A-3 3% agar bottom room temp. 3% agar top evaporated A-4 3% agar bottom freezer 3% agar top room temp.

Table 5.2: Agar phantoms without contrast prepared to calculate agar transverse relaxation time T2 as a function of concentration, and the stability of 3% agar gel T2 under various conditions.

28 Figure 5.2: Contrast agent sample: multiple layers of agar containing varying concentrations of magnetic particles.

29 Layer Coating Particle A [C] Particle B [C] Particle C [C] Particle D [C] (mg/mL) (mg/mL) (mg/mL) (mg/mL) Phantom 1 Phantom 2 Phantom 3 Phantom 4 1 SDS 0.00 0.00 0.00 0.00 2 SDS 0.388 0.315 0.355 0.339 3 SDS 0.129 0.105 0.118 0.113 4 SDS 0.0430 0.0352 0.0394 0.0375 5 SDS 0.0144 0.0117 0.0131 0.0125 6 SDS 0.00479 0.00391 0.00436 0.00418 7 SDS 0.00160 0.00130 0.00146 0.00139 8 SDS 0.000534 0.000434 0.000485 0.000464 9 SDS 0.00 0.00 0.00 0.00 Phantom 5 Phantom 6 Phantom 7 Phantom 8 1 AOT 0.00 0.00 0.00 0.00 2 AOT 0.315 0.318 0.357 0.309 3 AOT 0.105 0.106 0.119 0.103 4 AOT 0.0350 0.0354 0.0397 0.0343 5 AOT 0.0116 0.0118 0.0133 0.0115 6 AOT 0.00389 0.00393 0.00442 0.00381 7 AOT 0.00130 0.00131 0.00147 0.00126 8 AOT 0.000434 0.000436 0.000491 0.000423 9 AOT 0.00 0.00 0.00 0.00 Table 5.3: Magnetic material and concentration used in the phantoms containing contrast agents.

approximately 10 mg of magnetic material in 1 mL of distilled water with approximately 1 mg of SDS or AOT in a 1.5 mL centrifuge tube labeled C1. These are common surfactants used to help suspend and separate individual particles. A sonicator was used to agitate the suspension and coat the particles with the surfactant for t > 10 min. This aqueous concentration was then successively diluted by transferring (pipetting) 500 µL of the prior concentration to a new centrifuge tube, adding 1 mL of distilled water and then sonicating the suspension. In this way C2-C8 were made. The initial aqueous concentration C1 is not placed in agar and is set aside after this process, the seven dilutions C2-C8 are used in the agar layers. After the aqueous concentrations are made they are placed in a rack and set aside. Third, in a separate rack 9 disposable culture tubes labeled L1 through L9 are placed, the rack is then set in a heat bath at 90◦C. 1 mL of agar from the main batch is placed in each tube. To assist with accuracy a separate sample tube with 1 mL of water is used to

30 create a marker on the agar tubes. Prior to adding any layers, the phantom sample tube was marked at 500 µL incre- ments. The first layer placed in the phantom was 500 µL of L1, pure agar. It is allowed to fully set before another layer is added to the phantom. The second layer was 500 µL of L2 infused with 100 µL of C2. C2 was sonicated immediately before being placed in the agar of L2. It was then dispersed in the agar by quickly voiding and filling the pipette repeatedly. Afterwards, 500 µL of the suspension were placed in the phantom on top of the first layer of the phantom. The above method was used for each layer containing particles. The top layer was pure agar from L9. Parafilm was used to seal all contrast agent loaded phantoms.

5.3 System Setup

5.3.1 Machine and Software

The test system for this research was a Bruker Biospin 7 T, 20 cm horizontal bore magnet MRI machine with a working free space of 8.6 cm for imaging. The software platform provided for the machine was Paravision 4.0 MRI/MRS installed on a personal computer. The entire system is located in a laboratory with a stable room temperature. The MRI machine is enclosed in a faraday cage to protect the computers and software running the machine.

5.4 Imaging Sequence

5.4.1 Multiple Slice Multiple Echo

MR images were obtained using the software suite Paravision 4.0, which has a predeﬁned multiple slice multiple echo (MSME) imaging sequence. It was the imaging sequence used on

31 Echo Time Repetition Time π pulses Field of View Slice Thickness T E T R n FOV SI (ms) (ms) (mm) 10 2000 16 8.0/4.0 cm 1.00

Table 5.4: Main parameters of the MSME imaging sequence. all phantoms to collect data for the transverse relaxations of agar with and with out contrast agents. MSME is a CPMG spin echo sequence with the form

◦ ◦ 90x − [−TE/2 : 180y − TE/2 : echo−]n − TR, (5.1) where TE is the echo time, TR is the repetition rate of the sequence, and n is the number of π pulses used to create the echo train. The π/2 and π pulses are given in degrees, and their directional eﬀect (in a rotating reference frame) on the water protons is denoted by the subscripts. The main properties of the sequence are provided in table 5.4. The echo time was 10 ms and the repetition time was 2000 ms. There are n = 16 rephasing π pulses in the MSME sequence. Signal intensity is monitored at each echo occurring at the intervals of TE.

5.5 Imaging

Imaging was limited to a single slice directly through the middle of the phantom sample tube as shown in ﬁgure 5.3. The slice is 1 mm thick and is centered on the widest part of the sample tube. The ﬁeld of view plane is 8.0 cm by 4.0 cm with the sample tube located at the center of the plane. For each sequence 16 images are formed corresponding to the intervals of TE.

32 (a) (b) (c)

Figure 5.3: Image slice shown from the (a) front, (b) side, and (c) top.

5.6 Imaging Analysis

To determine the T2 times for phantoms containing magnetic particles a square region of interest (ROI) was chosen for each phantom layer. Multiple square ROIs were also used to analyze the agar phantoms not containing contrast agents. The areas of the ROIs varied from 0.17 cm2 to 0.38 cm2 and were adjusted to be as large as possible within the layer (similar areas were used for phantoms A-1 through A-4). The signal intensity of the ROIs for each phantom at intervals of the echo time were the collected raw data.

33 Chapter 6

Results and Discussion

6.1 Particle Characterization

The particle dimensions were confirmed via SEM and are given in table 6.1, along with the density ρ and saturation magnetization MS values reported by Rosensweig and Coey [6,7]. Images of the particles are provided in figures 6.1- 6.5. In the table, c refers to one half of the magnetic easy axis length of the particle, and a refers to one half of the hard axis. The aspect ratio α is greater than one for prolate (football shaped) particles, and less than one for oblate (flat disc) particles. The volume (not given in the table) in all cases is given by V = 4/3πa2c. The effective anisotropy was calculated as the sum of the magnetocrystalline anisotropy and shape anisotropy [7]. The values of table 6.1 were used in the following analysis and calculations. The mean radius of particle 1 is 6 nm. The patent of Massart [1] indicates that the mean radius should be around 5 nm. The mean radius of particle 2 is well below 50 nm, the value stated by Sigma Aldrich. The dimensions of particles 3 and 4 were in agreement with those published by Kumar and Koltypin et al. [2], and Matsui [3], respectively. For particle 5 length a was found to be 100 nm, twice the size as reported by Aldrich (2a < 100 nm).

34 Particle Composition c a α = c/a ρ MS Keff kg kA 3 (nm) (nm) ( m3 )( m ) (kJ/m ) 1 Fe3O4 6 6 1.00 5180 [6] 446 [6] 13.5 [7] 2 Fe3O4 16 16 1.00 5180 [6] 446 [6] 13.5 [7] 3 Fe3O4 24 7 3.43 5180 [6] 446 [6] 65.6 [7] 4 Fe3O4 225 15 15.0 5180 [6] 446 [6] 83.5 [7] 5 BaFe12O19 10 100 0.10 5278 [7] 380 [7] 258 [7] Table 6.1: Physical properties of sample particles: c is on half of the magnetic easy axis length, a is one half of the magnetic hard axis length, α is the aspect ratio, ρ is particle density, MS is the saturation magnetization of the material, and Keff is the effective particle anisotropy. Values of ρ and MS for magnetite are those reported by Rosensweig [6], those of barium ferrite are taken from Coey [7]. The effective particle anisotropy is the sum of the magnetocrystalline anisotropy and shape anisotropy taken from Coey [7]. Any particle dimension that could not be confirmed via SEM is denoted with an asterisk.

Length c was observed as 10 nm. No average thickness was given by Sigma Aldrich, but a

radius of an equivalent sphere was provided (RES = 25.4 nm). The observed mean radius of

2 (1/3) an equivalent sphere was RES = (a c) = 46.5 nm, which is roughly twice the expected value. The values of table 6.1 were used in the following analysis and calculations.

6.2 Inherent Transverse Relaxation of Agar Gels

The relaxation times of the pure agar phantoms are given in table 6.2. The transverse relaxation times were found by fitting the signal intensity decay for the defined regions of interest (ROIs) of each phantom. The results show that agar is a stable material with a well defined transverse relaxation time T2. Agar phantoms A-1 and A-2 were prepared to analyze the transverse relaxation rate of agar as a function of concentration. The relaxation rates

R2 for phantoms A-1 and A-2 are in agreement with other published results [4] as can be seen in ﬁgure 6.6. The line shown is that calculated by Davies et al., whose experimental work determined a linear dependence of agar R2 as a function of agar concentration [4]. The results of A-1 and A-2 match, giving conﬁdence to the R2 values of the agar gels used in our phantoms.

35 Figure 6.1: SEM image taken of particle type 1. It is made of magnetite, and has a spherical shape (mean radius ≈ 6 nm). This particle type was synthesized using the Massart Method [1].

36 Figure 6.2: SEM image taken of particle type 2. It is made of magnetite, and has a spherical shape (mean radius ≈ 16 nm). This particle type was purchased (Sigma Aldrich).

37 Figure 6.3: SEM image taken of particle type 3. It is made of magnetite, and has a prolate shape with half axes lengths a = 7 nm (magnetic hard axis) and c = 24 nm (magnetic easy axis). This particle type was synthesized using a method similar to that of Kumar and Koltypin et al. [2].

38 Figure 6.4: SEM image taken of particle type 4. It is made of magnetite, and has a prolate shape with half axes lengths a = 15 nm (magnetic hard axis) and c = 225 nm (magnetic easy axis). This particle type was synthesized using a method proposed by Matsui [3].

39 Figure 6.5: SEM image taken of particle type 5. It is made of barium ferrite, and has an oblate shape with half axes lengths a = 100 nm (magnetic hard axis) and c = 10 nm (magnetic easy axis). This particle type was purchased (Sigma Aldrich).

40 Agar phantoms A-3 and A-4 were made to compare the eﬀects of the gel setting under diﬀerent physical environments. The relaxation times of phantoms A-3 and A-4 are

in agreement with that of phantom A-2, T2 = 41.8 ± 1.44 ms. The top layer of phantom A-3 was 3% agar allowed to evaporate on the hot plate for thirty minutes. This is approximately how long it takes to create an individual agar phantom, and was done to determine whether the T2 of each phantom layer containing contrast would vary due to evaporation. It was expected that the agar would be more concentrated if some of the water evaporated. There was a slight decrease in T2 indicating this does happen, but it did not produce a large change in the T2 time. The bottom layer of phantom A-4 was placed in a freezer. The freezer was pumping cold air onto the sample as it was allowed to set for ten minutes. Again there was a slight decrease in the T2 time, but not a signiﬁcant eﬀect.

Phantom Composition Solidifying T2 Enironment (ms) A-1 2% agar room temp. 61.1 ± 3.43 A-2 3% agar room temp. 41.8 ± 1.44 A-3 3% agar room temp. 41.4 ± 1.44 3% agar evaporated 40.2 ± 1.32 A-4 3% agar freezer 40.3 ± 1.17 3% agar room temp. 42.3 ± 1.32

Table 6.2: Observed relaxation rate of agar under various conditions.

