The Role of Diffusion in NMR Proton Relaxation Enhancement by Ferritin

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By Michael A. Boss, B.S. Physics, M.S. Physics Graduate Program in Physics

The Ohio State University 2010

Dissertation Committee: Professor P. Chris Hammel, Advisor Professor Klaus Honscheid Professor Thomas Lemberger Professor Nandini Trivedi c Copyright by Michael A. Boss 2010 Abstract

By using binary solutions of water and glycerol, we controlled diffusion so as to better understand its role in the relaxation rate enhancement of protons in the vicinity of ferritin at 7 tesla. The slower diffusion rates and higher external magnetic field used in these experiments are more consistent with the conditions expected in MRI experiments. New data was obtained on the diffusion coefficients of water and glycerol in binary solutions with relatively dilute amounts of glycerol. The effects of chemical exchange in such systems was also quantified. Two main relaxation mechanisms have been proposed for protons in the vicinity of ferritin: an outer-sphere mechanism (OS) in which spins diffuse past ferritin and experience a changing Larmor frequency by moving through the ferritin’s magnetic field, and a proton exchange dephasing mechanism (PE), where protons temporarily reside on the surface of the ferritin core and sample a single, enhanced, Larmor frequency. At high-field, the OS mechanism becomes increasingly important because of a quadratic dependence on field strength, versus linear for the competing mechanism involving proton exchange. It was found that the relaxation enhancement of protons of both water and glycerol in the presence of ferritin was inversely proportional to their diffusion coefficients, in agreement with the OS model of relaxation enhancement. The strength of the relaxation enhancement on inverse diffusion coefficient was weaker for slow-diffusing glycerol than for water: glycerol molecules spent more time in a weaker magnetic field, indicating that glycerol did not approach the ferritin core as closely as water,

ii potentially answering questions about molecular intake into the ferritin structure. The results of these experiments have important implications for the quantiﬁcation of brain iron in vivo.

iii Para mis abuelos.

iv Acknowledgments

I have many people to thank for many reasons. I begin with my parents, because they began me, and have been behind me during all the trials and tribulations of graduate school. Thank you to my advisor, Chris Hammel, who took me in when things were looking most dire for continuing my research in NMR, and has guided me in understand what it means to be a physicist and a good scientist. Thank you to Denis Pelekhov for his assistance throughout the years, and to Yuri Obukhov for many fruitful discussions of NMR, in particular regarding chemical exchange. Thank you to Susan Olesik for providing me with the opportunities to teach outside of OSU, giving me valuable experience both professionally and in terms of social awareness, while also giving me valuable ﬁnancial support to continue my studies. I am grateful to the many students with whom I have worked on problems, both in class and in the lab, and there are more than I can name here. To Seongjin Choi, I owe much for the many conversations about NMR and MRI we had over the years. Thanks to Don Burdette and Jeﬀ Stevens for helping maintain my sanity by making me laugh when times were tough. To KC Fong, Steve Avery, and James Morris for

general physics discussions and help with LATEX. To my hundreds of FEH students who let me practice explaining physics for hundreds of hours. To Kay Chapman for helping me ﬁnd references in the Health Sciences Library. And to all my fellow graduate students who have have been such good listeners and friends. Thank you.

v Vita

October 25, 1977 ...... Born - Bedford, Ohio

May 2000 ...... B.S. in Physics, Cum Laude, Case Western Reserve University, Cleveland, Ohio May 2002 ...... M.S. in Physics, University of Illinois, Urbana-Champaign September 2002-present ...... PhD student in Physics, The Ohio State University, Columbus, OH Fields of Study

Major Field: Physics

vi Table of Contents

Page Abstract ...... ii Dedication ...... iv Acknowledgments ...... v Vita...... vi List of Figures ...... x List of Tables ...... xii

Chapters

1 Introduction and Overview 1 1.1 Motivation: Non-invasive Detection and Quantiﬁcation of Iron . . . . 1 1.2 Nuclear Magnetic Resonance Overview ...... 3 1.3 Ferritin: Role, Physical and Magnetic Properties ...... 5 1.4 Ferritin-induced Relaxation Enhancement ...... 11 1.4.1 Outer-sphere mechanism ...... 11 1.4.2 Proton exchange mechanism ...... 13 1.4.3 Two relaxation mechanisms ...... 14 1.5 Chapter Outline ...... 16

2 Nuclear Magnetic Resonance 19 2.1 Signal Source- Equilibrium Polarization ...... 19 2.2 Excitation and Detection ...... 21 2.2.1 Excitation and the rotating frame ...... 22 2.2.2 Signal detection ...... 24 2.3 Magnetization: Equations of Motion ...... 24 2.4 Physics of Relaxation ...... 25 2.4.1 Autocorrelation function and spectral density ...... 26 2.5 Spectroscopy ...... 28 2.6 Experimental Techniques ...... 29 2.6.1 Spin echoes ...... 31 2.6.2 Inversion recovery ...... 33 2.6.3 Saturation recovery ...... 33

vii 2.6.4 Carr-Purcell ...... 35 2.6.5 Carr-Purcell-Meiboom-Gill ...... 36 2.7 Diﬀusion ...... 37 2.7.1 Pulsed Gradient Spin Echo (PGSE) ...... 38 2.7.2 Pulsed Gradient Stimulated Echo (PGStE) ...... 40

3 Magnetic Resonance Apparatus 42 3.1 Probe Overview ...... 42 3.1.1 Radiofrequency and gradient board assembly ...... 43 3.1.2 Electronics ...... 44 3.2 Radiofrequency Coils ...... 45 3.2.1 Construction ...... 45 3.2.2 Tuning ...... 47 3.3 Gradients ...... 48 3.3.1 Coil design and modeling ...... 48 3.3.2 Assembly material ...... 50

4 Diffusion Data 53 4.1 Introduction and Premise ...... 53 4.2 Discerning Mechanisms of Dephasing: The Role of Diffusion . . . . . 54 4.2.1 Relaxation mechanism dependencies ...... 55 4.2.2 In vivo environment ...... 55 4.3 Controlling and Measuring Diffusion ...... 57 4.3.1 Extracting diffusion coefficients ...... 57 4.3.2 Water-glycerol samples ...... 60 4.4 Gradient Calibration ...... 61 4.5 Experimental Diffusion Data ...... 64 4.5.1 Pure solvents: water and glycerol ...... 64 4.5.2 Diffusion in binary mixtures ...... 66 4.6 Summary of Experimental Results ...... 71

5 Relaxation Data 76 5.1 Introduction and Premise ...... 76 5.2 Methodology and Experimental Results ...... 77 5.2.1 Longitudinal relaxation and saturation recovery ...... 77 5.2.2 Transverse relaxation (R2) and fast CPMG sequence . . . . . 81 5.2.3 Distinguishing water and glycerol relaxation ...... 89 5.2.4 Chemical exchange and T2 dispersion ...... 89

6 Discussion 98 6.1 Relaxation Enhancement, ∆R2 ...... 98 6.2 ∆R2 as a Function of OH concentration ...... 101 6.3 ∆R2 as a Function of Diﬀusion ...... 102 6.3.1 Water enhancement ...... 102

viii 6.3.2 Glycerol enhancement ...... 102 6.3.3 Water vs glycerol- distance of closest approach ...... 104 6.4 Conclusions ...... 106

ix List of Figures

Figure Page

1.1 Perl’s stain and MRI of the brain stem of a 44 year old male...... 4 1.2 Physical structure of apoferritin ...... 6 1.3 Ferritin magnetic structure ...... 8 1.4 Ferritin- canted AFM sublattices and defects ...... 9 1.5 Ferritin- magnetization vs applied field ...... 10 1.6 R2 vs B0 ...... 15 2.1 Zeeman splitting ...... 20 2.2 Precession of the magnetic moment in an external field ...... 22 2.3 Magnetization in the rotating frame ...... 23 2.4 Spectral density plot ...... 28 2.5 NMR spectrum of glycerol ...... 29 2.6 Free induction decay of water...... 30 2.7 Spin echo pulse sequence ...... 31 2.8 Spin echo rephasing ...... 32 2.9 Inversion recovery pulse sequence ...... 33 2.10 Saturation recovery pulse sequence ...... 34 2.11 Carr-Purcell train ...... 36 2.12 CPMG pulse sequence ...... 37 2.13 Effects of imperfect 180◦ pulse and correction via CPMG sequence . . 38 2.14 PGSE sequence ...... 39 2.15 PGStE sequence ...... 41

3.1 Assembled NMR probe ...... 43 3.2 Assembled NMR probe ...... 44 3.3 Bare RF board ...... 46 3.4 RF board with electronic ...... 47 3.5 Gradient board wire geometry ...... 49 3.6 Gradient board magnetic ﬁeld ...... 50 3.7 Gradient board constructed with thermal epoxy ...... 52

x 4.1 3 dimensional representation of a glycerol molecule...... 58 4.2 Gradient calibration ...... 63 4.3 Diffusion of pure glycerol at room temperature ...... 67 4.4 Diffusion of pure water at room temperature...... 68 4.5 Water diffusion compared to literature ...... 69 4.6 Glycerol diffusion compared to literature ...... 70 4.7 Diffusion in a binary mixture of water and glycerol ...... 72 4.8 Diffusion in a binary mixture of water and glycerol with ferritin . . . 73 4.9 Water diffusion as a function of composition ...... 74

5.1 Saturation recovery data, water ...... 79 5.2 Saturation recovery data, glycerol ...... 80 5.3 Saturation recovery for a water-glycerol mixture ...... 82 5.4 CPMG results of water ...... 86 5.5 CPMG results of glycerol ...... 87 5.6 Eﬀects of long τCP on water-glycerol CPMG experiment ...... 90 5.7 Eﬀects of long τCP on water-glycerol CPMG experiment ...... 91 5.8 Transverse relaxation in water-glycerol with ferritin ...... 92 5.9 T2 dispersion of pure water ...... 95 5.10 Chemical exchange in a water-glycerol system ...... 97

