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Simulation of Magnetic Particle Imaging Using Rotational Brownian Dynamics Simulations

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Title:
Simulation of Magnetic Particle Imaging Using Rotational Brownian Dynamics Simulations
Creator:
Dhavalikar, Rohan D
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[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (54 p.)

Thesis/Dissertation Information

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Master's ( M.S.)
Degree Grantor:
University of Florida
Degree Disciplines:
Chemical Engineering
Committee Chair:
RINALDI,CARLOS
Committee Co-Chair:
DOBSON,JON P
Graduation Date:
5/3/2014

Subjects

Subjects / Keywords:
Diameters ( jstor )
Eggshells ( jstor )
Hydrodynamics ( jstor )
Magnetic fields ( jstor )
Magnetism ( jstor )
Magnetization ( jstor )
Magnets ( jstor )
Nanoparticles ( jstor )
Signals ( jstor )
Simulations ( jstor )
Chemical Engineering -- Dissertations, Academic -- UF
brownian -- imaging -- simulation
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Chemical Engineering thesis, M.S.

Notes

Abstract:
Magnetic Particle Imaging (MPI) is an emerging biomedical imaging technique which uses the non-linear magnetization characteristics of magnetic nanoparticle tracers to obtain millimeter scale temporal and spatial resolution images. Human tissue, being diamagnetic, does not produce a background signal, meaning that the images have near-perfect contrast. Theoretical models assume the magnetic dipoles of nanoparticles used as tracers instantaneously align with the applied field which is at odds with the experimental results showing that finite relaxation affects the signal and resolution of the image. We model this relaxation effect using rotational Brownian dynamics simulations to predict the effect on the resolution and account for the shift in image location. We model the behavior of nanoparticles in response to the rapid movement of a field free point (FFP), which leads to the flipping of the saturated nanoparticles inducing a voltage in the receiving coil. The magnetic dipoles lag behind the changing field due to relaxation and hence produce a significant difference in the signal properties. We also model the magnetic particle relaxometer which is used to measure properties of tracer material by applying an oscillating field to the suspension of nanoparticles The study demonstrates the effect of Brownian relaxation on the signal strength, resolution, and image shift distance. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (M.S.)--University of Florida, 2014.
Local:
Adviser: RINALDI,CARLOS.
Local:
Co-adviser: DOBSON,JON P.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-05-31
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by Rohan D Dhavalikar.

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UFRGP
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Applicable rights reserved.
Embargo Date:
5/31/2015
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908645580 ( OCLC )
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LD1780 2014 ( lcc )

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1 SIMULATION OF MAGNETIC PARTICLE IMAGING USING ROTATIONAL BROWNIAN DYNAMIC S SIMULATIONS By ROHAN DEEPAK DHAVALIKAR A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIR EMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2014

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2 2014 Rohan Deepak Dhavalikar

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3 To my parents Deepak Dhavalikar and Smita Dhavalikar

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4 ACKNOWLEDGMENTS Firstly, I would like to than k my advisor, Dr. Carlos Rinaldi for his tremendous support and advice throughout the course of the thesis. The encouraging talks by him always motivated me to do better and helped me accomplish my goals. I would like to thank my group members Ana Bohorque z Bhumika Sule Tapomoy Bhattacharjee Lorena Maldonado Camargo Melissa Cruz Acuna and Maria Hernandez for their friendship and patience during group meetings I would like to special ly thank Dr. Isaac Torres Diaz for being a re sourceful guide for my res earch and Lorena Maldonado Camargo for synthesizing and characterizing cobalt ferrite particles for use in the relaxometer experiment I would like to thank Dr. Patrick Goodwill, Research Associate in University of Ca lifornia at Berkeley for testing sampl es in the relaxometer and for informative discussions. I thank my family, especially my parents for providing me this opportunity to pursue graduate studies and my friends for their valuable support. Lastly, I would like to thank the Department of Chemical Engineering and University of Florida for the support and the infrastructure during the course of the thesis.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF FIGURES ................................ ................................ ................................ .......... 6 LIST OF ABBREVIATIONS ................................ ................................ ............................. 8 ABSTRACT ................................ ................................ ................................ ..................... 9 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 11 2 ROTATIONAL B ROWNIAN DYNAMICS ................................ ................................ 15 2.1 Theory ................................ ................................ ................................ ............... 15 2.2 Rotational Brownian dynamics simulations ................................ ....................... 16 2.3 Algorithm ................................ ................................ ................................ ........... 18 3 LINEAR MODEL FOR X SPACE MPI ................................ ................................ ..... 20 3.1 Non dimensional analysis ................................ ................................ ................. 23 3.2 Dimensional analysis ................................ ................................ ........................ 25 4 SIMULATION OF RELAXOMETER ................................ ................................ ........ 36 4.1 Particle characterization ................................ ................................ .................... 37 4.1.1 Dynamic light scattering (DLS) measurements ................................ ....... 37 4.1.2 Superconducting quantum interference device (SQUID) measurements ................................ ................................ .............................. 38 4.1.3 Transmission electron microscopy (TEM) measurements ....................... 39 4.1.4 AC susceptibility measurements ................................ ............................. 39 4.2 Berkeley relaxometer experiment ................................ ................................ ..... 40 4.3 Brownian dynamics simulations of relaxometer ................................ ................ 45 4.4 Simulation results ................................ ................................ .............................. 46 5 CONCLUSIONS ................................ ................................ ................................ ..... 50 LIST OF REFERENCES ................................ ................................ ............................... 52 BIOGRAPHICAL SKET CH ................................ ................................ ............................ 54

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6 LIST OF FIGURES Figure page 2 1 Particle model ................................ ................................ ................................ ..... 16 3 1 Response of magnetic nan oparticle to the movement of the field free point. ..... 20 3 2 Comparison of response according to adiabatic theory with simulations taking into account finite relaxation effects. ................................ ........................ 23 3 3 Dimensionless plots for the comparison of PSF with the arrow indicating the scan direction. ................................ ................................ ................................ .... 24 3 4 Comparison of signal strength and resolut ion for 5nm shell thickness at FFP scanning speed of 50m/s, field 15mT and gradient of 6T/m between adiabatic theory and the non adiabatic case.. ................................ ................................ .... 26 3 5 Rotational Brownian dynamics simulation predictions for scanning speed of 50m/s, field of 15mT and gradient of 6T/m ................................ ........................ 27 3 6 Adiabatic theory predictions for scanning speed of 50m/s, field of 15mT and gradient of 6T/m. ................................ ................................ ................................ 27 3 7 Predicted magnetization and signal for core diameter of 20nm at scan speed of 50m/s. ................................ ................................ ................................ ............. 29 3 8 Predictions for scan speed of 100 m/s at appli ed field of 15mT and gradient of 6T/m.. ................................ ................................ ................................ ............. 30 3 9 Adiabatic theory prediction for scan speed of 100m/s at applied field of 15mT and gradient of 6T/m.. ................................ ................................ ........................ 30 3 10 Simulation prediction for scan speed of 200m/s at an applied field of 15mT and gradient of 6T/m.. ................................ ................................ ........................ 31 3 11 Adiabatic theory prediction for scan speed of 200m/s at appli ed field of 15mT and gradient of 6T/m.. ................................ ................................ ........................ 31 3 12 Simulation predictions for variation in field gradient for particles with 10nm shell thickness subjected to an applied field of 15mT and field fr ee point scanning speed of 100m/s.. ................................ ................................ ................ 32 3 13 Rotational Brownian dynamics predictions for applied field of 5mT, scan speed of 100m/s and field gradient of 6T/m.. ................................ ..................... 33