6.3 Transverse Relaxation of Agar Containing Con-

trast Agents

The observed transverse relaxation times for each phantom layer ROI containing contrast agents are given in table 6.3. The results were similar for either surfactant. The results shown hereafter are those for particles coated with SDS. The relaxation times having a linear relationship (those of lower particle concentrations) were used to create the relaxation

41 Figure 6.6: Transverse relaxation rate of agar phantoms A-1 and A-2 versus the published results of Davies et al. for agar R2 versus concentration [4].

rate versus particle concentration plots in ﬁgure 6.7 in order to determine the relaxivity r2 of each type of particle using equation 4.8. The relaxivities r2 of the contrast agents are given in table 6.4. These results indicate that using the same amount (mass) of these magnetic materials the oblate barium ferrite spheroids have a greater eﬀect on the observed T2 than the spherical

or prolate magnetite nanoparticles. The transverse relaxivities r2 of the contrast agents coated with SDS or AOT both rank the contrast mediums used in the order of particle D, B, C, then A. The barium ferrite oblate ellipsoids (particle D) has the largest effect on the agar transverse relaxivity. The larger of the two spherical particles (particle B) also has a strong effect on the transverse signal intensity, but it is slightly less than that of the barium ferrite platelets. The smaller spheres of magnetite (particle A) provided the lowest relaxivity of the the agents studied. The prolate form of magnetite (particle C) has a relaxivity that lies between the smaller and larger sphere effects.

42 Layer Coating Particle A [C] T2 Particle B [C] T2 Particle C [C] T2 Particle D [C] T2 mg/mL ms mg/mL ms mg/mL ms mg/mL ms Phantom 1 Phantom 2 Phantom 3 Phantom 4 1 SDS 0.00 0.00 0.00 0.00 2 SDS 0.388 7.700 0.315 3.445 0.355 3.805 0.339 4.174 3 SDS 0.129 10.65 0.105 9.031 0.118 5.278 0.113 6.916 4 SDS 0.0430 17.16 0.0352 10.55 0.0394 11.45 0.0375 8.712 5 SDS 0.0144 23.69 0.0117 22.01 0.0131 17.52 0.0125 18.93 6 SDS 0.00479 34.19 0.00391 38.18 0.00436 35.85 0.00418 24.46 7 SDS 0.00160 35.16 0.00130 42.94 0.00146 38.70 0.00139 37.29 8 SDS 0.000534 32.54 0.000434 51.90 0.000485 48.95 0.000464 48.33 9 SDS 0.00 0.00 0.00 0.00 Phantom 5 Phantom 6 Phantom 7 Phantom 8 1 AOT 0.00 0.00 0.00 0.00 2 AOT 0.315 6.051 0.318 4.669 0.357 3.560 0.309 4.173 3 AOT 0.105 9.208 0.106 9.062 0.119 4.161 0.103 5.667 4 AOT 0.0350 20.51 0.0354 13.49 0.0397 9.865 0.0343 8.523 5 AOT 0.0116 24.62 0.0118 21.21 0.0133 18.75 0.0115 14.01 6 AOT 0.00389 30.95 0.00393 36.73 0.00442 26.06 0.00381 33.27 7 AOT 0.00130 32.32 0.00131 45.70 0.00147 38.45 0.00126 41.92 8 AOT 0.000434 34.73 0.000436 52.81 0.000491 49.70 0.000423 52.09 9 AOT 0.00 0.00 0.00 0.00

Table 6.3: The transverse relaxation time T2 of samples containing particles A, B, C, and D in varying concentrations, that are separated into layers. Two surfactants were used to coat the particles: SDS and AOT.

Figure 6.7: Transverse relaxation rates as a function of particle concentration. The slopes r2 are given in table 6.4.

43 Particle Axis r2 mL major/minor ( mg·s ) A 6 nm/6 nm 705.7 ± 80.67 B 16 nm/16 nm 2157 ± 46.82 C 24 nm/7 nm 1696 ± 226.2 D 10 nm/100 nm 2425 ± 153.8

Table 6.4: Transverse relaxavities of particles.

6.4 Shape and Size Eﬀects on Transverse Relaxivity

The four particles used as contrast agents have different shapes and volumes. Plots of the relaxivities as a function of particle shape or size independently produced no noticeable relationship. To account for both properties simultaneously, the relaxivities were plotted as a function of particle surface area (a particle parameter dependent upon shape) divided by volume (a particle parameter dependent upon size). The results of this are shown in figure 6.8. A strong linear relationship between the particle parameters and transverse relaxivity is apparent in the data. For both sets of data the transverse relaxivity increases as the ratio of surface area to volume decreases. For a particle of constant volume decreasing the surface area maximizes the relaxivity. Theoretical work on spherical particle contrast agents indicates that increasing the size of a spherical particle contrast agent above a certain radius should decrease the relaxation rate of the medium about the particle when using a CPMG spin echo imaging technique [11]. This is because the effects of diffusion about the particle should be reduced for larger particles. The water protons would not traverse large enough lengths to dephase due to the particle field gradient. The maximum relaxation rate for magnetite occurs at the critical radius Rc = 19 nm according to Yung [11]. The surface area to volume ratio for a sphere

8 with radius Rc is SA/V = 1.58 · 10 1/m. This ratio lies between those for samples d and b in ﬁgure 6.8 (dashed line). Yung has ﬁt others’ data in his paper [11] showing r2 is reduced for spherical particles with a radius R > Rc for spin echo imaging.

44 Figure 6.8: Particle parameter eﬀects on relaxivity r2.

The oblate barium ferrite particle has a SA/V ratio smaller than that of a sphere with radius Rc and a much larger volume, yet has a stronger influence on the relaxation rate than spheres with a mean radius R = 16 nm close to the critical radius for magnetite (particle B). The spherical model of contrast agents cannot explain why the larger oblate spheroid has a significant effect on the transverse relaxation rate of agar where spheres of this size should not produce such large effects. There is a definite association between particle shape and size affecting transverse relaxivity. The strong linear relationship in the data of figure 6.8 shows this dependence. The oblate barium ferrite platelet has a SA/V equivalent to a sphere with radius R = 20 nm, but a volume 12.5x larger. The prolate magnetite spheroid has a SA/V equivalent to a sphere with radius R = 8.62 nm, but a volume 1.84x larger. The fact that the prolate and oblate spheroid samples follow the same trend as the spherical samples validated by Yung’s model is a significant result that supports the idea that shape is an important parameter to consider for contrast agents. It also indicates that Yung’s model is valid for spheroids in general, and not just spheres. These results suggest larger non-spherical particles can be

45 formed to produce strong relaxation effects that spherical particles of equivalent size cannot produce using spin echo imaging sequences. Gillis stated the general trend for spherical contrast agents is that given the same amount (mass) of magnetic material a smaller quantity of larger particles provides a greater effect on contrast than an abundance of smaller particles [29]. He attributes this to fewer larger particles being more dispersed within the containing medium so that their field gradients do not overlap. This decreases the probability that water protons can diffuse between particles, and thus move back into a region of similar field strength which negates the diffu- sional dephasing. In this view it is better to have fewer larger particles with field gradients that envelope a larger volume provided that the field strength still reduces appreciably over small distances about the particle so that the water protons appreciably dephase due to diffusion. Non-spherical particles have field gradients that extend farther out in the directions parallel to the particle magnetization and are more closely condensed about the particle in the perpendicular directions compared to a spherical particle of similar volume [30, 31]. This change in field gradient is causing a desirable effect (for the purposes of MRI using CPMG spin echo techniques) on water protons diffusing near these types of particles. They are increasing the volume in which particle diffusion is causing irreversible dephasing in CPMG spin echo MRI. It is likely that there are not individual particles suspended in the agar phantoms used for this research, but that there are dispersions of multiple particle aggregates within the agar. The particles used are ferromagnetic and were injected using a pipette, causing them to come in close proximity to one another as they were injected. This may have some influence on the research results. Further research should be conducted to determine the effects of particle aggregations. However, the strong linear relationship of the ratio of SA/V indicates that the particle shape is influencing the relaxation rate of the agar gel regardless

46 of particle aggregation.

47 Chapter 7

Conclusions

We have analyzed the effect of various shaped (spheroidal) contrast agents on the transverse relaxation rates of agar gels, and submitted the results for patenting. A linear dependence between the ratio of particle surface are to volume and the particle relaxivity was observed. The results indicate that particle shape influences the relaxation rates of water protons in agar gels, in addition to particle volume. Larger oblate barium ferrite spheroids provide equivalent if not better contrast enhancement than smaller spherical magnetite particles, and smaller prolate magnetite particles. The current theoretical models of transverse relaxation indicate that the oblate barium ferrite spheroid is causing a greater effect on the transverse relaxation rate than should be possible for a sphere of equivalent volume. A study of oblate spheroids with various sizes, but the same aspect ratio (and therefore the same SA/V ) should be completed. It is logical that there is still a volume dependence among these types of particles that must be known in order to further examine the effect of non-spherical contrast agents on transverse relaxivity in the spin echo case. Similar tests should be conducted on prolate spheroids as well, and non-spheroidal geometries. Research should also analyze the effects of particle aggregation. Larger multiple- particle structures should produce varying field gradients that can affect the water protons

48 diffusing about them. It may be beneficial to create larger phantoms in order to determine these effects, or develop a better phantom preparation technique that more thoroughly agitates the agar and particles as the agar sets. Additionally, creating larger phantoms with a single concentration of magnetic material dispersed within it would provide a larger region of interest to analyze. This would improve the accuracy of the results found in this study.

49 Part II

Magnetic Nanoparticle Shape and Size Eﬀects on Power Absorption from an Alternating Magnetic Field

50 Chapter 8

Magnetic Particle Hyperthermia

“Those diseases which medicines do not cure, iron cures (the knife); those which iron cannot cure, ﬁre cures; and those which ﬁre cannot cure, are to be reckoned wholly incurable.” - Hippocrates

Hyperthermia is a condition of elevated body temperature. Biological cells are responsive to changes in temperature. They react to cope with the strain of deviations from their ambient temperature, but if the they sustain a high elevated temperature for a prolonged period of time the cells become impaired and are unable to survive. It has been shown that cancer tumors are more susceptible to local temperature fluctuations than healthy tissues of the body [14]. Elevating the temperature of a tumor above T = 44 (◦C) for a period of one hour effectively kills tumor cells while leaving the surrounding healthy tissue unphased, or minimally injured [14, 49]. This corresponds to a temperature difference of ∆T = 7 (◦C). Hyperthermia is an appealing form of therapy for treating deep-seated tumors, where the risks and impacts of surgery become large and other therapies (radiation and chemotherapy) have large impacts on both healthy and cancerous cells. Radiation therapy is the use

51 of external high-energy electromagnetic waves or internal high-energy electromagnetic waves emitted from ionized particles to damage the DNA of cells within their field, which ter- minates the entire cell [50]. This method destroys both healthy and cancerous cells in the targeted area of the patient. Localization and beam focusing are the only methods to control the effects of this therapy. Chemotherapy is the use of chemical compounds that are detrimental to rapidly dividing cells [50]. While chemotherapy is more selective than radiation therapy there are many healthy cells that undergo division more rapidly than others (blood, hair follicle, and digestive tract lining cells) that are harmed in this treatment [16, 50]. Hyperthermia has the potential to be a true cell-specific therapy since tumor cells are more sensitive to temperature differences. In order for this to be realized it must be possible to control the locality of the temperature difference and the magnitude of the temperature difference. Depending on the sustained temperature difference from ambient temperature cells can be eliminated with or without rupturing them. These two types of cell death are often referred to as apoptosis and necrosis, but necrosis is really an aftereffect of ruptured cells and not the actual method by which they die [51]. When a cell ruptures its materials can be cytotoxic to cells around it causing a chain of cell deaths. It also causes immune system responses, unlike apoptosis, such as inflammation and angiogenesis [18] to stop the chain of tissue cell death. Some works distinguish heating methods as two separate treatments depending on how the cells are destroyed: hyperthermia is considered the practice of inducing apoptosis, and tumor ablation the practice of inducing cell rupturing. However, both treatments are often labeled as hyperthermia in many published works [14, 27, 52, 53, 54, 55]. I will refer to all heat therapy as hyperthermia in this work. Currently, hyperthermia is often used in tandem with radiation or chemotherapy because straining tumor cells with heat enhances the effectiveness of the other therapies [50]. However, modifying hyperthermia to be a standalone tumor therapy would be a major advancement in the fight against cancer and limit the impacts of tumor therapy on patients.