6.1 ∆R2 of water and glycerol vs 1/D ...... 103

xi List of Tables

Table Page

5.1 OH T1 as a function of composition and iron concentration ...... 83 5.2 CHx T1 as a function of composition and iron concentration . . . . . 83 5.3 Water R2 as a function of composition and iron concentration . . . . 92 5.4 Glycerolx R2 as a function of composition and iron concentration . . 93

6.1 Water ∆R2 as a function of composition and iron concentration . . . 100 6.2 Glycerol ∆R2 as a function of composition and iron concentration . . 100

xii Chapter 1 Introduction and Overview

1.1 Motivation: Non-invasive Detection and Quan- tiﬁcation of Iron

Non-invasive quantification of iron in vivo is an outstanding challenge for biomed- ical research. It is important to be able to measure iron concentrations, as the iron level in various organs can be indicative of disease. Magnetic Resonance Imaging (MRI) is a non-invasive imaging modality, based on Nuclear Magnetic Resonance (NMR), that is sensitive to the presence of iron in the body, notably in the brain. The goal of this dissertation is to clarify the mechanisms responsible for the relaxation of the magnetization of protons in the vicinity of the iron-storage protein ferritin. In particular, and in contrast with previous work in this area, we present experimental measurements of the proton relaxation rate enhancement as a function of diffusion rate that clarifies these mechanisms. Iron is an element that plays a crucial role in many processes in the human body. It is a component of hemoglobin, responsible for transporting oxygen via blood vessels, and is an integral part of other proteins and cellular structures. Maintaining proper iron levels in the body is important for health: low iron levels can lead to oxygen deprivation in the tissues of the body, while high levels can lead to the formation of free radical molecules that are damaging to DNA. Certain diseases can be indicated

1 by an abnormal iron level: hemochromatosis can be revealed by excess iron levels in the liver in the form of ferritin (an iron-storage protein) or hemosiderin (a complex of whole and denatured ferritin proteins),[1] anemia by insufficient iron in the blood in the form of hemoglobin. Iron also plays a role in neurodegenerative diseases such as Parkinson’s and Alzheimer’s, appearing in the senile plaques associated with the latter disorder.[2] Quantifying iron levels in the blood is a relatively straightforward procedure that is minimally invasive for most patients. However, measuring the local iron concentration of tissues and organs can be difficult, requiring invasive procedures such as biopsy. Furthermore, biopsy is not always feasible, such as when dealing with brain tissue. For this reason, iron levels in the brain are typically measured postmortem using techniques involving sectioning tissue into thin slices and employing chemical stains, such as Perl’s stain[2, 3], though they can also be detected using electron microscopy techniques[4]. Nevertheless, this makes measurement of iron levels for the purpose of disease diagnosis very difficult, if not impossible. What is needed is a non-invasive means of detecting iron in vivo. Fortunately, in the last three decades the medical community has found such a means: MRI[3, 5]. MRI is based on NMR, which relies on the nuclear spins of atoms precessing about an external magnetic field, in a manner analogous to a gyrating top precessing about a gravitational field. These spins precess coherently and generate an oscillating magnetic flux, which can be inductively detected. While MRI does not directly detect iron, the presence of iron has an effect on the magnetic behavior of the nuclear spins of the hydrogen atoms that are the source of the MRI signal, serving in general to shorten the time that the nuclear spins are coherently aligned in a plane transverse to an externally applied magnetic field (and thus generating detectable signal), enhancing the local transverse relaxation rate. These regions of shortened signal lifetime will

2 appear in an MRI as hyperintense areas, i.e. dark spots. The more iron there is, the shorter the local signal lifetime, and the greater the signal attenuation. While it is not overly difficult to detect the presence of iron with MRI, as seen in Fig. 1.1, it remains an outstanding problem to be able to quantify how much iron is in a given region. In order to accurately do this, it is necessary to have a solid understanding of the NMR relaxation mechanisms by which the magnetic fields generated by iron particles affect hydrogen atoms. By this means, one can measure the relaxation time of the MRI signal and translate it into a local iron concentration. The main focus of this thesis will be to examine the proposed relaxation mechanism models in order to more accurately determine iron levels via MRI. Special attention will be paid to the role of diffusion in ferritin-induced relaxation enhancement, with diffusion being controlled through a novel technique of using water-glycerol mixtures, which serves to mimic the different diffusion coefficients seen in vivo. In addition, we will provide details of the NMR experiments that enable discernment of the relaxation mechanisms, as well as improvements made to the home-built NMR apparatus used to collect the data presented in this thesis.

1.2 Nuclear Magnetic Resonance Overview

Before discussing ferritin and the mechanisms by which it affects protons, it is necessary to present a brief background in NMR, specifically in the source of the NMR signal and the processes that limit its lifetime (a more formal description of NMR will be given in Chapter 2). In nuclear magnetic resonance, nuclei with a net magnetic moment are polarized by an external magnetic field such that at thermal equilibrium the moments align with the field, either parallel or anti-parallel. One of these orientations will be energetically preferable, so there will be a net magnetic moment for the spin ensemble: in the case of protons, this net moment points parallel

3 Figure 1.1: Perl’s stain and MRI of the brain stem of a 44 year old male. The highlighted regions (substantia nigra, SN, and red nucleus, RN) show the presence of iron. On the left, the Perl’s stain causes iron-rich regions to become blue. On the right, there is very close correlation of a T2-weighted MRI image with the Perl’s stain. The presence of iron causes local tissues to have a shorter MRI signal lifetime, resulting in a lower intensity when compared to surrounding tissue. From Ref. [3]

to the ﬁeld. By applying a radiofrequency pulse of suﬃcient power and duration at the Larmor frequency (ωL = γB0, where γ is the gyromagnetic ratio for the nuclei of interest), this moment can be rotated into the transverse plane, where the

spins will precess coherently about the external field direction at ωL. This oscillating magnetization generates a oscillating magnetic flux which can be inductively detected via a pickup coil, turning the NMR signal into a measurable electric signal. This precession does not continue indefinitely. Spins experience fluctuations in

their local magnetic ﬁeld, and thus ωL, due to their local magnetic environment as well as the inﬂuence of neighboring spins. Fluctuations in the locally experienced

ﬁeld at ωL can cause quantum mechanical transitions which in aggregate return the spin ensemble to thermal equilibrium. The time for this longitudinal relaxation is

known as T1. At the same time, diﬀerences in the local Larmor frequency cause

4 the spin ensemble to become incoherent, reducing the amplitude of the transverse

component of the magnetization (on top of the T1 eﬀect). The characteristic time for the transverse component of the ensemble’s magnetization to return to thermal equilibrium (where it is zero) is called T2. These relaxation times are of key interest in the use of MRI to quantify iron in vivo.

1.3 Ferritin: Role, Physical and Magnetic Proper- ties

The primary role of ferritin is to sequester and store iron in the body.[6] The ferritin molecule is very old in the sense that virtually all eukaryotic organisms contain some version of it.[7] This is not surprising given that iron atoms left free will react with many diﬀerent components of a living organism, potentially causing much damage.[8] The solution provided by evolution is to use a roughly spherical shell, apoferritin, to enclose a mineral iron core. In human beings, the apoferritin shell is approximately 13 nm in diameter, while the core diameter varies in size depending on the loading factor (LF) of the ferritin protein, or the number of irons atoms contained inside each protein. The maximum diameter is 7-8 nm, while the maximum LF is 4500 iron atoms. The apoprotein shell consists of 24 subunits. Channels form at the junctures of these subunits, and the size of these channels is on the order of 1-4 A,[9,˚ 10, 11, 12], as seen in Fig. 1.2. There are two types of channels formed, depending on whether the juncture is between 3 or 4 subunits: iron enters via 3-fold channels and leaves the protein via the 4-fold channels. Research indicates that water enters the cavity via the same channel as the iron, owing to the hydrophobic nature of the 4-fold channels.[12, 13, 14, 15, 16] The mineral core of ferritin is in the form of ferrihydrite,

5Fe2O3·9H2O. Easily hydrated, hydroxyl (OH) functional groups reside on the surface

5 Figure 1.2: Physical structure of the ferritin protein shell, apoferritin. The protein measures approximately 13 nm in diameter: unshown is the ferritin core. Of particular interest is the channel at the juncture of three subunits (an example is highlighted): it is through these 3-fold channels that iron, as well as water is believed to be taken into the apoferritin cavity, where they aggregate in the form of ferrihydrite. This mineral core has an imperfect AFM magnetic ordering which gives rise to a net SPM behavior.

6 of the core, and possibly deeper within the structure of the core. Ferrihydrite is known to be prone to defects such as vacancies in its structure, and X-ray diffraction studies show that it is not well described by a single crystal structure[17, 18, 19]. These physical properties have important consequences for the magnetic ordering of the core, as well as its effect on protons. The magnetic properties of ferritin have been extensively studied.[20, 21, 22, 23, 24, 25] Brooks et al. divided the ferrihydrite core into two regions: the surface and inner core. The Néeltemperature differs for the surface of the core compared to its interior: surface iron atoms behave paramagnetically due to a lower TN (below room temperature), while the inner atoms have a much higher TN [24]. Fig 1.3 demonstrates this type of structure. In essence, though, the core is antiferromagnetic (AFM), but behaves superparamagnetically due to vacancies in the crystal structure and canting of the AFM sublattices[26]. The canting is caused by the external magnetic field, and is perpendicular to the easy axis. Vacancies lead to uncompensated spins, also contributing to a net magnetic moment, as demonstrated in Fig. 1.4. The magnetic moment of ferritin depends on how large the core (the loading factor, LF ), as well as how large the external magnetic field is. Because the core behaves superparamagnetically, the moment increases with field, and most interestingly, does not saturate at the field strengths used in MRI, on the order of a few tesla. The field-dependence of the moment is linear at room temperature, shown in Fig. 1.5, as reported by several different researchers.[24, 23, 22, 21] The moment in general is on the order of tens to hundreds of µB,[24, 26] depending on the applied field and the average loading factor of the proteins.

7 Figure 1.3: The magnetic structure of ferritin. An apoprotein shell (apoferritin) surrounds an inner mineral core. The shell has a diameter of 13 nm, while the core has a diameter that varies depending on the loading factor, but which scales as LF1/3, up to a maximum of 8 nm. In this model of the magnetic ordering by Brooks et al., only the interior of the core contributes to the net moment of the particle. Figure adapted from Ref. [24].