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7 3 14 Predicted magnetization and signal for 20nm core diameter nanoparticle and a shell thickness of 5nm in a field of 5mT, gradient of 6T/m and scan speed of 100m/s showing the peak a t the edge of the time window. ............................ 34 3 15 Predictions from simulations for applied field of 10mT at a scan speed of 100 m/s and gradient of 6T/m.. ................................ ................................ .................. 35 3 16 Predictions from simulations for applied field of 15mT at a scan speed of 100 m/s and gradient of 6T/m.. ................................ ................................ .................. 35 4 1 DLS measurements of Oleic acid coated nanoparticles ................................ ..... 37 4 2 TEM measurements.. ................................ ................................ ......................... 39 4 3 AC susceptibility measurements for cobalt ferrite nanoparticles suspended in 1 octadecene and mineral oil. ................................ ................................ ............. 4 0 4 4 Pulse sequence in the relaxometer. ................................ ................................ ... 41 4 5 Representative figures showing the raw signal and PSF obtained from the relaxometer.. ................................ ................................ ................................ ....... 42 4 6 Reconstructed positive velocity PSF for particles subjected to a drive field frequency of 4.5kHz and varying amplitudes of 15, 30 and 45 mT. .................... 42 4 7 Recon structed positive velocity PSF for particles subjected to a drive field amplitude of 30mT. ................................ ................................ ............................. 43 4 8 Reconstructed PSF for particles suspended in mineral oil for variation in frequency of drive fi eld and amplitude of 30 mT. ................................ ................ 44 4 9 Predicted simulation results for a drive field frequency of 1.6 kHz and amplitude of 45mT. ................................ ................................ ............................. 47 4 10 Simulation predictions for drive field frequency of 4.5 kHz and amplitude of 45mT.. ................................ ................................ ................................ ................ 47 4 11 Brownian dynamics simulation predictions for drive field frequency of 9.3 kHz and amplitude of 45 mT.. ................................ ................................ ..................... 48 4 12 Simulation predictions for drive field frequency of 25 kHz and amplitude of 45 mT. ................................ ................................ ................................ ..................... 48 4 13 Effect of variation in the drive field amplitude on the signal strength. ................. 49

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8 LIST OF ABBREVIATIONS CKD Chronic Kidney Disease. DLS Dynamic Light Scattering. FOV Field Of View. FWHM Full Width at Half Maximum. MPI Magnetic Particle Imaging. MRI Magnetic Resonance Imaging. P FOV Partial Field Of View. PRF Particle Response Function. PSF Point Spread Function. RBD Rotational Brownian Dynamics. SPIO Super Paramagnetic Iron Oxide. SQUID Superconducting Quantum Interference Device. TEM Transmis sion Electron Microscopy.

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9 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science SIMULATION OF MAGNETIC PARTICLE IMAGING USING ROTATIONAL BROWNIAN DYNAMIC S SIMULATIONS By Rohan Deepak Dhavalikar May 2014 Chair: Carlos Rinaldi Major: Chemical Engineering Magnetic Particle Imaging (MPI) is an emerging biomedical imaging technique which use s the non linear magnetizati on characteri stics of magnetic nanoparticle t r acers to obtain millimeter scale temporal and spatial resolution images H uman tissue being diamagnetic does not produce a background signal, meaning that the images have near perfect contrast. Theoretical models assume t he magnetic dipoles of nanoparticles used as tracers instantaneously align with the applied field which is at odds with the experimental results showing that finite relaxation affect s the signal and resolution of the image. We model this relaxation effect using rotational Brownian dynamics simulations to predict the effect on the resolution and account for the shift in image location. We model the behavior of nanoparticles in response to the rapid movement of a field free point (FFP) which leads to the fl ipping of the saturated nanoparticles inducing a voltage in the receiving coil. The magnetic dipoles lag behind the changing field due to relaxation and hence produce a significant difference in the signal properties We also model the magnetic particle re laxometer which is used to measure properties of tracer material by applying an oscillating field to the suspension

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10 of nanoparticle s The study demonstrates the effect of Brownian relaxation on the signal strength resolution, and image shift distance.

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11 CH APTER 1 INTRODUCTION Magnetic Particle imaging (MPI) is a ne w tomographic imaging technique which utilizes static and oscillating magnetic fields to map the spatial distribution of magnetic nanoparticles used as tracers. This technique overcomes the disadv antage of host tissue background signal encountered in MRI as it directly images the tracer particles It is also a safer imaging alternative, especially for Chronic Kidney Disease (CKD) patients for whom the iodine contrast agents are toxic [1] Gleich and Weizenecker [2] first reported this real t ime imaging technique in 2005, proposing use of the nonlinear magnetization characteristics of ferromagnetic nanoparticles. The use of a strong static field gradient creates a point where the magnetic field magnitude is weaker than the saturati on field of s uper paramagnetic iron oxide (SPIO) This point is called the Field Free Point (FFP) [3] and it can be moved to scan a sample space. The change in the magnetization of the SPIO is detected by an induction receiver coil as a change in voltage. The FFP can be shifted mechanically and/or electronically So called Frequency domain [2] MPI superposes an alternating magnetic field to generate a harmonic spectrum of the magnetization signal that is used for image reconstruction. An oscillating magnetic field called a modulation field, is super posed on a time independent selection field. A signal will be produced in the receiver coil only if magnetic particle s are present at the position of the FFP The remaining particles will not respond to the AC field since they are in a state of saturation. The FFP is moved through the field of view of the sample space and the image is constructed using the magnitude of harmonics that are generated by the Fourier transformation of the induced signal. Matrix inversion is carried out for image reconstruction but is error prone since

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12 the number of elements is large. Also, since the MPI signal is small as compared to the signal generated by the excitation field the detection of MPI harmonics is not possible at the fundamental frequency. Recently, Conolly and co worker s dev eloped X space [1] M PI exploiting the same characteristics of super p aramagnetic iron oxide ( SPIO ) nanoparticles to obtain the signal without need of an oscillatory field. As the FFP moves through the field of view (FOV), the change in direction of the local applied field creates a torque on the SPIOs causing them to flip o r rotate. The change in magnetization induces a voltage in the receiver coil which is utilized for image construction. The obtainable spatial resolution is in the millimeter scale with a temporal resolution in the millisecond scale [4] MPI has the potential to detect nanogram quantities of magnetic tracers deep in the body for angiography, cell tracking, cancer imaging, and imag ing major organs and the finer coronary arteries in real time [4, 5] Current theoretical models commonly assume that the tracer nanoparticles instantaneously align with the applied magnetic field [6] However, experimental observations [7] indicate that there is a deviation from this assumption and the effect of finite magnetic relaxation of the tracer nanoparticles on the signal and resolution has bee n considered by authors using simplified models [7] .As the properties of tracer particles affect the performance of MPI imaging it is required to understand more fully the effect of finite relaxation. Typically, the tracer nanoparticles used in MPI are single do main magnetic nanoparticles with an iron oxide core and coated with a polymer or surfactant to prevent agglomeration of the particles. These particles respond to time varying magnetic field s

