MODELING THE RESPONSE OF MAGNETIC NANOPARTICLES TO MAGNETIC FIELDS IN MAGNETIC PARTICLE IMAGING

By

ROHAN DEEPAK DHAVALIKAR

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2017

© 2017 Rohan Deepak Dhavalikar

To my parents Deepak Dhavalikar and Smita Dhavalikar

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ACKNOWLEDGMENTS

First, I would like to thank my advisor Dr. Carlos Rinaldi for giving me this

opportunity to pursue doctoral studies and guiding me through the process. I appreciate

his efforts to keep me motivated throughout my graduate studies and help me grow as a

researcher and a teacher. I was fortunate to take the Continuum Basis class with him as

a student and also work with him as the teaching assistant. Thank you for your support,

guidance, and motivation to push boundaries and explore new areas of research.

I would like to express my sincere gratitude to my committee members for the

discussions which motivated interesting studies and helped answer interesting

questions. I would like to thank Dr. David Arnold for the fruitful discussions during the

MPI meetings and for explaining complex electrical engineering concepts in simple

terms. I’m thankful to Dr. Jon Dobson for discussions on topics in magnetism and Dr.

Tony Ladd for encouraging me to think beyond my research topic.

I would like to acknowledge collaborators at the UC Berkeley Imaging Systems

Laboratory for many interesting and stimulating discussions on upcoming MPI

applications. I was fortunate to visit their lab in 2016 and enjoyed learning more about x-

space MPI from Dr. Steven Conolly, Dr. Daniel Hensley, and Zhi Wei Tay. I’m also glad

to have had the opportunity to interact and work with Dr. Patrick Goodwill. I would

specially like to thank Dr. Hensley for testing particles in the relaxometer, working with

me on the x-space signal reconstruction algorithm, and conducting the spatial heating

experiments at UC Berkeley. I would also like to thank Dr. Nicoleta Baxan from Bruker

BioSpin MRI GmbH for testing particles in the magnetic particle spectrometer.

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Graduate studies are incomplete without the mentoring experience gained from

working with undergraduate students. For this deeply rewarding and learning

experience, I’m grateful to Steven Ceron, Daniel Prestridge, and Justina Chan.

I would like to thank the members of the Rinaldi Lab and my colleagues at UF for

the scientific discussions in the lab and the fun times during conferences. I would like to

specially thank Dr. Lorena Maldonado-Camargo for synthesizing and characterizing the

cobalt ferrite particles for use in experiments, Ishita Singh for synthesizing iron oxide

nanoparticles, and Nicolas Garraud for interesting discussions on MPI.

I would like to thank my family, especially my parents and my brother for their

valuable support and encouragement to pursue higher education. A special thanks

goes to Dr. Ana Bohórquez for constantly encouraging and motivating me to do better,

and for supporting me in my endeavors.

Lastly, I would like to thank the Department of Chemical Engineering and the

University of Florida for the infrastructure during the course of my doctoral studies, and

National Institutes of Health for financial support.

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TABLE OF CONTENTS page

ACKNOWLEDGMENTS .................................................................................................. 4

LIST OF FIGURES .......................................................................................................... 8

ABSTRACT ................................................................................................................... 12

CHAPTER

1 INTRODUCTION .................................................................................................... 14

1.1 Magnetic Particle Imaging ................................................................................. 14 1.2 MPI Scanner Construction ................................................................................ 14

1.3 Signal Reconstruction Methods ........................................................................ 15

1.4 MPI Tracers ...................................................................................................... 16 1.5 Theoretical Models............................................................................................ 18 1.6 Dissertation Structure ....................................................................................... 23

2 FERROHYDRODYNAMIC MODELING OF MAGNETIC NANOPARTICLES IN A MAGNETIC PARTICLE SPECTROMETER AND A MAGNETIC PARTICLE RELAXOMETER ..................................................................................................... 26

2.1 Background ....................................................................................................... 26

2.2 Theory ............................................................................................................... 28 2.3 Particle Characterization ................................................................................... 33

2.3.1 Characterization Techniques ................................................................... 33

2.3.2 Characterization Results ......................................................................... 34 2.4 Response of Particles to Magnetic Fields in a Magnetic Particle

Spectrometer ....................................................................................................... 34 2.4.1 Simulation Parameters ............................................................................ 34 2.4.2 MPS Measurements ................................................................................ 35

2.4.3 Results .................................................................................................... 35 2.5 Response of Particles to Magnetic Fields in a Magnetic Particle

Relaxometer ........................................................................................................ 40 2.5.1 Simulation Parameters ............................................................................ 40 2.5.2 MPR Measurements ................................................................................ 40

2.5.3 Results .................................................................................................... 40 2.6 Conclusions ...................................................................................................... 44

3 SPATIAL CONTROL OF MAGNETIC NANOPARTICLE HEATING IN MAGNETIC FLUID HYPERTHERMIA USING MAGNETIC PARTICLE IMAGING FIELD GRADIENTS ............................................................................... 58

3.1 Background ....................................................................................................... 58

3.2 Theory ............................................................................................................... 60

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3.3 Simulation Parameters ...................................................................................... 63 3.4 Methods ............................................................................................................ 64 3.5 Results .............................................................................................................. 66

3.5.1 SAR Distribution in a Bias Field ............................................................... 66 3.5.2 Dynamic Hysteresis Loops ...................................................................... 67 3.5.3 Spatial Focusing of SAR using Selection Field Gradient ......................... 69 3.5.4 Spatial Control of Heating using a Field Gradient in Experiments ........... 70

3.6 Conclusions ...................................................................................................... 71

4 MODELING THE RESPONSE OF NÉEL PARTICLES FOR USE IN MAGNETIC PARTICLE IMAGING .............................................................................................. 83

4.1 Background ....................................................................................................... 83

4.2 Theory ............................................................................................................... 85 4.2.1 Landau-Lifshitz-Gilbert Equation ............................................................. 85 4.2.2 Determination of Relaxation Time ........................................................... 88

4.3 Simulation Parameters ...................................................................................... 88 4.4 Results .............................................................................................................. 89

4.4.1 Response of Particles to a DC Field ........................................................ 89 4.4.2 Particle Response in the Absence of a Magnetic Field ............................ 92 4.4.3 Response of Particles to a Sinusoidal Magnetic Field ............................. 95

4.5 Conclusions ...................................................................................................... 98

5 SUMMARY ........................................................................................................... 125

LIST OF REFERENCES ............................................................................................. 128

BIOGRAPHICAL SKETCH .......................................................................................... 137

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LIST OF FIGURES

Figure page 1-1 Illustration of MPI physics for generation of MPI signal.. .................................... 25

2-1 Schematic illustration of the construction of a magnetic particle spectrometer (MPS) and a magnetic particle relaxometer (MPR). ........................................... 46

2-2 Illustration of the applied field profile in an MPR.. ............................................... 46

2-3 Cobalt ferrite particle characterization.. .............................................................. 47

2-4 Simulation results showing a comparison between the Langevin function, Sh equation and the MRSh equation.. ..................................................................... 48

2-5 Effect of shell thickness on the harmonic spectra predicted by the Langevin function, MRSh and the Sh equation. ................................................................. 49

2-6 Effect of core diameter on the harmonic spectra at a constant hydrodynamic diameter. ............................................................................................................ 50

2-7 Field dependence of parallel relaxation time for various core diameters at a constant hydrodynamic diameter of 70 nm. ........................................................ 51

2-8 Effect of drive field frequency on the harmonic spectra predictions of the Langevin function, MRSh and the Sh equation................................................... 51

2-9 Effect of the drive field amplitude on predictions of the Langevin function, MRSh and Sh equation. .................................................................................... 52

2-10 Comparison of simulation predictions with measurements of cobalt ferrite particles in a Bruker magnetic particle spectrometer. ......................................... 52

2-11 Magnetization response of particles to an applied field in a magnetic particle relaxometer......................................................................................................... 53

2-12 PSF comparison between the simulation predictions and experiments. . ........... 54

2-13 Dependence of PSF peak location with the drive field amplitude for a frequency of 25 kHz. ........................................................................................... 55

2-14 PSF peak position at 20 mT amplitude and different drive field frequency. ........ 55

2-15 Full width at half maximum (FWHM) obtained from point spread functions (PSFs) for condition of 25 kHz and variable drive field amplitude....................... 56

2-16 Simulation results from Langevin function at 25 kHz frequency and field amplitude upto 60 mT. . ...................................................................................... 56

9

2-17 Effect of drive field frequency with an amplitude of 20 mT on the FWHM as predicted using models and from experiments. .................................................. 57

3-1 Applied field profile for selective heating of magnetic nanoparticles. .................. 73

3-2 Distribution of SAR values as a function of the bias field. ................................... 73

3-3 Dynamic hysteresis loops for representative bias fields. .................................... 74

3-4 Magnetization response with time for magnetic field frequencies used in MPI in the presence of a bias field. ........................................................................... 75

3-5 Effect of bias field on the magnetization response of particles at frequencies typically used in MPI and MFH.. ......................................................................... 76

3-6 Magnetization response of particles to alternating field frequencies employed in MFH for bias field of 50 mT, 25 mT, and 0 mT. .............................................. 77

3-7 Comparison of the area of dynamic hysteresis loops and its product with frequencies for representative conditions of bias field and frequencies used in MPI and MFH.. ................................................................................................ 78

3-8 Dependence of maximum achievable SAR in the FFR on the field amplitude of the excitation field for frequencies employed in MPI and MFH.. ..................... 79

3-9 Use of selection field gradient to achieve millimeter scale spatial focusing of heat deposition. .................................................................................................. 79

3-10 Comparison of simulation results of SAR distribution with experiments conducted in the UC Berkeley setup. ................................................................. 80

3-11 Demonstration of spatial focusing of heating by magnetic nanoparticles using a selection magnetic field gradient.. ................................................................... 80

3-12 Determination of field gradient. ........................................................................... 81

3-13 Experiments showing spatial focusing of heating region with a change in the selection field gradient. ....................................................................................... 82

4-1 Response of 5 nm magnetic nanoparticles to an applied DC field of 100 mT in the z direction. .............................................................................................. 100

4-2 Response of 10 nm magnetic nanoparticles to a constant DC field of 100 mT in the z direction. .............................................................................................. 101

4-3 Response of 15 nm magnetic nanoparticles to a constant DC field of 100 mT.. .................................................................................................................. 102

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4-4 Plots showing the distribution of individual moments after application of a 100 mT field in the z direction for 15 nm particles at representative time points. .... 103

4-5 Comparison of equilibrium magnetic moment values with the predictions of the Langevin function for a 5 nm particle. ......................................................... 104

4-6 Comparison of equilibrium magnetic moment values obtained from simulations of 10 nm particles with the predictions of the Langevin function.. .. 104

4-7 Comparison of simulations obtained using 15 nm particles with the predictions of the Langevin function. ................................................................ 105

4-8 Simulation results for 20 nm particles in comparison with the predictions of the Langevin function showing significant deviation in the non-linear region. ... 105

4-9 Simulation of 20 nm particles subject to a constant DC field of 40 mT.. ........... 106

4-10 Simulation results of the average dimensionless magnetic moment in the z direction for 15 nm particles to determine the alignment time with the field based on the slope of the fitted line to the portion of the decaying curve. ........ 107

4-11 Comparison of alignment time obtained at different field strengths for particles with 5, 10, 15 nm, and 20 nm diameter with damping constant equal to 0.1. ............................................................................................................... 107

4-12 Response of 15 nm particles to a 100 mT DC field with a damping constant = 0.01. .............................................................................................................. 108

4-13 Simulation results for 15 nm particles subjected to a 100 mT DC field in z

direction having a damping constant =1. ........................................................ 109

4-14 Effect of on the alignment time for 15 nm particles at different field strengths. .......................................................................................................... 109

4-15 Simulation results for 10 nm particles showing the decay in the average dimensionless magnetic moment after an applied DC field of 100 mT has been switched off. ............................................................................................. 110

4-16 Plots showing the distribution of moments obtained from simulation of 1000 particles after an applied field of 100 mT is removed at representative time points.. .............................................................................................................. 111

4-17 Simulation results for 10 nm particles with easy axes aligned in the z direction, same as the direction of the applied field, showing decay in the z direction magnetic moment after removal of applied magnetic field. ................ 112

11

4-18 Magnetic moment distribution plots for 1000 particles with easy axis in z direction showing the decay of moments after an applied field of 100 mT is switched off....................................................................................................... 113

4-19 Moment decay after removal of applied field for different diameter particles. .. 114

4-20 Comparison of relaxation time obtained from simulations with the predictions

of the Néel-Arrhenius equation using a damping constant of 0.1. ................. 114

4-21 Effect of variation in damping constant on the relaxation time of particles obtained from simulations. ............................................................................... 115

4-22 Simulation results of 10 nm particles at varying anisotropy constants showing a slower decay in the magnetic moment as the anisotropy constant value increases. ......................................................................................................... 115

4-23 Comparison of the relaxation time obtained from a fit to decaying magnetic moment curves for particle diameter between 5-10 nm and anisotropy constant values between 5-50 kJ/m3. ............................................................... 116

4-24 Response of 5 nm particles to a sinusoidal excitation field of frequency 25 kHz and amplitude 25 mT.. ............................................................................... 117

4-25 Simulations showing the response of 10 nm particles subjected to an alternating field of 25 kHz frequency and 25 mT amplitude. ............................. 118

4-26 Simulation of 15 nm particles exposed to a sinusoidal excitation field (25 kHz, 25 mT) showing a delay in the magnetic moment response to the alternating field. .................................................................................................................. 119

4-27 Magnetic moment distribution plots at representative time points for 15 nm particles exposed to an alternating magnetic field.. .......................................... 120

4-28 Effect of excitation field amplitude on the average dimensionless magnetic moment for 15 nm particles with damping constant 0.1. ................................... 121

4-29 Response of 15 nm particles to a sinusoidal magnetic field at different field

amplitudes with damping constant equal to 0.01.. ......................................... 122

4-30 Effect of magnetic field amplitude on the response of 15 nm particles to an alternating field of 25 kHz frequency with damping constant equal to 1.. ......... 123

4-31 Comparison of simulation of 15 nm particles with a damping constant = 1 with the predictions of the Langevin function on application of a field with 25 kHz frequency and 25 mT amplitude.. .............................................................. 124

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

MODELING THE RESPONSE OF MAGNETIC NANOPARTICLES TO MAGNETIC

FIELDS IN MAGNETIC PARTICLE IMAGING

By

Rohan Deepak Dhavalikar

December 2017

Chair: Carlos Rinaldi Major: Chemical Engineering

Magnetic particle imaging (MPI) is a tomographic tracer imaging technology

which makes use of the nonlinear magnetization response of magnetic nanoparticles for

image generation. It is a safer alternative to radioactive imaging modalities like positron

emission tomography (PET) and single photon emission computed tomography

(SPECT) due to the use of biocompatible magnetic nanoparticles as tracers. Due to its

high sensitivity and quantitative nature, MPI has been employed to perform

cardiovascular imaging, stem cell tracking, and cancer imaging.

Superparamagnetic nanoparticles are a key component in MPI and greatly

influence the achievable resolution of the imaging modality. Thus, apart from interest in

synthesizing a variety of customized nanoparticles for MPI applications, there is a

growing interest in modeling their response to the magnetic fields encountered in MPI.

Initial studies made use of the Langevin function which assumes instantaneous

response of particles to the applied magnetic field. However, in practice, finite magnetic

relaxation or response time has been implicated in blurring of the image, even limiting

the attainable resolution in MPI.