8 Figure 1.4: Canted AFM sublattices in ferrihydrite. The anisotropy axis is indicated by the dashed lines. The external ﬁeld causes the AFM sublattices to have a slight canting, leading to a net moment in a direction perpendicular to the easy axis, rep- resented by the double-lined arrows. Defects (D) also contribute to the net magnetic moment of ferrihydrite, in addition to the canting. Figure adapted from Ref. [26].

9 Figure 1.5: Ferritin magnetization vs external field strength. At room temperature and typical NMR/MRI field strengths, the field-dependence is linear. Figure adapted from Ref. [23].

10 1.4 Ferritin-induced Relaxation Enhancement

Protons in the vicinity of ferritin experience the magnetic field of the ferrihydrite core. This field is in addition to the external field, leading to an enhancement of the relaxation rates of nearby protons. In particular, the transverse relaxation rate of

protons (R2 = 1/T2) is very sensitive to the presence of ferritin. Two mechanisms have been proposed to explain this sensitivity: an outer-sphere mechanism in which spins diffuse past the protein, and a proton exchange mechanism, where protons temporarily reside on the core surface. In both mechanisms, dephasing occurs as spins experience different Larmor frequencies during the time they are either diffusing past the ferritin (on the order of several nanoseconds), or residing on the core’s surface (for approximately 30 ns).[26] The dependencies on diffusion coefficient and external field strength differ for these two mechanisms, and will be described in further detail: in short, only the outer-sphere mechanism is diffusion dependent, while proton exchange is not. Measuring the relaxation rate enhancement of protons as a function of diffusion or external field strength provides information on the contributions of the relaxation mechanisms.

1.4.1 Outer-sphere mechanism

One way protons are affected by the presence of ferritin is by diffusing through the dipolar field created by the core, known as the outer-sphere mechanism. This has been described in detail previously: as the spins diffuse near the protein, the magnetic field they experience changes, which causes them to precess at different frequencies. This enhanced dephasing manifests itself as an increased transverse relaxation rate.[27, 28, 29, 30] Gossuin et al.[31] recently described the outer-sphere relaxation of protons in aqueous solution by ferritin in terms of explicit variables:

11 1 16 µ2γ2N µ2 = 0 F (1.1) T2,OS 135π RD

Here γ is the gyromagnetic ratio of hydrogen, NF is the number of ferritin proteins per unit volume, µ is the magnetic moment of the ferrihydrite core with radius R, and D is the diffusion coefficient of water. The model presented in Eq. (1.1) has evolved from arguments first put forth by Solomon[32] and invoked by Gueron[33] in studies of relaxation due to paramagnetic ions, and then refined by Gilles, Roch, and Brooks[34] with application towards SPM nanoparticles. Eq. (1.1) is the end product of these treatments, which invoke the spectral density functions of protons near paramagnetic, and in the case of ferritin, superparamagnetic spheres. Magnetic field correlation functions for spins diffusing in the dipolar field of such particles are Gaussian.[33] We will discuss correlation functions and spectral density more in Chapter 2.4; in the meantime, the relaxation rates are explicitly depend on the square of the moment that generates the field which the protons are experiencing and inversely proportional to the diffusion coefficient. A simple model to understand the diffusion mechanism’s dependence on µ2, and by

2 extension, on B0 , is to imagine it as a random walk along the surface of a sphere with radius R, where the z-component of the dipole magnetic field varies as a function of angle with respect to the external field direction. We first imagine the mean-squared phase accumulation of spins moving on this surface:

2 2 2 ∆φrms = ∆ωrmsτ N (1.2) where is N is the number of steps taken in the random walk, equal to T/τ, i.e. the number of times a spin interacts with a ferritin protein. In this simple model, the time T2 can be deﬁned as the time it takes for a given spin to dephase by 1 radian

12 relative to spins that are not aﬀected by ferritin. If we set Eq. (1.2) equal to unity,

2 let τ = τD = R /D, and solve for T (=T2), we obtain

2 1 2 R ≈ ∆ωrms (1.3) T2 D

∆ω is the Larmor frequency shift due to the dipole ﬁeld, which is proportional to µ.

1.4.2 Proton exchange mechanism

The ferrihydrite core is coated with hydroxyl functional groups. These groups may serve as exchange sites, whereby protons from aqueous solution exchange with the hydroxyl protons for some residence time, τPE.[35, 36] The distribution of ∆ω for these

2 2 sites has been determined by to be Lorentzian,[37, 38] described by K/[π (∆ωrms + K )], where K is the half-width at half maximum of the distribution, in units of radians/sec. During this exchange time, the protons experience a constant z-component of magnetic ﬁeld, which is large given the proximity to the core, leading to larger dephasing than occurs just by diﬀusive motion. They then exchange back into the pool of bulk water protons. The relaxation rate predicted from this physical process[26] is given by:

1 QC = K F (1.4) T2,P E COH

Here, K is as described previously, Q is the number of exchange sites per ferritin core

(in the hundreds for typical LF), CF is the concentration of ferritin proteins in mols, and COH is the concentration of hydroxyl protons, 111 M for water. K is proportional to the external magnetic field: as B0 increases, µ scales linearly, and the field created by the ferritin core at the hydroxyl sites increases proportionally. Thus ∆ωL at each site will also increase linearly with field.

13 In the limit that τD < τPE, this process is diffusion-independent. While it is true that diffusion brings in new spins to be dephased by this mechanism, it does not play a role in the actual dephasing as the protons are static relative to the core’s field during the time they reside on the ferrihydrite surface. By the time a given proton is released back into the bulk, the protons around the core have been completely refreshed, so that they have not been strongly affected by being close to the core via the outer sphere mechanism, while at the same time ensuring that each spin’s dephasing events are unaffected by previous ones. Because there is exchange between the bulk water and the hydroxyl sites on the core, the relaxation caused by this mechanism is weighted by the number of hydroxyl sites on the ferrihydrite surface compared to the number of hydroxyl protons in the bulk, reflected by QCF /COH .

1.4.3 Two relaxation mechanisms

The two relaxation mechanisms described exhibit two fundamental differences: their field-dependence (linear for proton exchange, quadratic for outer sphere), and their dependence on diffusion (explicit for outer sphere, unimportant for proton exchange). Researchers devoted much effort into exploring the field-dependence of ferritin-induced proton relaxation in aqueous solution: the evidence pointed strongly towards a proton exchange mechanism because R2 = 1/T2 varied linearly with field.[35] However, these experiments were conducted under conditions that do not accurately reflect the in vivo environment and the latest MRI technologies. In particular, the latest generation of MRI apparatus operate at fields of 7T, with active development of systems operating at even higher field strengths. Additionally, in the brain, where much of effort to quantify iron using MRI has been focused, the diffusion coefficients are considerably smaller than those used in vitro.[39]

OS 2 PE Since R2 ∼ B0 /D, while R2 ∼ B0, working at higher ﬁelds with diﬀusion

14 Figure 1.6: R2 vs external ﬁeld strength for aqueous solutions of ferritin in vitro. The concentration of iron is 100mM, with an average loading factor of 1720 iron atoms. ◦ The solution was maintained at a temperature of 37 C. The ﬁeld-dependence of R2 provides evidence for the proton exchange mechanism, albeit under conditions not representative of in vivo ferritin. Adapted from Ref.[35].

15 coefficients more like those seen in the brain makes the OS mechanism more likely to noticeably contribute to the proton relaxation rate enhancement. It is relatively straightforward to take in vitro ferritin in aqueous solutions and perform experiments at high-field. What is less clear is how one can change the diffusion coefficient easily in order to explicitly test the diffusion dependence of ferritin-induced proton relaxation, and more definitively determine whether or not the OS mechanism can be ignored, as seems to be evidenced by the field dependence. The solution we employed was to use mixtures of water and glycerol. There are other means of slowing the diffusion rate of water molecules. Certainly one can use gels such as agarose, albumin, or gelatin. Unfortunately, these do not create a homogeneous environment, and add structure to the solution, making it less clear what part of the relaxation enhancement comes from the presence of ferritin, and what part comes from the overall structure of the solution. Water and glycerol on the other hand are highly miscible, and can create a relatively homogeneous binary mixture, without any kind of long range structure as seen in gels. The heart of this dissertation is to test the diffusion dependence of ferritin-induced relaxation enhancement by controlling diffusion by adding more or less glycerol as need be. The details of this method will be provided in Chapters 4 and 5.

1.5 Chapter Outline

In Chapter 2, we will examine the basics of nuclear magnetic resonance (NMR), the underlying phenomenon that enables MRI. We will discuss the source of the NMR signal, as well as how that signal is excited and detected. We will describe the processes of relaxation phenomenologically, as well as how NMR can discern between different types of molecules. We will demonstrate how an NMR signal can carry information about diffusion rates, as well as the different types of experiments use to

16 characterize relaxation rates. We made significant changes and improvements to our homebuilt NMR apparatus, which will be explained in Chapter 3. In particular, we will describe modifications made to our radiofrequency (RF) coil, a solution to overheating issues in our magnetic field gradient coils, and a redesign of a duplexer to enable the use of a single RF coil for both excitation and detection of the NMR signal. We will also examine modeling of the gradient coils. Diffusion data will be presented in Chapter 4. The specific NMR experiments chosen to quantify diffusion will be discussed, as well as the rationale behind them. We will discuss the importance of discerning water and glycerol diffusion, and how this was accomplished with just one experiment. New data for dilute water-glycerol solutions will be presented, and compared to existing data from the literature. Chapter 5 contains relaxation data of water and glycerol protons, both with and without ferritin at varying concentrations. We will discuss the difficulties in obtaining good relaxation data in water-glycerol systems due to short relaxation times, and especially due to the unanticipated process of chemical exchange, whereby protons from water exchange with the hydroxyl protons of glycerol. This process can obfuscate the relaxation enhancement induced by ferritin, but by employing methods that allowed the acquisition of data in a regime where chemical exchange could be disregarded we obtained inherent relaxation rates. New data for chemical exchange in water-glycerol systems will be presented, and we will also examine relaxation data for the same systems with the addition of ferritin, a novel measurement. An in-depth discussion and analysis of data will occur in chapter 6. We will paint a physical picture of how exactly ferritin causes relaxation in water-glycerol systems. The results we obtained will be compared and contrasted with those in the literature. Specifically, we will examine the diffusion dependence of ferritin relaxation

17 enhancement, a phenomenon which has to date not been explicitly tested by any other experimenters. In addition, possible future work and implications of this dissertation will be brieﬂy outlined.