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13 either by one or by a combination of two relaxation mechanisms. I n Brownian relaxation, the magnetic dipole moment is blocked inside the particle and the dipole aligns with the applied field by physical rotation of the particle opposed by hydrodynamic drag [8] ,with a characteristic time given by (1 1) w here is the viscosity of the surrounding fluid, D is the magnetic core diameter, is the non magnetic shell thickness and is the thermal energy. I n Nel relaxation the magnetic dipole is free to rotate internally and aligns with the applied field without the particle physically rotating. The characteristic time is given by Equation (1 2) and depends on the thermal energy magnetocrystalline anisotropy and the magnetic core diameter with as the characteristic attempt frequency [8] . (1 2) In general, both mechanisms take place simultaneously with the faster one dominating. The effective magnetic relaxation time is the re ciprocal of Nel and Brownian relaxation time represented as Equation (1 3). (1 3) Since both mechanisms depend primarily on the core and hydrodynamic diameters of the particles, a threshold diameter can be determined below wh ich the particle s relax by the Nel mechanism and above which they relax by the Brownian mechanism.

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14 To obtain sub millimeter resolution, particle diameters in the range of 20 60 nm are proposed for use in MPI [6] P articles in this range would clearly relax by the Brownian mechanism and hence the effect of re laxation on the resolution and signal needs to be assessed in order to explain experimental observations [7] We analyze the effect of relaxation by applying Rotational Brownian Dynamics (RBD) simulati ons to model the dynamic magnetization of suspensions of non interacting magnetic nanoparticle s by modifying algorithms previously developed by Rinaldi and co workers [9, 10]

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15 CHAPTER 2 ROTATIONAL BROWNIA N DYNAMICS 2.1 Theory A dilute suspension of nanoparticles in a fluid can be treated as a collection of magnetic dipoles because of their small size Application of a magnetic field to the suspension of nanoparticles exerts a torque on the nanoparticles t ending to align the magnetic dipole moment of the particles with the field. In the presence of a strong magnetic field, the dipole moments align in the direction of the applied field and the suspension becomes magnetically saturated. This phenomenon can be modelled using the Langevin equation, a stochastic differential equation in which the motion of the Brownian particle is described by the summation of the forces and torques due to hydrodynamic effects magnetic effects and Brownian motion The forces due to hydrodynamic effects are due to the viscous drag experience d by the particle and the random or Brownian force is due to the random collisions of the molecules of the liquid on the particle. In the case of a spherical particle as illustrated in Figure 2 1, the physical and magnetic properties are generally expressed in the c artesian co ordinate system with axes aligned with the principal axis (primed axi s) rather than relative to the laboratory space coordinate axes (unprimed axis) The dipole moment of the particle is directed along the axis and the magnetic field is applied in yz plane. Equations (2 1) and (2 2) describe the motion of the particle. (2 1) (2 2)

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16 where is the angular velocity of the particle, is the rotation vector, the unpe rturbed velocity, and the moment of inertia Figure 2 1. Particle model 2.2 Rotational Brownian d ynamics simulations The rotational Brownian dynamics simulations are based on the integration of the stochastic angular momentum e quation in the non inertial limit to obtain the orientation of the particle with time. Moment of inertia is negligible because of the small sizes and low masses of the nanoparticles. Therefore, the time scale where is the rotational hydrodynamic resistance dyadic, a constant in the case of spherical particles ; for the angular velocity correlation is short as compared to the natural observation time for Brownian motion. Hence, the inertial term in the equ ation of motion can be neglected and Equations (2 1) and (2 2) are simplified to (2 3) (2 4)

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17 The equations are written with respect to particle locked coordinates and hence the vectors are represent ed as prime d As there is no interaction between the particles, the translational and rotational equations can be decoupled. Since we are interested only in the rotational motion of the particle, we evaluate only Equation (2 4) The hydrodynamic torque on t he nanoparticles assuming a quiescent fluid is given by (2 5) The viscosity of the surrounding fluid is given by the hydrodynamic resistance dyadic and the angular velocity of the particle is given by .The magnetic torque due to an applied field is given by (2 6) Here is the vacuum permeability and is the applied field transformed to particle axis using the transformation matrix .A transformation matrix is an operator to transform the components of a vector from one coordinate system to another coordinate system. Eul er angles consist of a set of three coordinates necessary to specify the orientation of the particle. However, the use of Euler angles present difficulty for use in numerical solutions as the trigonometric functions involved could lead to a singular proble m [11] In order to remove the singularities from the algorithm the transformation matrix is expressed in terms of the quaternion parameters and as (2 7) In which the quaternion parameters must satisfy the relation

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18 Equation s (2 6) and (2 5) are substituted in Equation (2 4) to obtain an expression for the angular velocity of the particles (2 8) Solving Equation (2 8) for and integrating from time to time using the first order forward Eulers method we get (2 9) Applying the fluctuation dissipation theorem [12] to the Brownian term in Equation (2 9) (2 10) Here and is a random vector characterized by a Gaussian distribution with mean and variance In order to reduce the number of variables in the angular momentum equation, time is non dimensionalized with respect to the r otational diffusion coefficient and the dimensionless vector variables are given by (2 11) Using the non dimensional va riables, Equation (2 10) can be written as, (2 12) where is the Langevin p arameter 2.3 Algorithm The algorithm proceeds from a n initial random configuration of the particles After each time step, the change in orientation of the particles is calculated using Equation (2

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19 12) and the change in orientation of the particles in terms of quaternion parameters is obtained by (2 13) The quaternion parameters of each particle are normalized afte r each time step and the average magnetization in the direction is obtained by Equation (2 14). (2 14) Using the above derived equations, the rate of change of magnetization with time i s calculated for the model described in Chapter 3 and for the relaxometer experiment described in Chapter 4.

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20 CHAPTER 3 LINEAR MODEL FOR X SPACE MPI To explain the behavior of particles in the changing magnetic field used in X Space MPI, we have adopted t he Eulerian point of view of a collection of particles at a fixed location in space Figure 3 1 shows the various parameters releva nt in und erstanding the effect of relaxation on the performance of the magnetic nanoparticles. From the point of view of the particles, the scanning of the field free point is seen as a magnetic field which is initially high and constant in one direction and varies linearly. As the field free point approaches the location of the particles, the field switches direction. Figure 3 1 Response of magnetic nanoparticle to the m ovement of the field free point. The signal peak shows a shift in the scan direction due to relaxation effect The model [13] can be mathematically expressed in terms of a uniform magnetic field far from the field free point, time the time at which the field free point Reprinted with permission from R. Dhavalikar, C. Rinaldi, Journal of Applied Physics 2014, 115.