13

In this work, ferrohydrodynamic modeling was used to investigate the effect of

field-dependent relaxation on predictions of the point spread function (PSF) in a

magnetic particle relaxometer (MPR) and the nanoparticle harmonic spectra in a

magnetic particle spectrometer (MPS). These instruments are often employed to assess

the performance of magnetic nanoparticle tracers prior to their use in MPI. Through this

study, qualitative and quantitative agreement between the model predictions and

experiments was obtained without the need of empirical parameter fits. The

ferrohydrodynamic magnetization relaxation equation by Martsenyuk, Raikher and

Shliomis (MRSh) was also applied to calculate specific absorption rate (SAR) in

magnetic fluid hyperthermia (MFH) under the influence of a MPI selection field gradient.

Through simulations and experiments, spatial control of SAR and temperature was

achieved by tuning the MPI selection field gradient. Another study was conducted to

model the response of immobilized particles relaxing by the Néel mechanism using the

Landau-Lifshitz-Gilbert (LLG) equation. Through this study, the choice of damping

constant was found to significantly affect the response of magnetic moment to field

conditions used in MPI.

14

CHAPTER 1 INTRODUCTION

1.1 Magnetic Particle Imaging

Magnetic particle imaging (MPI) is an emerging biomedical tracer technology first

introduced in 2005 by Gleich and Weiznecker [1]. The technology utilizes biocompatible

superparamagnetic iron oxide nanoparticles for imaging and exploits the nonlinear

magnetization characteristics of these particles to a scanned gradient field. This

technology is a safer alternative to radioactive tracer technologies like single photon

emission computed tomography (SPECT) and positron emission tomography (PET),

both of which use radioactive tracers to achieve molecular imaging. MPI falls in the

category of tracer or molecular imaging technology because the image is obtained only

from the location where the magnetic nanoparticles are present, while the surrounding

tissue is not visualized in the image. The nanoparticles are thus called tracers, instead

of contrast agents. Due to this feature, MPI provides an excellent signal to noise ratio

(SNR) but lacks ability to image anatomical structures unless labeled by the tracer. The

strength of the MPI signal is linearly proportional to the particle concentration and hence

can provide quantitative information. Till date, MPI has been utilized for real-time

cardiovascular imaging [2], stem cell tracking and imaging [3], preclinical brain trauma

imaging [4], lung perfusion imaging [5], and cancer detection [6] while several other

applications are currently under development.

1.2 MPI Scanner Construction

An important feature of the MPI scanner is its ability to make use of a field free

region (FFR). The FFR is a region of negligible magnetic field to prevent saturation of

the magnetic nanoparticles. The FFR is obtained by combining two permanent magnets

15

with opposing poles. The FFR can also be created by using electromagnetic coils with

currents flowing in opposite directions. The field produced by these coils is called a

selection or bias field. The FFR can either be moved mechanically or electronically.

Another solenoid coil located inside the field of view (FOV) is used to generate an

alternating magnetic field. This alternating magnetic field is referred to as the drive field.

The drive field is superposed on the selection field. Particles located in the FFR are free

to respond to this time-varying sinusoidal magnetic field, whereas particles located

further away from the FFR continue to remain in the state of saturation. The

magnetization of particles situated in the FFR shows large oscillations in response to

the alternating magnetic field, whereas minor oscillations are observed in the

magnetization response of particles in the saturated region. These changes in the

magnetization induce a voltage in another solenoid coil which lies within the drive coil.

This coil is called the receive coil and registers the signal (induced voltage) as a

function of time. An illustration describing this signal generation process is shown in

Figure 1-1. So far, two companies, Philips in partnership with Bruker in Germany and

Magnetic Insight, Inc. in the United States of America have successfully developed and

installed preclinical scanners suitable for testing small animals.

1.3 Signal Reconstruction Methods

The acquired MPI signal is at present processed by two reconstruction methods:

system function or harmonic space reconstruction method [7, 8] and x-space

reconstruction method [9, 10]. In a system function approach, the acquired time domain

signal undergoes Fourier transformation to obtain the signal harmonics in the frequency

domain. In order to determine the concentration of nanoparticles in an actual scan, a

system calibration needs to be performed prior to the experiment. During the system

16

calibration, a nanoparticle sample with a known concentration is placed at distinct

locations in the region of scanning and the signal harmonics are acquired for each pixel.

The acquired signal harmonics at each pixel form the system matrix. This calibration

scan consumes a large amount of time. Thus, to reconstruct an image from an actual

scan, matrix inversion of the system matrix needs to be employed, which can become

computationally intense. Also, as the system matrix is obtained by scanning a

nanoparticle sample suspended in water, it might lead to image reconstruction artifacts

when used for in vivo applications, because water is a simple fluid that deviates from a

complex biological fluid environment such as blood.

On the other hand, the x-space reconstruction approach does not rely on a

system matrix for image reconstruction. It is a fast and robust method which just

requires knowledge of the instantaneous MPI signal and location of the FFR. However,

it relies on certain assumptions to facilitate reconstruction. The assumptions state that

the location of the FFR is unique at all times, the particles respond instantaneously to

the change in the field, and the lost first harmonic signal due to filtering of the acquired

signal can be fully recovered. These assumptions are not encountered in the system

function approach as the system matrix takes into account the characteristics of the field

and particles. So far, both approaches have been equally employed for various

applications to show the potential of MPI and establish MPI as a promising biomedical

imaging technology.

1.4 MPI Tracers

Although hardware plays an important role in MPI, the properties of tracers have

an equally important role. Ferucarbotran, also known as Resovist, is a

superparamagnetic iron oxide nanoparticle formulation coated with carboxydextran and

17

currently the gold standard in MPI. This nanoparticle formulation was primarily

developed to improve contrast in liver lesions for enhanced magnetic resonance

imaging (MRI) [11] but has been widely used for MPI applications [12-14] due to being

commercially available and FDA approved. These particles are small in size (~4 nm),

have a wide size distribution due to formation of clusters, and are known to have a short

in vivo circulation time. Theoretically, according to calculations using the Langevin

function [9] , MPI resolution and signal strength can be increased using large diameter

particles [15]. Thus, several efforts to synthesize particles with narrow size distribution

and large diameters have been undertaken [16-18]. The thermal decomposition

synthesis method [19] was employed to synthesize magnetic nanoparticles with narrow

size distribution. To enhance blood circulation time through increased colloidal stability,

the use of polyethylene glycol (PEG) coatings has been explored [20, 21]. These

coatings are charge neutral and have been shown to improve the colloidal stability of

the particles, thus finding use in vascular applications. Khandhar et al. [22] observed

that changing the PEG molecular weight had a significant effect on the colloidal stability

and the blood circulation time. For targeted delivery of nanoparticles to specific sites,

the particle surface needs to be modified further with specialized targeting ligands. For

example, to achieve molecular imaging of brain gliomas, particles were conjugated with

lactoferrin. These particles showed enhanced internalization into C6 glioma cells [23].

To obtain phase pure particles, post-synthesis oxidation was carried out by Kemp et al.

[24] to transform the phase of synthesized particles from wüstite to magnetite.

Prevention of magnetic dead layer formation during synthesis has always been a

challenging aspect of particle synthesis. However, a recently developed protocol by

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Unni et al. [18] involving controlled addition of molecular oxygen during synthesis shows

evidence of diminished magnetic dead layer and similar physical and magnetic

diameters. Thus, continued research to improve particle properties is expected to

ultimately result in improved MPI resolution and SNR.

1.5 Theoretical Models

To speed up development of MPI, understanding the effect of various parameters

affecting the resolution and image properties is important. This can be achieved through

models which can describe the behavior of the nanoparticles in the magnetic fields

encountered in MPI and using computer simulations to predict the response of these

nanoparticles. Initial models were based on the Langevin function [9, 25, 26] which

assumed that magnetic nanoparticles responded instantaneously to the applied

magnetic fields. The Langevin function also assumes that the particles are non-

interacting, single domain, and of the same size. The magnetization of the particles

using the Langevin function is given by Equation 1-1.

0

( , )( , ) ( ) coth ( , )

( , )

BH

B

m t k Tt M t

k T m r t

H rM r r e r

H, (1-1)

where 0 ( )M r is the saturation magnetization, m is the magnetic moment, ( , )tH r is

the total magnetic field at position r and time t , T is the absolute temperature, and

( , )H te r is the unit vector in the field direction. In a study by Weaver et al. [27], they used

the Langevin function to predict the response of particles in the presence of a bias field,

and found that the Langevin function could predict the general shape of the signal as a

function of bias field but failed to reproduce the exact shape. They attributed this to size

distribution of nanoparticles in the sample.

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To account for the effect of size distribution of particles on the signal and to

explain the signal from Resovist (ferucarbotran), Eberbeck et al. [28] used a bimodal

size distribution model. They showed that a simple model of a monomodal lognormal

size distribution, given by Equation (1-2), was not adequate to explain the behavior of

ferucarbotran which shows presence of aggregates.

3

0

1( ) ( , , ) ( , , , )

6s sM H M f d d L d M T H ddV

, (1-2)

where, is the volume fraction of magnetic solids, sM is the saturation magnetization,

V is the volume of the core, L is the Langevin function, T is the temperature, H is the

external field strength, f is the lognormal size distribution function with median and

geometric dispersion parameter . Since the calculated magnetization curve using a

monomodal lognormal distribution did not provide a good fit to the experimental data,

aggregates were taken into account through a bimodal size distribution, with 2 being

the normalized fraction of particles belonging to the larger distribution 2f . The

magnetization response was then modeled as

2 1 2 2( ) (1 ) ( ) ( )M H M H M H . (1-3)

Although, the use of a bimodal size distribution model yielded a satisfactory fit for

the data of Resovist, this study made use of fitting parameters.

In another set of experiments conducted in the UC, Berkeley magnetic particle

relaxometer, the Langevin function was unable to explain the shift in the x-space point

spread function and blurring in the acquired MPI images [29, 30]. These observations

were attributed to finite relaxation, the time required by the magnetic moment to

respond to a change in an applied magnetic field. Particles where the magnetic moment

20

is fixed are said to relax by the Brownian mechanism and the particles in which the

moment is free to rotate internally to a change in the magnetic field are said to relax by

the Néel mechanism. To account for finite relaxation in MPI, Rauwerdink and Weaver

[31] proposed the use of a complex susceptibility (Debye model). In this case, the

magnetization of the particles can be described in terms of a real in-phase component

' , and an imaginary out-of-phase component " and is given as

0( ) ( 'cos( ) "sin( ))M t H t t , (1-4)

where,

' "

0 02 2

1,

1 ( ) 1 ( )

. (1-5)

Here, 0H is the applied field, is the field frequency, is the relaxation time and 0 is

the initial susceptibility. Although this model takes into account the relaxation time, it is

only valid in the case of small field amplitudes and frequencies, and not applicable in

MPI where the field frequencies and amplitudes are much higher. To explain the shift in

the PSF, Goodwill et al. [29, 30] proposed the use of an effective relaxation time model

where the magnetization response is given by

0

eff

1( ) ( ( ) ( ))M t M t M t

. (1-6)

Here, eff is the effective relaxation constant and 0 ( )M t is the equilibrium magnetization

determined from the Langevin function. They showed that by using appropriate fitting

parameters, the calculated PSF using the effective relaxation time model gave

considerable agreement with the PSF obtained from experiments. Although, the

agreement was quite remarkable, this model required identification of fitting parameters

and does not take into account the field dependence of relaxation time.

21

Effect of relaxation time on MPI signal and resolution was demonstrated by

Dhavalikar and Rinaldi [32] using rotational Brownian dynamics simulations, which have

been demonstrated to take into account the magnetic field dependence of the relaxation

time [33, 34]. In this work, the magnetization response of non-interacting particles was

determined by solving a stochastic angular momentum equation obtained from a

balance of hydrodynamic, magnetic, and Brownian torques acting on the particle. A

significant difference between the predictions of the Langevin function and the results

from rotational Brownian dynamics simulations was observed. A phenomenological

model to account for the field dependence of relaxation time was proposed by Schmidt

et al. [35]. The total magnetic moment of an ensemble of magnetic nanoparticles was

described by a differential equation given by

( )

( ) ( ( )) ( ( ))dm t

m t VM H t H tdt

, (1-7)

with

( 0)

( )1 b

HH

aH

. (1-8)

Here, m is the magnetic moment, V is the sample volume, H is the magnetic field

strength, is the relaxation time, and a and b are fitting parameters. Reasonable

agreement was obtained with experiments by determining the fitting parameters through

correlation with experimental data. However, the fitting parameters were not universal

across all magnetic nanoparticle formulations and required experimental data for a

correlational analysis to obtain the nanoparticle specific parameters. Another approach

to understand the field dependence of relaxation time was proposed by Deissler et al.

[36] which involved numerically solving the Fokker-Planck equation for relaxation.

22

The behavior of magnetic nanoparticles in MPI has also been described by

modeling single particle dynamics using the Landau-Lifshitz-Gilbert (LLG) equation [37-

42]. The LLG equation given by Equation (1-9) describes the precession of the

magnetic moment in the nanoparticle in the presence of crystalline and shape

anisotropy, thermal fluctuations, and an external field.

det det2 2

,

( ( )) ( ( ))1 (1 )

ii fl i i fl

i s

d

dt m

mm H H m m H H . (1-9)

Here, im is the magnetic moment, is the gyromagnetic ratio, is the damping

constant, ,i sm is the saturation value of the magnetic moment, detH is the deterministic

field, and flH is the fluctuating field.

The models described in this section are a step in the right direction, but some of

them either make use of fitting parameters and empirical equations or utilize Euler

angles which are prone to singularities for describing the rotation of the nanoparticles in

response to the applied magnetic fields. In some models, the relaxation time is

independent of the applied field, but in MPI where the applied field changes

continuously, the applied field would have a significant effect on the relaxation time of

the nanoparticles and hence needs to be taken into account. Thus, overcoming these

shortcomings and building better models to predict the response of particles in MPI

remains a challenge. Developing better models will not only help explain the complex

behavior of nanoparticles in MPI but also aid in providing insights regarding particle and

MPI scanner properties to explore.

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1.6 Dissertation Structure

Chapter 1 provided an introduction to magnetic particle imaging and the physics

behind MPI signal generation. It also highlighted the importance of modeling the

response of magnetic nanoparticles in MPI and reviewed existing models, emphasizing

the need of improved models to better understand the response of particles in MPI.

Chapter 2 compares the predictions from phenomenological ferrohydrodynamics

magnetization relaxation equations at field conditions typically employed in MPI. The

Martsenyuk, Raikher and Shliomis(MRSh) magnetization relaxation equation [43] which

incorporates field dependent relaxation time was solved numerically to determine the

magnetization response of the particles to a high frequency and amplitude magnetic

field. The MRSh equation was applied to predict the response of magnetic particles to

magnetic fields applied in a magnetic particle spectrometer (MPS) [25, 44] and a

magnetic particle relaxometer (MPR) [29, 45]. These equipment lack the spatial imaging

component of an MPI scanner but are often used to characterize the magnetic

nanoparticles and assess their feasibility for use in MPI. Through this study qualitative

and to some extent quantitative agreement of simulation predictions with experiments

was observed without the need of fitting parameters [46, 47].