18 Chapter 2 Nuclear Magnetic Resonance

Nuclear Magnetic Resonance (NMR) is a phenomenon in which the nuclei of certain atoms with a net magnetic moment line up and generate a coherent, precessing magnetization which can be detected, usually via inductive methods. This net magnetic moment is related to the nuclei’s intrinsic angular momentum, J, by µ = γJ, where γ is the gyromagnetic ratio, diﬀerent for each nucleus. In the case of hydrogen, γ = 2.675 × 108 rad/s·T. ˆ In the presence of an external magnetic ﬁeld (by convention, B0 = B0k), Zeeman

splitting occurs, with the familiar Hamiltonian of H = −µ · B0. For the sake of simplicity, we will use a spin-1/2 particle with positive γ, of which hydrogen (the nuclei of choice in this dissertation, the cited NMR literature, and in MRI) is an example. Two eigenstates arise, with eigenvalues E± = ∓γ~B0/2, where + refers to an alignment of the moment parallel with the ﬁeld, and - antiparallel. However, these spin states are not equally populated, leading to an overall magnetic moment for the ensemble of spins.

2.1 Signal Source- Equilibrium Polarization

At thermal equilibrium, an ensemble of spins has a net magnetization. Because the eigenstates are unequally populated, there will be a diﬀerence in the number of

19 Figure 2.1: Zeeman splitting for spin-1/2 nuclei.

spins aligning with ﬁeld and those aligning against it. The lower energy states will have higher populations than the higher energy states; this population diﬀerence is what will ultimately generate the NMR signal. The population of the spin states for a nuclei with spin I can be described using Boltzmann statistics. The net magnetization is given by:

I X m (1 + β~γB0m) M = nγ m=−I (2.1) th ~ I X (1 + β~γB0m) m=−I where n is the number of spins per unit volume. Invoking properties of summations, this can be simpliﬁed to:

I (I + 1) M = nγ2 2βB (2.2a) 0 ~ 0 3 nγ2 2B 1 = ~ 0 , for I = (2.2b) 4kBT 2

20 The magnetization is proportional to the strength of the external magnetic field, and inversely proportional to temperature. This makes sense in terms of a competition between the Zeeman energy and the thermal energy: at high temperatures, the latter is big enough to cause transitions into the higher energy state. At room temperature, and fields on the order of tesla, the difference between up and down spins in an ensemble of hydrogen nuclei is very small, such that only 1 in every 105 nuclei is contributing to net magnetization. Each individual spin’s magnetic moment has both longitudinal and transverse components. As explained in Slichter[40] and other classic NMR texts,[41, 42, 43] at thermal equilibrium the expectation value of the transverse component of the moment oscillates at the Larmor frequency, ωL = γB0. Classically, this is equivalent to a magnetic moment µ which rotates about the z-axis at ωL. In effect, the external field exerts a torque on the moment, causing it to precess, as seen in Fig. 2.2. The transverse components of the spins are not coherent across the ensemble, so the net magnetization in the transverse plane is zero at thermal equilibrium.

2.2 Excitation and Detection

The application of a magnetic ﬁeld oscillating at the Larmor frequency and ori- ented in a plane transverse to B0 induces transitions between the states. This has the eﬀect of reducing Mz to zero, while at the same time causing the spins’ magnetic moment to have coherent expectation values in the transverse plane, i.e. the the net magnetization points in the transverse plane and precesses about B0 at ωL. It is easiest to imagine this process classically by examining the torque exerted on the net magnetization, and to do so from a rotating frame of reference.

21 Figure 2.2: Precession of an individual magnetic moment in an external ﬁeld. The precessional rate is the Larmor frequency, ωL = γB0.

2.2.1 Excitation and the rotating frame

We move to a frame of reference rotating at frequency ω. In this frame of reference, the eﬀective magnetic ﬁeld is given by

ω B = B − kˆ + B î (2.3) eff 0 γ 1 where B1 is the maximum amplitude of the transversely applied magnetic field. If the frame rotates at the Larmor frequency, the effective field lies only in the transverse plane. This will exert a torque on the net magnetization of the spin ensemble, causing it to rotate. By applying a pulse of sufficient power and duration, the magnetization can be rotated completely into the transverse plane. This is depicted in Fig. 2.3. Typically this excitation is accomplished using a solenoidal coil that surrounds the sample. For protons, frequencies in the RF spectrum are typical, ranging from tens

22 Figure 2.3: Magnetization in the rotating frame. The frame is rotating at ωL relative to the lab frame. An oscillating transverse magnetic ﬁeld, B1 is applied, which in the rotating frame appears static; the longitudinal component of Beff is zero because of the ω = ωL. For the sake of simplicity, this has been shown in the -x direction in order to cause a rotation to the +y axis. The angle of rotation depends on the strength of B1 and its duration, θ = γB1tpulse.

23 up to hundreds of MHz for external ﬁeld strengths on the order of several tesla.

2.2.2 Signal detection

In the lab frame, this magnetization precesses about the external field at the Larmor frequency. In effect, and precessing magnetic flux is created by the spin ensemble. This oscillating flux induces an EMF in a surrounding solenoidal coil (typically the same one used to excite the sample) which can be measured. The maximum amplitude of this EMF is:

ε = ωLb1MV (2.4)

where b1 is the magnetic field per unit current in the solenoid. Hoult and Richards [44] further described the signal-to-noise ratio in terms of the coil geometry, the gyromagnetic ratio of the nuclei of interest and the external field strength, taking into account such phenomena as skin depth and the proximity effect.

2.3 Magnetization: Equations of Motion

We have brieﬂy outlined how the NMR signal is generated and detected. As described in Chapter 1, the signal does not last indeﬁnitely. Its return to thermal equilibrium after excitation is characterized by a longitudinal and a transverse re-

laxation time, T1 and T2, respectively. Phenomenologically, the behavior of the net magnetization is described by the Bloch equations[45], in eﬀect the equations of motion for the magnetization vector. In the rotating frame, these equation can be written

24 as:

dMz Mz − M0 = γ (M × B)z − (2.5a) dt T1 dMx Mx = γ (M × B)x − (2.5b) dt T2 dMy My = γ (M × B)y − (2.5c) dt T2

Under the presence of a transverse magnetic field oscillating at the Larmor frequency, the magnetization will rotate about the oscillating field direction (which in the rotating frame appears as static). In a typical NMR experiment, this rotation will be allowed to continue to 90◦, at which point the excitation RF will be turned off, leaving the magnetization in the transverse plane. At that point, the magnetization will begin its return to thermal equilibrium according to the second terms in Eq. (2.5). This return is characterized by exponential behavior for both the longitudinal and transverse magnetization.

2.4 Physics of Relaxation

The causes of relaxation are changes in the magnetic field a nuclear spin experiences as a function of time. To be brief, T1-processes are caused by transverse oscillating magnetic fields at the Larmor frequency which induce transitions in the eigenstates of the spins. This of course will also affect T2: if the observable magnetization has relaxed back to the longitudinal axis, there will be no remaining transverse component to generate the changing flux that is inductively detected. In addition, static longitudinal fields affect T2, by changing individual spins’ Larmor frequencies: this induces dephasing as the spins no longer precess at the same rate, and over time fall increasingly out of phase with one another. Because T1 processes involve a transfer of energy out of the spin system, it is often times referred to as spin-lattice

25 relaxation, whereas T2 is known as spin-spin relaxation. These fields are generated by the local environment of the spin. Specifically, the thermally induced motion of the spins creates the oscillating fields responsible for longitudinal relaxation via dipole-dipole interaction between nearby spins. In the case of water, this interaction usually takes the form of the hydrogen pair on a single water molecule, tumbling through the liquid in such a way that the field experienced by one proton due to the other is modulated by the tumbling frequency. We will briefly examine the physics of longitudinal relaxation, which as stated above limits

T2 as well.

2.4.1 Autocorrelation function and spectral density

One way to quantify this process is to look at the autocorrelation function of the magnetic ﬁeld a given spin experiences, deﬁned as:

G(τ) = hB(t)B(t + τ)i (2.6)

Here B refers to the transverse component of the field a spin experiences, and τ is the interval between examining the field. As τ increases, the autocorrelation function will tend to 0, as the spin is less and less likely to keep experiencing the same field strength. If we assume random fluctuations, we can approximate the autocorrelation function as having an exponential form:

G(τ) = B2e−τ/τc (2.7)

where τc is the correlation time, the characteristic time for the field a spin experiences to change, i.e. a measure of how fast the fluctuations are occurring. Since it is the frequency of these fluctuations that is important, we will look at the Fourier transform

26 of the autocorrelation function, the spectral density:

Z ∞ J(ω) = G(τ)e−iωτ dτ (2.8) −∞

This will tell us how many ﬂuctuations occur at a given frequency. Plugging in our expression for an exponentially decaying autocorrelation function, Eq. (2.7), yields a Lorenztian:

2 τc J(ω) = 2B 2 2 (2.9) 1 + ω τc

For long τc, the spectral density function is narrow, and accordingly broad for short

τc. A sharp transition in the density is seen at τc, as seen in Fig. 2.4. It can be shown

[40, 43] that the longitudinal relaxation rate, R1 = 1/T1, is related to the spectral density by:

1 1 2 2 2 τc = γ J(ω) = γ B 2 2 (2.10) T1 2 1 + ω τc

As ωL approaches 1/τc, the thermally-induced fluctuations become more effective at inducing transitions, leading to enhanced relaxation. In the case that 1/τc ωL, the relaxation rate will be proportional to the correlation time. At the correlation times typically found in the liquids examined in this dissertation, and the Larmor frequencies encountered in this thesis, this approximation is valid. Because the density of fluctuations responsible for inducing transitions is small at ωL, relaxation is weak in these systems, and as a result T1 is fairly long. Further details of both longitudinal and spin-spin relaxation via the dipolar interaction between two spins were developed by Bloembergen, Purcell and Pound, the famous BPP theory of relaxation [46], and further developed by Solomon [32]. Clear synopses of these works can be found in the classic texts by Slichter[40] and Levitt[43].