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21 passes over the location of the particles and the speed of the field free point is related to .Equation (3 1) shows the mathematical representation of the model. (3 1) The as sumed linear magnetic field gradient is given by Equation (3 2) (3 2) The non linear response of the particles to the applied magnetic field is show n in Figure 3 1 and the signal which is recorded by the pickup coil in terms of voltage, is proportional to the rate of change of magnetization with time. The signal curve shows a peak identifying the location of the image. T he difference represents the shift in location of the image of the nanoparticles du e to relaxation effects. Ideally, if the particles align instantaneously with the changing magnetic field, the signal peak will be observed at time and indicates the presence of the particles at that particular location. In this situation the location of particles and the image coincide with each other. However, in the case where relaxation is taking place, the signal peak would have shifted by a distance in the direction of mot ion of the field free point indicating that the location of particles appears to have changed. This distance is referred to as the image shift distance. Croft et al [7] have demonstrated this shift in peak through experimental ob servations. The shift in peak also has an effect on the resolution that can be achieved with the particles. The resolution is related to the full width at half maximum (FWHM) [1] and

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22 hence in our case, the resolution is given by The performance of the particles till date has been represented in te rms of a point spread function (PSF) [7] or particle response function (PRF) [14] where the signal is plotted against the applied field .In our simulations, the PSF is represented as a function of time since we found that for the particles having longer relaxation times the interpretation of the PSFs was difficult This will be elaborated in section 3.1in which simulations were run in non dimensional form. The adiabatic theory for X space MPI developed by Goodwill et al [1] assumed instantaneous magnetization of the nanoparticles in the applied field and the non linear response of the particles was modelled using the Langevin function. (3 3 ) where is the Langevin parameter and is a function of the applied field The deviation i n the magnetization curve and signal in comparison with the adiabatic theory is depicted in Figure 3 2 As seen in Figure 3 2 A the magnetization is zero in the case of the adiabatic theory when the field free point is at the location of the magnetic nano particles. In contrast, the non adiabatic magnetization lags behind the response predicted by the adiabatic theory. The signal obtained from the magnetization c urves are shown in Figure 3 2 B with a symmetric adiabatic signal centered about time and an asymmetric non adiabatic signal showing a distinct shift in the scan direction. This shift is evident in experimental PSFs as reported by Croft et al [7] The incorporation of f inite relaxation effect also leads to a loss in signal strength and broadening of the signal The wide ning of the signal curve affects the FWHM and ultimately the resolution

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23 of the acquired image The signal predicted by the adiabatic theory is higher sinc e the particles align instantaneously with the changing magnetic field. Figure 3 2 Comparison of response according to adiabatic theory with simulations taking into acco unt finite relaxation effects. A ) Non linear magnetization response of nanopartic les to a linearly changing field. B ) Comparison of adiabatic signal with signal obtained from simulation which takes into account finite relaxation showing a disti nct shift in the scan direction 3.1 Non dimensional analysis The algorithm as derived in Cha pter 2 is formulated in terms of dimensionless parameters. The algorith m incorporates the relation given by Equation 3 2 to obtain the effect of linearly changing field in terms of Langevin parameter and the speed of the field free point in terms of The variation in represents the change in field strength and the variation in represents the speed at which the field free p oint passes over the particles, w ith lower values of indicating faster motion of the field free point. The signal is obtained as the rate of change of magnetization with time. The simulations were run for 10 6 non interacting spheric al particles The particles are permanently magnetized in the direction. The runs were performed with a time A) B)

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24 step of for different values of the Langevin parameter and variab le values of delta .The performance of particles was compared on the basis of the PSF which is plotted as a function of Figure 3 3 A shows the shift in signal peak in the scan direction for and variation in Larger values of indicate slower scanning speeds. Figure 3 3 B predicts the effect on the signal peak for and varying The x axis is rescaled with respect to in order to visualize the shift in the peaks . Figure 3 3 Dimensionless plots for the comparison of PSF with the arrow indicating the scan direction. A) Effect of varying on the signal at B) Effect on signal for v ariation of at fixed As can be seen from Figure 3 3 A at a fixed and varying a shift in the peak is observed in the scan direction. For smaller values of the peak signal is higher and the shift distance greater since it corresponds to faster scanning speeds. The faster scanning speeds lead t o the particles lag ging behind the changing applied field due to relaxation effects. I ncreasing tends to reduce the peak shift distance. The explanation to this phenomenon lies in the fact that at larger values of the particles alig n with the changing field faster than expected for purely Brownian relaxation It can A B

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25 be also observed in Figure 3 3 B that the PSF corresponding to remain incomplete and the particles continue to relax beyon d the time window of for change in m agnetic field. This leads to difficulties while analyzing th e PSF to obtain the resolution as the FWHM cannot be calculated. Hence, to overcome the situation of incomple te PSF, we plot our PSF with respect to time on the x axis instead of the applied field. In order to make the findings easy to comprehend, the parameters were dimensionalized and results of image shift distance and resolution are presented in units of mill imeter 3.2 Dimensional analysis S imulations were carried out for particle core diameters in the range of 20 50nm and for shell coating thi ckness of 5 15nm.Magnetic fields with a strength of 5,10 and 15 mT was applied to the suspension of particles having a linear gradient of 2,4 and 6T/m near the FFP. The scanning speed of the FFP in the range of 50 200 m/s was used. These parameters are representative of those reported in the literature for MPI [1, 7, 15] Since Brownian relaxation depends on the particle diameter, the effect on the signal, resolution and image shift distance was analyzed by varying the particle core diameter and the hydrodynamic diameter. Figure 3 4 demonstrates the comparison between the adiabat ic and non adiabatic signal and resolution for a constant shell thickness of 5nm FFP scan speed of 50m/s and field conditions of 15mT with a gradient of 6T/m. It should be noted that the adiabatic theory does not predict a shift in the image since it is assumed that the signal is generated at the exact location of the partic les and at the instant the FFP passes over the sample. The signal strength will be large in comparison with the non adiabatic theory since the particles align instantaneously with the changing

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26 field. Also, a fine resolution can be achieved since the PSF will be narrow and symmetric. Figure 3 4. Comparis on of signal strength and resolution for 5nm shell thickness at FFP scanning speed of 50m/s, field 15mT and gradient of 6T/m betwee n adiabatic theo ry and the non adiabatic case. A ) Signal predictions of the adiabatic theory are higher in comparison with predicti ons from non adiabatic theory. B ) The resolution for the non adiabatic case is worse due to broadening of the PSF Figure 3 4 shows a distinct deviation in predictions of the adiabatic theory and our rotational Brownian dynamics simulations. Due to finite relaxation effects, a loss in signal strength and a worsening of the resolution in comparison with the adiabatic theory predi ctions is seen in Figure 3 4. The deviation goes on increasing with increasing hydrodynam ic diameter which is in agreement with the fact that the Brownian relaxation time is proportional to the cu be of the hydrodynamic diameter, and hence the effects of fi nite relaxation will be more pronounced as the particle size increases. Predictions from simulations are given for FFP scanning speed of 50 m/s for an applied field of 15mT and a linear gradient of 6T/m i n Figure 3 5 .Adiabatic theory predictions for the si gnal and resolution are provided as a reference in Figure 3 6 to facilitate comparison with the predictions from the rotational Brownian dynamics simulations. A B