Chapter 3 describes theoretical and experimental studies to achieve spatial

control of heating in magnetic fluid hyperthermia (MFH) using MPI field gradients. MFH

[48] is a cancer treatment method where heat dissipated by magnetic nanoparticles in

the tumor on the application of an alternating magnetic field is used to disrupt and kill

cancer cells. However, the administered particles also accumulate in non-targeted

healthy organs, thus posing a damage risk during whole body hyperthermia. One

potential solution to this problem is the use of a magnetic field gradient. In Chapter 3,

24

the MRSh equation is used to obtain the magnetization response of the particles to a

combination of a static and alternating magnetic field, thus allowing for calculations of

energy dissipation rates from magnetic nanoparticles. Experiments were also conducted

to show spatial control of SAR and temperature under the influence of a selection

magnetic field gradient. Through this study, the potential of the MPI field gradients to

spatially control heating and the ability to tune this heating region to couple of

millimeters was realized [49].

Finally, Chapter 4 describes the Landau-Lifshitz-Gilbert (LLG) equation to model

the response of particles relaxing by the Néel mechanism, with their Brownian motion

suppressed. Here, the magnetic moment rotates internally to align with the direction of

the applied magnetic field. Chapter 4 explores the effect of change in the equation

parameters on the response of Néel particles. Through this work, a foundation was laid

for further studies on Néel particles in MPI and fundamental understanding of the

behavior of Néel particles was obtained.

25

Figure 1-1.Illustration of MPI physics for generation of MPI signal. Nanoparticles located in the field free region created by two opposing magnets are free to respond to an alternating drive field, whereas the motion of the nanoparticles in the saturated region is restricted due to a strong selection field. With the application of an alternating field, the change in magnetization response of the particles is highest in the field free region, resulting in a sharp signal. But, a negligible signal is obtained from nanoparticles in the saturated region.

26

CHAPTER 2 FERROHYDRODYNAMIC MODELING OF MAGNETIC NANOPARTICLES IN A

MAGNETIC PARTICLE SPECTROMETER AND A MAGNETIC PARTICLE

RELAXOMETER

2.1 Background

Initial efforts in modeling the response of magnetic nanoparticles in MPI relied on

the use of the simple Langevin function [25, 27, 50] and the Debye model [31, 51]. The

Langevin function is commonly used to describe the magnetization response of the

magnetic nanoparticles at equilibrium, whereas the Debye model is applicable in the

case of small field strengths. The Langevin function also assumes that particles align

instantaneously with the applied magnetic field, thus neglecting the response or

relaxation time of the magnetic nanoparticles. However, the MPI system is dynamic and

does not reach an equilibrium state and the fields employed are also large in

magnitude, thus these models might not be suitable to describe and capture the

complex behavior of the particles in MPI. Also, in practice, finite relaxation time was

found to affect the image properties leading to image blurring and a shift in the point

spread function (PSF) which was obtained from the x-space reconstruction of the

acquired signal [29, 30].

Ferrohydrodynamic equations developed to understand the behavior of

ferrofluids, a suspension of magnetic nanoparticles in a carrier fluid have been widely

explored in the ferrofluid literature [52]. The magnetization of the ferrofluids can be

Figures reproduced with permission from R. Dhavalikar, L. Maldonado-Camargo, N. Garraud, C. Rinaldi, Ferrohydrodynamic modeling of magnetic nanoparticle harmonic spectra for magnetic particle imaging, J. Appl. Phys., 118 (2015) 173906.

Figures reproduced with permission from R. Dhavalikar, D. Hensley, L. Maldonado-Camargo, L.R. Croft, S. Ceron, P.W. Goodwill, S.M. Conolly, C. Rinaldi, Finite magnetic relaxation in x-space magnetic particle imaging: Comparison of measurements and ferrohydrodynamic models, J. Phys. D: Appl. Phys., 49 (2016) 305002.

27

obtained by the magnetization relaxation equations, which is part of the set of

ferrohydrodynamic equations. One of the simplest and widely used magnetization

relaxation equation in the ferrofluid literature was proposed by Shliomis [53]. This

equation, herein referred to as the Sh equation takes into account the relaxation time of

the magnetic nanoparticles. However, this equation was found to give accurate results

for small amplitude and frequency alternating magnetic fields [33, 34, 54]. Another

magnetization relaxation equation developed by Martsenyuk et al. [43] was found to be

applicable for high field and frequency alternating magnetic fields. This equation, herein

referred to as the MRSh equation takes into account the field dependence of the

relaxation time. Other models which have been developed make assumptions regarding

the relaxation time [29] or make use of fitting parameters to estimate the field dependent

relaxation time [35]. The Sh and the MRSh equations are simple phenomenological

equations and could be utilized to predict the behavior of magnetic nanoparticles

without the use of fitting parameters. These equations have not been explored in depth

for MPI and Chapter 2 will explore their application in MPI to provide insights through

simulations.

To assess the quality of nanoparticles in MPI with high-throughput, an MPI

scanner lacking the spatial imaging component is typically used. This equipment is

called the magnetic particle spectrometer (MPS) [25, 55] and the magnetic particle

relaxometer (MPR) [29, 30, 45]. An illustration showing the construction of the MPS and

MPR is given in Figure 2-1. The MPS and MPR consist of a concentric set of solenoid

coils. In a MPS, the outermost coil called the excitation coil carries an alternating current

to generate a high frequency alternating field, typically of the order of 25 kHz. The

28

second coil called the receive coil serves to record the induced voltage (signal) due to

the change in magnetization of the particle sample located in the probe chamber. The

alternating magnetic field generated by the excitation coil leads to an alternating

magnetization response from the particles in the probe chamber. This change in the

magnetization leads to an induced voltage (signal) in the receive coil which surrounds

the probe chamber. The signal is further analyzed through finite Fourier transform to

obtain the harmonic spectra. The resulting harmonic spectrum is compared with that of

the standard (Ferucarbotran) to facilitate in the particle selection process. In the MPR,

the entire setup remains the same, with an addition of a bias coil. This bias coil forms

the outermost coil and generates a slow ramping DC magnetic field. The ramping rate is

much slower than the frequency of the excitation field, thus the DC field could be

assumed to be constant over multiple excitation field cycles. The acquired signal due to

the combination of the alternating and slow ramping bias field is further processed by

the x-space reconstruction approach to acquire a point spread function (PSF). The

resulting PSF is compared with that of the standard (Ferucarbotran) to determine the

feasibility of the particles for use in MPI.

In Chapter 2, ferrohydrodynamic equations are applied to the case of

unidirectional field profiles encountered in both the MPS and MPR. The results obtained

from simulations are compared with those from experiments to obtain insights and

explore the potential of these equations in modeling the response of magnetic particles

in MPI.

2.2 Theory

Magnetic nanoparticles in suspension are in a state of randomness in the

absence of a magnetic field. Their net magnetization at equilibrium is thus zero as

29

magnetic moments are oriented in random directions. When magnetic nanoparticles are

subjected to an external magnetic field, the magnetic moments align in the direction of

the field. If allowed to reach a state of equilibrium, all the magnetic moments align in the

direction of the field, thus reaching magnetic saturation. This equilibrium magnetization

can be described by the Langevin function [50] given by Equation (2-1)

0 1coth ( ),

s

ML

M

(2-1)

where BmH k T is the Langevin parameter. Here,

0M is the equilibrium

magnetization, sM is the saturation magnetization, m is the magnetic moment and H

is the applied field. The Langevin function is commonly used in MPI [9, 25] but recent

experiments have shown that it might be inaccurate [29, 30], thus indicating that finite

relaxation effects might play an important role in MPI.

Magnetic nanoparticles when subjected to a change in the magnetic field,

respond by two mechanisms, Brownian relaxation or Néel relaxation. A moment fixed

within the particle is said to be blocked and responds to a change in the direction or

magnitude of the applied field by physical rotation of the particle. These particles are

referred to as Brownian particles and their characteristic relaxation time for a negligibly

small field is given by

B

3 h

B

V

k T

(2-2)

where B is the Brownian relaxation time,

hV is the hydrodynamic volume, is the

medium viscosity, Bk is the Boltzmann’s constant, and T is the absolute temperature.

For the situation where the magnetic moment is free to rotate internally within the

30

particle, the particles are referred to as Néel particles. Their characteristic relaxation

time is given by

N 0 exp m

B

KV

k T

(2-3)

where N is the Néel relaxation time,

Bk is the Boltzmann’s constant, T is the absolute

temperature,0 is the attempt time(~ 10-9 s), K is the particle magnetocrystalline

anisotropy, andmV is the volume of the magnetic core. It should be noted that Equations

(2-2) and (2-3) are valid in the situation of non-interacting dilute suspension of particles

and in negligible external field.

To account for finite relaxation, Shliomis [53] developed a macroscopic

phenomenological magnetization equation, herein referred to as the Sh equation. This

equation has been widely used in the ferrofluid literature for its simplicity and accurate

description of the behavior of magnetic nanoparticles at low frequency and amplitude

magnetic fields. Equation (2-4) gives the mathematical form of the Sh equation and

incorporates a constant relaxation time.

0( ) ( ),

6B

d

dt

M MM M M HΩ M (2-4)

where Ω is the flow vorticity, is the volume fraction, is the medium viscosity, B is

the Brownian relaxation time, 0

M is the equilibrium magnetization, andM is the

magnetization of the suspension under the field H . For unidirectional magnetic field and

no bulk flow conditions, Equation (2-4) reduces to

0( )

B

M MdM

dt

(2-5)

31

For fields comparable to those encountered in MPS and MPR, a magnetization

equation derived microscopically from the Fokker-Planck equation by Martsenyuk et al.

[43] takes into account the field dependence of relaxation time. Using an effective field

method, they were able to provide closure of the first moment of magnetization,

resulting in

0

2 2,

d

dt H H

H H M M H M HM

Ω M (2-6)

where ln ( ) 2 ( )

, ln ln

B B

d L L

d

are the parallel and perpendicular relaxation

times. For unidirectional magnetic field and no flow condition, Equation (2-6) reduces to

1e

B

HHdM

Mdt

. (2-7)

Here, eH is the so-called effective field. Magnetization never quite reaches a state of

equilibrium, so for small departures from equilibrium, this non-equilibrium magnetization

is considered to be in equilibrium with this effective field. This magnetization then

relaxes to its equilibrium value as the effective field tends to the true field.

The variables in these equations are non-dimensionalized by their magnitudes in

order to reduce the number of variables. The magnetization is non-dimensionalized by

the saturation magnetizationsM and the time by the Brownian relaxation time

B as

given in Equation (2-8).

, s B

M tM t

M . (2-8)

Equations (2-1), (2-5) and (2-7) are non-dimensionalized using the parameters in (2-8)

to give

32

0

1cothM

(2-9)

0

dMM M

dt (2-10)

[1 ] , e

dMM

dt

(2-11)

where ee

B

mH

k T . In the case of an MPS, a time varying excitation or drive field is used

and is given by

d( ) sin(2 )H t H ft (2-12)

where f is the drive field frequency and dH is the drive field amplitude. In the case of

an MPR, the applied field is a combination of the drive field and a slow ramping bias

field as shown in Figure 2-2. The field is expressed mathematically as

b d( ) ( ) ( )H t H t H t (2-13)

where b,0

b b,0

scan

2( )

H tH t H

t and

d d( ) sin(2 )H t H ft . Here, scant is the scan time for a

measurement, t is the time and b,0H is the magnitude of the bias field at time 0t . The

frequency is non-dimensionalized by the relaxation time to give dimensionless

frequency 2 Bf .

The resulting dimensionless differential equations are solved numerically using

MATLAB’s ODE45 function to obtain the dimensionless magnetization as a function of

dimensionless time. The negative of the change of magnetization with time is recorded

as the nanoparticle signal. For MPS measurements, the signal undergoes a finite

Fourier transform (FFT) to obtain a harmonic spectrum. The odd harmonics up to the

33

49th harmonic in the dimensional form are then used for analysis. Whereas, for MPR

measurements, the signal is reconstructed using an x-space reconstruction algorithm

developed by UC Berkeley [9, 10, 56] to obtain a point spread function (PSF). The basic

components of the x-space algorithm include phase correction and alignment for the

drive field and received signals, filtering of the raw data, gridding of the time-domain

signal to the image domain using the known trajectory of the field-free region, recovery

of spatial DC information through use of a continuity algorithm and overlapping partial

fields-of-view (pFOVs), and stitching of the DC-recovered pFOVs into a single output

PSF. The features of the PSF like the full width at half maxima, the peak strength, and

the peak shift distance are used for further analysis and comparisons.

2.3 Particle Characterization

For experiments in the MPS and MPR, cobalt ferrite nanoparticles were used as

representative particles as they primarily respond by the Brownian relaxation

mechanism. These particles were synthesized in the Rinaldi lab via the thermal

decomposition method at 320 C [19]. These particles were coated with oleic acid to

prevent aggregation and suspended in a compatible organic solvent toluene.

2.3.1 Characterization Techniques

A JEOL 200CX transmission electron microscope was used to determine the

core diameter of the particles. The hydrodynamic diameter of the particles was obtained

by the dynamic light scattering technique using a Brookhaven Instruments

ZetaPALS/BI-MAS. Dynamic magnetic susceptibility (DMS) measurements were carried

out using a DynoMag (Acreo) to confirm the relaxation mechanism.

34

2.3.2 Characterization Results

Dynamic light scattering(DLS) measurements as shown in Figure 2-3a using a

Brookhaven Instruments ZetaPALS/BI-MAS provided the volume weighted average

hydrodynamic diameter of 18nm and geometric deviation of ln = 0.0658. Some

aggregates were also observed in these measurements around 45-55 nm.

Transmission electron microscopy images in Figure 2-3b show individual particles

having a spherical shape. The number weighted average physical diameter of the

individual particles was found to be 14 nm with a geometric deviation of ln = 0.122

after fitting the size distribution histogram to a log normal size distribution as seen in

Figure 2-3c. DMS spectrum shown in Figure 2-3d indicated that the particles relax

primarily by the Brownian mechanism based on the Debye model [57]. Using the

relation 1 and the peak frequency in the DMS spectrum, the hydrodynamic

diameter was estimated to be 17 nm, in agreement with the DLS measurements.

2.4 Response of Particles to Magnetic Fields in a Magnetic Particle Spectrometer

2.4.1 Simulation Parameters

In a magnetic particle spectrometer (MPS) employed by Bruker, drive field

frequency of 25 kHz and field amplitude of 25 mT is typically used. For simulations, the

field amplitude was varied between 5-25 mT and the drive field frequencies were varied

between 1.5 kHz - 25 kHz. According to theory, large particles are expected to provide

better resolution; hence particle diameters in the size range of 20-50 nm were used in

simulations. For comparison with experiments, a particle size of 14 nm was used in

simulations. Particles are typically coated with an organic coating to prevent

aggregation. For simulations, shell coating thickness was varied between 5-20 nm while

35

for comparison with experiments a shell thickness of 1.5 nm was used in simulations.

The scan time was 1 second and the temperature was assumed to be 298 K.

2.4.2 MPS Measurements

MPS measurements were carried out in a magnetic particle spectrometer at

Bruker BioSpin MRI GmbH, Germany. A sample volume of 5L was used in the

measurements. A drive field frequency of 25 kHz and field amplitude of 25 mT was

utilized. A harmonic spectrum containing odd harmonics starting with the 3rd harmonic

was recorded and used for comparison with simulations. The data acquisition time was

10 seconds.