52 One promising non-invasive hyperthermia therapy is magnetic particle hyperthermia (MPH). MPH therapy is the use of oscillating applied magnetic fields to alter the magnetizations of small magnetic nanoparticles (particle dimension are typically on the nanometer scale) internally, which causes the particles to absorb energy from the field and then change this acquired energy into heat, which elevates the temperature of the particles [14, 15]. For therapy purposes, the magnetic particles can be localized in a tumor region via local injections, or non-local injections if chemical compounds that interact more regularly with cancer cells are placed on the surface of the particles so that they tend to attach or infiltrate cancer cells [14, 56, 57] . The ability to locate particles only near cancer cells and control the temperature difference via the applied field and particle temperature allows MPH to have significant impact on tumor cells and little impact on the surrounding healthy cells with out the use of surgery, anticancer chemical compounds, or radiation. Researchers are investigating the usefulness of MPH by studying the ability of external oscillating applied magnetic fields to influence the internal magnetization of magnetic particles as well as their physical movements when suspended in viscous mediums theoretically and experimentally [6, 15, 27, 53, 54, 55, 58, 59]. The current standard of measurement (efficiency) to rank and classify particles of various magnetic materials is to compare the theoretically predicted maximum amounts of power absorbed per mass of magnetic material (from the applied field) in the sample to the power per mass of magnetic material calculated from the calorimetric temperature difference of experimental heating trials on concentrations of magnetic particles in aqueous suspensions. Both of these power to mass ratios are clas- sified in the literature as specific absorption rates (SARs), although the one derived from the calorimetric temperature differences is the difference of the input power per mass of the particles to the output power per mass of the magnetic material. These two power ratios (absorption and calorimetry) have been assumed to have no dependence upon particle shape, but to have a dependence upon particle volume since mass and volume are related by the

53 density of the magnetic material. It has also been assumed that the observed calorimetric power output ratio of particles is the net sum of non-interacting individual particle contributions, and not dependent upon multi-particle structures that arise in these types of physical systems. This research examines the possibility that particle shape in addition to size has an influence on the ability of a magnetic material to absorb energy from an applied field, and the resulting temperature difference of the material. To test the importance of particle shape I use the available theoretical models with prolate (football shaped) and oblate (flat disc shaped) spheroidal particles, in addition to spheres. I also conduct experimental calorimetric heating trials on spheroidal particles I was able to fabricate or obtain from chemical suppliers. For the experimental work I constructed a high power solenoid electrical circuit that uses a large alternating current. To create a low stable solenoid chamber (the interior volume of the solenoid) temperature I fabricated a solenoid cooling system in order to actively cool the solenoid coils, which generate large amounts of heat due to passing large currents through the long narrow wire comprising the solenoid. This was necessary in order to monitor time dependent temperature differences of aqueous based samples of the various magnetic particles available occuring due to the applied field generated by the solenoid. An additional consideration in this research is the supposed patient-safety limit on the sinusoidal applied magnetic field magnitude H0 and frequency f used for MPH. The

8 stated safety limit is H0 · f = 4.85 × 10 (A/m·s) [60, 61]. If this limit is valid, it limits the applied ﬁelds applicable for MPH, and thus limits the ability to generate large temperature diﬀerences in short amounts of time using magnetic particles. I account for this in my research, but discuss the possibility of MPH in the absence of this limit as well. I used

8 a near maximum valid applied ﬁeld with H0 · f = 4.5 × 10 (A/m·s) for my theoretical and experimental work, and discuss the implications of this limit on the heating ability of magnetic particles in general.

54 I find that particle shape in addition to particle volume has an impact on the ability of a given magnetic material to absorb and emit power, which impacts the temperature difference of the magnetic material. Non-spherical shape reduces the ability of ferromagnetic material with cubic crystal structure to absorb power from an applied field, which limits the temperature difference of the material. The results indicate that prolate forms of magnetite absorb power from the applied field, which is not theoretically predicted. They also indicate that the heating rate and power absorption of magnetic particles in an aqueous suspensions fluctuate, which may be due to particle interactions, particle movement, or both. The following discussion begins with a historical overview of hyperthermia. This is followed by the theoretical principles applicable to MPH research, and the available models and methods used in MPH. Next, a description of the materials and experimental methods used in this research are given. Descriptions of the theoretical models considered are presented in the materials and methods section as well. The results are presented and discussed hereafter. A brief talk of future works is then provided before the conclusions.

55 Chapter 9

Historical Overview

The use of heat as a treatment for health related issues is long and extensive. The ancient Romans, Greeks, and Egyptians all used heat on protruding masses of the body [14]. Ayurvedic physicians in India practiced hyperthermia as far back as 3000 BC. In that era heated stones and water bladders were used to locally heat breast masses and swollen livers. Whole body heat therapy was conducted by sprinkling water over heated rocks in a small room. Modern day heating pads and saunas are remanences of these ancient practices. In 1868 a German, W. Busch, found that the induced fever from erysipelas bacterial infection caused cancer tumor regression [62]. This discovery was based on the observations of a patient having a soft tissue sarcoma located on the neck that fell ill with a high fever. Years later an American, William Coley, also observed a regression in a soft tissue sarcoma in a patient stricken with erysipelas [63]. Because of this ﬁnding Coley began injecting strepto- cocci strands at tumor sites in patients and studying the eﬀects. From this he concluded that serratia marcenscens along with streptococcus enhanced tumor regression even via non-local injections. This combination was named “Coley’s toxins [64].” It was given intravenously until a high fever (40 ◦C) was induced in the patient. The fever was maintained for weeks or months as a whole body hyperthermia therapy.

56 From 1893 to 1962 Coley’s toxins was administered to patients outside of clinical procedures in the United States of America. The Food Drug Administration of the United States of America deemed Coley’s concoction as a new drug in 1962, limiting it only to clinical trials. The FDA raised standards for drug therapy across the board after thalidomide, a common nausea drug, was found to cause major birth defects and neuropathy in a large volume of patients. Because of this chemotherapy and radiation became the dominant cancer therapies assisted in some cases by non-bacterial hyperthermia therapy. The first symposium pertaining to hyperthermia cancer therapy took place in 1975 and was held in Washington, DC [65]. By this time the use of electromagnetic waves was realized as a potential candidate for hyperthermia. What was lacking at this time was a well defined clinical trial using solely hyperthermia as the cancer therapy. There was also a lack of knowledge about the different effects of whole body hyperthermia (WBH) and local hyperthermia on the patient. WBH methods such as thermal chambers, electromagnetic waves, contact heating with pads or water baths, infrared radiation, and extracorporeal heating of the blood were practiced in clinical trials. They produced major fatal side effects in some patients. These side effects included liver failure and necrosis, cardiovascular stress and fatigue, and pulmonary edema [66]. All of these side effects were the result of heating the entire body rather than selectively targeting only the tumorous region. Local hyperthermia was completed using electric probes and heat pump probes to show tumor regression, but required surgery. Focused ultrasound and near infrared has also been used, but monitoring thermal dose and the heating region is delicate and complicated. Today, the use of magnetic nanoparticles as local heating agents is a realization. They can be administered via local or non-local injections. Non-local injections require attaching cancer-specific surface chemistry agents to the particle. The presence of the particles at the tumor is verified using magnetic resonance imaging and the field strengths and frequencies used to heat the particles can be limited to not adversely affect other parts of the patient.

57 Heat is generated as the particles absorb energy from the oscillating magnetic field as it switches back and forth rapidly, and through brownian motions if the particles vibrate or rotate in the magnetic field. The first feasibility study of cancer thermotherapy using magnetic nanoparticles took place between March 2003 and July 2004. The first nanotherapy center was established at CharitéUniversity Hospital in Berlin, Germany. Their patented therapy, MagForce Nanotherapy, is the first commercially accepted nano-cancer-therapy in the EU. It was accepted in October of 2011.

58 Chapter 10

Power Absorption of Ferromagnetic Particles From an Oscillating Applied Field

In MPH research single domain particles of ferromagnetic material are placed in a viscous liquid medium and subjected to a large oscillating applied field. Single domain particles of ferromagnetic material are used because these types of particles have large anisotropic magnetizations MS, which are the saturation magnetization of the particle material, and require large energies to flip the particle magnetization along its easy axis [7, 15, 67]. The preferred orientation of the particle magnetization is called the magnetic easy axis or simply the easy axis. The easy axis of ferromagnetic particles is strongly dependent upon the crystal structure of the material and the shape of the particles made of the material [7]. The easy axis of prolate particles is the major axis of the particle (the longer axis) because this minimizes the demag field created by surface charges (the longer distance separating opposite charges weakens the opposing internal field). For very small particles crystal stress due to surface charges on the particle become important to the particle magnetization [68].

59 Figure 10.1: Schematic of a prolate particle with with half axes lengths c and a and aspect ratio α = c/a suspended in a viscous medium with viscosity η whose magnetization M makes an angle θ with respect to the particle easy axis (a), and particle easy axis makes an angle φ with respect to the applied ﬁeld direction.

There are two ways a particle magnetization can align with the applied field in a viscous liquid: the particle magnetization can rotate with respect to the easy axis, or the particle can physically rotate, which changes the angle the particle easy axis makes with the applied field. In this work I denote θ as the angle the particle magnetization makes with respect to the easy axis, and φ as the angle the particle easy axis makes with the applied field. A schematic of a prolate particle with half axes lengths c and a and aspect ratio α = c/a whose easy axis is a, which is suspended in a viscous medium with viscosity η, is given in figure 10.1. To achieve the main goal of MPH research (large temperature differences that occur in moderate time frames using minimal amounts of magnetic material) requires finding fer-

60 romagnetic particles whose magnetization responses to an oscillating applied field (∆θ and ∆φ) cause the particle material to absorb the maximum amount of energy possible from one oscillation of the applied field. This type of energy absorption is magnetic hysteresis. The magnitude of magnetic hysteresis is found by plotting the time dependent particle magnetization component in the applied field direction Mk(t) as a function of applied field strength

µ0H(t) in the (µ0H, Mk) plane for one period of the applied field. A schematic of a typical hysteresis loop for coherent magnetization reversal in a single domain ferromagnetic particle whose easy axis makes some small angle φ with respect to the applied field direction is provided in figure 10.2. The loop is characterized by the particle remanent magnetization

MR (the aligned component of the particle material saturation magnetization at zero applied

field strength), and the coercive field µ0HC (the applied field strength required to remove the remanent magnetization of the particle). The enclosed area of the loop is

A = 4µ0MRHC . (10.1)

The total area A enclosed by the magnetization response to one oscillation of the applied field is the magnetic hysteresis, which is the energy absorbed per volume of magnetic material and has units (J/m3). This energy absorption density by the particle material occurs at a rate of the applied field frequency f, which makes the power absorption density of the particle P = Af (W/m3). The particle power absorption rate is often changed into the particle specific absorption rate SARP (power/mass of magnetic material) in MPH research,

P Af SAR = = , (10.2) P ρ ρ where ρ is the density of the particle material ρ, and the units (W/g) are commonly used [6, 15].

61 Figure 10.2: Schematic of a typical hysteresis loop for coherent magnetization reversal in a single domain ferromagnetic particle whose easy axis makes some small angle φ with respect to the applied field direction. The loop is characterized by the particle remanent magnetization MR (the aligned component of the particle material saturation magnetization at zero applied field strength), and the coercive field µ0HC (the applied field strength required to remove the remanent magnetization of the particle). The enclosed area of the loop is A = 4µ0MRHC .