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27 Figure 3 5. Rotational Brownian dynamics simulation predictions for scanning speed of 50m/s, field of 15mT and gradient of 6T/m. A) Signal, B) resolution and C) i mage shift distance for variation in core diameter and shell thickness. Figure 3 6 Adiabatic theory predictions for scanning speed of 50m/s, field of 15mT and gradient of 6T/ m. A) Adiabatic signal and B) ad iabatic resolution. As seen in Figure 3 5 A the signal strength appears to increase cubica lly with hydrodynamic diameter for a fixed shell thickness whereas for a fixed core diameter, the signal appears to increase linearly with h ydrodynamic diameter. A similar trend is seen in the predictions from the adiabatic theory in Figure 3 6 A with the signal strength significantly higher than that predicted by rotational Brownian dynamics simulations. Th e resolution in Figure 3 5 B appears to improve i.e. lower R values with increasing core and hydrodynamic diameter. For a fixed shell thickness, the resolution appears to improve cubically with the increasing core diameter whereas for a fixed core diameter A B C A B

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28 the resolution worsens w ith increasing shell thickness. The adi abatic theory as shown in Figure 3 6 B predicts a resolution less than 0.5 mm This resolution can be achieved because the PSF shows a sharp and symmetric peak without distortion which is possible only in the case where relaxation is not taken into account The image shift distance in Figure 3 5 C shows increasing or decreasing trends depending on the shell thickness. For particles with shell thickness of 5nm, the image shift distance increases with increase in hydrodyna mic diameter. For shell thickness in the range of 7.5nm to 10nm the dependence of image shift distance on the hydrodynamic diameter appears to be weakly linked. However, for shell thickness of more than 10nm, the image shift distance decreases with increa se in the hydrodynamic diameter. We attribute th e difference in trend of the image shift distance to the case in which a significant portion of the signal occurs outside the time window of change in the magnetic field between an d .This is illustrated in Figure 3 7 for a core diameter of 20nm and various shell thickness at a constant scan speed of 50m/s. For a shell thickness of 5nm as shown in Figure 3 7 A the magnetization curve reaches the new equil ibrium value within the time window of and hence the signal peak is generated within the interval. In contrast, for she ll thickness of 15nm(Figure 3 7 B ), representative of the case where the image shift distance decreases with i ncrease in core diameter, the magnetization curve remains unsaturated w ithin the time window of change in the magnetic field. This leads to the signal peak being formed at the far end of the time interval with the particles continuing to relax at a constan t field. T he

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29 predicted signal peak shows a distinct shift in the scan di rection which is in accordance with the experimental observations Figure 3 7. Predicted magnetization and signal for core diameter of 20nm at scan speed of 50 m/s. A) 5nm and B ) 15 nm shell thickness. The predictions for scan speed of 100 m/s are shown in Figure 3 8 along with the adiabatic theory predictions for purpose of comparison in Figure 3 9. The trend in the behavior of particles is similar to that predicted for scan speed of 50m/s. The difference s being that the signal strength is higher, the resolution is worse and the ima ge shift distance has increased when compared to predictions shown in Figure 3 6. The signal increases because the rate of change of magnetization becomes faster a s the FFP passes over the particles. The increased scanning speed also leads to particles not aligning with the changing magnetic field due to relaxation and lagging behind the instantaneous applied field. The increase in the predicted image shift distance can be attributed to the fact that the particles do not respond instantaneously to the changing A B

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30 magnetic field. Hence, the equilibrium magnetization flips in the opposite direction at a finite time after the field free point has passed the locatio n of the particles. Figure 3 8. Predictions for scan speed of 100 m/s at applied field of 15mT and gradient of 6T/m. A) Signal, B) resolution and C ) Image shift distance for varying core and hydrodynamic diameter. Figure 3 9. Adiabatic theory prediction for scan speed of 100m/s at applied field of 15mT and gradient of 6T/m. A) Adiabatic signal and B) a diabatic resolution. Predic tions are also obtained for sca n speed of 200m/s and the trend in the behavior of the particles is similar to that sho wn in Figure 3 5 and Figure 3 8. The predictions from the simulations are shown in Figure 3 10 and those from adiabatic theory are presented in Figure 3 11. A B C A B

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31 Figure 3 10. Simulation prediction for scan speed of 200m/s at an applied field of 15mT an d gradient of 6T/m. A) Signal, B) resolution and C) i mage shift distance for various core and hydrodynamic diameters. Figure 3 11. Adiabatic theory prediction for scan speed of 200 m/s at applied field of 15mT and g radient of 6T/m. A) Adiabatic signal an d B) a diabatic resolution. It can be seen from the above discussion that scan speed defini tely affects the signal, resolution and image shift distance produced by the particles Hence, i gnoring relaxation effects could lead to assigning the location of particl es incorrectly in an image as well as overestimating the resolution and the size of the particles used in the MPI system. Apart from scan speed the effect of field gradient on the signal, resolution and image shift distance was also assessed. Figure 3 12 shows the effect of variation in A B C A B

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32 field gradient for a constant scan speed of 100 m/ s and applied field of 15mT on the performance of particles with a shell thickness of 10nm. Figure 3 12. Simulation predictions for variation in field gradient for particles with 10nm shell thickness subjected to an applied field of 15mT and field free point scanning speed of 100m/s. Comparison of A) signal, B) resolution and C ) image shift distance to assess the effect of variation in field gradient. As seen in Figu re 3 12 A the signal strength appears to increase cubically with increase in hydrodynamic diameter. Since Brownian relaxation time depends on the cube of the hydrodynamic diameter, finite relaxation effects will become increasingly significant The increa se in field gradient leads to an increase in signal strength because the rate of change of magnetization is faster translating to a higher voltage induced in the pickup coil. Figure 3 12 B shows an improvement in the resolution as the hydrodynamic diamete r increases. This is in agreement with the theoretical resolution reported by Goodwill et al [1] in which the larger sized particle provide a better resolution. The FWHM for the PSF becomes finer with increase in gradient strength and cube of the diameter, leading to improvement in the resolution. From Figure 3 12 C it can b e seen that t he image shift distance predicted by the simulations is the least for a field gradient of 6T/m and becomes larger as the field gradient decreases. However, there appears a weak dependence on the hydrodynamic diameter. A B C

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33 Another important par ameter that should be taken into account to assess the effect on the MPI signal, resolution and image shift distance is the applied field strength. Simulations were carried out for a field free point scanning speed of 100 m/s, field gradient of 6T/m and ap plied field strength of 5mT, 10mT and 15mT. Figure 3 13 shows the effect of an applied field of 5mT on the performance of the nanoparticles. Figure 3 13. Rotational Brownian dynamics predictions for applied field of 5mT, scan speed of 100m /s and field gradient of 6T/m. A ) Signal, B) resolution and C ) i mage shift distance for variation in core diameter and shell thickness. As seen from Figure 3 13 A the signal increases cubically with the increase in core diameter but appears to remain constant with inc rease in shell thickness. The strength of the applied field being low, the net average magnetization of the suspension of magnetization is a lso low. Hence the rate of change of magnetization is slower leading to a small signal from the nanoparticles. The r esolution predicted by the simulations as shown in Figure 3 13 B shows improvement (i.e. lower R values) with increa se in the hydrodynamic diameter The image shift dist ance in Figure 3 13 C shows no change for 20nm core diameter particles but an increase in the shift as the hydrodynamic diameter increases. As explained earlier, the shift distance depends on whether the magnetization changes from one equilibrium value to another within the time window of change of field or there is a significant portion of the signal that occurs A B C