2.4.3 Results

Figure 2-4 illustrates the effect of finite relaxation on the performance of particles

in a MPS through comparison of the predictions from the Langevin function, Sh

equation, and the MRSh equation. Figure 2-4a shows a comparison between the

dynamic hysteresis loops obtained using the three models on the application of an

alternating magnetic field. The Langevin function does not show a broadening of the

loop since the moments align instantaneously with the change in the field. However, the

hysteresis loops obtained from both, the Sh and the MRSh equations show a

considerable broadening of the loop. This can be attributed to the lag or delay in the

response of particles to the applied magnetic field. The delay can be observed in Figure

2-4b, c where the dimensionless magnetization and dimensionless signal shows a

difference in shape between the magnetization curves and signal peaks. A considerable

difference in the signal strength is also observed as compared to the predictions of the

Langevin function when finite relaxation is taken into consideration. Finally, FFT of the

signal in Figure 2-4d shows a quick decay in the harmonics for predictions from the Sh

36

and the MRSh equation along with a drop in the magnitude of each odd harmonic. For

the various simulation parameters outlined earlier, the effect of finite relaxation will be

analyzed through comparison of harmonic spectra.

To understand the effect of varying shell thickness on the harmonic spectra,

cobalt ferrite particles with a core diameter of 30 nm were subjected to a magnetic field

of frequency 4.5 kHz and 25 mT amplitude. Figure 2-5 shows the comparison between

the predictions of the Langevin function, the Sh and the MRSh equation. The Brownian

relaxation time changes cubically with the hydrodynamic diameter of the particles, thus

even a small change in particle thickness will result in a significant change in the

relaxation time. Figure 2-5 shows a significant drop in the harmonic spectra for the Sh

and the MRSh equations at a shell thickness of 20 nm. These large magnetic

nanoparticles physically rotate to align with the applied field and hence respond slowly

to a change in the applied field, ultimately resulting in a slow change in their

magnetization. As the signal is proportional to the rate of change of magnetization with

time, a signal with a reduced strength is obtained. Thus, with an increase in the shell

thickness, the signal strength decays rapidly when finite relaxation is taken into account.

However, in the case of the Langevin function, where particles are assumed to align

instantaneously with the applied field and does not have a dependence on the

relaxation time, the effect of shell thickness is insignificant as seen in Figure 2-5 where

the harmonic spectrum for different shell thickness overlap.

Next, the hydrodynamic diameter is kept fixed at 70 nm while the core diameter

is varied. Thus, the Brownian relaxation time remains constant in this case. When a field

of 4.5 kHz and 25 mT is applied, Figure 2-6 shows a significant difference between the

37

predictions of the models with varying core diameter. The Langevin function predicts the

highest signal for a 50 nm particle and slower rates of decay. The Sh equation

predictions show an overlap as the relaxation time remains constant and is independent

of the Langevin parameter. However, the harmonic spectra obtained using the MRSh

equation shows an increase in the signal with an increase in the core diameter. This

can be attributed to the decrease in the parallel relaxation time ( ) with the applied field

as plotted in Figure 2-7. The parallel relaxation time decreases with the increase in core

diameter, thus leading to a quicker response of magnetic nanoparticles to the applied

field. As the signal is proportional to the rate of change of magnetization with time, the

magnitude of the signal is greater. However, the Brownian relaxation time remains

constant for varying field strength and is presented in Figure 2-7 to serve as a

reference.

Now, the effect of drive field frequency on the harmonic spectra is studied.

Simulations for 30 nm core diameter with a 5 nm thick shell and subjected to a drive

field with amplitude of 25mT and varying frequencies were carried out. Figure 2-8

shows an increase in the signal magnitude for all the models. Although the amplitude of

the drive field is the same, the increase in frequency leads to field maximum in a short

amount of time. As the signal depends on the rate of change of magnetization with time,

the quick change in the field and thus the magnetization, leads to increased signal

strength. As the Langevin function assumes instantaneous alignment of the moments

with the applied field, the magnitude of higher harmonics continues to be significant and

the spectrum shows a slow decay. In contrast, incorporation of finite relaxation results in

a significant drop in the harmonics, with the magnitude of the higher harmonics being

38

close to the signal floor. This drastic drop in the harmonic decay can be attributed to the

response of particles lagging behind the applied field. The inability of particles to

respond to the rapidly changing magnetic field results in the particles reaching a lower

magnetization, resulting in a decrease in the rate of change of magnetization and

ultimately in lower signal strength. However, the harmonics decay much slower in the

case of predictions by the MRSh equation as compared to the predictions of the Sh

equation. This could be explained by the shortening of the relaxation time due to the

magnetic field amplitude dependence of relaxation time accounted in the MRSh

equation. So, even with the increase in the frequency of the drive field, the

magnetization response predicted by the MRSh equation closely tracks the rapid

change in the magnetic field.

Next, simulations for nanoparticles with 30 nm diameter and 5 nm shell thickness

were run for varying drive field amplitudes. The harmonic response was studied for

drive field frequency of 1.5 kHz and 25 kHz. Figure 2-9 compares the harmonic

response at various drive field amplitudes for both the frequencies. It can be seen that

all the models predict an increase in the signal magnitude. This can be attributed to the

effective increase in the rate at which the field magnitude changes, primarily because

the field amplitude increases at a constant frequency. The magnetization achieved

depends on the magnitude of the applied field, thus a greater change in magnetization

results in a higher signal. For the MRSh model in the case of 25 kHz (Figure 2-9b), the

first harmonic is quite sensitive to the applied field amplitude as compared to the other

models due to the dependence of the relaxation time on the field strength. Whereas, for

39

the Sh equation predictions, not much variation is observed in the harmonic spectra with

a change in the field amplitude.

Finally, simulation predictions were compared with the experimental

measurements obtained using a Bruker spectrometer for cobalt ferrite nanoparticles

described earlier. The independently determined properties of the particles were used in

all the models to obtain the harmonic spectrum. Figure 2-10 shows a comparison of the

harmonic spectra starting with the 3rd harmonic. In the data recorded by a Bruker

spectrometer, the fundamental harmonic also contains information of the drive field as a

result of direct feedthrough; hence the fundamental harmonic is discarded and not used

for comparison. Also, to facilitate comparison between the models and the experiments,

all the harmonics are normalized with the magnitude of their respective 3rd harmonic. All

models show excellent qualitative agreement with the decaying trend observed in the

experiments up to the 19th harmonic. Quantitatively, the prediction from the Langevin

function shows the largest deviation, while the predictions of the Sh and the MRSh

equation show better agreement with the experiments. This agreement is remarkable as

the parameters used in the simulations were independently determined and the

simulations did not make use of fitting parameters. However, due to the small size of the

particles used in the experiment, this experiment could not differentiate between the Sh

and the MRSh equation. This can be explained from the predictions seen earlier where

a significant difference was observed between the models for particles with large

diameters.

40

2.5 Response of Particles to Magnetic Fields in a Magnetic Particle Relaxometer

2.5.1 Simulation Parameters

The MPR developed by UC Berkeley typically uses a drive field frequency of 25

kHz and field amplitude of 20 mT. For simulations, the drive field frequency was varied

from 1.6 kHz to 25 kHz and the field amplitude was varied from 10 mT to 60 mT. The

bias field was slowly ramped from -75 mT to +75 mT in 0.25 seconds. The simulations

made use of particles having a 14 nm diameter and a shell thickness of 1.5 nm. These

particle parameters were obtained from independent particle characterization

measurements of particles synthesized in the Rinaldi lab as given in the Particle

Characterization section.

2.5.2 MPR Measurements

MPR measurements were carried out in a relaxometer developed at University of

California, Berkeley. Cobalt ferrite particles suspended in toluene were tested at drive

field frequencies of 1.6 – 25 kHz and amplitudes of 10-40 mT. The bias field had a

magnitude of 75 mT. The scan time for each measurement was 0.25 seconds and the

temperature was maintained at 293 K during the measurements. Signal from the

nanoparticle sample was processed by the same x-space algorithm which was used to

process signal from simulations.

2.5.3 Results

Simulations were carried out at conditions at which experiments were performed.

The particle properties were determined through independent methods and used in the

simulation without fitting parameters. The particles were subjected to a sinusoidal

ramping magnetic field to assess the effect of finite relaxation on their response. Figure

2-11 shows the magnetization response predicted by the Langevin function, the Sh

41

equation and the MRSh equation for a representative set of parameters (25 kHz, 20 mT

amplitude) at a) strong bias field, b) moderate bias, and c) low bias. As the Langevin

function assumes instantaneous response of magnetic moments to a change in the

applied field, the predictions in all the cases are in-phase with the applied field.

However, with the incorporation of finite relaxation, a slight delay is observed in the

magnetization response predicted by the Sh and the MRSh equation. In Figure 2-11a,

the strong bias field aligns the moments in the direction of the field, thus leading to the

magnetization to reach saturation. Oscillations are observed in the magnetization

response because of the superimposed drive field but are restricted to a small region,

resulting in a small change in the magnetization amplitude. With the decrease in the

bias field, as is the case in Figure 2-11b, the particles respond freely to the alternating

drive field. However, instead of a sinusoidal appearance, the magnetization response

has a jagged profile. This could be explained by the combination of magnetic torque

acting on the particles due to the bias field and the drive field, either accelerating or

decelerating their alignment based on the direction of both the fields. As the bias field is

lowered further to an extent that the alternating field is the dominant field, the particles

are free to respond to the change in the alternating field. This response is shown in

Figure 2-11c where the magnetization response is sinusoidal. It can also be observed

that the magnetization obtained from the Langevin function crosses zero when the

applied field becomes zero. However, a delay is observed in the magnetization obtained

from the Sh and the MRSh equation as they both incorporate finite relaxation.

Now, to facilitate comparison between the simulations and experiments,

nanoparticle signal obtained from the rate of change of magnetization with time is

42

processed by x-space reconstruction algorithm developed by UC Berkeley MPI lab to

obtain the point spread function (PSF). Representative PSFs are shown in Figure 2-12

for different conditions. These PSFs are normalized to facilitate comparison between

the simulations and the experiments. Figure 2-12a shows a negligible difference

between the models as well as the experimental measurements. This is to be expected

as at 1.6 kHz particles can track the change in the applied field without finite relaxation

affecting their behavior. With increase in the drive field frequency, a shift in the

experimental PSF peak is observed. This is observed in Figure 2-12b where a drive

field frequency of 12.2 kHz is used. The shift is also predicted by the Sh and the MRSh

equation. Figure 2-12c shows a further change in the shape of the experimental PSF

along with an increase in the peak shift distance. Again, the Sh and the MRSh equation

show a better agreement with the experiments than the Langevin function. The previous

PSFs were for measurements at field amplitude of 20 mT. With a further increase in the

drive field amplitude to 40 mT, as in the case of Figure 2-12d, the MRSh equation

shows excellent agreement with the experiment PSF in terms of the shape and the peak

location. This can be attributed to the field dependence of relaxation time in the MRSh

equation.

Next, the peak position of the PSFs was determined for a fixed frequency of 25

kHz but various field amplitudes. Figure 2-13 shows the comparison of simulation

predictions from the models with the experiments. As expected from theory, the

Langevin function should have a minimum shift in the peak location as the

magnetization response is in phase with the applied field. Thus, a negligible shift is

observed with the field amplitude. For the predictions from the Sh model, a gradual shift

43

in the peak location is observed due to relaxation. However, the peak shift in

experiments is much greater. The simulation predictions of the MRSh model show

excellent agreement with the peak location in the experiments. This agreement could be

attributed to field dependence of relaxation time in the MRSh equation as compared to a

fixed relaxation time in the Sh equation and an absence of a relaxation time in the

Langevin function.

Next, the effect of field frequency at constant field amplitude of 20 mT on the

peak position was studied. As shown in Figure 2-14, the Langevin function does not

show a change in the peak location as the field frequency is increased. This is due to

the instantaneous response of the particles to the applied field as assumed in the

Langevin function. Thus, irrespective of the frequency, the particles will continue to track

the change in the applied field as finite relaxation does not hinder the alignment

process. However, the predictions of the MRSh equation show a gradual shift in the

peak location and show good qualitative as well as quantitative agreement with the

experiments. The Sh equation predictions also show an increase in the peak location,

thus providing qualitative agreement with the trend observed in the experiments.

Along with the peak position, the point spread functions can be evaluated to

obtain the full width at half maximum (FWHM). The FWHM is commonly used to assess

the achievable resolution in MPI with the particles under consideration for a specific field

gradient. Figure 2-15 shows the trend in the FWHM calculated from the PSF with

respect to the field amplitude at 25 kHz drive field frequency. A dip in the FWHM and

then a gradual rise was observed in the experiments. Interestingly, this trend was

captured by all three models, thus providing little insight in their differences. The

44

apparent minimum in the FWHM predictions of the Langevin function at constant

frequency was due to changes in the shape of the signal as shown in Figure 2-16. A

significant difference between the Sh and the MRSh equation was also not observed in

the FWHM plot.

Next, to understand the effect of field frequency on the FWHM, comparisons

were done at field amplitude of 20 mT and various frequencies. Figure 2-17 shows the

predictions from models as well as the experiments, with the experiments showing an

increasing trend. The Langevin function predictions show some fluctuations but remain

more or less constant. However, the Sh and the MRSh equations show a gradual

increase as seen with the experiments, but do not show quantitative agreement with

experiments and nor a significant difference between themselves. Although, on the

basis of the FWHM there is a lack of quantitative agreement between the simulations

and the experiments, previous results indicate excellent agreement between the

predictions of the MRSh equation and the experiments based on the PSF shape and

the peak location. A significant difference may not have been observed between the

MRSh and Sh models due to the particle diameter being small. This is based on the

previous observations from the simulations of the magnetic particle spectrometer where

the use of large particle diameter showed a considerable difference between the model

predictions.

2.6 Conclusions

The studies shown in Chapter 2 provide evidence that ferrohydrodynamic

magnetization relaxation equations can potentially be used to model the response of

nanoparticles in MPI. The assumption that particles respond instantaneously to applied

magnetic fields as in the case of the Langevin function was shown to be inaccurate

45

through comparison between the simulations and the experiments. Also, the use of Sh

equation which has a constant relaxation time and the MRSh equation which has a field

dependent relaxation time to predict the response of particles in MPS and an MPR was

demonstrated. In the case of response of particles in a MPS, the number of harmonics

contributing to the signal was found to be greater for the MRSh equation as compared

to the Sh equation. However, both the equations predict that harmonics decay quicker

than those obtained using the Langevin equation. The effect of core diameter, shell

thickness, drive field amplitude and frequency on the harmonic spectra obtained from all

three models was studied. Through comparison of simulations with experimental

measurements in a Bruker MPS, better qualitative as well as quantitative agreement up

to the 19th harmonic between the predictions of the Sh equation and the MRSh equation

and the experimental measurements was observed. Similarly, in the studies involving

the response of particles to field applied in a MPR, comparison of simulation PSFs with

experiments shed light on the effect of finite relaxation on particle performance. Of

particular interest was the excellent agreement between the shapes of the PSF

obtained using the MRSh equation and the shape of the experimental PSF. This is

indeed remarkable as the particle properties were determined independently and the

simulations did not make use of fitting parameters. Also, the MRSh equation was able to

closely track the shift in the PSF peak as obtained from experimental measurements

while both the MRSh and the Sh equation were able to predict the increasing trend in

the FWHM with field frequency. These results show promise in predicting the response

of particles for use in MPI without the need of fitting parameters and the potential of

ferrohydrodynamic modeling in MPI.

46

Figure 2-1. Schematic illustration of the construction of a magnetic particle spectrometer

(MPS) and a magnetic particle relaxometer (MPR). a) A MPS consists of concentric excitation and receive coil with the probe chamber located in the center and b) a MPR has an additional bias coil which applies a slow ramping field.

Figure 2-2. Illustration of the applied field profile in a MPR. The slow ramping bias field

(left) is superimposed with an alternating drive field (center) to mimic the field profile experienced by magnetic nanoparticles by the movement of the FFR. Here, for illustration purposes the drive field frequency is small.