27 JHΩL 1. ´ 10-12

8. ´ 10-13

6. ´ 10-13

4. ´ 10-13

2. ´ 10-13

Τc 1010 1011 1012 1013 1014

Figure 2.4: Spectral density vs τc. A sharp transition in the spectral density can be −12 seen at the correlation time, τc = 10 s in this plot.

2.5 Spectroscopy

One of the more useful facets of NMR is chemical shift. The local environment of a nuclear spin is heavily influenced by the magnetic moments of electrons from neighboring atoms. This causes slight differences in Larmor frequencies, known as chemical shifts. By taking the time domain NMR signal and subjecting it to a Fourier transform, one obtains the frequency spectrum of the NMR signal, reflecting these

diﬀerences in ωL. In systems with multiple molecules, relaxation times and diﬀusion rates can be determined separately by integrating the area underneath resolved peaks, rather than looking at the overall amplitude of the NMR signal.[41] Fig. 2.5 shows an example of chemical shift for the glycerol molecule. Glycerol has 3 hydroxyl protons (bonded with oxygens) and 5 protons associated with carbon atoms. These protons

experience diﬀerent Larmor frequencies. The expected ratio of the OH and CHx peaks’ areas is 5/3, or 1.67. Because glycerol is hydrophilic, it is likely that some atmospheric water was absorbed in the sample, increasing the overall proportion of

28 OH protons, and decreasing the ratio to 1.58.

6 3.5x10

Glycerol NMR Spectrum CHx Peak 3.0 OH Peak 2.5

2.0 Relative Peak Area: A /A =1.58 CHx OH 1.5

1.0

0.5

Signal Amplitude [Arb. Units] [Arb. Amplitude Signal 0.0 -1000 -500 0 500 1000 Frequency Shift [Hz] Figure 2.5: NMR spectrum of glycerol obtained at 7 tesla. The expected ratio of the OH and CHx peaks is 1.67, the data is in fairly good agreement. This sample may exhibit some contamaination with water, upping the proportion of hydroxyl protons.

2.6 Experimental Techniques

In modern practice, the NMR signal is usually generated by applying an RF pulse of sufficient power and duration to rotate the magnetization vector into the transverse plane where it generate an alternating magnetic flux which is inductively detected. This signal will not last indefinitely due to relaxation. This signal is known as a 29 free induction decay (FID). Under ideal conditions, the envelope of the FID would decay exponentially with a time constant equal to T2. In practice, there are other

∗ effects that cause the signal to decay in a faster time, T2 . Usually this is due to field inhomogeneities which cause spins in different regions of the sample to precess at different rates. The effects are seen in Fig. 2.6.

5000

-5000

FID Amplitude [Arb. Units]

-3 0 10 20 30x10 Time [s]

Figure 2.6: Free induction decay of water. While water has a relatively long T2 of about 2.5 s at 7 tesla, due to the eﬀects of external ﬁeld inhomogeneities, the FID decays on a timescale closer to 1 ms. The envelope of the FID remains exponential.

30 2.6.1 Spin echoes

There is a solution to the problems posed by ﬁeld inhomogeneities: spin echoes. First developed by Hahn[47], spin echoes are formed by the application of a 180◦ pulse after the initial 90◦ excitation pulse, where the 180◦ is simply twice as long as the 90◦ at the same power.

τ τ

◦ ◦ 90−x 180y Echo

Figure 2.7: Spin Echo pulse sequence. The 90◦ and 180◦ pulses are separated by a time τ. After 2τ an echo forms, which is recorded in the same manner as an FID would be.

As spins lose their coherence due to locally diﬀerent Larmor frequencies, the transverse magnetization responsible for the NMR signal attenuates. Some of the spins precess faster, and some slower. In the rotating frame, this can imagined as a spread about the reference Larmor frequency, which remains stationary. After some time τ has passed, the situation in the left of Fig. 2.8 will have occurred, where the fastest spins have moved counter-clockwise in the rotating frame, and the slowest spins clockwise. We assume an initial 90◦ pulse that caused the magnetization to lie

31 along the y-axis in the rotating frame. By applying a 180◦ pulse about the y-axis in the rotating frame in this case (accomplished through control of the RF phase), the phases of the spins will be mirrored relative to the y-axis. However, the direction of their dephasing remains the same. In other words, the fast spins continue to dephase counter-clockwise, the slow spins in a clockwise manner. After another time interval equal to the ﬁrst between the 90◦ and 180◦ pulses, 2τ from the initial excitation, the dephasing will have been reversed, and the spins will have momentary coherence. This is known as a spin echo, and is commonly used in NMR pulse sequence to recover transverse magnetization lost to changes in the magnetic ﬁeld spins experience on a timescale longer than τ.

180◦ f s

s f

Figure 2.8: Recovery of magnetization using spin echo. Spins experiencing a higher Larmor frequency move counter-clockwise relative to spins with ωL,0 = γB0, while those with slightly lower ωL move clockwise in the rotating frame. By application of a 180◦ pulse about the y-axis at a time τ after the initial excitation, the fast spins are moved to the opposite side of the y-axis, but still precess counter-clockwise, and the slow spins clockwise. This allows for them to rephase, recovering a maximum signal amplitude at time 2τ.

32 2.6.2 Inversion recovery

The inversion recovery sequence is a technique used to measure T1. Rather than an initial 90◦ pulse, a 180◦ is used to invert the spin population, as seen in Fig. 2.9. After

180◦ 90◦ FID

Figure 2.9: Inversion recovery pulse sequence. Used to measure T1, spins are ﬁrst inverted by a 180◦ pulse, and allowed to return to equilibrium for a time t, when they are exposed to a 90◦ pulse, and the resulting FID is measured.

a time t, the spins are subjected to a 90◦ pulse, and the resulting free induction decay is recorded. By increasing τ, the spins are allowed to recover to thermal equilibrium. By plotting the amplitude of the FID versus t, the magnetization can be seen to go from

−M0 to +M0 on a timescale of T1. Assuming perfect inversion, the magnetization recovers to the thermal equilibrium according to

−t/T1 M(t) = M0 1 − 2e (2.11)

2.6.3 Saturation recovery

One of the limitations of the inversion recovery sequence is that it can be time- consuming to acquire data: in between each acquisition, suﬃcient time must be allowed for the spins to return to thermal equilibrium, on the order of 5 T1. For-

33 tunately, a solution exists in the form of the saturation recovery pulse sequence. In contrast to the previously mentioned pulse sequences, the ﬁrst step is to scramble the spins such that they have zero net magnetization, longitudinal or transverse. In other words, the goal at the start is to destroy any coherence the spins might have, making the need for a long delay between scans disappear. There are many ways

◦ RFsat RFsat 90 FID

tevolve trecovery

n repetitions

Figure 2.10: Saturation recovery pulse sequence. Used to measure T1, spins are ﬁrst inverted by a 180◦ pulse, and allowed to recover to equilibrium for a time t, when they are exposed to a 90◦ pulse, and the resulting FID is measured.

of accomplishing this saturation: one common method is to take advantage of the inhomogeneities present in the external ﬁeld by applying a train of RF pulses, for instance n 90◦ pulses. Imagining a classical picture in the rotating frame, by allowing for a delay between pulses for spins to become incoherent, tevolve, and then applying more RF, the spins are continuously rotating about the same axis, but from diﬀerent orientations, such that as time goes on, both their net transverse and longitudinal moments are zero. The spins are then allowed to recover for a time t, after which an FID is recorded. It does not matter if the spins have not fully reached thermal equilibrium, the next repetition of the experiment can begin immediately because the spins will be driven away from thermal equilibrium anyway. This allows for much

34 faster acquisition rates, important in liquids NMR and MRI, where T1 is often on the order of seconds. The magnetization will recover according to:

−t/T1 M(t) = M0 1 − e (2.12)

2.6.4 Carr-Purcell

The standard Hahn spin echo recovers magnetization lost due to differences in the Larmor frequencies of the spins caused by the external field inhomogeneties. However, as shown by Carr and Purcell, it is sensitive to the effects of diffusion. Spins that move from one region of the sample to another during the time τ between RF pulses will not perfectly rephase after the 180◦ pulse because the ”rephasing” angle will not equal the ”dephasing” angle due to different Larmor frequencies. A solution to this problem was first formulated by Carr and Purcell.[48]. Rather than using one 180◦, their variation on the spin echo was to use a train of 180◦ pulses. In this way, dephasing is constantly being undone; as long as the frequency of the pulses is high enough, the effects of diffusion are undone, and transverse magnetization is recovered. In this way, the only dephasing that occurs is due to the intrinsic spin-spin relaxation. This comes from the dipolar interactions of neighboring spins: as their moments are inverted, their relative orientations are preserved. In this way, the intrinsic T2 can be measured by looking at the envelope of the echo amplitude magnitude, as seen in Fig. 2.11. In practice this is done by acquiring echoes in between pulses, or by using a variable number of pulses and acquiring a single echo at the end of the train. This is useful in cases where spectroscopic information is desired, as explained in Chapter 2.5.

35 τCP τCP τCP ◦ ◦ ◦ ◦ 90y 180y 180y 180y

Figure 2.11: Carr-Purcell pulse sequence. Used to measure T2, spins are ﬁrst excited by a 90◦ pulse. As they dephase due to ﬁeld inhomogeneities, magnetization is recovered by the application of a train of 180◦ pulses. Any remaining signal attenuation is due to spin-spin relaxation, i.e. T2 processes. The echo envelope depicted falls as e−t/T2 , where t is the experiment time as measure from the initial excitation.