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34 outside this window. In this situation, the signal peak is at the edge or beyond the time window of change of field and the particles continues to relax against a constant field as seen in Figure 3 14. At lower magnetic fields, the t orque due to Brownian forces is stronger than the torque exerted by the magnetic field and hence the effects due to B rownian relaxation are predomina nt. Figure 3 14. Predicted magnetization and signal for 20nm core diameter nanoparticle and a sh ell thickness of 5nm in a field of 5mT, gradient of 6T/m and scan speed of 100m/s showing the peak at the edge of the time window The effect of higher field strength shows a behavior consistent with that discussed before with respect to variable scan spee ds. Figure 3 15 shows the effect of an applied field of 10mT and the influence of an applied field of 15mT is shown in Figure 3 16. On comparing the two figures the difference in signal strength is negligible but the signal continues to increase cubically with the hydrodynamic diameter for a fixed shell thickness and linearly for fixed core diameter. The difference in the resolution is minor and the image shift distance is in the range of 0.5 2mm. However, on comparison with the predictions obtained for th e conditio n of 5mT shown in Figure 3 13, a significant difference is observed in the signal strength and the resolution of the MPI system. t 0 t 0 + Magnetization Mag netic Field Scan Direction Signal

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35 Figure 3 15. Predictions from simulations for applied field of 10mT at a scan speed of 100 m/s and gradien t of 6T/m. A) Signal, B) resolution and C ) image shift distance for variation in shell thickness and core diameters. Figure 3 16. Predictions from sim ulations for applied field of 15 mT at a scan speed of 100 m/s and gradient of 6T/m. A) Signal, B) reso lution and C ) image shift distance for variation in shell thickness and core diameters. The significant difference in the resolution and signal for higher field strength along with the discussion on the effect of variation of parameters provided an underst anding of the essential parameters affecting the MPI system. A B C A B C

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36 CHAPTER 4 SIMULATION OF RELAXOMETER The Berkeley magnetic nanoparticle relaxometer [16] is used to measure the one dimensional MPI point spread function and helps to meas ure the core diameter and relaxation constant of magnetic nanoparticles. Unlike the MPI spectrometer [17] the information is recorded in time domain. I t consists of a concentric assembly of electrom agnetic coils. The outer most coil is the b ias coil, followed by the t ransmit coil and the coil closest to the center surrounding a sample holder is the r eceive coil. The bias coil is used to linearly ramp the field in order to simulate moving a point sourc e sample in a gradient field. Fields up to mT are used. The transmit coil also referred to as the drive coil, generates an oscillating sinusoidal magnetic field with a peak to peak amplitude in the range of 10 200 mT and at freq uency ranging from 1.5 kHz to 11.5 kHz [7] The superposition of t his oscillating field o n the bias field results in the final field to which the nanoparticle sample is subjected The signal from the na noparticles is received by the inductive receive c oil and t he signal is reconstructed to obtain the PSF Ex perimental evidence reported by Goodwill et al. [16] shows a distinct shift in the PSF along the scanning direction. This is att ributed to the particles undergoing relaxation. To explain the shift in the PSF, we developed an algorithm using rotational Brownian dynamics to incorporate the finite relaxation effect. To test the relevance of the model, cobalt ferrite particles synthesi zed in the Rinaldi lab were sent to University of California at Berkeley for analysis in the Berkeley MPI relaxometer to obtain experimental PSFs. The characterization o f the particles was carried out by Lorena Maldonado Camargo a graduate student in Rina ldi Lab to obtain relevant information

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37 regarding the particle size to be used in the simulations. The characterization techniques used for the particles are described in short in the following section. The experimental conditions and results at which the p article samples were tested will be discussed later. 4.1 Particle characterization The cobalt ferrite particles coated with oleic acid synthesized via the thermal decomposition method in the Rinaldi lab were suspended in 1 Octadecene and mineral oil for th e purpose of characterization. The particles we re characterized using dynamic light scattering (DLS) technique, superconducting quantum interference d evice (SQUID), t ransmission electron microscope (TEM) and AC susceptometer. The techniques provided inform ation regarding the hydrodynamic, magnetic and core diameters to be used in the simulation. 4.1.1 Dynamic light s cattering (DLS) measurements Dynamic light scattering was used to obtain the hydrodynamic diameter of the p articles and the observed main size of the particles was 21nm as shown in Figure 4 1. Figure 4 1. DLS measurements of Oleic acid coated nanoparticles

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38 4.1.2 Superconducting quantum interference d evice (SQUID) measurements Magnetic characterization of the nanoparticles was conducted using a superconducting quantum interference device (SQUID) magnetometer. The magnetic volume median diameter of the nanoparticles was estimated using the method of Chantrell et al. [18] from equation (4 1). (4 1) and the standard de viation is given by (4 2) where the initial susceptibility is obtained from the slope of M vs H curve at low field, the particle concentration is obtained from the M intercep t of the M vs curve at high field and from the intercept. The estimated magnetic core diameter was calculated to around 9nm. This value was used in the simulations to calculate the and the Langevin parameters for the bias field and the excitation field. Although the SQUID measurements help to estimate the magnetic core diameter, it is necessary to conduct TEM measurements to confirm that the magnetic core diameter does not exceed the particle core diameter. This also helps to estimate the thickness of the magnetically dead layer present in the particles. Accurately determining the magnetic core diameter is necessary since the signal in MPI depends on the core diameter of the tracer particles.

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39 4.1.3 Transmission electron m icroscopy (TEM) measurements The core diameter was determ ined from the TEM image in Figure 4 2 A and the size distribution is shown in Figure 4 2 B. The mean diameter of the particles is 14nm The TEM measurements estimate a particle size bigger than that predicted from the SQUID analysis and hence confirms the presence of a magnetically dead layer. Figure 4 2. TEM measurements A) TEM image with a s cale bar of 50nm.B) Size distribution plot to obtain mean diameter. 4.1.4 AC s usceptibility measurements To assess whether the particles relax by the Brownian relaxation mechanism AC susceptibility measurements were carrie d out. A ma ximum peak was observe d for particles suspended in 1 octadecene at 47815.04 rad/s (7610 Hz) and at 4146.98 rad/s (660 Hz) for particles suspended in mineral oil The formation of a peak indicates that the nanoparticles relax by the Brownian mechanism. Also, the shift in the pea k is due to the difference in viscosity of the carrier fluids with mineral oil being more viscous than 1 octadecene. As the frequency is inversely proportional to the viscosity of the carrier fluid, the peak for mineral oil is observed at one order of m agn itude lower than the peak for 1 octadecene. The AC susceptibility measurements are shown in Figure 4 3. According to the Debye model, the main diameter is found to be 17 nm. A B

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40 Figure 4 3. AC susceptibility measurements for cobalt ferrite nanoparticles suspe nded in 1 octadecene and mineral oil. 4.2 Berkeley relaxometer experiment The characterized particle samples suspended in 1 octadecene and mineral oil were sent to University of California at Berkeley for testing in the MPI relaxometer. Both samp les were measured at four frequencies (1.6, 4.5, 9.3 and 25 kHz) at varying amplitudes (15, 30 and 45mT). The bias coil generated a linearly varying offset field in the range of mT. These field s were applied for a scan time of 1 second. T he nanoparticle signal is measured from a gradiometric receive coil that detects the change in magnetization of the nanoparticles through induction. The phase sniffer coil measures the activity of the drive coil. The signal with time data is obtained by su btracting the baseline signal(no sample in holder) from the received signal (sample in holder) The point spread function i s reconstructed from the signal and the information from the phase sniffer coil is used to eliminate any phase acquired from hardware imperfections. The nanoparticle responses obtained in the positive velocity direction are