47

Figure 2-3.Cobalt ferrite particle characterization. a) Hydrodynamic diameter determined using dynamic light scattering (DLS) technique and b) transmission electron microscopy images showing spherical particles. c) Histogram obtained from analysis of TEM images to determine the physical diameter and d) DMS spectrum showing particles relaxing by the Brownian mechanism based on the location of the out of phase peak.

48

Figure 2-4.Simulation results showing a comparison between the Langevin function, Sh

equation and the MRSh equation. a) Dynamic hysteresis loops show broadening of loops for models which incorporate finite relaxation. b) Dimensionless nonlinear magnetization response of particles with time and c) comparison of signal shape showing a peak shift. d) FFT of acquired signal showing a quick decay in the harmonic spectrum for the Sh and the MRSh equation.

49

Figure 2-5.Effect of shell thickness on the harmonic spectra predicted by the Langevin function, MRSh and the Sh equation.

50

Figure 2-6. Effect of core diameter on the harmonic spectra at a constant hydrodynamic diameter.

51

Figure 2-7. Field dependence of parallel relaxation time for various core diameters at a constant hydrodynamic diameter of 70 nm.

Figure 2-8. Effect of drive field frequency on the harmonic spectra predictions of the Langevin function, MRSh and the Sh equation.

52

Figure 2-9. Effect of the drive field amplitude on predictions of the Langevin function, MRSh and Sh equation. Simulations modeled for field frequency of a) 1.5 kHz and b) 25 kHz.

Figure 2-10. Comparison of simulation predictions with measurements of cobalt ferrite particles in a Bruker magnetic particle spectrometer.

53

Figure 2-11. Magnetization response of particles to an applied field in a magnetic particle relaxometer. Magnetization predictions for particles in the region of a) strong bias field, b) moderate bias field, and c) low bias field showing variations in the shape and magnitude. Here, the symbols are a guide to the eye.

54

Figure 2-12. PSF comparison between the simulation predictions and experiments. The field conditions are a) 1.6 kHz, 20 mT, b) 12.2 kHz, 20 mT, c) 25 kHz, 20 mT, and d) 25 kHz, 40 mT. The markers are a guide to the eye while the lines represent the positive scan PSF.

55

Figure 2-13. Dependence of PSF peak location with the drive field amplitude for a frequency of 25 kHz.

Figure 2-14. PSF peak position at 20 mT amplitude and different drive field frequency.

56

Figure 2-15. Full width at half maximum (FWHM) obtained from point spread functions (PSFs) for condition of 25 kHz and variable drive field amplitude.

Figure 2-16. Simulation results from Langevin function at 25 kHz frequency and field

amplitude up to 60 mT. a) Dimensionless signal showing increase in signal strength with increase in field strength and b) signal PSF at different field amplitude showing narrowing of PSF at 20 mT.

57

Figure 2-17. Effect of drive field frequency with amplitude of 20 mT on the FWHM as predicted using models and from experiments.

58

CHAPTER 3 SPATIAL CONTROL OF MAGNETIC NANOPARTICLE HEATING IN MAGNETIC

FLUID HYPERTHERMIA USING MAGNETIC PARTICLE IMAGING FIELD

GRADIENTS

3.1 Background

Magnetic nanoparticles dissipate heat on application of a high frequency

alternating magnetic field [58]. This property of magnetic nanoparticle finds application

in a cancer treatment therapy called magnetic fluid hyperthermia (MFH) [59, 60]. In

MFH, magnetic nanoparticles are delivered to the site of tumor either through direct

injection or intravenously. If delivered intravenously, the particles accumulate in the

tumor passively due to the enhanced permeation and retention (EPR) effect [61, 62].

The application of an alternating magnetic field results in the rotation of the magnetic

moments, thus leading to nanoscale dissipation of heat. Cancer cells being susceptible

to temperature in the hyperthermia range (42-45 C) undergo cell death when the

temperature in the tumor reaches the hyperthermic range after application of an

alternating magnetic field [63-70]. Human tissue is permeable to the magnetic field and

thus magnetic field can pass through the tissue without obstruction. Typically, an

alternating magnetic field in the range of 100-500 kHz and amplitude of 10-30 mT is

used. The magnetic nanoparticles are also cleared from the tumor by blood or taken up

by the reticuloendothelial system leading to accumulation in the liver and spleen [71-73].

In this case, the application of the magnetic field to the entire body, can lead to

significant damage due to heating in the non-target organs like liver and spleen. Pre-

clinical studies in mice subjected to whole body hyperthermia have shown collateral

Figures reproduced with permission from R. Dhavalikar, C. Rinaldi, Theoretical predictions for spatially-focused heating of magnetic nanoparticles guided by magnetic particle imaging field gradients, J. Magn. Magn. Mater., 419 (2016) 267-273.

59

damage to healthy tissue [74]. Similarly, undesired side effects may arise due to

magnetically actuated drug release when exposed to whole body alternating magnetic

field. Thus, there is a need to develop approaches and technology to achieve spatially

selective heating and prevent undesired heating.

Application of a static magnetic field has been explored to achieve spatial heating

[75-77], but lack the ability to dynamically select the desired region of heating. To

overcome this limitation, the selection field gradient in MPI can be put to use. The

selection field gradient in MPI produces a field free region (FFR) which can be

translated easily either mechanically or electronically. The application of an alternating

magnetic field leads to free rotation of the magnetic moments in the FFR, whereas the

rotation of the magnetic moments in the saturated region of the selection field is

restricted due to the alignment of the magnetic moments in the direction of the selection

field. This feature of MPI can be utilized to actuate heating in the FFR, whereas

restricting heating in the saturated region. This approach has been explored by other

research groups [78-80], but currently there is a lack of models to accurately model the

selective heat dissipation.

A couple of models are currently available to describe the heat dissipation in a

uniform alternating magnetic, one of which describes the dependence of the heat

dissipation rate on the particle and field properties [58]. This model by Rosensweig

provides an explanation of the heat dissipation mechanism and potential applications of

heat dissipating magnetic nanoparticles. Other theoretical models have been also

developed to describe the effect of a strong DC magnetic field on the heat dissipation

rate [81-84]. More recently, an empirical model was developed by Murase et al. based

60

on Rosensweig’s model to calculate the heat dissipation rate of nanoparticles [80]. This

heat dissipation rate is also referred to as the specific absorption rate (SAR) in the MFH

literature [85]. However, the Rosensweig’s model as well as the empirical model by

Murase are valid in small field amplitude and do not take into account the field

dependence of relaxation time which would become relevant at field amplitudes and

frequencies typically employed in MFH. As shown in Chapter 2, the magnetization

relaxation equation developed by Martsenyuk, Raikher and Shliomis [43] which takes

into account the field dependence of relaxation time, has potential to be utilized for

modeling the behavior of the particles in high frequency and amplitude alternating fields.

Thus, Soto-Aquino and Rinaldi [86] made use of the MRSh equation to determine the

heat dissipation rate and study its effect on the SAR in an alternating magnetic field.

In Chapter 3, the MRSh equation is employed for the case of combined

alternating field and selection or bias field to determine the effect of field dependent

relaxation time on the SAR. Theoretical predictions showing the effect of field gradient

on the heating region are presented along with experiments showing tuning ability of

selection field gradient on the heating region.

3.2 Theory

The application of an alternating magnetic field results in the heat dissipation

from magnetic nanoparticles due to relaxation losses. The relaxation losses arise due to

the delay caused by moments relaxing by the Brownian mechanism which involves the

rotation of the moment fixed within the particle or due to moments relaxing by the Néel

mechanism which involves internal rotation of the moment [87, 88]. Based on

thermodynamic laws, the work done by the magnetic field on the nanoparticles is

61

dissipated as heat and the average rate of energy dissipation per cycle of period 2 p is

given by [86]

2

0

1,

2

pdH

Q M dtp dt

(3-1)

where is the magnetization and is the instantaneous magnetic field amplitude.

The instantaneous field for this case is composed of an alternating magnetic field and a

bias field, given by Equation (3-2).

( ) ( ).b exH t H H t (3-2)

Here, bH is a time independent bias field but dependent on the position of the FFR,

whereas the excitation field is a sinusoidal field given by cos( )exH t with field amplitude

exH and frequency . An illustration of the applied field and the associated

magnetization response of particles is given in Figure 3-1. As stated earlier the selection

field gradient creates a region of low bias field called the field free region (FFR). The

particles in the FFR are free to rotate in response to an applied alternating magnetic

field. This response is shown in the illustration as a sinusoidal magnetization response

between +M and –M, ultimately resulting in maximum heat dissipation in this region. For

particles in a strong bias field away from the FFR, the particles are aligned in the

direction of the field and hence the alternating field produces minor oscillations. This

results in reduced heat dissipation in the saturated region away from the FFR.

Using the applied field from Equation (3-2) in Equation (3-1) gives

2

0

sin( ) .2

p

exHQ M t dt

p

(3-3)

M H

62

To reduce the number of variables in the integration, the variables are non-

dimensionalized. Here, the magnetization is non-dimensionalized with the saturation

magnetization (sM ), whereas the frequency and time are non-dimensionalized with the

relaxation time ( )B . The non-dimensional variables are given by

; ; .s B BM M M t t (3-4)

Using the nondimensional terms, Equation (3-3) can be represented as

2

0

sin( ) .2

p

ex dH MQ M t dt

p

(3-5)

Here,B

p , dM is the domain magnetization and is the particle volume fraction.

Chapter 2 showed the potential of the MRSh equation [43] in modeling the response of

magnetic nanoparticles for conditions of high field amplitude and frequency, thus in this

study the magnetization response in Equation (3-5) was also modeled using the MRSh

equation. The simplified one dimensional MRSh equation for the conditions of collinear

field and no flow conditions in nondimensional terms is given by

[1 ] ,e

dMM

dt

(3-6)

where the Langevin parameter associated with the applied field is B

mH

k T and the

Langevin parameter associated with the effective field is ee

B

mH

k T . The differential

equation is solved numerically using MATLAB’s ODE45 function to obtain the

dimensionless magnetization at discrete time intervals. Numerical integration is carried

out using the trapezoidal rule to obtain the average rate of energy dissipation. The

energy dissipation rate is expressed in terms of specific absorption rate (SAR) given by

63

Q

SAR

, (3-7)

where is the particle volume fraction and is the density of the particles.

Experimentally, SAR is calculated using the initial temperature rise method given

by

exp

w w p p

i p

C m C mTSAR

t m

. (3-8)

Here, T is the initial change in temperature over a time interval t after application of

the field. wC and pC is the specific heat capacity of suspension fluid and the magnetic

nanoparticles respectively. wm and pm is the mass of water and iron oxide in the

sample respectively.

The selection field gradient G can be utilized to assign a SAR value for a given

bias field to a location relative to the FFR. Thus, the distance from the FFR is given by

bFFR

Hd

G (3-9)

3.3 Simulation Parameters

Simulations were carried out for iron oxide nanoparticles having a core diameter

of 20 nm and suspended in water. The particles were assumed to be non-interacting

and colloidally stable due to the presence of an organic shell. The shell thickness of the

particles in this study was selected as 2 nm. Simulations for frequencies typically used

in MPI in the range of 1-25 kHz were considered [15]. Higher frequencies in the range of

100- 500 kHz are typically employed in MFH for which simulations were also carried out

[89]. The bias field strength was 50 mT and the excitation field amplitude was varied

64

from 10-40 mT. SAR values were evaluated for these conditions and for selection field

gradients of 1-6 T/m.

For simulation of commercial particles (nanoMag-MIP) which were used in a

customized MPI-MFH setup constructed at the UC Berkeley Imaging Systems

Laboratory [90], a core diameter of 19 nm was used based on their reported value. The

hydrodynamic diameter of the particles was reported to be in the range of 20-100 nm.

For the purpose of simulations, a shell thickness of 8 nm was assumed as determined

from varying the dextran shell thickness so as to obtain SAR values comparable to

those in experiments. The particles were suspended in water and held at room

temperature. As in the experiment, a viscosity of 0.89 mPa.s and temperature of 298 K

were assumed in the simulations. The simulations were carried out at 353 kHz and 20

mT amplitude excitation field, and a field gradient of 2.35 T/m was used to match the

experimental setup.

3.4 Methods

A customized setup capable of generating a field gradient of 2.35 T/m and

applying a high frequency excitation field of 353 kHz was constructed at UC Berkeley

Imaging Systems Laboratory for testing the concept of spatially focused heating. A vial

containing 100 L of dextran coated iron oxide nanoparticles (nanoMag-MIP) was used

for the experiments. This vial was kept in a 3D printed sample holder which was placed

inside the FFR of the setup. A water cooled excitation coil surrounding the sample was

used to apply an excitation field of 20 mT amplitude and 353 kHz frequency for 10

seconds. The temperature rise due to magnetic particle heating was recorded using an

optical temperature probe inserted in the vial. The vial was moved along a single axis in

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1 mm steps where the same excitation field was applied. The recorded rise in

temperature at each point was then used to determine the SAR distribution as a

function of distance in one dimension.

Another set of experiments to show spatially focused heating and tuning using a

selection field gradient was conducted in a customized setup at the University of Florida

(UF). Concentrated peptized iron oxide nanoparticles synthesized in the Rinaldi lab by

the aqueous coprecipitation method [91, 92] were suspended in DI water for use in the

experiments. 80-100 L of particle solution was pipetted from a stock solution

(120mgFe/mL) and dispensed in each well within a central portion of a 384 well plate.

An Ambrell EASYHeat 8310 LI induction heating system was modified to generate an

alternating field. The induction heater head was fitted with an Ambrell three turn coil to

generate an alternating field of 321 kHz frequency and 16mT amplitude. A FLIR A300

series infrared thermal imaging camera mounted on a tripod was used for recording

temperature of the plate during experiments. A customized magnet rig was designed

using OpenSCAD and 3D printed to house repelling NdFeB magnets as well as a 384

well plate. Repelling NdFeB permanent magnets (BY0Y08-N52, K&J Magnetics) were

utilized to create a selection field gradient. Fiber optic temperature probes connected to

a Neoptix temperature sensor were employed to monitor the temperature of magnets

during experiments to prevent overheating. Field was applied for 45 seconds during

experiments to allow particle heating and observe a temperature change. The plate,

magnets, and particle sample were kept at room temperature before each experiment

run. A LakeShore gaussmeter coupled with a transverse Hall probe was used to

66

measure the magnetic field between the opposing magnets in order to characterize the

field gradient.

3.5 Results

3.5.1 SAR Distribution in a Bias Field

First, the effect of bias field on the SAR value distribution was studied through

simulations. Here, the bias field was varied from -50 mT to +50 mT and excitation field

frequencies typically employed in MPI and MFH were superimposed on the bias field.

The excitation field amplitude was held constant at 30 mT. The predictions shown in

Figure 3-2 indicate that a bias field and thus a selection field gradient can have a

significant effect on the SAR distribution. The magnitude of SAR values for MPI

frequencies is relatively lower than the SAR values for MFH frequencies as the

frequencies used in MPI are smaller than that used in MFH treatment. Also, as the

residence time of the FFR in any given region is small, in the order of milliseconds, the

total heat deposited which is a product of the SAR and the residence time will ultimately

be minute. Thus, minimum heat will be deposited during a MPI scan and the effects of

nanoscale heating during the scan will be negligible. On the other hand, for MFH

treatment, once the location of the tumor has been identified through MPI imaging, the

residence time of the FFR can be further increased to delve longer at the region of

interest so as to deliver a therapeutic heat dose. The nanoparticles accumulated in the

tumor will thus be able to dissipate sufficient heat to cause damage to cancer cells.