62 10.1 Theoretical Methods of Calculating The Particle

Speciﬁc Absorption Rate

Three theoretical methods that assume coherent magnetization rotation are used in MPH research to calculate the magnetic hysteresis (equation 10.1) of non-interacting ferromagnetic particles in a viscous medium, which is then used to calculate the particle power absorption rate (equation 10.2). Coherently means the particle magnetization changes orientation like a single dipole: a constant magnitude is maintained, but the orientation angle changes. The first available method, and most difficult and time consuming, is to do computer calculations of θ(t) and φ(t) from the particle energy using a time varying applied field

µ0H(t)[69]. This is challenging because φ(t) depends on the viscosity and temperature of the medium used, and no analytical solution for time exists if µ0H is time dependent. I did not do this in my research. The particle energy used to deﬁne the equations of particle motion and particle magnetization orientation necessary to calculate particle power absorption are discussed below.

However, there is an analytical solution for time, θ, and φ if µ0H is constant, but is not large enough to cause magnetization state switching across the easy axis. The orientation times of a magnetic particle in a viscous medium in a static external field has been calculated by Newman and Yarbrough [70] and Lissberger and Comstock [5] for field strengths that cause magnetization state switching across the easy axis and those that do not, respectively. In this work I use the method of Lissberger and Comstock to find an upper limit of the orientation time for spheroidal magnetic particles making various initial angles with respect to a static applied field. The equations of motion and time for prolate spheroids derived by Lissberger and Comstock are discussed along with the particle energy. The second method used to calculate the particle power absorption is the Stoner Wohlfarth Theory [67], which determines the hysteresis loop areas of systems of fixed particles

63 due to an oscillating applied field from the particle internal energy. The analytical solutions predicted by this theory are given below. Although this method is specific to fixed particles, it is applicable to our system if particle rotations (changes to φ) are slow compared to changes in the applied field. The third method of calculating particle power absorption is the Linear Response Theory proposed by Rosensweig [6]. This theory assumes the particles are non-interacting and suspended in a viscous medium, and neglects the particle anisotropy energy term (defined below) of the particle internal energy. To achieve its analytical power absorption density equation the LRT also assumes the susceptibility of the particle system is a constant, meaning the system magnetization is a linear function of the the applied field and is zero in the absence of an applied field [6]. This assumption restricts the use of LRT to particles with rapidly fluctuating magnetizations on the time scale of changes to the applied field. This type of particle behavior is defined as superparamagnetic, and is only applicable to very small particles which have small anisotropy energies. The LRT analytical power absorption equation is given after the Stoner Wohlfarth Theory.

10.1.1 Power Absorption and Particle Orientation Calculations

From Particle Energy

To change the magnetization orientation of a ferromagnetic particle away from the particle easy axis (θ) requires a certain amount of energy that is dependent upon the volume of the particle V , the crystal structure anisotropy KC , the shape anisotropy KSH , and the stress anisotropy KST [7]. The anisotropy constants are the energy densities (energy per particle volume) required to change the magnetization orientation with respect to the particle easy axis for each type of magnetization anisotropy. It is typically assumed spheroidal particles are defined by KC and KSH , and the effective particle anisotropy is taken to be KEff = KC +KSH

64 [5, 15, 70]. There is debate about whether these anisotropies are truly additive since they are direction-speciﬁc, but the particle shape anisotropy grows large rapidly as the aspect ratio of the particle is increased or decreased from that of a sphere (α = 1), which has no shape anisotropy [7, 15]. Under this assumption the amount of energy required to overcome the particle magnetization anisotropy, called the particle anisotropy energy EA, is [67, 70]

2 EA(θ, φ) = KEﬀV sin (θ). (10.3)

It is a maximum for θ = 90◦ since increasing θ further causes the magnetization to fall back onto the particle easy axis in the opposite direction. In MPH an applied magnetic field is the energy source affecting the particle anisotropy energy (as well as thermal energy and the magnetization of other particles, which are ignored for now). The particle energy due to an applied magnetic field with strength µ0H acting on the particle magnetization MS is [7, 67, 70]

EZ (θ, φ) = −µ0MSVH cos(θ − φ), (10.4)

where µ0 is the permeability of free space (assumed to be the permeability of ferromagnetic material), V is the particle volume, θ is the angle between the particle easy axis and the particle magnetization, and φ is the particle easy axis orientation with respect to the applied field direction. The particle energy due to the applied field is a function of both θ and φ, and is a minimum when the particle magnetization and particle easy axis are aligned with the applied field (θ = φ = 0◦), and a maximum when the particle magnetization is anti-aligned with the applied field and the easy axis is still aligned (θ = 180◦, φ = 0◦). The particle magnetization will always chose a θ that minimizes the total particle energy for any φ and µ0H. The particle internal energy components considered in MPH theories that are dependent upon the magnetization orientation are EA and EZ [15, 67]. If a

65 spheroidal particle is free to rotate, then it will also have a kinetic energy term EKE, which Newman and Yarbrough take to be (neglecting any precession behavior) [70]

1 E = I φ˙2, (10.5) KE 2 pf

where Ipf is the moment of inertia of the particle and attached fluid. To use this term the motions of the particle must be slow enough to characterize the fluid motion around the particle as laminar flow (non-turbulent). The particle energy E(θ, φ) due to these components is 1 E(θ, φ) = I φ˙2 + K V sin2(θ) − µ M VH cos(θ − φ). (10.6) 2 pf Eff 0 S

The combinations of θ and φ that minimize E(θ, φ) are found by taking the ﬁrst and second partial derivatives of E(θ, φ) with respect to θ. In addition to the critical points of the particle energy, the ﬁrst partial derivative gives an equation of constraint on the allowed combinations of θ and φ,

KEﬀ sin(2θ) = −µ0MSH sin(θ − φ) (10.7)

The second partial derivative is used to determine whether the critical points of the first derivative of the particle energy are a maxima, minima, or inflection point of the particle energy. In the absence of an applied field there are two particle energy minima corresponding to the two possible magnetization orientations along the easy axis of the particle (θ = 0◦ or θ = 180◦) for any φ. As applied field strength is increased the energy minimum (energy well) of the anti-aligned particle magnetization orientation is reduced, and then removed at a specific field strength. At this field strength only one particle energy minimum exists, and it is assumed that the particle magnetization then switches states to that of the aligned

66 energy well minimum. To do computer calculations of particle magnetization response requires an equation of motion in addition to the equation of constraint. The equation of motion for a particle under the assumptions and omissions of Newman and Yarbrough NY (as well as Lissberger and Comstock) is [5, 70] µ M HV φ˙ = − 0 S sin(θ − φ), (10.8) 2kBT τB where kB is the Boltzmann’s constant, T is the temperature of the suspension medium, and

τB is the rotational Brownian relaxation time of the spheroidal particle, which is dependent upon the aspect ratio α of the particle and the suspension medium viscosity η and temperature. A discussion of τB is provided below.

To calculate θ(t) and φ(t) using a sinusoidal applied field µ0H(t) requires one to choose a varying time-stepping rate that corresponds to some incremental change of the applied field and determine the initial and final combinations of θ and φ for each step in applied field strength. For the initial time and field strength the initial θ0 is found using the equation of constraint at the fixed applied field strength and supplying an initial φ0. The change in the angles are then calculated using an algorithm such as the fourth-order Runge- Kutta method (used by Newman and Yarbrough). The final angles of the prior time step are then passed on to be the initial angles of the next time step for a new applied field strength value. This is done for one full oscillation of the applied field. After this the time dependent component of the particle magnetization aligned with the field is calculated at each time time step. This allows one to graphically map (µ0H,Mk) in order to determine MR and HC . This was beyond the scope of this work, and was not performed. However, the methods of Lissberger and Comstock were used to determine the easy axis rotations of spherical and prolate particles in a static weak field (one unable to flip the particle magnetization along the particle easy axis), which are presented in the results section. Oblate particles were not

67 modeled because at the time it was not realized that equation 10.8 is an equivalent form of the φ˙ equation presented in the Newman and Yarbrough and Lissberger and Comstock papers (which were presented as prolate models).

Brownian Relaxation Time τB

The Brownian relaxation time τB is the time it takes for an ensemble of like particles to lose their net magnetization by physically rotating due to thermal collisions if an aligning field is removed [68]. It is proportional the particle-fluid surface drag, making it a timescale for physical rotations of the particles in the system due to thermal collisions or the applied field. For spheroidal particle with half axes lengths a = b, c, and aspect ratio α = c/a, the Brownian relaxation time for end-over-end rotations (those causing magnetization rotation) in a viscous medium (ignoring precession behavior) is [71]

3ηVH τB = · FP , (10.9) kBT

where η is the ﬂuid viscosity, VH is the hydrodynamic volume of the particle (taken to be

2 VH = 4π/3(a c) = VP ), and FP is the Perrin shape factor (measure of increased particle-ﬂuid surface drag due to changing the aspect ratio of a spheroid). For prolates, with aspect ratio α > 1, (1 − α−4) F = 4 [aS(2 − α−2) − 2], (10.10) P 3α−2 where 2 S = (1 − α−2)−1/2 ln[(1 + (1 − α−2)1/2)α]. (10.11) a

For oblates, with aspect ratio α < 1,

(1 − α−4) F = 8 [α−2aS(2α−2 − 1) + 2], (10.12) P 3

68 where 2 S = (α−2 − 1)−1/2 tan−1[(α−2 − 1)1/2]. (10.13) a

The Perrin shape factor is FP = 1 for an aspect ratio α = 1. Koenig makes plots of these functions in his paper [71]. They show increasing (decreasing) the aspect ratio of prolates (oblates) increases the Brownian relaxation time.

10.1.2 Stoner Wohlfarth Theory

The Stoner Wohlfarth Theory is used to calculate the energy absorption density (magnetic hysteresis area A) of fixed orientation (φ constant) non-interacting single domain magnetic particles due to irreversible magnetization state switching along the particle easy axis using the particle internal energy (equation 10.6 with out EKE). The SWT assumes the particle magnetization reverses coherently. The original work of Stoner and Wohlfarth [67] modeled particle magnetization orientation with respect to an applied field for a fixed particle in the limit of T = 0 or f = ∞. They found, in this limit, particles having their easy axis aligned with the magnetic field (φ = 0) produce the largest hysteresis loop. The loop created for φ = 0 is the shape of a square, with area

A = 4µ0HC (φ = 0)MS = 4µ0HK MS = 8KEﬀ, (10.14)

for particles with a coercive ﬁeld equal to the particle anisotropy ﬁeld µ0HC = µ0HK =

2KEﬀ/MS, where MS is the particle saturation magnetization. In this unique case the particle

remanent magnetization is MR = MS. As φ is increased from zero, the hysteresis loop area A decreases, and is null for φ = 90◦. The decrease in A is do to reductions of the coercive ﬁeld strength, and the remanent magnetization. For an ensemble of randomly oriented uni-

axial ﬁxed particles, Stoner and Wohlfarth determined the average coercive ﬁeld is µ0HC (φ

random)= 0.48µ0HK , and the average remanent magnetization is MR(φ random)= 0.5MS.

69 The resulting hysteresis area for a random distribution of uni-axial particles is

A = 2µ0HC MS = 0.96µ0HK MS = 1.92KEﬀ. (10.15)

The Stoner Wohlfarth Theory (SWT) was expanded by Usov et al. [72] to include temperature T , applied field strength µ0H0 and frequency f, and particle volume V dependence for the coercive field through a dimensionless parameter κ. They found the coercive field for aligned systems of uni-axial particles (φ = 0) is

1/2 µ0HC = µ0HK (1 − κ ), (10.16) where k T k T κ = B ln B . (10.17) KEﬀV 4µ0H0MSV fτ0

−9 τ0 is a time constant taken to be τ0 = 1×10 (s) [6, 15]. It is explained within the discussion on the N´eelRelaxation time. For randomly oriented systems of uni-axial particles they found the coercive ﬁeld is

n µ0HC = 0.48µ0HK (b − κ ). (10.18)

Usov et al. determined the coefficients to be b = 0.9 and n = 1 using the assumption that the coercive field is always equal to the switching field. Carrey et al. found b = 1 and n = 0.8 by including deviations between the coercive and switching fields for particles at large angles with respect to the applied field [15]. Carrey et al. determined equations 10.16 and 10.18 are both valid for κ < 0.7 [15]. The method of Usov et al. is preferred to that of Garcia-Otero [73] because it relies on the physical parameters of the system alone, and not a measurement time that causes the coercive field to vary depending on modeling terms such as the time-step value [15]. The coercive field functions of Usov et al. [72] give major

70 hysteresis loops areas of

1/2 1/2 A = 4µ0HC MS = 4µ0HK (1 − κ )MS = 8KEﬀ(1 − κ ) (10.19) for aligned systems of uni-axial particles (φ = 0), and

n n A = 2µ0HC MS = 0.96µ0HK (b − κ )MS = 1.92KEﬀ(b − κ ) (10.20) for randomly oriented systems of uni-axial particles (φ random).