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41 separated from those acquired in t he negative velocity direction to give positive and negative scan direction PSFs. An example of total applied field is shown in F igure 4 4 due to the super position of an oscillating field with frequency 10Hz (l ow for illustration purpose) and amplitude of 30 mT on a bias field in the range of mT Figure 4 4. Pulse sequence in the relaxometer. A) Oscilla ting field with frequency 10Hz and amplitude 30 mT. B) Bias field in the range of mT. C) Total app lied field by superposition of oscillating and bias field. A representative raw signal for particles suspended in 1 octadecene is s hown in Figure 4 5 A The lack of signal towards both ends of the time series is because the excitation field is oscillating in the region of saturation of the Langevin curve of the nanoparticles. As the bias field starts varying linearly, the particles re spond to the oscillating magnetic field and the change in magnetization is measured in the receive coil due to induction The signal magnitude increases gradually and shows distinct peaks at the center. The point spread function is obtained from processing the signal with time data using a reconstruction algorithm developed by Goodwill et al. [15] The algorithm reconstructs the PSF by stitching together data from different pa rtial field of view (pFOV) scans of the sample. A representative PSF showing a shift in the positive and negative scan direction is shown in Figure 4 5 B. A B C

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42 Figure 4 5. Representative figures showing the raw signal and PSF obtained from the relaxometer A) Raw signal showing distinct peak at the center. B) Reconstructed PSF from the raw signal showing a shift in the scan direction. Figure 4 6 shows the effect of change in field amplitude at oscillating frequency of 4.5 kHz for particles suspended in 1 o ctadecene For the purpose of comparison, we have used the PSF in the positive scan direction. Figure 4 6. Reconstructed positive velocity PSF for particles subjected to a drive field frequency of 4.5kHz and varying amplitudes of 15, 30 and 45 mT. The s hift in the PSF in the scan direction is evident in Figure 4 6 with the magnitude of the signal increasing with increas ing amplitude of the drive field. The A B

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43 increase in signal strength is attributed to the increase in the rate of change of magnetization du e to increasing effects of magnetic torque acting on the particles with blocked magnetic moments. The shift in peak is due to the relaxation effect of the nanoparticles as they can rotate only through physical rotation It can also be observed that the shi ft increases with the increasing amplitude of the applied field. Another comparison to assess the effect of the changing frequency of oscillating field at amplitude of 30mT is shown in Figure 4 7. The shift in the PSF is al so observed for the variation in the frequency. The PSF for a frequency of 25 kHz on rec onstruction was incomprehens ible as shown in Figure 4 7 and hence was not included in the comparison. A possible reason might be that t he applied frequency of 25 kHz is very high as compared to the fre quency at which a Brownian peak was observed from the AC susceptibility measurements. Figure 4 7. Reconstructed positive velocity PSF for particles subjected to a drive field amplitude of 30mT.A) Effect of variation in frequency. B) PSF for a freque ncy of 25kHz. The oscillating frequency of 9.3 kHz also being higher than the peak frequency measured by the AC susceptometer leads to the reduction in the signal strength and the emergence of a second peak. These two comparisons ga ve us a brief idea about the A B

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44 effect of relaxation affecting the signal strength and peak shift. The PSF obtained for other set s of parameters were similar to those shown above The comparison of PSF for particles suspended in mineral oil at varying frequencies of oscillating fiel d and amplitude of 30mT is shown in Figure 4 8. As seen in the figure below, the PSF shows a behavior similar to that shown in Figure 4 7 B for frequency much higher than peak frequency of 660 Hz measured in AC susceptometer. Mineral oil being highly visco us, the relaxation time of the particles increases due to the increasing effects of hydrodynamic torque. Figure 4 8. Reconstructed PSF for particl es suspended in mineral oil for variation in frequency of drive field and amplitude of 30 mT. In order to p redict the shift in PSF caused due to the relaxation of the particles a n algorithm was developed which incorporates the fin ite relaxation effects. R otational Brownian dynamic s simulations were used to predict the response of single domain magnetic nano par ticles to application of magnetic field pulse sequence used in the Berkeley relaxometer. The predicted signal from the simulations will be used to obtain the PSF and a comparison will be made wit h the experimental observations.

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45 4.3 Brownian dynamics simula tions of relaxometer Brownian dynamics simulations are based on the integration of the stochastic angular momentum equation to obtain the orientation of each particle. Equa tion (4 3 ) gives the summation of the torques acting on the particle in a non inerti al limit. (4 3 ) T he torques acting on the particle are represented with respect to particle locked coordinates with being the hydrodynamic torque, due to magnetic effects an d due to Brownian motion. The torque due to hydrodynamic effects is given by (4 4 ) where is the viscosity of the surrounding fluid is the rotatio nal hydrodynamic resistance coefficient and is the angular velocity of the particle. The magnetic torque is given by (4 5 ) where is the vacuum permeability and is the applied field. The applied field is a summation of the bias field and the excitation field Here is the time and is the time o f the scan. In order to reduce the number of variables, time was non dimensionalized with respect to the rotational diffusion coefficient and the vector variables were non dimensionalized with respect to their corresponding magnitudes. Substituting Equatio n (4 4) and (4 5) in Equation (4 3 ) and setting and integrating from time to using the first order Euler method and applying the fluctuation dissipation theorem to the Brownia n term we get (4 6 )

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46 The vector is a random vector which follows the Gaussian distribution with mean a nd variance given by (4 7 ) The field is fr equency dependent through the sinusoidal excitation fie ld and is given by Equation (4 8 ). (4 8 ) where the Langevin parameters are and The algorithm proceeds from a starting configuration for a system of 10 5 particles and calculates the orientation at each time step. All runs were performed with a time step of The dimensionless parameters were obtained for the experimental conditions at which the p articles suspended in 1 octadecene were tested. The system is assumed to consist of non interacting single domain spherical particles. The results from the simulation are presented in the following section. 4.4 Simulation results The results from simulatio ns will be used to show the effect of relaxation on the signal. However, a quantitative comparison with the experimental observations was not carried out since the PSF reconstructed from the experimental data by the Berkeley Lab is obtained through applica tion of various filters and other signal processing techniques The simulations predict the average magnetization for the system of particles and the derivative of this magnetization with tim e provides the predicted signal Figure 4 9 shows the predicted r esults from the simulation for the application of drive field with frequency 1.6 kHz and amplitude 45mT. The bias field in all the simulations is mT. The magnetization curve shows a shape similar to the Langevin curve. The oscill ation in the curve increases as the bias field nears zero. This leads to the signal showing a peak