However, in both the cases, the shape of the SAR distribution resembles a bell shape,

i.e. the SAR values or heat dissipation is the highest when bias field is zero and

minimum when the bias field has a maximum value. Also, the maximum achievable

SAR is predicted to increase with an increase in the frequency.

67

3.5.2 Dynamic Hysteresis Loops

The shape of the SAR distribution curve can be explained by the dynamic

hysteresis loops shown in Figure 3-3. In Figure 3-3a, the magnetization response of

particles to the applied alternating magnetic field at frequencies typically employed in

MPI is shown. The plots show the effect of representative bias fields on the dynamic

hysteresis loop for conditions of strong, moderate and zero bias field. At zero bias field,

the area of the loop at 25 kHz is the largest and as SAR is given by the product of

frequency and the area of the loop, increase in frequency leads to increase in SAR

values. Also, higher frequencies lead to broadening of the loop, primarily due to the

inability of the particles to track the change in the magnetic field. The delay due to finite

relaxation leads to a lag in the magnetization response of these particles to the applied

alternating field. The magnetization versus time plots in Figures 3-4, 3-5 and 3-6 show a

significant delay in the magnetization response. However, for MPI frequencies, the

magnetization response reaches values of unity. In contrast, for MFH frequencies, the

dynamic hysteresis loop as shown in Figure 3-3b become compact with increase in the

frequency. This can be attributed to the delay caused by the finite relaxation time and

the inability of the particles to follow the rapid change in the alternating field. As shown

in Figures 3-5b and 3-6, the failure of particles to respond instantaneously to the

alternating magnetic field, leads to reduced magnetization response where

magnetization changes between -0.5 to +0.5. However, SAR being proportional to the

product of frequency and the area of the loop, the decrease in area is compensated with

an increase in the frequency, ultimately leading to higher SAR values than those

obtained for MPI frequencies.

68

Next, as the bias field is increased to 25 mT, the dynamic hysteresis loop shows

an asymmetry. This asymmetry originates from the magnetization response with time

where the bias field and the alternating field apply a competing torque when they are in

the opposite direction and an additive torque when they are in the same direction.

Because of the competing nature of the torques acting on the particles, at a given

frequency the area of the dynamic hysteresis loop also decreases, ultimately leading to

a drop in the SAR. The drop is significant in the case of MFH frequencies since the area

is much smaller. The magnetization versus time plots in Figures 3-4, 3-5 and 3-6 show

the response of the particles in the presence of a bias field and provide further evidence

to explain the change in magnetization over short intervals.

Now, when the bias field is further increased to 50 mT, most of the particles are

aligned in the direction of the bias field. The presence of the alternating magnetic field

leads to minor oscillations in the magnetization, but the oscillations are restricted to a

small region, with the magnetization reaching saturation. These changes in the

magnetization response of the particles result in the dynamic hysteresis loop to become

flat and thus result in a reduced loop area. The extreme reduction in the loop area leads

to a further drop in the SAR. The simulations for MPI and MFH frequencies show this

type of behavior. Figure 3-7 shows the calculated dynamic hysteresis loop areas and its

product with frequencies for the various frequencies considered in this study, thus

providing further quantitative evidence to support the arguments made earlier.

The effect of field amplitude on the SAR value for various frequencies was also

studied. Here, the field amplitude was changed between 10 mT to 40 mT. The

maximum achievable SAR i.e. when the bias field is zero was compared at different

69

field amplitudes. Figure 3-8 shows the SAR value obtained from simulations for the

FFR. For MFH frequencies, a sharp rise in the SAR was observed beyond 25 kHz. The

SAR curves reach a plateau for a fixed amplitude condition indicating that the product of

frequency and area of dynamic hysteresis loop becomes constant. As shown in the

figure inset, the SAR continues to rise for frequencies typically employed in MPI due to

a broadening of the dynamic hysteresis loop as well as an increase in applied excitation

field frequency. Also, with an increase in the field amplitude, there is a faster alignment

of the moments with the applied field due to the stronger magnetic torque experienced

by the moments, ultimately resulting in attainment of higher magnetization values and

thus SAR.

3.5.3 Spatial Focusing of SAR using Selection Field Gradient

The selection field gradient was varied to determine its effect on the distribution

of SAR. Figure 3-9 shows simulation predictions for SAR at selection field gradients of 1

up to 6 T/m. The SAR was presented as a function of the distance from the FFR to

show the tuning ability of the field gradient. With the increase in the field gradient, it was

observed that the SAR distribution became narrower and sharper. This observation

paves way to tune the region of heating and avoid non-specific heating damage. Also,

depending on the area of the region to be treated, the field gradient can be adjusted to

target the desired region of heating. The increase in field gradient implies that the bias

field changes quickly within a given distance. This change can be achieved by either

increasing the strength of the field producing coils or by adjusting the distance between

the coils. The simulation predictions show the possibility to tune the region of heating

up to millimeter scale and provide an innovative way to achieve spatial control over

heating in MFH.

70

3.5.4 Spatial Control of Heating using a Field Gradient in Experiments

A comparison between the SAR values obtained from simulation with the SAR

values obtained from experiments conducted in the UC Berkeley setup is shown in

Figure 3-10. A good qualitative agreement was observed between the simulations and

experiments. In experiments as well as simulations, maximum heat dissipation was

observed when the bias field was zero and minimum heat dissipation as the bias field

was increased. This agreement between simulations and experiments could potentially

lead to simulations being used as a pre-planning tool for MFH treatment.

To further study the effect of field gradient on nanoparticle heating, another

experimental setup, as shown in Figure 3-11a, was constructed at UF. Here, the

infrared thermal camera was used to record the temperature of the plate filled with

nanoparticle sample and the generated image was used to compare the results

obtained from the experiments in the presence and absence of the permanent magnets.

Figure 3-11b shows uniform heating of all filled wells when an alternating field is applied

with a temperature change of 7C. This is to be expected as there is no field gradient

and all wells are subjected to the same alternating field. However, when opposing

NdFeB magnets are introduced in the setup, as shown in Figure 3-11c, non-uniform

heating is observed across the particle filled region of the plate. Maximum temperature

is achieved in the region corresponding to the field free region (FFR) generated by the

opposing NdFeB magnets whereas minimum temperature is obtained in wells closest to

magnets. In this experiment, the presence of a field gradient leads to a temperature

distribution across the plate as a consequence of differential heat generation from the

particles. The field values were measured at equidistant points along the central line

71

joining the magnets, as shown in Figure 3-12a, and were used to calculate the field

gradient (2.72 T/m) from a linear fit to the measured values, as shown in Figure 3-12b.

To explore the effect of change in field gradient on heating area, the distance

between the magnets was varied. For this purpose, a 3D printed magnet rig with the

capability to manually adjust distance between the magnets was manufactured as

shown in Figure 3-13. In order to obtain a higher field gradient, the distance between

the magnets was reduced by bringing the opposing magnets closer to each other. As

shown in Figure 3-13, at low field gradient when the magnets are separated by a

distance of 222 mm, uniform heating is observed in all the particle filled wells. This is

similar to the case where the field gradient is completely absent and results as a

consequence of the magnets being far apart. As the magnets are brought closer and

the distance is reduced to 186 mm, wells closest to the magnets reach lower

temperatures as compared to the wells located at the center. Finally, when the distance

between the magnets is reduced further to 151 mm, the resulting high field gradient

further reduces the region of heating. A change in the field gradient through a change in

distance between the magnets was shown to affect the heating area and thus provide

experimental proof of spatially controlling the heating area using magnetic field

gradients.

3.6 Conclusions

The MRSh magnetization relaxation equation was utilized to determine the

magnetization response of particles to an applied field consisting of a bias field and an

alternating magnetic field. The bias field was found to have a significant effect on the

SAR distribution, thus resulting in reduced heat dissipation under the influence of a

strong bias field and maximum heat dissipation in the FFR. The presence of a bias field

72

was found to affect the shape of the dynamic hysteresis loop, ultimately affecting the

calculated SAR. Experiments using a 384 well plate filled with magnetic nanoparticles

exposed to an alternating magnetic field in a selection field gradient showed maximum

temperature in the field free region, while reduced temperatures was observed in wells

located in the saturated region. This difference in temperature was found to be a

consequence of reduced heat dissipation in the presence of a field gradient. The

simulations also showed the potential of using the selection field gradient to tune the

region of heating, where an increase in field gradient was found to theoretically reduce

the region of heating to a distance of millimeters. In experiments, the field gradient was

varied by changing the distance between opposing magnets and a narrowing of heating

region was observed as the gradient strength was increased. The qualitative agreement

observed between simulation and experiment SAR distribution could be used for pre-

planning MFH treatment in the future. Through this entire study, consisting of

simulations and experiments, the potential use of MPI selection field gradients in

spatially focusing heating in MFH was highlighted and a platform for developing image

guided heat delivery systems was realized.

73

Figure 3-1. Applied field profile for selective heating of magnetic nanoparticles.

Figure 3-2. Distribution of SAR values as a function of the bias field. Simulation

predictions at excitation field frequencies used in a) MPI and b) MFH at 30 mT amplitude show a bell shaped distribution.

74

Figure 3-3. Dynamic hysteresis loops for representative bias fields. Increase in the bias field strength shows reduction in the dynamic hysteresis loop area for a) MPI frequencies and b) MFH frequencies.

75

Figure 3-4. Magnetization response with time for magnetic field frequencies used in MPI in the presence of a bias field. Magnetization response in comparison with the applied field at an alternating field frequency of a) 5 kHz and b) 15 kHz, and field amplitude of 30 mT.

76

Figure 3-5. Effect of bias field on the magnetization response of particles at frequencies typically used in MPI and MFH. Magnetization curves for a) MPI frequency of 25 kHz and b) MFH frequency of 100 kHz are compared with the applied field.

77

Figure 3-6. Magnetization response of particles to alternating field frequencies employed in MFH for bias field of 50 mT, 25 mT, and 0 mT. a) Magnetization response at 300 kHz frequency and b) magnetization response at 500 kHz frequency.

78

Figure 3-7. Comparison of the area of dynamic hysteresis loops and its product with frequencies for representative conditions of bias field and frequencies used in MPI and MFH. a) Area of dynamic hysteresis loop for 5, 10, and 25 kHz alternating magnetic field, b) area of dynamic hysteresis loop for 100, 300, and 500 kHz alternating magnetic field, c)product of frequency and dynamic hysteresis loop area at MPI frequencies, and d) product of frequency and dynamic hysteresis loop area at MFH frequencies.

79

Figure 3-8. Dependence of maximum achievable SAR in the FFR on the field amplitude of the excitation field for frequencies employed in MPI and MFH. Inset- MPI frequency SAR predictions for excitation field amplitude between 10 and 40 mT.

Figure 3-9. Use of selection field gradient to achieve millimeter scale spatial focusing of

heat deposition.

80

Figure 3-10. Comparison of simulation results of SAR distribution with experiments

conducted in the UC Berkeley setup.

Figure 3-11. Demonstration of spatial focusing of heating by magnetic nanoparticles

using a selection magnetic field gradient. a) Experimental setup b) Uniform heating of all particle-filled wells in the absence of the magnetic field gradient. c) Spatially focused heating of particle-filled wells in the presence of the magnetic field gradient.

81

Figure 3-12. Determination of field gradient. a) Field values measured at equidistant

points along the central line joining the magnets in the magnet rig. The letters were used to help keep track of measurement points. b) Field gradient was calculated from a linear fit to the measured field values.

82

Figure 3-13. Experiments showing spatial focusing of heating region with a change in the selection field gradient.

83

CHAPTER 4 MODELING THE RESPONSE OF NÉEL PARTICLES FOR USE IN MAGNETIC

PARTICLE IMAGING

4.1 Background

The ferrohydrodynamic models described in Chapter 2 and Chapter 3 are strictly

applicable to particles relaxing by the Brownian mechanism. However, most magnetic

nanoparticles used for MPI respond to changes in magnetic fields by the so-called Néel

relaxation mechanism [50]. In these particles, the magnetic moment is free is rotate

internally without the physical rotation of the particle itself and can change its orientation

relative to the particle crystallographic axes. Resovist, also known as Ferucarbotran, an

FDA approved contrast agent for detecting lesions in the liver [11] was adopted for use

as an MPI tracer [1, 2, 12, 13, 93, 94]. It consists of clusters of iron oxide nanoparticles

which are in the size range of 3-5 nm and primarily respond by the Néel mechanism.

They are currently also considered as gold standard in MPI and used in performance

assessment of synthesized nanoparticles. There is a growing interest to synthesize MPI

tailored particles with large diameters primarily responding by the Néel mechanism, to

increase the MPI signal and resolution [18]. Thus, understanding and modeling their

behavior in applied magnetic fields in MPI is crucial.

Most studies use the Langevin function to model the response of the magnetic

nanoparticles [8, 9, 25]. However, the Langevin function is only strictly applicable in the

case of systems in equilibrium and does not take into account the dynamic state of an

MPI system. In another study, a bimodal distribution model proposed by Eberbeck et al.

[28] was shown to fit the Resovist results well, suggesting that the magnetic

nanoparticle clusters behave like single core magnetic nanoparticles of about 20 nm.

However, this model does not account for the response of individual particles and hence

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cannot be utilized to explain the effect of properties of individual particles and particle-

particle interactions on the MPI signal. Thus, models which can predict the response of

individual particles would be beneficial in estimating properties of tailored nanoparticles

to be synthesized for MPI applications.

The Landau-Lifshitz-Gilbert (LLG) equation [95-97] has been used to model the

response of individual particles with internally rotating magnetic moments. This equation

takes into account the precession of the magnetic moment in a magnetic field and the

convergence of the magnetic moment in the direction of the field. This equation has

been employed to study the effect of shape anisotropy and crystal anisotropy on the

MPI performance of magnetic nanoparticles relaxing by the Néel mechanism [38, 98].

These studies neglected the Brownian motion responsible for the change in the

orientation of the particle easy axis due to thermal fluctuations of the surrounding fluid in

a suspension of magnetic nanoparticles in a carrier fluid. The easy axis was also fixed

in the direction of the applied field. In another study by Reeves and Weaver [41] , a

combined Néel and Brownian algorithm was developed to account for the rotation of the

easy axis and the internal rotation of the moment to an applied magnetic field for

randomized easy axis orientations. To determine the orientation of the magnetic

moments, these studies made use of Euler angles, which are prone to singularities.

Although these studies highlight the potential of the LLG equation in modeling the

response of Néel particles, the effect of change in the various parameters of the LLG

equation on the particle response is explored to a minimal extent.

In Chapter 4, an LLG algorithm is developed to understand the response of

immobilized Néel particles at different parameters. The algorithm makes use of

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quaternion parameters to determine the orientation of the moments so as to avoid

singularities encountered by the use of Euler angles. Results from application of a

unidirectional static and alternating magnetic field, at conditions typically employed in

MPI, on the response of particles for the case of uniaxial crystal anisotropy are

presented and discussed.