Time Constant τ0

The time constant τ0 is the inverse of the attempt frequency proposed by N´eel[74]. It is

−9 typically taken to be τ0 = 1 × 10 (s). This is the value used in all calculations for this research. In reality τ0 also inherently depends on the applied ﬁeld parameters, and the particle magnetization, which will change the particle energy [15, 75, 76, 77].

10.1.3 Linear Response Theory

The Linear Response Theory (LRT) is a method of calculating the particle power absorption for a system of non-interacting single domain particles suspended in a viscous medium that was proposed by Rosensweig [6]. The LRT gets its name from its assumption that the particle magnetization is “linearly” responsive to the applied field, implying a constant susceptibility (measure of magnetization response to applied field force) to an oscillating applied field. In this system the particles can physically rotate, and the magnetization can coherently rotate. Physical rotations may occur due to Brownian motion, which are the result of thermal collisions, or may occur due to magnetic torque. Rosensweig determined the power absorption per volume of magnetic material of a

71 system of particles with a constant susceptibility is

ωτ P = f∆U = πµ χ H2f , (10.21) 0 0 0 1 + (ωτ)2

H where ∆U = −µ0M dH = A is the energy per volume of magnetic material absorbed by the particles from the applied field in one period (1/f), χ0 is the static susceptibility, ω = 2πf is the angular frequency of the applied field, and τ is the effective particle relaxation time τ τ τ = B N , (10.22) τB + τN where τB is the Brownian relaxation time, and τN is the Néelrelaxation time. The Néel relaxation time is discussed below. For LRT, the static susceptibility χ0 is taken to be

φMS µ0MSH0VP kBT χ0 = coth − , (10.23) H0 kBT µ0MSH0VP

where φ is the ratio of the volume of particles VP to the volume of the particles-medium system VS = VP + VM (φ = VP /(VP + VM )).

N´eelRelaxation Time τN

The N´eelrelaxation time is the time it takes the particle magnetization to change direction along the easy axis of a particle due to thermal energy. It uses an average time constant τ0, which is the inverse of the attempt frequency of the particle magnetization to change states

[15, 74]. The N´eelrelaxation time weights this average τ0 as an exponential function of the ratio of the particle anisotropy energy and the system thermal energy. Rosensweig uses

√ KEﬀVP π e kB T τN = τ0 q , (10.24) 2 KEﬀVP kB T

72 −9 with τ0 = 1 × 10 (s). Other authors use the form [15, 55]

KEﬀVP k T τN = τ0e B , (10.25)

−9 with τ0 = 1 × 10 (s).

73 Chapter 11

Calorimetric Measurement of SAR

Many of the reported heating rates and saturation temperature differences that similar masses of various shaped and sized magnetic particles can instill into a surrounding viscous medium of constant mass are measured using calorimetry [15, 61]. Typically samples are prepared by placing a given mass of magnetic particles in a known mass of viscous liquid (water is typically used) and then sonicating (use ultrasonic vibrations to agitate) the solution [61]. This is done to disperse the particles within the liquid. Afterwards, the time dependent temperature of the sample is measured as the applied magnetic field acts on the system. The initial linear heating rate is used to calculate the power per mass of magnetic material [61, 78]. It is assumed that the heating rate given by this linear region of data is the net sum of non-interacting individual particle contributions within the system. Under this assumption, a mass of particles that cause a greater initial heating rate (higher SAR) are deemed more appropriate for MPH because the high constant heating rate implies each particle contributes a larger power contribution in the system, which should result in a higher overall temperature difference. This temperature difference should scale linearly with particle concentration.

74 The particle SAR calculated using the calorimetric temperature diﬀerence is [61, 78]

∆T mS SARPC = C , (11.1) ∆t mP where C is the mass weighted sample speciﬁc heat capacity (for constant pressure),

m C + m C C = P P W W , (11.2) mP + mW

mS is the total mass of the sample, mP is the mass of particles in the sample, CP is the

particle material speciﬁc heat capacity (for constant pressure), mW is the mass of water in

the sample (or other suspension ﬂuid), and CW is the speciﬁc heat capacity of water (for

constant pressure). I denote this power per mass ratio as SARPC to distinguish it from the

theoretical particle speciﬁc absorption rate SARP .

75 Chapter 12

Materials and Methods

12.1 Theoretical Models

12.1.1 Particle Rotation in a Viscous Medium

Lissberger and Comstock have solved the dynamic magnetization response to weak static applied fields (those that cannot flip the particle magnetization along the particle easy axis) of single domain uniaxial particles (of prolate and spherical shapes in their original model) suspended in a viscous medium when both particle and magnetization rotations can occur simultaneously [5]. Although this model is based on a static field, it allows us to determine if single domain particles can rotate on a macroscopic scale within the time that the applied field is in a single direction. It is assumed in this method that the effective anisotropy is the sum of magnetocrystalline and shape anisotropies. Lissberger and Comstock use a static magnetic field in the minusx ˆ direction. They choose θ as the angle between the particle magnetization and the plusx ˆ direction, and φ as the angle between the particle easy axis and the plusx ˆ direction. Using these angles for the particle energy they solved for the equation of constraint and the equation of motion and

76 found 1 φ = θ − sin−1(D sin θ), (12.1) 2

where µ M H D = 0 S , KEﬀ and θ " θ !# 1 tan 2 ∆t = ln D , (12.2) DE [tan(θ − φ)] 2 θ0 where K V E = Eﬀ . 2kBT τB

To determine φ as a function of time t requires inputing an initial magnetization angle

θ0, which is used to determine the initial particle easy axis angle φ0. The angle θ is then incremented by small steps ∆θ, followed by using equation 12.1 to determine φ, and then the angles θ and φ for each increment are used to calculate the time required for φ to respond to the change in θ using equation 12.2. The code used to model particle orientation as a function of time is provided in Appendix A.

12.2 Sample Preparations

12.2.1 Non-ferromagnetic Sample

A non-ferromagnetic sample was made to test the solenoid chamber temperature stability. The sample consisted of 1 mL distilled water placed in a 1.5 mL centrifuge tube (Fisher- brand). A small hole was bored into the top of the cap of the tube to allow the optical thermometer probe tip to be placed inside the sample container and suspended in the water.

77 12.2.2 Ferromagnetic Samples

Samples of aqueous suspensions of ferromagnetic material were prepared using the ﬁve available types of ferromagnetic particles (particles a, b, c, d, and e). Individual samples were made by placing 2 mg of each particle into a separate 1.5 mL centrifuge tubes and adding 1 mL of distilled water. This process was repeated using 20 mg of each particle. A sonicator (Branson 1510 Ultrasonic Cleaner) was used to disperse the magnetic material in the water prior to conducting heat trials.

12.3 Magnetic Particle Hyperthermia Testbed

Magnetic particle hyperthermia requires large oscillating magnetic fields to produce heat from magnetic nanoparticles. Solenoids are often used for these purposes because they produce a known uniform field strength within their chambers by passing an alternating current through them. To achieve high field strengths requires a combination of high current, and high winding density of solenoid coils. High winding densities require narrow diameter, high gauge wire. Decreasing the wire diameter increases the resistance of the wire, which typically generates large amounts of heat when a high current passes through the wire. This makes temperature measurements in the solenoid chamber difficult since it is necessary to isolate the heat generated by the particles from the heat generated by the solenoid. To maintain a stable chamber temperature requires a large open chamber for ventilation, or actively cooling the field generating solenoid. I have chosen the second option for this research because the necessary power and materials required to create a large open solenoid capable of high magnetic fields is not feasible.

78 (a)

Figure 12.1: High power magnetic field system. Subfigure (a) is the system wire circuit. Subfigure (b) is a schematic of the system.

12.3.1 Magnetic Field System

The AC magnetic heating testbed is an inductor-capacitor (LC) circuit. It is driven by a sinusoidal function generator (Simpson 420) driving a power amplifier (Hafler 9505) providing a maximum power of 750 W into a resistance of 8 Ω. Figure 12.1 is a model of the system. A bank of mica capacitors (Cornell/Dublier and Verovox) with high voltage/current ratings is used to tune the resonance frequency of the circuit by adding or subtracting capacitors connected in parallel. The output signal of the amplifier is monitored using an oscilloscope (Tektronix 2235). The inductor in the circuit is a single layer solenoid. It is 4.06 cm in length and has a diameter of 3.534 cm. The handmade solenoid was wound using high temperature insulated 31 gauge copper wire and modeled to match the 8 Ω impedance of the power amplifier.

79 There are 167 turns of wire comprising the solenoid. The solenoid is supported and cooled with a custom made water-jacket (figure 12.2) made of plexiglass. To model the inductor to match the load requirement of the power amplifier the resistance of solenoids made of various wire gauges were calculated based on the predetermined size of the solenoid chamber required to perform experiments. The dimensions chosen were a length of 4.060 cm and diameter of 3.536 cm. These dimensions were found to fit both the body of experimental mice and aqueous samples. The resistance was calculated as that of a long thin wire ρL R = (12.3) A

with resistivity ρ, length L, and cross-sectional area A. The resistivity used for the wire was that of copper (ρ = 16.8 x 109 Ω · m) at room temperature [79]. The length of wire was calculated for the radius of the solenoid and two 9 cm lead ends using

L = 2πrsolenoidN + 2 · Xleads (12.4) for N turns of wire. The number of turns comprising the solenoid was found by dividing the length of the solenoid by the diameter of wire to be used. Wire diameter was measured using a vernier caliper. The wire was assumed to be perfectly cylindrical, having a circular cross-sectional area A = πr2. The wire used was coated with a high temperature resistant material, making the radius used to calculate the number of turns and the radius used to calculate the cross-sectional area slightly diﬀerent. The radius used to calculate the cross- sectional area was that given by the American Wire Gauge chart for 31 gauge wire. Using both radii for their respective calculations it was found that insulated 31 gauge copper wire most closely matched the 8 Ω resistance required. The calculated resistance for this wire was Rcalc = 7.88 Ω. The number of turns of wire at this gauge was N = 167 to comprise a single layer solenoid 4.06 cm long with a diameter of 3.536 cm. The calculated inductance

80 Figure 12.2: High power magnetic ﬁeld system solenoid and water-jacket.

of the solenoid was found to be Lcalc = 0.591 mH using the formula for a short solenoid with one layer of windings [80] given by

µ L = f 0 AN 2, (12.5) l where l is the length of the solenoid, A is the cross-sectional area of the solenoid, and f is the form factor taken to be 1 f = r , (12.6) 1 + l where r is the radius of the solenoid.

12.3.2 Solenoid Cooling System

The solenoid cooling system is shown in ﬁgure 12.3. Cold water is continuously pumped through the solenoid water-jacket by a pump (Little Giant CP6500T) placed in a water- bath to dissipate the heat generated by the coils. The water-bath holds 10 L of water that is actively chilled using a cooling solenoid (FTS Systems LLC-40). An aquarium airpump (Optima) is used to agitate the water and prevent the cooling solenoid from icing over when

81 Figure 12.3: Cooling system for testbed solenoid. operating. The water pump (Little Giant CP6500T) circulates the water between the bath and the solenoid water-jacket through 0.375” inner diameter clear PVC tubing (Fisherbrand) at 6.57 L/min.