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47 at the center in Figure 4 9 B. The bias field being stronger than the drive field ,nanoparticles are saturated at the beginning of the scan sequence. Fi gure 4 9. Pre dicted simulation results for a drive field frequency of 1.6 kHz and amplitude of 45mT. A) Dimensionless average magnetization. B) Signal Figure 4 10 shows predictions for a drive field frequency of 4.5 kHz and amplitude of 45 mT. The PSF sh own in Figure 4 10 B shows a small shift in the peak. The signal predicted by the simulations is higher than that predicted for the application of the drive frequency of 1.6 kHz as shown in Figure 4 9. This behavior is consistent with that observed in the experiment. Figure 4 10. Simulation predictions for drive field frequency of 4.5 kHz and amplitude of 45mT A) Signal and B) PSF A B A B

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48 Simulation predictions for a drive field oscillating with frequency of 9.3 kHz and amplitude of 45 mT are shown in Figure 4 11. A significant widening of the PSF is observed as shown in Figure 4 11 B. The width of the PSF appears to be larger as compared to the one observed for frequency 4.5 kHz, as shown in Figure 4 10 B, and still shows prominent peaks at the center. F igure 4 11. Brownian dynamics simulation predictions for drive field frequency of 9.3 kHz and amplitude of 45mT. A) Signal and B) PSF showing a distinct shift from the center. Figure 4 12. Simulation predictions for drive field frequency of 25 kHz and amplitude of 45 mT. A) Signal and B) PSF with a deviation in the direction of scan. As seen in Figure 4 12, the PSF for frequency 25 kHz shows a completely distorted shape This is because the applied frequency is very high as compared to the A B A B

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49 peak frequen cy measured in the AC susceptometer. It can be seen from the above results that the signal shape from the simulations shows a shape similar to tha t obtained from the experiments .Although, Brownian dynamics simulations predict a shift in the PSF as seen in the experimental observations in order to obtain a fine PSF similar to the one reconstructed in the experiments, the signal data needs to be processed further using compl ex signal processing techniques. Figure 4 13. Effect of variation in the drive fi eld amplitude on the signal strength. Figure 4 13 shows the comparison of signal strength for variation in the amplitude of drive field An increase in the signal strength is predicted for increase in the drive field amplitude and frequency. This predictio n agrees with the experimental results shown in Figure 4 6.From the simulations it can be inferred that a higher frequency and large amplitude is suitable to obtain better signal from the nanoparticles. The simulations also predict the irregular shape of t he PSF for frequencies higher than 9.3 kHz as seen in the experiments along with the shift in the PSF which is due to the effect of particle relax ation incorporated in the simulation algorithm.

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50 CHAPTER 5 CONCLUSIONS The primary aim of the study was to elu cidate the effect of Brownian relaxation on the performance of the magnetic nanoparticles used in Magnetic Particle Imaging. Rotational Brownian dynamics simulations were used to predict the behavior of the particles relaxing by the Brownian mechanism The rotational Br ownian dynamics simulations gave an insig ht about the effect of finite magnetic relaxation of magnetic nanoparticles on the signal strength shift in the point spread function and ultimately on the resolution of the MPI system. The simulation predictions show significant deviation from the adiabatic theory and help to explain the experimental observations. T o improve the MPI performance of magnetic nanoparticles a set of optimized parameters (core diameter, shell thickness, scan speed, gradien t etc ) can be obtained from the predictions that compromise gains in signal strength and resolution with reduction in the image shift distances. We also concluded that, since t he predicted image shift distances are in the order of millimeter s, not account ing for the relaxation effects in MPI scanners would lead to significant error while assigning the location of particles in an image. The simulation results also suggest ed use of magnetic nanoparticles with large diameters to obtain better signal responses and improve resolution in MPI at the expense of increased image shift distances due to Brownian relaxation. The results from the simulation illustrate d how finite relaxation can have a profound effect on the image properties that can be achieved with the imaging technique. The simulation predictions of relaxometer experiment were found to be comparable with the experimental observations and aid ed in explaining the shift observed in the point spread function obtained from the experiments conducted in the

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51 B erkeley relaxometer. The study also helped in analyzing the effect of variation in the drive field frequency and amplitude on the particle response The simulations predict an increase in the signal with increase in the frequency as well as the amplitude. However, f urther work is needed to elucidate the potential effects on the response of the particles in the relaxometer to variation in bias field.

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52 LIST OF REFERENCES [1] P. W. Goodwill, E. U. Saritas, L. R. Croft, T. N. Kim, K. M. Kri shnan, D. V. Schaffer, S. M. Conolly, Advanced Materials 2012, 24, 3870. [2] B. Gleich, R. Weizenecker, Nature 2005, 435, 1214. [3] J. Borgert, J. D. Schmidt, I. Schmale, J. Rahmer, C. Bontus, B. Gleich, B. David, R. Eckart, O. Woywode, J. Weizenecker, J. Schnorr, M. Taupitz, J. Haegele, F. M. Vogt, J. Barkhausen, Journal of Cardiovascular Computed Tomography 2012, 6, 149. [4] J. Weizenecker, B. Gleich, J. Rahmer, H. Dahnke, J. Borgert, Physics in Medicine and Biology 2009, 54, L1. [5] P. W. Goodwill, J. J. Konkle, B. Zheng, E. U. Saritas, S. M. Conolly, I EEE Transactions on Medical Imaging 2012, 31, 1076. [6] P. W. Goodwill, S. M. Conolly, I EEE Transactions on Medical Imaging 2010, 29, 1851. [7] L. R. Croft, P. W. Goodwill, S. M. Conolly, I EEE Transactions on Medical Imaging 2012, 31, 2335. [8] R. Rosensweig, Ferrohydrodynamics Dover, New York 1997. [9] J. H. Sanchez, C. Rinaldi, Journal of Magnetism and Magnetic Materials 2009, 321, 2985. [10] D. Soto Aquino, C. Rinaldi, Physical Review E 2010, 82. [11] H. Goldstein, C. Poole, J. Safko, Classical Mechanics Addison Wesley, San Francisco 2002. [12] E. Dickinson, J.Chem.Soc, Faraday Trans. 2 1985, 81, 591. [13] R. Dhavalikar, C. Rinaldi, Journal of Applied Physics 2014, 115. [14] H. Arami, R. M. Ferguson, A. P. Khandhar, K. M. Krishnan, Medical physics 2013, 40, 071904. [15] P. W. Goodwill, K. Lu, B. Zheng, S. M. Conolly, The Review of scientific instruments 2012, 83, 033708. [16] P. W. Goodwill, A. Tamrazian, L. R. Croft, C. D. Lu, E. M. Johnson, R. Pidaparth i, R. M. Ferguson, A. P. Khandhar, K. M. Krishnan, S. M. Conolly, Applied Physics Letters 2011, 98. [17] S. Biederer, T. Knopp, T. F. Sattel, K. Ludtke Buzug, B. Gleich, J. Weizenecker, J. Borgert, T. M. Buzug, Journal of Physics D Applied Physics 2009, 42

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53 [18] R. W. Chantrell, J. Popplewell, S. W. Charles, I EEE Transactions on Magnetics 1978, 14, 975.

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54 BIOGRAPHICAL SKETCH Rohan Dhavalikar received his Bachelor of Engineering degree in chemical e ngineering from University of Pune (Formerly University of Poona) India in May 2012. In fall 2012, he joined University of Florida to pursue a Master of Science degree in chemical engineering.