2.6.5 Carr-Purcell-Meiboom-Gill

In the original Carr-Purcell experiment, the phase of the pulses, the initial 90◦ pulse and the subsequent 180◦ pulses, was all the same. This causes the alternating polarity of the echo as seen in Fig. 2.11. Of greater concern is a consequence of this design: imperfections in the 180◦ pulse lead to cumulative errors as the spins are not perfectly rotated back into the transverse plane. Because it is difficult to generate a perfect B1, there will always be regions of the sample for which a given pulse duration is not 180◦. Meiboom and Gill came up with an ingenious solution: by controlling the phase of their pulses, and rotating the 180◦ pulses about an axis perpendicular to the initial 90◦ pulse’s rotation axis, the effects of pulse imperfection can be eliminated.[49] The 180◦ pulses are applied repeatedly in a Carr-Purcell train, with the effect that every other echo causes the spins to return entirely to the transverse plane (outside of T1 relaxation). The combined technique is known as Carr-Purcell-Meiboom-Gill, or CPMG, and is extremely common in magnetic resonance experiments- an example

36 pulse sequence can be seen in Fig. 2.12. An example of the eﬀects of an imperfect 180◦ and the resulting correction by a CPMG pulse train can be seen in Fig. 2.13.

Odd echoes are in-plane and recorded τCP τCP τCP ◦ ◦ ◦ ◦ 90x 180y 180y 180y

Figure 2.12: CPMG pulse sequence. Imperfections in the 180◦ pulse are mitigated by applying 180◦ pulses that are ±90◦ out of phase with the initial excitation pulse. The pulse train is then sampled every other echo, i.e. the odd echoes, which have the spins in-plane.

2.7 Diﬀusion

One of the more important uses of NMR is to measure diffusion rates. Carr and Purcell demonstrated a method for measuring diffusion coefficients, D, using spin echoes that relied on a constant (for the duration of the experiment) field gradient.[48] However, it proved insufficient in practice for many cases, especially in the case of slow diffusion. In order to measure smaller diffusion coefficients, and increasingly

∗ larger gradient was needed: this had the eﬀect of leading to a much shorter T2 , making the SNR of the experiment very low, while also requiring higher RF strengths for excitation, B1, in order to excite the entire sample: in order to have suﬃcient bandwidth, the pulse would have to be much shorter than normal, thus requiring a

37 Figure 2.13: Eﬀects of imperfect 180◦ pulse (a) and correction via CPMG sequence (b). Adapted from Ref. [49].

greater magnitude. Thus measuring D via spin echoes with a constant ﬁeld gradient proved very limiting.

2.7.1 Pulsed Gradient Spin Echo (PGSE)

Fortunately, a solution to the problems raised by using constant field gradients was discovered by Stejskal and Tanner in the form of pulsed field gradients.[50]. Rather than applying a gradient for the duration of the experiment, which causes short signal lifetimes and necessitates high power RF, field gradient pulses are applied in pairs for a short duration before and after the 180◦ pulse in a spin-echo sequence, as seen in

Fig. 2.14. These field gradients are linear, i.e. Bgrad,z (r) = Gzr is a linear function of position in a given direction. For convenience, will assume a gradient in the y- direction of the lab frame, i.e. the strength of the z-component of the magnetic field varies linearly with position in the y-direction. The first pulse serves to label the

38 τ τ

90◦ 180◦ Echo

∆

δδ G

Grad Figure 2.14: Pulsed Gradient Spin Echo sequence. The echo signal varies as e−bD, 2 2 2 δ where b = γ G δ ∆ − 3 . b is varied by changing the gradient pulse duration, the diﬀusion time, or the gradient strength.

spins, such that the spins initial position is recorded in terms of the dephasing angle in the rotating frame, equal to γGyy1δ, where δ is the duration of the gradient pulse,

and Gy is the gradient strength in tesla per meter. The spins are then allowed to diffuse for a time ∆, during which time a 180◦ is applied in order to generate an echo. The second pulse serves to undo the effects of the first pulse, causing a dephasing

γGyy2δ. If y1 6= y2 due to the eﬀects of diﬀusion, this rephasing will not be perfect, and the echo amplitude will be attenuated as a result. The resulting amplitude will be:

−γ2G2δ2(∆− δ )D M(2τ) = M0e 3 (2.13)

where τ is the time between initial excitation and formation of the echo. The argu-

2 2 2 δ ment of the exponential is oftentimes expressed as bD, where b ≡ γ G δ ∆ − 3 .

39 By varying the value of b by changing δ, ∆, or G, several diﬀerent data points can be acquired. In essence, the signal depends on b, and plotting ln (M(b)/M0) vs. b yields a straight line whose slope is D.

2.7.2 Pulsed Gradient Stimulated Echo (PGStE)

One drawback to PGSE experiments is that if diffusion is slow, as in the case of viscous liquids, a relatively long diffusion time will need to be used. Logistically, it is difficult to simply employ higher gradient strengths due to hardware limitations. In addition, if T2 is short, the SNR of the resulting spin echo can be very low, since τ will always be at least on the order of ∆. Short T2 often accompanies samples with low

D, or in samples that have fast diffusion but short T2 due the presence of magnetic entities such as contrast agents. The solution to this problem is to employ stimulated echoes. First described by Hahn in his seminal paper on echoes,[47] they were first used in the context of measuring diffusion coefficients by Tanner.[51] Stimulated echoes do not employ 180◦ pulses, but instead use three 90◦ pulses, as seen in Fig. 2.15. The first excites the sample, the second stores the transverse magnetization back along the longitudinal axis, and the third retrieves it back into the transverse plane. By putting a gradient pulse in between the first two RF pulses, the spins are labelled. By applying the second gradient after the last RF pulse, the spins are rephased, minus diffusion effects. While the magnetization is stored along the longitudinal axis, the spins are immune to the effects of transverse relaxation, allowing for the determination of diffusion coefficients in samples with short T2.

40 ◦ ◦ ◦ 90x ττ 90x 90x Echo

∆

δδ G

Grad Figure 2.15: Pulsed Gradient Stimulated Echo sequence. The echo signal varies exactly as in a PGSE experiment. However, during the diﬀusion time ∆ the spins are ”stored” along the longitudinal axis, where they are not aﬀected by transver relaxation. The stimulated echo forms at a time τ after the third 90◦ pulse.

41 Chapter 3 Magnetic Resonance Apparatus

The NMR apparatus used for this thesis consisted of a 7T Oxford Instruments 300/89 superconducting magnet, a Tecmag Apollo NMR spectrometer, Techron gradient ampliﬁers, and several homebuilt components: the probe, including RF circuitry, gradient coils, and duplexer to allow the use of one coil for both excitation and detection of the NMR signal. Several modiﬁcations were made to the equipment which will be described in this chapter, notably to the duplexer design, RF coil construction and electronics, and especially gradient coil design.

3.1 Probe Overview

The NMR probe was constructed from aluminum, and had a cylindrical design: from the probe head, cables for the RF and each of the three gradient coils extended to the base, where the RF coil, gradient coils, and sample were enclosed in an cylindrical can. Cables were isolated by the use of aluminum conduit along the length of the probe, which helped prevent cross-talk between gradients. Due to repeated failure of BNC connectors on the gradient lines as well as gradient cross-talk, machined aluminum blocks and teﬂon spacers were installed in order to accommodate larger N-type connectors and better isolate the gradient lines from one another. The RF electronics resided on a single PCB board, which was linked to the probe

42 via an SMA connector. All gradient coils were in a single assembly (also linked via SMA connectors) which surrounded the RF board: together, these two were joined to a plastic base with spacers that attached to the actual probe assembly with cables. The combined assembly can be seen in Fig. 3.1. This assembly was surrounded by an aluminum can containing Fluorinert (3M FC-43), a thermally conductive and electrically insulating liquid. This ﬂuid was used to both maintain temperature due to gradient coil overheating and also to help with susceptibility-matching near the sample, where diﬀerent materials were used.

Figure 3.1: Assembled NMR Probe. The RF and gradient assembly sits in an aluminum can ﬁlled with 3M Fluorinert, FC-43, which helps in maintaining temperature and susceptibility-matching. N-type connectors for the tri-axial gradient system, as well as a BNC line used for RF can be seen at the top of the probe.

3.1.1 Radiofrequency and gradient board assembly

After loading of the sample into the RF coil, the RF board was inserted inside of the gradient board, and the combined assembly screwed onto an acrylic base which was then mated to the rest of the probe via threaded aluminum rods. A closeup of the probe head can be seen in Fig. 3.2. The probe head was then immersed in the FC-43 inside the aluminum can and the fully-assembled probe lowered into the

43 superconducting magnet.

Figure 3.2: The copper colored RF board sits inside of the grey thermal epoxy gradient board. SMA connectors and cabling can be seen leading to the top of the probe. This assembly sits in an aluminum can ﬁlled with 3M Fluorinert, FC-43, which helps in maintaining temperature and susceptibility-matching.

3.1.2 Electronics

In the experiments performed for this dissertation, a single RF coil was employed both for excitation and detection of the NMR signal. Because the transmitter and receiver circuits both use the same coil, it is necessary to isolate them from one another to minimize noise and protect sensitive circuitry. Previously, a PIN diode

44 switch was used to accomplish this, details of which can be found in Ref. [52]. This was done because the typical transmitter powers used were very low in microcoil MRI, necessitating a design diﬀerent from the standard crossed-diode duplexer. Because of the higher transmitter powers used in our experiments due to relatively larger sample (and thus RF coil) sizes, a more robust solution to receiver/transmitter isolation was necessary. A classic crossed-diode setup was used to isolate the transmitter branch of the spectrometer. At the same time, a λ/4 cable was used to isolate the receiver branch. Details of this type of duplexer can be found in Fukushima.[53]

3.2 Radiofrequency Coils

The RF electronics of our homebuilt system reside on a piece of PC board, sand- wiched between two layers of copper. The copper was etched away on one side to leave only a small trace of metal, onto which was soldered the solenoid coil and capacitors to make a resonant tank circuit. The board is seen prior to installation of the RF electronics in Fig. 3.3.