12.3.3 Magnetic Field Probe Calibration

An external magnetic field sensor (Magnetic Sciences MC162) connected to an oscilloscope was calibrated at low solenoid magnetic field strengths to monitor the solenoid field when operating the testbed at high power outputs. For this a high power variable resistor was set to a resistance R = 100 Ω and placed in series with the capacitor bank and solenoid to create a resistor-inductor-capacitor (RLC) circuit driven by a function generator running through a power amplifier. The input frequency signal was tuned to the resonance frequency of the circuit manually by monitoring the output voltage of the external field probe while sweeping the frequency back and forth using the function generator frequency dial for a constant amplitude sinusoidal signal. After this the output voltage of the external magnetic

82 field probe and the voltage drop across the resistor were measured for various input signal amplitudes. A 10x probe connected to the oscilloscope was used to monitor the voltage drop across the resistor. This process was repeated using a resistance R = 50 Ω. The current of the circuit for each data set was calculated from the resistance and voltage drop across the resistor. The magnetic field strength was calculated using the current and turn density of the solenoid. The calculated magnetic field strength was then plotted against the external magnetic field sensor readings to obtain a slope relating the probe voltage to the solenoid field strength. This slope was used to create a conversion chart relating the external magnetic field sensor reading to the internal field strength of the solenoid when operating the system at higher currents.

12.3.4 Testbed Characterization

The field strength of the testbed solenoid was characterized as a function of frequency. To accomplish this high voltage/current mica capacitors were used to vary the effective capacitance of the testbed inductor-capacitor (LC) circuit from 1 to 70 nF. The resonance frequency of the circuit was found manually by varying the frequency of the signal from the function generator for a constant amplitude input signal while monitoring the solenoid field strength using the external magnetic field probe . The maximum field strength for each capacitance value was recorded.

12.4 Heating Trials

12.4.1 Non-ferromagnetic Sample Heating Trials

Heating trials were conducted on a non-ferromagnetic sample to test chamber temperature stability. For these heating trials the applied ﬁeld and solenoid cooling system were turned

83 on and ran continuously for ninety minutes with the non-ferromagnetic sample present in the solenoid chamber. A styrofoam insert was used to hold the sample within the center of the chamber, and further isolate the sample from the heat of the solenoid. An optical thermometer (Luxtron 3100) probe was suspended in the sample water to monitor temperature changes of the sample, and an alcohol thermometer was used to monitor temperature changes of the water cooling the solenoid. Time and temperature data were recorded manually once every minute. The magnetic field strength was monitored using the field probe placed at the opening of the chamber and recording the observed signal magnitude on the oscilloscope. The probe output voltage was converted to magnetic field strength using table 13.1. The frequency of the signal was calculated from the magnetic probe signal. Data was collected for a forty minute period of time.

12.4.2 Ferromagnetic Sample Heating Trials

Calorimetric heating trials were performed on individual aqueous suspension samples of particles a, b, c, d, and e having concentrations of 2 (mg/mL) and 20 (mg/mL) that were held within the solenoid chamber using a styrofoam insert. To suspend the particles within the given sample to be used for the heat trial the sample was sonicated for 1 minute, and then to allow the temperature of the sample to reach a value near the solenoid chamber temperature a time tFS ≈ 90 (s) or tFS ≈ 180 (s) occurred prior to the beginning of the individual heat trial. However, in one case a sample containing 20 (mg/mL) of particle b was hand-shaken for one minute, and the heating trial began immediately following this agitation. The sample temperature during each heating trial was monitored with an optical thermometer. Time and temperature data were recorded manually. The magnetic field strength was monitored using the field probe placed at the opening of the chamber and recording the observed signal magnitude on the oscilloscope. The probe output voltage was converted to magnetic field strength using table 13.1. The frequency of the signal was

84 calculated from the magnetic probe signal on the oscilloscope.

85 Chapter 13

Experimental Testbed Characterization

13.1 Magnetic Field Probe Calibration

The magnetic field probe (Magnetic Science MC162) voltage signal was found to have a linear relationship to the field strength H0 calculated from voltage measurement drops across a resistor. The results are given in figure 13.1 for low field measurements across a 50 Ω and 100 Ω resistor. The data has a slope m = 5.832 (kA/(m·V)), and intercept y = 0 (kA/m). This slope was used to create a conversion table, table 13.1, which is used to determine the field strength of the solenoid when operating the magnetic field testbed.

13.2 Operating Range

The solenoid magnetic ﬁeld strength as a function of frequency is given in table 13.2. The largest ﬁeld strength obtainable by our system is H0 = 53.0 (kA/m) at a frequency f = 28 (kHz) for a solenoid with a turn density n = 4113 (turns/m). This corresponds to a current

86 Figure 13.1: Calculated solenoid magnetic ﬁeld strength H0 from voltage drop measurements across a resistor as a function of magnetic ﬁeld probe (MC162) voltage.

MC162 Probe Irms Hrms H0 (V) (A) (kA/m) (kA/m) 1 0.51 2.06 2.91 2 1.02 4.12 5.83 3 1.52 6.19 8.75 4 2.03 8.25 11.7 5 2.54 10.3 14.6 6 3.05 12.4 17.5 7 3.56 14.4 20.4 8 4.06 16.5 23.3 9 4.57 18.6 26.3 10 5.08 20.6 29.1 11 5.59 22.7 32.1 12 6.09 24.7 34.9 13 6.60 26.8 37.9 14 7.11 28.9 40.9 15 7.62 30.9 43.7 16 8.13 33.0 46.7 17 8.63 35.1 49.6 18 9.14 37.1 52.5 19 9.65 39.2 55.4 20 10.2 41.2 58.3

Table 13.1: Magnetic ﬁeld strength conversion chart for peak to peak voltage readings of the magnetic ﬁeld probe.

87 with Irms = 9.12 (A). The maximum output current of the amplifier, with a power rating of 750 (W), is Irms = 9.68 (A) for a solenoid with resistance R = 8 (Ω). While we can acheive 94% of our amplifier power at this frequency, the applied field strength is reduced as frequency is increased.

f Hrms H0 kA kA (kHz) ( m )( m ) 28 37.5 53.0 30 35.1 49.6 33 32.0 45.3 38.5 27.8 39.3 46.5 21.7 30.7 91 7.42 10.5 154 2.47 3.49

Table 13.2: Largest obtainable magnetic ﬁeld strength H0 as a function of frequency f for our laboratory system.

13.3 Solenoid Chamber Temperature Stability

The time dependent temperature response of a non-ferromagnetic water sample subjected to an applied field with field strength H0 = 45 (kA/m) and frequency f = 30 (kHz) within our solenoid chamber is shown in figure 13.2 along with the temperature of the water used to cool the solenoid coils. In this test the applied field and solenoid cooling mechanism were running continuously for ninety minutes prior to data collection. The temperature of the non-ferromagnetic water sample and the temperature of the water cooling the solenoid were stable over forty minutes of data collection. The temperature of the non-ferromagnetic water sample was T = 28.8 ± 0.1 (◦C) and the temperature of the water cooling the solenoid was T = 16.6 ± 0.1 (◦C). The laboratory system is acceptable for performing calorimetry heating trials with an applied field of H0 = 45 (kA/m) and frequency f = 30 (kHz), after the laboratory system stabilizes.

88 (a) (b)

Figure 13.2: Nonmagnetic sample heating trial.

In the results below I first present and discuss all of the experimental and theoretical data pertaining only to particles a, b, c, d, and e. I then present and discuss the theoretical results that apply to magnetic particles in general. From here I discuss what types of particles are useful for MPH. All particle models used to create figures and tables use the values provided in table 6.1, an applied field with H0 = 45 (kA/m) and f = 30 (kHz), a temperature T = 37 (◦C), and a fluid viscosity of η = 0.001 (Pa·s) (unless otherwise noted). The viscosity value used here is that typically used in other MPH related papers [15, 55]. Any other important parameters are given where needed.

89 Chapter 14

Results and Discussion

14.1 Theoretical Results for Laboratory Particles

Specific absorption rates (SARP , equation 10.2) and coercive fields (in units kA/m) of the laboratory particles predicted using SWT are provided in table 14.1 for an applied field with

H0 = 45 (kA/m) and f = 30 (kHz). To obtain these results the dimensionless parameter κ (equation 10.17) of Usov et al. [72], which accounts for applied field and system temperature effects on the coercive field, was plotted as a function of effective particle anisotropy KEff

2 1/3 and volume (denoted as the radius of an equivalent sphere RES = (a c) ) to determine the validity of the temperature and field dependent SW equations for coercivity and hysteresis losses, equations 10.16- 10.20, for particles a, b, c, d, and e. The plot of κ is shown in figure 14.1 for temperature T = 37 (◦C). The criterion for using the temperature and field dependent equations, as pointed out by Carrey et al. [15], is that κ < 0.7. Only particle a did not satisfy this condition. To calculate the aligned and random particle distribution SARP of particle a, equations 10.14 and 10.15 were substituted into equation 10.2, respectively.

For particle a, the aligned orientation coercive field is HC = 2KEff/µ0MS, and the random orientation coercive field is HC = 0.48(2KEff/µ0MS). These equations do not explicitly

90 depend on temperature or the magnetic ﬁeld. The asterisks next to the values of particle a in table 14.1 denote that a separate set of equations were used for the calculations compared to the other particles. Values followed by a star imply that the predicted coercivity is greater than that of the laboratory applied ﬁeld strength.

Particle Keff κ SARP (aligned) SARP (random) HC (aligned) HC (random) (kJ/m3) (W/g) (W/g) (kA/m) (kA/m) a 13.5 2.58 223* 150* 4.82* 2.31* b 13.5 8.16 e-2 447 130 34.4 20.0 c 65.4 7.52 e-2 2200? 635? 169 97.9 d 83.1 4.61 e-4 3770? 922? 290 142 e 258 5.45 e-5 11900? 2870? 915 442

Table 14.1: Specific absorption rates SARP and coercive field strengths HC of particles a, b, c, d, and e for an applied field with H0 = 45 (kA/m) and f = 30 (kHz). Temperature is T = 37 (◦C). For particles where κ < 0.7, equations 10.19 and 10.20 (using b = 1 and n = 0.8 for the random case) were used in equation 10.2 to calculate SARP for aligned and random distributions of particles, respectively. For these particles, equations 10.16 and 10.18 were used to calculate HC for aligned and random distributions of particles, respectively. For particles where κ > 0.7, equations 10.14 and 10.15 were used in equation 10.2 to calculate SARP for aligned and random distributions of particles, respectively. For these particles the aligned orientation coercive field is HC = 2KEff/µ0MS, and the random orientation coercive field is HC = 0.48(2KEff/µ0MS). The SARP and HC values for particles having κ > 0.7 are denoted with asterisks. Values followed by a star imply that the predicted coercivity is greater than that of the laboratory applied field strength.

Predicted values of SARP using the LRT, where power absorption density is given by equation 10.21, are shown in table 14.2 for an applied field with H0 = 45 (kA/m) and f = 30 (kHz). Temperature was taken to be T = 37 (◦C), and the viscosity of water η = 0.001 (Pa·s). This viscosity is slightly higher than that for room temperature water, and is typically used in MPH research [55]. Predicted initial heating rates are also provided in table 14.2, and were calculated by setting equations 10.21 and 11.1 equal and solving for the heating rate. The easy axis orientation φ response, predicted using the methods of Lissberger and Comstock [5], of particles a, b, c, and d suspended in water with viscosity η = 0.001 (Pa·s) due to a static applied field with H0 = 10 (kA/m) in an opposing direction are given in figure

91 Figure 14.1: Plot of κ, equation 10.17, as a function of eﬀective particle anisotropy Keff 2 1/3 and volume, denoted as the radius of an equivalent sphere RES = (a c) . An applied ﬁeld with H0 = 45 (kA/m) and frequency f = 30 (kHz) was used. Temperature is taken to be T = 37 (◦C). Equations 10.16- 10.20 are valid for κ < 0.7.