4.2 Theory

4.2.1 Landau-Lifshitz-Gilbert Equation

For immobilized single domain magnetic nanoparticles, the change in orientation

of the internal magnetic moment is given by the Landau-Lifshitz-Gilbert equation [38,

41]

det det2 2

,

( ( )) ( ( ))1 (1 )

ii fl i i fl

i s

d

dt m

mm H H m m H H . (4-1)

Here, im is the magnetic moment, is the gyromagnetic ratio, is the damping

constant, ,i sm is the saturation value of the magnetic moment, detH is the deterministic

field, and flH is the fluctuating field. The first term in the equation describes the moment

precession while the second term describes the convergence of the precession

trajectory to the direction of the field. The deterministic field is given by Equation (4-2)

and consists of contributions by the external fields like the DC field, alternating or

excitation field, and the anisotropy field.

det dc ex an H H H H , (4-2)

where sin(2 t)ex ex fH H and 2

,

2 p

an i

i s

KV

m H m n n . Here, f is the frequency of

excitation field, K is the uniaxial crystalline anisotropy constant, and n is the direction

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of the easy axis. The fluctuating field contribution to the total field arises due to thermal

fluctuations and is given by [38]

,

,

2( ) 0; ( ) ( ') ( ')

fl fl fl

i i j Bi j

i s

k Tt t t t t

m

H H H , (4-3)

where ,i j represent the Cartesian directions. A Stratonovich interpretation of the

integral is used to evaluate the stochastic differential equation. The numerical form of

the resulting equation is

det det2 2

,

2 2

,

( ) ( )1 (1 )

( ) ( ) ,1 (1 )

i i i i

i s

i fl i i fl

i s

t tm

t tm

m m H m m H

m H m m H

(4-4)

with

,

2 Bfl

i s

k T

m

H W . (4-5)

Here, W consists of random numbers with standard normal distribution.

To reduce the number of variables in the equation, variables in Equation (4-4)

were made dimensionless. Time was non-dimensionalized using the gyromagnetic ratio

and the saturation magnetization of the suspension0 sM , and the vector variables

with respect to their corresponding magnitudes. Here, 0 is the permeability of vacuum.

detdet 0'

det ,

; ; ;fli

i fl s

i s fl

t t MH m H

HH m

H m H (4-6)

Substituting the dimensionless variables in Equation (4-4) reduces Equation (4-4) to

det det

2 2 2 2

det det(1 ) ( ) (1 ) ( )

( ) ( ) ,fl fl

i i i i

i i i

t t

t t

m m H m m H

m H m m H (4-7)

where

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, det

det2

, 0

1; ; 2 .

(1 )

i sB

i s s B

m Hk TW

m M k T

(4-8)

The LLG equation is transformed to particle coordinates using the transformation

matrix A , given by

0

2 2 2 2

1 2 3 1 2 0 3 1 3 0 2

2 2 2 2

1 2 0 3 0 1 2 3 2 3 0 1

2 2 2 2

1 3 0 2 2 3 0 1 0 1 2 3

2 2 2 2

2 2 2 2 ,

2 2 2 2

q q q q q q q q q q q q

A q q q q q q q q q q q q

q q q q q q q q q q q q

(4-9)

where 0 1 2 3, , , and q q q q are quaternion parameters with 2 2 2 2

0 1 2 3 1q q q q . The

transformed dimensionless magnetic moment, deterministic field, and fluctuating field

are given by Equations (4-10), (4-11), and (4-12) respectively.

'

i iA m m (4-10)

'

det detA H H (4-11)

'

fl flA H H (4-12)

Here, the primed variables indicate variables in the particle coordinate system. Thus,

the transformed LLG equation is given by

det det

' 2 2 ' ' 2 2 ' ' '

det det

' ' ' ' '

(1 ) ( ) (1 ) ( )

( ) ( ) .fl fl

i i i i

i i i

t t

t t

m m H m m H

m H m m H (4-13)

The change in the individual magnetic moment at each time step is calculated

numerically using the forward Euler method to determine the magnetic moment at

discrete time points. Magnetic moments are averaged after each time step to determine

the net dimensionless magnetic moment in each direction.

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4.2.2 Determination of Relaxation Time

For determining the response of magnetic moments to an applied DC field in the

z direction, the magnetization relaxation equation is given by

( )

.eqm mdm

dt

(4-14)

Here, is the alignment or the relaxation time and eqm is the equilibrium dimensionless

magnetic moment. For the situation of applied magnetic field, Equation (4-14) can be

simplified using the initial condition, 0m at 0t , to give

ln(1 ) .eq

m t

m (4-15)

Here, eqm is the equilibrium dimensionless magnetic moment after the field has been

applied for a long duration. Similarly, in the case when an applied magnetic field is

turned off or external field is absent, the response of magnetic moment can be given by

.dm m

dt (4-16)

Using the initial condition ( 0) eqm t m , Equation (4-16) can be written as

ln( ) .eq

m t

m (4-17)

Here, eqm is the equilibrium dimensionless magnetic moment at 0t under an applied

external magnetic field.

4.3 Simulation Parameters

Spherical iron oxide nanoparticles with diameters between 5-20 nm were used in

the simulations. The particles were assumed to be immobilized, i.e. their Brownian

motion was restricted. The saturation magnetization of the particles was considered to

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be 446 kA/m, corresponding to the domain magnetization [50]. The effect of damping

constant in the range 0.01 – 1 [40, 98] was explored. The uniaxial anisotropy constant

was considered to be 15 kJ/m3 [52] and the effect of varying anisotropy constant

between 5-50 kJ/m3 [58, 99] on the particle response was also studied. The

gyromagnetic ratio used in the simulations was 1.3 GHz/T [40]. The temperature in the

simulations was 298 K. The particle easy axis direction was assumed to be in the [0 0 1]

particle crystallographic direction. The DC field was applied in the z direction and varied

between 10-200 mT. An alternating field of 25 kHz frequency and amplitude between 5

and 25 mT was applied in the z direction. In order to obtain statistically reliable results,

the number of particles used in the simulations was chosen to be 1000 in some

simulations and as large as 10,000 in others. The dimensionless time step size used in

simulations was 0.01(2.18 ps) and 0.001(0.22 ps).

4.4 Results

4.4.1 Response of Particles to a DC Field

Simulations to assess the response of particles to a constant DC field were

executed for different particle diameters. Figure 4-1 shows the response of 1000

particles to a 100 mT DC field in the z direction, for particles with a size of 5 nm. Here,

the easy axes are randomly distributed and the damping constant was chosen to be

0.1. The field is switched on after 105 time steps. It can be seen that the average

dimensionless magnetic moment in the z direction rises in response to the applied

magnetic field and reaches a constant value in ~1s. The magnetic field applies a

torque to the moments to align them in the direction of the field. An equilibrium state is

reached after a long duration due to a balance between the torques exerted by the

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applied field, the fluctuating field, and the anisotropy field trying to align the moments in

the direction of the easy axis. Similarly, as shown in Figures 4-2 and 4-3, for particle

diameters of 10nm and 15 nm, a similar trend is observed. However, the magnitude of

the average dimensionless magnetic moment is greater than that obtained for 5 nm

particles, indicating increased alignment of the magnetic moments in the direction of the

field, ultimately reaching values close to unity. Representative moment orientation plots

for 15 nm particles at different time points are shown in Figure 4-4, to visualize the

moment orientation process in the application of a 100 mT DC field. Here, red points

represent the randomly oriented easy axis of each particle and the blue points represent

the magnetic moment associated with the particles. Figure 4-4a shows the uniform

distribution of the moments over the sphere in the absence of an external magnetic

field. The application of a constant DC field in the z direction results in the alignment of

moments in the direction of the field, as shown in Figures 4-4b, c. After applying the

constant DC field for a long time, the moments are preferably aligned in the direction of

the field as shown in Figure 4-4d.

Next, the equilibrium average magnetic moment values in a constant DC field of

varying field strength (10-200 mT) were obtained from simulations and compared with

the predictions of the Langevin function. The average magnetic moment values were

time averaged over the last 1000 time steps in each simulation to determine the

equilibrium value. Figure 4-5 shows the comparison of the time averaged dimensionless

magnetic moment values at equilibrium with the predictions of the Langevin function for

a simulation with 5 nm particles. Good agreement is obtained with the predictions of the

Langevin function, indicating that the particles reach a state of equilibrium if a constant

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field is applied for long durations. For 10 nm particles in Figure 4-6, the equilibrium

values obtained from simulations show a similar trend but with some deviation in the

non-linear region of the Langevin curve. However, a considerable deviation from the

Langevin curve is observed in the nonlinear portion for the case of 15 nm particles, as

shown in Figure 4-7. A much larger deviation is observed in the nonlinear portion for the

case of 20nm particles, as shown in Figure 4-8. This could be explained by the increase

in the anisotropy field, which is proportional to the cube of the particle diameter,

hindering the process of alignment of the magnetic moments in the direction of the

applied field. This ultimately leads to a reduced magnetic moment achievable for field

values in the nonlinear region at equilibrium. However, higher DC fields are able to

overcome the resistance offered by the anisotropy field to show agreement with the

predictions of the Langevin function. Figure 4-9 provides evidence that simulations with

20 nm particles exposed to a DC field of 40 mT reach equilibrium but do not reach an

equilibrium value predicted by the Langevin function.

The simulations with the field switch on condition were analyzed further to

determine the time required to align the magnetic moments in the direction of the field.

Figure 4-10 shows simulation results of the response of 15 nm particle size magnetic

moments to the applied field according to Equation(4-15). At low field strengths, the

slope of the curve is small indicating that a longer time is required for alignment with the

field. However, as the field strength is increased, the magnetic moments align much

faster in the direction of the field as observed from the sharp drop in the curve. The

slope of the line fitted to the initial portion of the curve was used to determine the

alignment time. Figure 4-11 shows the alignment time for different diameters at various

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field strengths. The alignment time for 5nm particles shows considerable fluctuations

and does not display a clear trend as seen for higher sized particles. The 20 nm

particles show shorter alignment time than the 10 nm particles for the same field

strength. A significant difference is obtained between the alignment time calculated at

low field and at high field strength with a decreasing trend observed in the case of 10,

15, and 20 nm particles with increasing field strength.

The effect of damping constant on the response of particles to an applied field

was studied. Simulation results of 15 nm particles to 100 mT DC magnetic field with a

damping constant of 0.01 and with a damping constant of 1 are shown in Figure 4-

12 and Figure 4-13 respectively. The simulations show a slower response to the applied

field in the case of 0.01 , as compared to the response in the case of 1 . An

intermediate response was observed in the case of 0.1 as was seen in Figure 4-3.

However, in all the cases the equilibrium dimensionless magnetic moment was found to

reach unity on the application of a DC field for a long duration. As the value of

determines the rate at which the moments align with the applied field, the change in its

value did not affect the achievable equilibrium values. However, almost two order of

magnitude difference was observed in the alignment time for simulations at =0.01 and

=1 as shown in Figure 4-14. This result suggests that the choice of the damping

parameter can significantly affect the response of particles to an applied field.

4.4.2 Particle Response in the Absence of a Magnetic Field

The response of particles in the absence of a magnetic field was assessed

through simulations of 10,000 particles with randomized easy axis. Figure 4-15 shows a

representative plot of the decay of average dimensionless magnetic moment for 10 nm

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particles when a DC magnetic field in z direction is switched off. The damping constant

was considered as 0.1. The initial rise in the z direction average dimensionless

magnetic moment is due to the application of a DC magnetic field in the z direction as a

result of the alignment of magnetic moments in the direction of the field. The instant the

DC magnetic field is removed, the magnetic moments snap to align with their easy axis,

resulting in a sudden drop in the average magnetic moment. The average magnetic

moment continues to decay to zero as thermal fluctuations randomize the direction of

the magnetic moments. Magnetic moment distribution plots at representative time points

from simulation of 1000 particles after removal of an applied field are shown in Figure 4-

16. At time equals zero, Figure 4-16a shows the moments aligned in the direction of the

applied field. Once the field is removed, the moments relax to align with their easy axis

as shown in Figures 4-16b, c. After sufficient time is allowed to pass, the moments are

randomized due to thermal fluctuations as shown in Figure 4-16d. In order to

understand the relaxation process, another simulation with the easy axes of all the

particles initially aligned in the same direction as the applied field direction was carried

out. Thus, once the field is removed, the direction of the moments is randomized by the

thermal fluctuations leading to decay in the average dimensionless magnetic moment in

the z direction as shown in Figure 4-17. This decay is shown at different time points in

Figure 4-18. As shown in Figure 4-18a, the easy axis of all particles points in the same

direction as the applied field and the moments are aligned in the direction of applied

field. Once the field is switched off, thermal fluctuations randomize the orientation of the

moments with time as shown in Figures 4-18b, c, and d.

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Simulations with particles of different diameters with their easy axes aligned in

the direction of the applied field were executed to understand the effect of particle size

and volume on the relaxation time. To facilitate calculation of relaxation time, the

magnetic moment decay was plotted as lneq

m

m

versus time based on the simplified

Equation (4-17). Figure 4-19 shows the decay in the magnetic moment for various

diameters and it can be observed from the plots that the 5nm particle size magnetic

moment decays faster than the 10 nm particle size magnetic moment. This can be

explained by the anisotropy field or anisotropy energy preventing the thermal

fluctuations from randomizing the moment orientation in the case of 10 nm particles.

The slope of the decaying curve was determined through a degree one polynomial fit to

the initial portion of the decaying magnetic moment curve. The relaxation time

determined from simulations was compared with the relaxation time obtained using the

Néel-Arrhenius equation given by Equation (2-3). As shown in Figure 4-20, the

predictions from simulations show a linear trend in good qualitative agreement with the

predictions of the Néel-Arrhenius equation. However, quantitative agreement was not

observed with that of the Néel-Arrhenius equation.

To understand the influence of damping constant on the particle response in

the absence of a magnetic field, simulations with damping constant of 0.01 and 1 were

carried out. Figure 4-21 shows the dependence of the relaxation time on the damping

parameter. Although, a trend similar to that of the Néel-Arrhenius equation was

observed for all values of , qualitative and quantitative agreement was best observed

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for =1. Thus, this result provides further evidence that choice of the damping

parameter in simulations can affect the response of particles.

Now, the effect of varying the anisotropy constant was studied at damping

constant of 1. Figure 4-22 shows the decay in the magnetic moment of 10 nm particles

when an applied DC field is switched off for anisotropy constants reported for iron oxide

nanoparticles in the literature. Here, the decay is observed to be slower when the

anisotropy constant was higher. This is to be expected as the increase in the anisotropy

constant would prevent fast randomization of the magnetic moment due to thermal

fluctuations. The relaxation time obtained from a linear fit to the decaying curve for

particle diameters between 5-10 nm at various anisotropy constants is given in Figure 4-

23. An increasing trend is observed in the plot at all values of the anisotropy constant.

For a fixed particle diameter the relaxation time was seen to increase with the

anisotropy constant.

4.4.3 Response of Particles to a Sinusoidal Magnetic Field

To understand the dynamic response of particles, simulation of 1000 magnetic

nanoparticles with randomized easy axes to a sinusoidal excitation field was carried out.

Initial simulations made use of =0.1 as the value of the damping constant. The choice

of this damping constant was based on previous work by other research groups [38, 96,

98]. The dimensionless average magnetic moment versus time response for particles

with diameter 5 nm, 10 nm, and 15 nm to an alternating field of 25 kHz and amplitude

25 mT in the z direction, typical field conditions employed in MPI, is shown in Figure 4-

24, 4-25 and 4-26, respectively. The simulation was run for 80 s with a dimensionless

time step of 0.01(2.18 ps) to obtain the response of particles over two periods. The

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normalized excitation field exH is plotted along with the magnetic moment response to

facilitate comparison and understand the dynamic response. As seen in the plots, the 5

nm and 10 nm particles show negligible deviation from the applied field i.e. the average

dimensionless magnetic moment drops to zero when the sinusoidal magnetic field is

zero, indicating that the moments track the change in the magnetic field quite well.

However, for 15 nm particles, a significant delay in the magnetic moment response is

observed. This can be attributed to the large anisotropy working to prevent the

alignment of the moments in the direction of changing magnetic field. Magnetic moment

distribution at representative time points in one cycle of the alternating magnetic field is

shown in Figure 4-27 to visualize the particle response to an alternating magnetic field.