3.2.1 Construction

In contrast to previous work done in the lab, our coils were designed to be reused with many samples. In the past, microcoils used in MRI were wound on a per sample basis, and glued directly to micropipettes or capillaries. In order to examine diﬀerent samples, the coil would have to be destroyed, or a new one created on a separate RF board. Because there was a need to look at a great many samples, this was not a feasible solution. A new way of winding coils was needed. The solution was to take 1 mm outer-diameter (OD) glass rods to use as a form factor, and coat them in a thin layer of paraﬃn prior to coil winding. After the solenoid was wound onto the rod, it was coated in a layer of UV epoxy (Norland

45 Figure 3.3: Bare RF board, prior to tank circuit installation.

46 Optical Adhesive #68) to ﬁx it to the RF board, and cured under a UV lamp. The whole assembly was gently heated to melt the paraﬃn, allowing removal of the glass form factor, leaving behind the solenoid in a rigid sheathing of epoxy, attached to the RF board.

3.2.2 Tuning

Each RF coil needed to be tuned prior to use. Because of the limited space in and around the RF board when placed inside the gradient assembly, it was not feasible to use variable capacitors. In addition, commercially available variable capacitors in the range of capacitance needed were invariably magnetic, leading to complications in acquiring NMR data. For this reason, the coils were tuned to the Larmor frequency at √ 7T of 300 MHz by hand using chip capacitors, using ω = 1/ LC. A fully assembled board, complete with chip capacitors can be seen in Fig. 3.4.

Figure 3.4: RF board with electronics. The glass capillary has an outer diameter of 1 mm. The reusuable RF coil can be seen in the immediate foreground. In parallel with the RF coil are two chip capacitors. Coil inductances are on the order of nanohenries; the chip capacitors used for turning are on the order of pF.

47 3.3 Gradients

In order to perform MRI experiments or measure diffusion coefficients using pulsed gradient methods, magnetic field gradients are needed. A tri-axial gradient system is employed to enable these types of experiments. In the course of this research, two significant improvements were made to our gradient coils: the encasing material and the ability to accommodate larger samples, specifically seeds.

3.3.1 Coil design and modeling

Field gradients are created in the x, y, and z directions, where x lies along the axis of the sample capillary. The x and y gradients have similar coil geometries, differing only in the planar spacing d, as depicted in Figs. 3.5 and 3.7. In a single plane of coil, two inner pairs of wires carry current in the same direction (spaced as close together as possible), while two outer wires have currently flowing in the opposite direction. This pattern of current is repeated with the same orientation on the opposite coil plane. The z gradient coil is an anti-Helmholtz coil. The planar spacing d is on the order of 3.2 mm, differing slightly for each plane of wires: it is as small as possible to maximize the gradient strength, which increases with decreasing d, while still being able to fit around the RF assembly. Gradient coil performance was modeled using Mathematica, and found to be in good agreement with experimental results in terms of gradient strength per unit current (within 2%). The optimal spacing of the x and y inner versus outer wires was √ found to be 3d, in agreement with previous work on gradient coil design.[54]. A plot of the magnetic field as a function of position for the y-coils is seen in Fig. 3.6.

48 Figure 3.5: Gradient board wire geometry. Long current carrying wires are used to generate ﬁeld gradients in the z-direction along the x and y axes. An anti-Helmholtz geometry is used to create the same type of gradient along the z-axis.

49 y HmmL

-0.2 0.0 0.2

0.0005

0.0000

By HTeslaL z HmmL

0.2 0.0

-0.0005 -0.2

Figure 3.6: A plot of magnetic field versus y and z for the y-gradient coil, using a current of 5 amps, generating a gradient of 1.6 T/m. The field and thus magnitude of the gradient are proportional to the current flowing through the coil.

3.3.2 Assembly material

One of the major improvements made to the NMR apparatus was changing the gradient coil assembly material from an acrylic to a thermally conductive (but electrically insulating) epoxy, Duralco 132. This material ultimately afforded the same, if not greater structural strength than acrylic, while at the same mitigating the effects of gradient coil heat buildup. Previously, gradient board failures were frequent due to overheating issues: wires would get hot due to resistive heating, which would in turn weaken the acrylic. As the acrylic’s structural integrity decreased, Lorentz forces acting on the gradient coils during pulses became sufficient to move the wire violently enough that they would break. Without gradients, diffusion measurements became impossible.

50 Duralco 132 is a difficult material to work with. It is machineable, but special care must be taken in its preparation for the purpose of holding gradient coil wires. In particular, even after thorough mixing, the material is prone to having large void defects, caused by air bubbles. Gradient boards manufactured with Duralco 132 initially failed along the wires due to these bubble defects. However, it was found that by pumping on the epoxy immediately after mixing, but before curing, it was possible to significantly diminish the size and number of these defects. This allowed for high currents to be used to generate large field gradients without fear of device failure. The ultimate design of the gradient board, machined out of Duralco 132 can be seen in Fig. 3.7, as well as Fig. 3.2.

51 Figure 3.7: Gradient board constructed with thermal epoxy.

52 Chapter 4 Diffusion Data

4.1 Introduction and Premise

In the previous chapters, we saw that one of the fundamental differences between the outer sphere and proton exchange relaxation mechanisms was their dependence on the local diffusion coefficient. We demonstrated how by using NMR techniques, diffusion coefficients can be measured, and how rates for different chemical species can be distinguished by chemical shift. The ability to make these measurements allows for the effects of diffusion on ferritin-induced relaxation enhancement to be measured, as long as the diffusion rate can somehow be controlled. This is an approach that has not been explored by other researchers, and provides important insight into the relaxation mechanisms’ relative contributions to the dephasing of protons by ferritin. A solution to the problem of controlling diffusion is to use a binary mixture of water and glycerol. These two substances are highly miscible, have similar hydroxyl proton densities, densities that are not overly dissimilar, and diffusion coefficients that differ by 3 orders of magnitude. This last fact means that adding relatively small amounts of glycerol can have noticeable effects on the diffusion coefficient of water molecules. The diffusion coefficient of glycerol molecules can be determined by tracking CHx protons, which have a distinct chemical shift from OH protons, easily resolved by NMR spectroscopy. There are some difficulties posed by this particular

53 choice of system: the hydroxyl protons of water and glycerol chemically exchange

with one another, which leads to an apparently higher transverse relaxation rate R2 in CPMG experiments when τCP exceeds a characteristic exchange time, τCE. This will be discussed further in Chapter 5. Furthermore, because the addition of glycerol changes the Brownian motion of water molecules as evidenced by their reduced diffusion rate, increasing the correlation time, the spectral density of ﬂuctuations in the magnetic ﬁeld experienced by a given spin will shift, changing the relaxation rate of each species independent of any changes induced by the presence of ferritin. Since the

Larmor frequency is well below 1/τC , this means that the density of ﬂuctuations that leads to relaxation most eﬀectively (those at the Larmor frequency) will increase, as will the relaxation rate itself as a function of increasing glycerol content. As shown

in Chapter 2, the relaxation rate is more or less linear with τC , so increasing this time increases the relaxation rate. Finally, the higher relaxation rates caused by the

addition of glycerol, and later by the presence of ferritin, lead to very short T2, necessitating the use of pulse gradient stimulated echo experiments to determine diﬀusion coeﬃcients.

4.2 Discerning Mechanisms of Dephasing: The Role of Diﬀusion

In order to distinguish between the relaxation mechanisms, it is necessary to change the independent variables and see what the effect on the relaxation rate is. If changing a given variable has no discernible effect, then it is unlikely that a relaxation mechanism upon which it depends plays an important role. We will briefly reexamine the dependencies of the outer sphere and proton exchange mechanisms and discuss how to contrast them experimentally.

54 4.2.1 Relaxation mechanism dependencies

Researchers have primarily conducted investigations into the concentration dependence and magnetic field dependence [55, 35] of ferritin-induced relaxation enhancement of protons because both primary relaxation mechanisms depend on external field strength, albeit in different ways, as seen in Eqns. (1.1) and (1.4): the proton exchange mechanism exhibits a linear dependence and the outer sphere mechanism is proportional to the square of the field. However, most of this research has been conducted at lower fields strengths than in this work; in addition, virtually all the data were acquired in simple aqueous solutions at elevated temperature (as high as 40◦C), resulting in high diffusion coefficients. Looking more closely at Eq. (1.1), we can see that both of these will cause the relative importance of the diffusive mechanism to be reduced: when spins diffuse more quickly, they do not remain in the vicinity of ferritin proteins long enough to be dephased to an appreciable degree. To be sure, the explicit diffusion dependence needs to be tested to determine the outer sphere mechanism’s contribution.

4.2.2 In vivo environment

There are other reasons by which diffusion’s role in ferritin-induced relaxation enhancement needs to be further explored. Since the primary motivation of this thesis is the detection and quantification of brain iron, we will focus on the environment of protons in the human brain. To first order, the human brain is a large volume of water. While that approximation may suffice for simple models and calculations, upon further examination we can see that the diffusion coefficient inside the brain does not match that of bulk water. Investigations into the diffusion coefficients in the brain show that water moves much more slowly than it does in bulk.[39] In addition, there may be a role played by restricted diffusion, where water

55 molecules move about relatively quickly, but are constrained to a small volume, creating an apparent diffusion coefficient that is again smaller than that of bulk water. If the compartmentalization of the water is on the order of microns , a typical length scale in brain cells, spins will experience the local field of a given ferritin more often than they would otherwise in the timescale of a typical NMR/MRI experiment, further enhancing the relaxation. As proof, let us look at the case of protons with a

−9 2 diffusion coefficient of DW = 2.3 × 10 m /s. In a typical MRI experiment, the time between RF or gradient pulses (for spin and gradient echo sequences respectively) is on the order of milliseconds; for the sake of argument we will say 20 ms. In most brain cells there is not spherical symmetry, but we will set that aside and imagine an environment such that spins can diffuse isotropically in 3 dimensions. In that case, √ the rms distance spins will travel is given by 6Dt, where t is the timescale of the MR experiment. Using these numbers, we see that spins will travel a distance of 16.6 µm, which at the very least is on the order of brain cell dimensions, if not exceeding it. This means that many spins will hit the cell boundary and resample the field generated by the ferritin protein’s core, further enhancing relaxation. In other words, the apparent diffusion coefficient is decreased as a result of the restrictions imposed by cell walls or other such barriers. In summary, the in vivo environment of protons is very different from that of many of the in vitro experiments conducted at present. Diffusion is slower than it is for pure water, and this can have an effect on the relaxation rate of protons in the presence of ferritin. In order to test this hypothesis, we need a “clean” system, one where all the constituent parts are well understood, and one where the diffusion coefficient can somehow be controlled. This is where the binary mixtures of water and glycerol come into play.