Particle Conc. dT/dt SARP (mg/mL) (K/s) (W/g) a 2 0.000311 0.651 b 2 0.0155 31.9 c 2 0.0174 36.4 d 2 0.0000584 0.122 e 2 0.000146 0.306 a 20 0.00315 0.660 b 20 0.1569 32.9 c 20 0.176 36.8 d 20 0.000591 0.124 e 20 0.00147 0.308

Table 14.2: Predicted speciﬁc power absorption rates using LRT, where power losses are given by equation 10.21, for particles a, b, c, d, and e. An applied ﬁeld with H0 = 45 (kA/m) and f = 30 (kHz) was used. Temperature was taken to be T = 37 (◦C), and the viscosity of water η = 0.001 (Pa·s). Initial heating rates were calculated by setting equations 10.21 and 11.1 (without the fraction mS/mP ) equal and solving for the heating rate.

92 14.2. Here, (a) depicts orientation time for spherical particles, and (b) and (c) are for prolate shapes having the aspect ratio of particles c and d, respectively. The method and equations of Lissberger and Comstock [5] were used to create a Matlab file, LCOrientation.m (provided in Appendix A), for the results. The Lissberger and Comstock model is only applicable to spheres and prolates. Particle e could not be modeled using their published methods. Particles a and b can align with the applied field used in the model within 8 µs from any initial orientation. Particle c can align with the static field within 16 µs from any initial orientation. Particle d aligns with the applied field from any initial orientation within 160 µs. The laboratory field has a magnitude over four times the magnitude of the static field used for the model. This should cause more rapid alignment of the particles. The laboratory field magnitude is also greater than the anisotropy field of particles a and b. These particles should align in an even shorter time due to their magnetization flipping along the particle easy axis, resulting in a smaller rotation angle required to align the particle easy axis with the laboratory field axis. Table 14.3 contains the relevant times of the experimental research: the time the

applied field is in a single direction direction τH , the Brownian rotation time τB of each particle, and the Néelrelaxation time τN of each particle. For calculations of τB and τN it is assumed fluid viscosity is η = 0.001 (Pa·s) and temperature is T = 37 (◦C).

Particle τH τB τN (µs) (µs) (µs) a 16.5 6.34 5.19 b 16.5 120 9.72 e23 c 16.5 95.5 1.23 e18 d 16.5 38400 > 9.72 e23 e 16.5 2348 > 9.72 e23

Table 14.3: Relevant times of the laboratory experiment: the time the applied field is in a single direction τH , the Brownian relaxation time τB of each particle, and the Néelrelaxation time τN for each particle. For calculations of τB and τN it is assumed fluid viscosity is η = 0.001 (Pa·s) and temperature is T = 37 (◦C).

93 (a)

(b)

(c)

Figure 14.2: Particle easy axis orientation φ versus time for particles a, b, c, and d suspended in water, η = 0.001 (Pa·s), and subjected to a static field in the minus x direction with H0 = 10 (kA/m). The method and equations of Lissberger and Comstock [5] were used for the results. Subfigure (a) depicts orientation time for spherical particles, which is applicable for particles a and b. Subfigure (b) and (c) are for prolates having the aspect ratio of particles c and d, respectively.

94 14.2 Experimental Results

The results (temperature diﬀerence ∆T on the left and calorimetric power absorption per

mass of magnetic material SARPC on the right) of the heating trials conducted on aqueous suspensions of magnetic nanoparticles a, b, c, d, and e in our laboratory system are given

in figures 14.3- 14.10. The applied field used had a strength H0 = 45 ± 2 (kA/m), and frequency f = 30 ± 1 (kHz). Figure 14.3 is a compilation of the individual heating results for 2 mg/mL samples of particles a, b, c, d, and e taken for heating trials starting tFS ≈ 180 (s) after sonicating the individual sample used for the heating trial. Figures 14.4 and 14.5 are compilations of the individual results for 20 mg/mL samples of particles a, b, c, d, and e for heating trials starting tFS ≈ 90 (s) and tFS ≈ 180 (s) after sonicating the individual sample used for the given heating trial, respectively. Figures 14.6- 14.10 are compilations of all individual heating trial results of samples containing particles a - e, respectively. In the figures the legend of each calorimetric power absorption graph also applies to the corresponding temperature difference graph. The shape and color family are held constant for each particle; using black squares to denote particle a, red spheres to denote particle b, blue upright triangles to denote particle c, downward-facing green triangles to denote particle d, and left-facing magenta triangles to denote particle e. The temperature difference values given in the figures are the observed temperature minus the initial temperature of the sample. This was done to show the temperature changes from a common reference temperature. The initial temperatures of the samples varied due to sonication prior to undergoing a heat trial. I tried to keep the deviations from solenoid chamber temperature minimal by starting the heating trial of each sample only when the sample temperature was near the ambient solenoid chamber temperature, making the initial

◦ temperature range Ti = 28 ∼ 33 ( C). This was done to minimize stray cooling and heating, which could alter the heating curves observed.

95 In the figures relating to only particle b, an additional heating trial is shown and denoted as “20 mg/mL, NS”. For this heating trial the sample was hand-shaken for one minute prior to the start of the heating trial. This was done to look at the effects of ferromagnetic particle agglomerates on the heating results of particle b, which is used to make a general conclusion about the effects of ferromagnetic particle agglomerates in general on the heating ability of single domain ferromagnetic particles. It is clear the experimental specific absorption rates of the particles match neither

the SWT predicted values nor the LRT predicted values. The experimental values SARPC fluctuate in value, and those causing large temperature differences are also fluctuating in value. Also, various overall temperature differences occurred for identical samples. This fluctuation of power absorption and overall temperature differences implies the particles are behaving dynamically in the sample. This immediately breaks the validity of the SWT, but not that of LRT. A number of theoretical simulations and experiments were done to try and explain this fluctuating power absorption behavior. First, the applied field and temperature dependent SWT was used to determine the effects of shape and size dispersions for fixed non-interacting particles. The coercive field strength HC and energy absorption density (A) of magnetite as functions of effective particle

anisotropy KEﬀ and particle volume, denoted as the radius of an equivalent sphere RES = (a2c)1/3, for aligned and randomly oriented particle systems using an applied ﬁeld with

H0 = 45 (kA/m) and f = 30 (kHz). These results are shown in ﬁgures 14.11 and 14.12, and are shown as colormaps corresponding to the magnitude of HC (A/m) and energy

3 absorption density A (J/m ). Any HC > 45 (kA/m) will remove the ability of irreversible magnetization orientation switching in the experimental data. Because of this only the coercive field strength data for HC values up to HC = 50 (kA/m) is shown in figure 14.11, and only the predicted energy absorption density A data for HC ≤ 45 (kA/m) is shown in figure 14.12.

96 (a) (b)

Figure 14.3: Time dependent temperature diﬀerence (a) and calorimetric heating (b) of particles a, b, c, d, and e using an applied ﬁeld strength H0 = 45 (kA/m) and frequency f = 30 (kHz). Each individual heat trial was initiated after a time tFS ≈ 180 (s) from when the sample being tested was removed from the sonicator.

(a) (b)

Figure 14.4: Time dependent temperature diﬀerence (a) and calorimetric heating (b) of particles a, b, c, d, and e using an applied ﬁeld strength H0 = 45 (kA/m) and frequency f = 30 (kHz). Each individual heat trial was initiated after a time tFS ≈ 90 (s) from when the sample being tested was removed from the sonicator.

97 (a) (b)

Figure 14.5: Time dependent temperature diﬀerence (a) and calorimetric heating (b) of particles a, b, c, d, and e using an applied ﬁeld strength H0 = 45 (kA/m) and frequency f = 30 (kHz). Each individual heat trial was initiated after a time tFS ≈ 180 (s) from when the sample being tested was removed from the sonicator.

(a) (b)

Figure 14.6: Time dependent temperature diﬀerence (a) and calorimetric heating (b) of samples containing particle a using an applied ﬁeld strength H0 = 45 (kA/m) and frequency f = 30 (kHz).

98 (a) (b)

Figure 14.7: Time dependent temperature diﬀerence (a) and calorimetric heating (b) of samples containing particle b using an applied ﬁeld strength H0 = 45 (kA/m) and frequency f = 30 (kHz).

(a) (b)

Figure 14.8: Time dependent temperature diﬀerence (a) and calorimetric heating (b) of samples containing particle c using an applied ﬁeld strength H0 = 45 (kA/m) and frequency f = 30 (kHz).

99 (a) (b)

Figure 14.9: Time dependent temperature diﬀerence (a) and calorimetric heating (b) of samples containing particle d using an applied ﬁeld strength H0 = 45 (kA/m) and frequency f = 30 (kHz).

(a) (b)

Figure 14.10: Time dependent temperature diﬀerence (a) and calorimetric heating (b) of samples containing particle e using an applied ﬁeld strength H0 = 45 (kA/m) and frequency f = 30 (kHz).

100 (a)

(b)

Figure 14.11: Coercive field strength HC (A/m) as a function of anisotropy and particle 2 1/3 volume, denoted as the radius of an equivalent sphere RES = (a c) , for aligned (equation 10.16) and randomly oriented particles (equation 10.18) using the temperature and frequency dependent SWT model. An applied field with H0 = 45 (kA/m) and f = 30 (kHz) was used for this data. Temperature is taken to be T = 37 (◦C). The white circles encapsulate the areas about the values taken to be those of particles a, b, c, and d. Particle e has an anisotropy over three times that of particle d. It is not shown in the figure.

101 (a)

(b)

Figure 14.12: Energy absorption density (hysteresis losses due to irreversible magnetization rotation along the particle easy axis) A (J/m3) as a function of effective particle anisotropy 2 1/3 Keff and particle volume, denoted as the radius of an equivalent sphere RES = (a c) , for aligned (equation 10.19) and randomly oriented particles (equation 10.20) using the temperature and frequency dependent SWT model. An applied field with H0 = 45 (kA/m) and f = 30 (kHz) was used for this data. Temperature is taken to be T = 37 (◦C). The white circles encapsulate the areas about the values taken to be those of particles a, b, c, and d. Particle e has an anisotropy over three times that of particle d. It is not shown in the figure.

102 The white circles within the figures of aligned (a) and random (b) particle distributions encapsulate the areas about the values taken to be those of particles a, b, c, and d. Particle e has an anisotropy over 3 times that of particle d. It is not shown in the figures. These results indicate only particle b should cause non-negligible heating due the applied field in the view of the applied field and temperature dependent SWT, and that shape and size effects could strengthen or weaken the performance of the particle. However, a value greater than that predicted for a random distribution for particle b was observed in the 2 mg/mL sample of particle b, suggesting the particles are aligning in the system. This alignment is then accompanied by some other phenomenon or phenomena occurring in the sample that causes the particles of the 2 mg/mL sample containing particle b to lose their ability to absorb power from the field at later times. While the SWT model predicts no heating for samples c, d, and e, heating is observed in all three samples. This may be due to either effects not considered in the model, such as particle interactions, clustering aggregation prior to being subjected (which was observed), and the formation of rope-like structures while subjected to the applied field (which was also observed), or to inaccurate assumptions, such as coherent magnetization reversal, static particles, or the use of a constant temperature. The fact that increasing sample concentration

reduces the observed SARPC indicates that either aggregation (of either kind), or particle interactions are affecting the ability of the material to absorb power. It is likely both effects influence the magnetic properties of the particles. Figure 14.13 gives the LRT predicted heating rates of spherical (a) and non-spherical (b) and (c) particles suspended in a viscous medium and the applied field of the laboratory

system. The speciﬁc absorption ratio (SARPC ) was set equal to the power absorption density (A · f) to determine the heating rates. These results indicate that particles with volumes greater than that of a sphere with radius R ≈ 12 (nm) have substantially reduced heating rates for any non-spherical shape. The LRT cannot explain the observed, non-negligible,

103 SARPC values of particles d and e. These results also show how sensitive the LRT model is to the viscosity of the medium. An order of magnitude change to the viscosity causes drastic changes to the model results. Particle a resides at a critical point in the LRT model. Size dispersions (larger particles) of this particle could appreciably heat according to the LRT, especially if the sample becomes more viscous. This may explain the observed heating of sample a. However, particle a has an anisotropy field that is easily overcome by the applied field, and it aggregates in the applied field. It should rapidly align with the applied field, and remain aligned. Only small deviations in the particle orientation should occur due to thermal energy because the applied field rapidly changes direction. This implies the useful time parameter of LRT is the Néel relaxation time. For particle a it happens that τB ≈ τN ≈ 1/ω, where ω is the angular frequency of the applied field, and this relationship provides a near maximum predicted power absorption in the LRT. This predicted maximum is not affected if the ability to rotate is lost by particle a, since the effective relaxation time would be heavily weighted by τN .