Figure 4-27a shows the moment distribution at time t=0 where the magnetic moments

are uniformly distributed. Figure 4-27b shows moments aligned in the direction of the

applied field. As the field continues to change and reaches zero, the moments get

distributed around the entire sphere, as shown in Figure 4-27c. Finally, as the field

switches direction, the moments are aligned in the new direction of the field as shown in

Figure 4-27d.

Next, a comparison between the simulations for different applied field strengths is

shown in Figure 4-28. The excitation field is normalized to facilitate in the comparison.

As expected, the average dimensionless magnetic moment showed an increase in

magnitude with an increase in the field amplitude, while the delay was seen to increase

at low field amplitude indicating that the small applied field strength was insufficient to

change the direction of the magnetic moment quickly. However, at large field strengths,

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the particles are able to better track the changes in the alternating magnetic field

resulting in a reduced lag.

As was seen in earlier results, the damping constant showed a significant effect

on the response of particles and hence simulations were also done at =0.01 and =1.

Figure 4-29 shows the simulation results for =0.01 at different field amplitudes for 15

nm particles. Here, a similar trend is observed when the field strength is changed but

the shift in the magnetic moment response is much larger as compared to the case of

=0.1. This is to be expected as the damping constant affects the rate at which the

moment aligns with the changing field and thus a smaller value of the damping constant

leads to the magnetic moments lagging behind the changing magnetic field. However,

at =1 the shift is much smaller and the shape of the magnetic moment curve is

symmetric, as shown in Figure 4-30. Here, the magnetic moment curve closely tracks

the change in the magnetic field as compared to the magnetic moment curves obtained

at smaller damping constant values. These results further highlight the importance of

choosing the damping constant carefully and suggest the use of =1 as a damping

constant in these simulations.

Lastly, the simulation results were compared with the predictions of the Langevin

function. Figure 4-31 shows a comparison of the average magnetic moment response to

a sinusoidal magnetic field (25 kHz, 25 mT) with the predictions of the Langevin function

for simulations with =1 and particles with 15 nm diameter. As the Langevin function

models the behavior of particles which respond instantaneously to a change in the field,

it can be seen that the dimensionless magnetic moment reaches zero when the field is

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zero. However, the simulations show a small lag in response to an alternating field

resulting in a net magnetic moment even when the applied field is zero.

4.5 Conclusions

The Landau-Lifshitz-Gilbert (LLG) equation was used to model immobilized

particles with a magnetic moment which is free to rotate internally. The LLG simulations

use quaternion parameters for determining the orientation of the moments and take into

account the uniaxial crystal anisotropy. Studies involving the use of DC magnetic field

showed good agreement with the predictions of the Langevin function for particles with

a diameter of 5nm and 10 nm. However, a considerable deviation was observed in the

time averaged dimensionless magnetic moment from the predictions of the Langevin

function for 15 nm and 20 nm particles. In studies involving removal of a DC field, the

magnetic moment was found to decay to a new equilibrium state because of thermal

fluctuations. The choice of damping parameter was found to affect the relaxation

behavior. Simulations at =1 showed better qualitative and quantitative agreement with

the Néel-Arrhenius equation as compared to the simulations at lower damping constant

values. In the presence of an alternating field, 5 and 10 nm particles were found to

closely track the changes in the alternating field, however, the average dimensionless

magnetic moment from simulations of 15 nm particles showed a significant lag. This

was attributed to the applied field being unable to achieve change in the moment

direction against a large anisotropy. The effect of damping constant was also explored

and it was found that simulations at =1 closely tracked the change in the applied

magnetic field. The damping constant value was found to affect the simulation results

significantly. In summary, this study showed the potential of using the LLG equation in

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modeling Néel particles and laid a foundation to extend the study to multi-directional

fields, incorporate Brownian motion and help understand the effect on MPI signal and

resolution.

100

Figure 4-1. Response of 5 nm magnetic nanoparticles to an applied DC field of 100 mT

in the z direction. The damping constant was equal to 0.1 in the simulations.

101

Figure 4-2. Response of 10 nm magnetic nanoparticles to a constant DC field of 100 mT

in the z direction. The damping constant was equal to 0.1 in the simulations.

102

Figure 4-3. Response of 15 nm magnetic nanoparticles to a constant DC field of 100

mT. The damping constant was equal to 0.1 in the simulations.

103

Figure 4-4. Plots showing the distribution of individual moments after application of a

100 mT field in the z direction for 15 nm particles at representative time points. Here, the red dots represent the easy axis of each particle and the blue dots represent the magnetic moments. The orientation of the moments is shown at time equal to a) zero, b) 2 ns, c) 6.5 ns, and d) 978 ns.

104

Figure 4-5. Comparison of equilibrium magnetic moment values with the predictions of the Langevin function for a 5 nm particle. The damping constant was equal to 0.1 in the simulations.

Figure 4-6. Comparison of equilibrium magnetic moment values obtained from simulations of 10 nm particles with the predictions of the Langevin function. The damping constant was equal to 0.1 in the simulations.

105

Figure 4-7. Comparison of simulations obtained using 15 nm particles with the predictions of the Langevin function. The damping constant was equal to 0.1 in the simulations.

Figure 4-8. Simulation results for 20 nm particles in comparison with the predictions of the Langevin function showing significant deviation in the non-linear region. The damping constant was equal to 0.1 in the simulations.

106

Figure 4-9. Simulation of 20 nm particles subject to a constant DC field of 40 mT. A

damping constant of 0.1 was used in this simulation. The simulations reach equilibrium but the equilibrium values are lower than those predicted by the Langevin function. This explains the observed deviation in the nonlinear region when compared with the Langevin function predictions.

107

Figure 4-10. Simulation results of the average dimensionless magnetic moment in the z direction for 15 nm particles to determine the alignment time with the field based on the slope of the fitted line to the portion of the decaying curve. The damping constant is equal to 0.1 in the simulations.

Figure 4-11. Comparison of alignment time obtained at different field strengths for particles with 5, 10, 15 nm, and 20 nm diameter with damping constant equal to 0.1.

108

Figure 4-12. Response of 15 nm particles to a 100 mT DC field with a damping constant

= 0.01.

109

Figure 4-13. Simulation results for 15 nm particles subjected to a 100 mT DC field in z

direction having a damping constant =1.

Figure 4-14. Effect of on the alignment time for 15 nm particles at different field

strengths.

110

Figure 4-15. Simulation results for 10 nm particles showing the decay in the average

dimensionless magnetic moment after an applied DC field of 100 mT has been switched off. A damping constant of 0.1 was used in this simulation.

111

Figure 4-16. Plots showing the distribution of moments obtained from simulation of 1000

particles after an applied field of 100 mT is removed at representative time points. Here, a red dot represents an easy axis of one particle and a blue dot corresponds to its magnetic moment. The orientation of the moments is shown at time equal to a) zero, b) 3.2 ns, c) 6.5 ns, and d) 934 ns.

112

Figure 4-17. Simulation results for 10 nm particles with easy axes aligned in the z

direction, same as the direction of the applied field, showing decay in the z direction magnetic moment after removal of applied magnetic field. A damping constant of 0.1 was used in this simulation

113

Figure 4-18. Magnetic moment distribution plots for 1000 particles with easy axis in z

direction showing the decay of moments after an applied field of 100 mT is switched off. Here, the red arrow represents the direction of the easy axis of every particle and a blue dot corresponds to its magnetic moment. The orientation of the moments is shown at time equal to a) zero, b) 10.9 ns, c) 65.5 ns, and d) 934 ns.

114

Figure 4-19. Moment decay after removal of applied field for different diameter particles.

A damping constant of 0.1 was used in these simulations.

Figure 4-20. Comparison of relaxation time obtained from simulations with the

predictions of the Néel-Arrhenius equation using a damping constant of 0.1.

5 nm

6 nm 7 nm

8 nm

9 nm 10 nm

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Figure 4-21. Effect of variation in damping constant on the relaxation time of particles

obtained from simulations. At =1, qualitative and to some extent quantitative agreement was observed between the simulation results and the predictions of the Néel-Arrhenius equation.

Figure 4-22. Simulation results of 10 nm particles at varying anisotropy constants

showing a slower decay in the magnetic moment as the anisotropy constant value increases. A damping constant of 1 was used in these simulations.

5 nm

6 nm

7 nm 8 nm

9 nm 10 nm

116

Figure 4-23. Comparison of the relaxation time obtained from a fit to decaying magnetic

moment curves for particle diameter between 5-10 nm and anisotropy constant values between 5-50 kJ/m3. The simulations were carried out with a damping constant of 1.

117

Figure 4-24. Response of 5 nm particles to a sinusoidal excitation field of frequency 25

kHz and amplitude 25 mT. The excitation field is normalized with the maximum value to facilitate comparison. The simulation was carried out with a damping constant of 0.1.

118

Figure 4-25. Simulation showing the response of 10 nm particles subjected to an

alternating field of 25 kHz frequency and 25 mT amplitude. The simulation was carried out with a damping constant of 0.1.

119

Figure 4-26. Simulation of 15 nm particles exposed to a sinusoidal excitation field (25

kHz, 25 mT) showing a delay in the magnetic moment response to the alternating field. The simulation was carried out with a damping constant of 0.1.

120

Figure 4-27. Magnetic moment distribution plots at representative time points for 15 nm

particles exposed to an alternating magnetic field. Here, a red point represents the easy axis of a particle while a blue point represents its magnetic moment. The alternating response of magnetic particles is given for

time equal to a) zero, b) 10 s, c) 20 s, and d) 30 s.

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Figure 4-28. Effect of excitation field amplitude on the average dimensionless magnetic

moment for 15 nm particles with damping constant 0.1. The magnetic moment curve shows a reduced delay with increase in the field amplitude. Here, the field is normalized to facilitate comparison with the magnetic moment curves.

122

Figure 4-29. Response of 15 nm particles to a sinusoidal magnetic field at different field

amplitudes with damping constant equal to 0.01. The magnetic moment curves show substantial deviation from the applied field curve but the shift is reduced with an increase in the field amplitude.

123

Figure 4-30. Effect of magnetic field amplitude on the response of 15 nm particles to an

alternating field of 25 kHz frequency with damping constant equal to 1. The magnetic moment curves closely track the change in the applied magnetic field and show minimum deviation from the applied field profile.

124

Figure 4-31. Comparison of simulation of 15 nm particles with a damping constant = 1 with the predictions of the Langevin function on application of a field with 25 kHz frequency and 25 mT amplitude. Here, the Langevin function predictions reach zero when the applied field is zero, but the simulations show a small deviation from the predictions of the Langevin function.

125

CHAPTER 5 SUMMARY

The response of magnetic nanoparticles relaxing by the Brownian mechanism

was modeled using the ferrohydrodynamic magnetization relaxation equations. The

effect of field dependent relaxation on the PSF in a MPR and on the harmonic spectra in

a MPS was studied under the conditions relevant to these instruments. Langevin

function predictions were unable to predict the response of particle in experimental

studies but excellent qualitative agreement and to some extent quantitative agreement

between the MRSh simulations and experiments was observed without the need of

fitting parameters. A significant difference was observed between the MRSh and Sh

equation predictions for conditions of large core diameter particles, high drive field

frequency and amplitude. The study also provided insight on choosing parameters to

limit the effect of relaxation on the MPI properties. The potential of ferrohydrodynamic

modeling was highlighted through this study, thus laying a foundation to extend it to

complex particle structures, higher dimensional magnetic field profiles, and potential

MPI applications.

Theoretical calculations of the SAR under the influence of MPI selection field

gradient were obtained using the MRSh equation at conditions relevant to MPI and

MFH. The application of a selection field gradient was found to suppress heating in the

strong bias field region while localizing nanoparticle heating in the FFR when an

excitation field was applied. The bell shaped curve of the SAR distribution was

explained from the shape and the area of the dynamic hysteresis loops. The increase in

bias field strength was found to make the hysteresis loop asymmetric and ultimately flat,

reducing the area of the loop, and thus the amount of dissipated heat. The gradient

126

strength was shown to influence the region of heating, with the ability to tune the

heating region from centimeter to millimeter scale. Higher field gradient was found to

narrow the SAR distribution while depositing the same amount of heat in the FFR. Good

qualitative agreement was observed between the SAR values predicted by simulations

and those obtained from experiments. Experiments in the presence of a selection field

gradient showed a temperature distribution in wells filled with magnetic nanoparticles.

The highest temperature was observed in wells located in the FFR whereas minimum

temperature was recorded in the wells closer to the NdFeB magnets as a consequence

of reduced heat dissipation in the saturated region closer to the magnets. Decreasing

the distance between the magnets was found to affect the number of wells that reach 45

C, thus providing experimental evidence of the tuning ability of selection field gradient

to achieve focused heating. Thus, through this study, the potential of spatially focused

heat dissipation in MFH using MPI field gradients was highlighted. In terms of future

work, the model can serve as a platform to estimate the temperature distribution in a

selection field gradient, for pre-treatment planning, and for image guided thermal

therapy using MPI, while the experimental setup can serve as a starting point for in vitro

and in vivo studies to further understand the effect of a field gradient in MFH.

Lastly, in order to understand the response of particles whose magnetic moment

can respond through internal rotation to a change in magnetic field, the Landau-Lifshitz-

Gilbert (LLG) equation was employed. The LLG simulations made use of quaternion

parameters to avoid singularities encountered through the use of Euler angles. The LLG

simulations showed the response of immobilized magnetic nanoparticles to a DC field

when the field is switched on and the relaxation of magnetic moment to equilibrium after

127

removal of the applied field. Qualitative and quantitative agreement was found between

the simulations and the Néel-Arrhenius equation for diameters up to 10 nm when using

a damping constant equal to 1. The dynamic response to a sinusoidal magnetic field

showed a significant delay in the magnetic moment curve for 15 nm particles and a

considerable deviation from the Langevin function predictions when a damping constant

of 0.01 was used. However, this delay was found to decrease when a damping constant

equal to 1 was utilized, allowing for moments to track the change in the applied field

better. The model developed was found to capture the dynamics of the particles through

comparison at various MPI relevant parameters. This study provided an understanding

of the LLG equation and provides a platform for further studies which could involve

introducing Brownian motion to assess the effect of change in the orientation of the

easy axis, magnetite’s cubic crystalline anisotropy, particle-particle interaction, and

assessing the effect on MPI signal and resolution.

128

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BIOGRAPHICAL SKETCH

Rohan Deepak Dhavalikar received his Bachelor of Engineering degree in

chemical engineering from University of Pune (Formerly University of Poona), India in

May 2012 with top honors. In fall 2012, he joined the University of Florida to pursue a

Master of Science degree in chemical engineering and conducted research in the

Rinaldi lab on magnetic particle imaging. He graduated in May 2014 with a thesis titled

‘Simulation of magnetic particle imaging using rotational Brownian dynamics

simulations’. He continued his research in the Rinaldi lab and enrolled in the doctoral

program in the department of chemical engineering at the University of Florida in

summer 2015. During his doctoral studies he conducted research to understand the

behavior of magnetic nanoparticles in magnetic fields employed for magnetic particle

imaging. He mentored several undergraduate and graduate students during his time as

a doctoral student. Rohan was actively involved in various student organizations on

campus and served as the president of the Indian Graduate Student Association

between 2016-2017. Rohan was awarded the ‘Outstanding International Student Award’

for his academic achievements at the University of Florida. He received his Doctor of

Philosophy degree in chemical engineering in fall 2017. After graduation, Rohan joined

Intel Corporation in Hillsboro, Oregon as a process engineer.