Particle c should perform well according to the LRT, and the initial SARPC value of the 2 mg/mL sample matches the predicted value. However, the power absorption of this sample is rapidly reduced over time, and remains low, such that only a small overall temperature difference is achieved. The higher concentration samples of particle c do cause an appreciable (useful) overall temperature difference, but they have a low power absorption rate that fluctuates. This indicates an initial high SARPC is not necessary to cause large temperature differences. The trend of reduced power absorption over time and reduced SAR at higher concentrations is also seen in the results of particles a and b, whose power absorption is due to magnetization rotations. This indicates particle rotation is not the primary cause of power absorption for sample c (or d and e). Internal magnetization state reversal must then be causing the observed hysteresis in particles c, d, and e, but the SWT fails to predict it.

104 (a) (b)

(c)

Figure 14.13: Heating rates predicted by the linear response theory proposed by Rosensweig [6] for magnetic particles subjected to an applied field with H0 = 45 (kA/m) and f = 30 (kHz). A temperature of T = 37 (◦C) was used in this data. Viscosity values of η = 0.001 and 0.01 (Pa·s) are used to determine LRT sensitivity to the particle environment. Sample concentration used is 20 (mg/mL), unless otherwise specified on a graph. In the figure (a) is the result of spherical SPM particles, and (b) and (c) are for prolate SPM particles in a fluid viscosity η = 0.001 and η = 0.01 (Pa·s), respectively.

105 A downfall of the SWT, in general, is that it assumes the applied field strength used is greater than the coercive field strength of the particles when φ = 0◦, which is the anisotropy field strength of the particle. This is a requisite for its random distribution prediction to be valid. It also assumes the particles have a single magnetic easy axis. It is clear that strongly anisotropic particles, such as particles c, d, and e, are exhibiting hysteresis due to magnetization state reversal along the particle easy axis in an applied field that is well below the anisotropy fields of the particles. The SWT fails to account for this hysteresis because it cannot predict the switching field strength of a uni-axial particle. It can only make use of the fact that if an aligned particle switches magnetization orientation due to an applied field, then non-aligned particles will also in the same applied field. Thus, the SWT predicted energy absorption density of random distributions of particles is only applicable for an applied field that can overcome the anisotropy energy of the particle in question when the particle easy axis is aligned. In reality, the switching field strength of non-aligned uni-axial particles is some value greater than the coercive field strength, but less than or equal to the particle anisotropy field strength for particles making some angle φ 6= 0◦. The switching field strength is a maximum for φ = 0◦ and φ = 90◦, and a minimum for φ = 45◦, where the anisotropy energy barrier separating the two possible magnetization orientation states of a particle with a single easy axis is reduced by 1/2. However, even this reduction of the particle energy barrier cannot explain the magnetization state switching occurring in particles c, d, and e by itself. The particle magnetization for these particles must be changing which crystal easy axis it is lying along in order to cause hysteresis. Magnetite has six easy axis planes for the magnetization to lie along. Barium ferrite is truly uni-axial. This is why particle e does not absorb power well in the laboratory applied field. This is also why particle b absorbs power well from the applied field, and particles a, c, and d perform moderately in the laboratory applied field.

106 In addition, all models neglect interaction between particles, which is however likely to be strong. Significant aggregation is observed in all of the samples. This interaction will affect both the local field experienced by particles as well as their ability to physically rotate, and so will likely have an effect on the measured specific absorption rate.

14.3 Theoretical Optimization of SAR and MPH

Magnetization rotation is the cause of power absorption in magnetic material that results in significant temperature differences in samples. Figure 14.12 shows that A is dependent upon both shape (Keff ) and volume (RES). The maximum A for an aligned distribution of FM magnetite particles for the applied field used is A = 100 (kJ/m3). This gives a maximum aligned SARP = 579 (W/g). The maximum A for a random distribution of FM

3 magnetite particles in this applied field is A = 50 (kJ/m ), with a corresponding SARP = 290 (W/g). These are the largest power losses our laboratory applied field can induce, and the largest power losses we can use if there is truly a safety patient threshold applied field of H · f = 4.85 × 108 (A/m·s). Since I have had my hands in and out of this field numerous times and felt no pain, I am not sure of the validity of this limit. While these results are for a specific applied field, our laboratory applied field (the supposed maximum applied field), the general shape of the data is applicable to any applied field. Because of this it is possible to obtain the same level of power absorption for a magnetic material using prolate and oblate shaped particles, which have larger effective anisotropies than spheres of a similar volume. Thus volume and shape are two parameters that can be controlled to alter the anisotropy of a particle to maximize power absorption of a magnetic material in an applied field. This has two impacts in terms of magnetic particle hyperthermia. First, smaller prolate and oblate magnetic particles can be used to achieve the same level of power absorption

107 as that of larger spheres. Decreasing the overall size of particles makes it easier to localize particles within a tumor because they can more easily pass through all the boundaries of the cellular environment. Second, because prolate and oblate particles have more surface area than equivalent volume spheres their rate of heating and heat transfer will be different. A larger surface area will result in better heat transfer, so for a given SAR, the tissue surrounding a nanoparticle will initially increase more rapidly in temperature. A smaller surface area will mean less effective heat transfer, so the surroundings will have a lower temperature difference, but the particle itself will get hotter, resulting in poorer magnetic properties. This is all happening in a system that has its temperature regulated, so the details of the temperature profile would be difficult to model. In addition, optimum temperature profile in space and time for causing apoptosis has not been characterized. However, it would seem in general that reduced surface area to volume ratio would be detrimental to effective MPH.

108 Chapter 15

Conclusions

Experimental analysis of the effects of particle shape and size on the ability of a magnetic material to absorb power from an applied field has found that magnetization rotation is the dominant mechanism of power absorption. The results of this research indicate dynamic particle behavior and particle interactions affect the ability of a magnetic material to absorb power from an applied field. These interactions affect both the local field experienced by particles as well as their ability to physically rotate. This likely has an effect on the measured specific absorption rates, which was observed to decrease in time. The temperature differences achieved in this research are well below the Curie temperatures of the magnetic material. To improve magnetic particle hyperthermia further research is required that study these effects. Dispersing small magnetic particles in larger solid structures may stabilize power absorption of the magnetic material and cause better heat transfer to the surrounding medium. Heating was observed for prolate and oblate particle samples with anisotropy fields greater than the applied field. This heating is not accounted for in the available theoretical models. This may be due to either effects not considered in the model, such as particle interactions, clustering aggregation prior to being subjected (which was observed),

109 and the formation of rope-like structures while subjected to the applied ﬁeld (which was also observed), or to inaccurate assumptions, such as coherent magnetization reversal, static particles, or the use of a constant temperature. Heating was observed in all of the magnetic particles tested in this research. Of these particles, a spherical particle having a radius of 16 nm caused large sample temperature diﬀerences (∆T > 7 ◦C) even at low concentrations. This particle is small enough to be used as a local heating agent for MPH.

110 Appendix A

LCOrientation.m

clear all a=input(’Input particle minor axis a [nm]: ’)*1e-9; %m c=input(’Input particle major axis c [nm]: ’)*1e-9; %m k=c/a; %Aspect ratio R=RES(c,a); %Radius of equivalent sphere [m] H=10e3; %Magnetic field strength [A/m] mu_0 = 4*pi*10^-7; %Permeability of free space [T*m/A] Keff= 13.5e3; %Effective anisotropy [J/m^3] Ms=446e3; %Saturation Magnetization [A/m] eta=0.001; %Dynamic viscosity [Pa*s] [Pa]=[J/m^3] D=mu_0*H*Ms/Keff; %Lissberger Comstock parameter D (in paper IsH/Keff) E=Keff*c^3/(eta*6*k^2*R^3); %Lissberger Comstock parameter E h=input(’Input theta step size: ’); for j=1:17 theta(j,1)=180-j*10;

111 phi(j,1)=LCPhi(theta(j,1),D); t(j,1) = 0; i = 2; while phi(j,i-1) < 179 theta(j,i)=theta(j,i-1)+h; phi(j,i)=LCPhi(theta(j,i),D); dt(j,i-1)=1/(D*E)*log(tand(theta(j,i)/2)/tand(theta(j,i) -phi(j,i))^(D/2))-1/(D*E)*log(tand(theta(j,i-1)/2)/tand(theta(j,i-1) -phi(j,i-1))^(D/2)); t(j,i)=t(j,i-1)+dt(j,i-1); i=i+1; end plot(t(j,:),phi(j,:)); hold on; end xlabel(’time (sec)’); ylabel(’Particle Orientation \phi (degrees)’);

function f=RES(c,a) k=c/a; if k==1 f = 1^(1/3)*c;

112 else f = (2/3*(k^2+1)/(k^3*(k^2-1/2)/(k^2-1)^(3/2) *log((k+(k^2-1)^(1/2))/(k-(k^2-1)^(1/2))) -k^4/(k^2-1)))^(1/3)*c; end

113 Appendix B

Temperature and Field Dependent SWT Code

clear all %Boltzmann’s Constant kb = 1.38066e-23; %J/K %Permeability of Free Space mu_0 = 1.2566e-6; %T m/A %Temperature T = 310.15; %K (98.6 degrees F, 37 degrees C) %Larmor Time Constant tau_0=1e-9; %s %Applied Field Magnitude Hap=45e3; %A/m %Applied Field Frequency f=30e3; %Hz %Ferrofluid Density

114 rho_p = 5180; %kg/m^3 %Domain Magnetization Md = 446e3; %A/m for i = 1:1000 Keff(i) = 100*i-100; %J/m^3 for j = 1:1000 R(j) = 0.05e-9*j-0.05e-9; %m V(j) = 4/3*pi*R(j)^3; %m^3 Kappa(i,j) = kb*T/(Keff(i)*V(j))*log(kb*T/(4*mu_0*Hap*Md*V(j)*f*tau_0)); if Kappa(i,j) > 1 Kappa(i,j) = 1; end Hcaligned(i,j) = 2*Keff(i)/(mu_0*Md)*(1-Kappa(i,j)^(1/2)); %A/m if Hcaligned(i,j) > 50e3 Hcaligned(i,j) = 50e3; end Hcrandom(i,j) = 0.48*2*Keff(i)/(mu_0*Md)*(1-Kappa(i,j)^(0.8)); %A/m if Hcrandom(i,j) > 50e3 Hcrandom(i,j) = 50e3; end Aaligned(i,j) = 8*Keff(i)*(1-Kappa(i,j)^(1/2)); %J/m^3 if Hcaligned(i,j) > 45e3 Aaligned(i,j) = 0; end Arandom(i,j) = 1.92*Keff(i)*(1-Kappa(i,j)^(0.8)); %J/m^3

115 if Hcrandom(i,j) > 45e3 Arandom(i,j) = 0; end end j = 1; end figure mesh(R*1e9,Keff/1000,Kappa) xlabel(’Radius (nm)’) ylabel(’Keff (kJ/m^3)’) figure mesh(R*1e9,Keff/1000,Hcaligned) xlabel(’Radius (nm)’) ylabel(’Keff (kJ/m^3)’) figure mesh(R*1e9,Keff/1000,Hcrandom) xlabel(’Radius (nm)’) ylabel(’Keff (kJ/m^3)’) figure mesh(R*1e9,Keff/1000,Aaligned) xlabel(’Radius (nm)’) ylabel(’Keff (kJ/m^3)’)

116 figure mesh(R*1e9,Keff/1000,Arandom) xlabel(’Radius (nm)’) ylabel(’Keff (kJ/m^3)’)

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