56 4.3 Controlling and Measuring Diﬀusion

At room temperature, the diffusion coefficient of water is on the order of 10−9 m2/s, while that of glycerol is on the order of 10−12 m2/s, a difference of three orders of magnitude. By adding glycerol to water, the self-diffusion coefficients of both molecules will be changed: water will diffuse more slowly, while the glycerol will move more quickly in aqueous solution than it would by itself. The diffusion rate of each molecule can be measured simultaneously due to chemical shift, which differentiates glycerol from water. We will discuss how to determine the water molecules’ diffusion coefficient from raw data, as well as how specific compositions of water-glycerol mixtures were chosen.

4.3.1 Extracting diﬀusion coeﬃcients

Glycerol has a chemical formula of C3H8O3. Structurally, it appears as in Fig. 4.1, with three protons bonded to each of the three oxygen atoms (the hydroxyl protons), and the other five distributed among the carbon atoms. Because of the difference in the local magnetic environment of the protons bonded to either an oxygen or a carbon (due to differences in the electronic structure), these protons have slightly different Larmor frequencies, creating a chemical shift. This allows for NMR measurements to be made on just the CHx protons. Since water protons are all hydroxyl protons, all quantities determined from the CHx protons pertain solely to glycerol. In this way, we can obtain the diffusion coefficient of just the glycerol molecules. In the NMR spectrometer used in this thesis, there was sufficient spectral resolution to distinguish hydroxyl and CHx protons. However, either due to insufficient ability to shim the external field, or more likely due to the effects of chemical exchange, which can cause the merging of peaks,[43] it was not possible to determine from which chemical species the OH signal originated. In other words, there was 57 Figure 4.1: 3 dimensional representation of a glycerol molecule. Carbon atoms are indicated in black, oxygens in red, and hydrogens in white. There is a total of three hydroxyl protons in this molecule: these have a distinct chemical shift from the CHx protons, enabling the use of spectroscopy to distinguish glycerol from water, which lacks these CHx protons.

58 a single OH peak representing all OH protons, from both water and glycerol. This means that quantities determined from the OH peak represent a weighted average from water and glycerol. Fortunately, by knowing the composition of the mixture, we can process the data to extract just the water diﬀusion coeﬃcient.

The measured OH diffusion rate, DOH , is composed of a weighted average of the individual self-diffusion rates of water and glycerol, DW and DG. While it is true that the hydroxyl protons of water and glycerol chemically exchange, the time scale for this process, τCE, is much less than the time between events in a pulsed gradient experiment. Therefore, a proton will spend a fractional amount of the experiment time diffusing with a water molecule, proportional to the ratio of water hydrogens to the total number of hydrogens in the system. The diffusion rate of hydroxyl protons is then given by:

DOH = FW DW + FGDG (4.1)

where FW and FG are the fraction of OH protons from water and glycerol respectively.

These can be determined from the molar fractions of water and glycerol, fW and fG, by:

2fW FW = (4.2a) 3fG + 2fW 3fG FG = (4.2b) 3fG + 2fW

Here, the molar fractions are determined by the molecular weights of water (MWW =

18.02 g/mol) and glycerol (MWG = 92.08 g/mol), and how much of each is added to

the binary solution, e.g. fW = mW /MWW . The factors of 2 and 3 in Eq. 4.2 represent

the two hydrogens next to the oxygen in water (H2O) and the three hydroxyl protons

in glycerol (alternatively written as C3H5(OH)3) respectively. Eq.(4.1) can be used

59 to determine DW :

DOH − FGDG DW = (4.3) FW

4.3.2 Water-glycerol samples

There are several ways to combine water and glycerol compositions in a systematic manner. One can mix solutions by mass or volume, as has been done in previous works involving binary solutions of water and glycerol. In this thesis, solutions were mixed according to the fraction of hydroxyl protons coming from water and glycerol. This was done because of the interest in understanding the relaxation mechanisms: the proton exchange mechanism can only occur between hydroxyl protons on the ferritin core surface and hydroxyl protons from either water or glycerol. Mixing by hydroxyl fraction made the diffusion (and relaxation) data more transparent and easier to interpret. The sample preparation method was simple: the ratios of water and glycerol necessary to obtain the necessary ratio of hydroxyl fractions was calculated, taking into account the desired concentration of ferritin. In samples containing ferritin, an amount of ferritin stock solution (Sigma-Aldrich F4503) was weighed on a digital balance, a step skipped for the 0mM samples. Glycerol was then added according to the correct ratio to the stock solution, and then water in the same manner. These were briefly mixed, and then NaCl was added to maintain a 0.15M NaCl level, as found in the stock ferritin solution. These were then mixed using a plastic stirrer to ensure homogeneity. The resulting solution was poured into either a 3 or 5mL syringe, and any air bubbles were removed by depressing the plunger in an inverted orientation (needle towards ceiling). A 1mm OD capillary (Sutter Instrument B100-75-10) with a 750 µm ID was sealed at one using a UV epoxy (Norland Optical Adhesive #68) in such 60 a way as to create a plug that did not increased the OD of the capillary at the sealed end. The capillary was placed on the needle of syringe in an inverted position, with the needle pointing upwards. The syringe was then pushed downwards, causing the solution to be injected into the capillary. This method was developed to eliminate air bubbles in the capillary, which reduce signal amplitude when inside the RF coil. When the capillary was filled without any bubbles, it was lifted from the syringe, and a small amount of solution was removed from the open end using a KimWipe to create a small airspace. UV epoxy was then applied to this end to completely seal the sample. The sealed capillary with solution was loaded into the RF coil clean end first, so as not to damage the RF coil: the second sealed end usually had some excess epoxy on the outside which was left to ensure a good seal and zero leakage.

4.4 Gradient Calibration

When pulsed gradient NMR experiments are used to determine diffusion coefficients, an accurate calibration of the gradients is paramount to the accuracy and precision of the measurement. This can be seen in the Stejskal-Tanner equation, Eq. (2.13). The argument of the exponent is proportional to G2D: if the value of the gradient is not correct, then this will cause a miscalculation of the diffusion coefficient. Furthermore, the current that flows through the gradient coils can change from experiment to experiment due to the use of different gradient coils, differences in the performance of the gradient amplifiers, and most commonly, simple use of different currents as chosen by the spectrometer operator. This can be done in order to compensate for low SNR induced by fast relaxation of spins, in which case strong gradient pulses will be used in combination with shorter inter-gradient pulse times (∆). It can also be done to account for slower or faster diffusion, where strong or weaker maximum gradient strengths are employed respectively. In essence, the same b-value,

61 where b ≡ (γG)2 δ2 (∆ − δ/3), will not necessarily result in the same signal-to-noise ratio in a pulsed gradient experiment because there is a range of τCP or τM , the time between 90◦ pulses in a stimulated echo experiment, that can be employed. Finally, there are effects associated with gradient pulse length- short pulses can be affected by the rate at which the gradient amplifiers ramp up to the desired current. These ramp up times are on the order of 50-100 µs, much less than the typical gradient pulse length of several milliseconds. This means that gradient pulses of equal magnitude but different duration can have a slightly different effective gradient strength. For these reasons, before each diffusion measurement, the gradients were calibrated. As previously described, each pulsed gradient experiment employed a method of incrementally changing the gradient strength in order to change the b-value. In order to calibrate the gradient, a pulse sequence was employed to create a 1-D image of the sample via frequency encoding, a profile of the cross section along the y-direction. The sample diameter was well-known: visual inspection by microscope of several different capillaries, as well as verification with the manufacturer showed that the inner diameter was 750 µm, with a 1% tolerance. All capillaries were completely filled, ensuring that the sample diameter (d) was the same in all experiments. By measuring the bandwidth (BW ) of the 1-D frequency encoded profile, one can determine the gradient strength in T/m by:

2πBW G = (4.4) γd

Each gradient strength value used in PGSE or PGStE experiments was calibrated: incremented gradient pulses with amplitudes and durations exactly equal to the values used in the pulsed gradient experiment were entered into the NTNMR pulse programmer, and used to acquire frequency encoded images. These 1-D profiles were systematically analyzed in Igor Pro to find the edges of the profile so as to determine

62 the bandwidth. This bandwidth was then plugged into Eq. (4.4) to obtain G. These calibrated values of G were then used in determining the b-value for each data point in the pulsed gradient experiments.

0.15

0.10

0.05

0.00

-0.05

-0.10

Gradient Strength [T/m] Strength Gradient

-0.15 -10 -5 0 5 10 Normalized Gradient Amplitude Figure 4.2: Gradient calibration. The horizontal axis shows the relative amplitudes of pulses sent by the spectrometer to the gradient amplifiers, while the vertical axis shows the resulting gradient strength in T/m. The spectrometer first sent a digital pulse to a digital-to-analog converter, which then passed an analog signal to the input of the gradient amplifiers. The gradient performance shows good linearity with increasing pulse amplitude/amplifier current.

63 4.5 Experimental Diﬀusion Data

As shown in Chapter 2, diffusion data for different species can be acquired simultaneously using pulsed gradient experiments. A typical experiment yielded data in the following manner: spins were excited into the transverse plane by an RF pulse, labelled via a gradient pulse, and allowed to diffuse (stimulated echo experiments stored the magnetization along the longitudinal axis after this gradient pulse). An RF pulse was used to refocus the spins, and a second gradient pulse of equal duration, magnitude and polarity was used to ”unlabel” the spins. The farther spins had diffused, the less refocused they became by the gradient pulse, as their local Larmor frequency no longer matched the one experienced during the initial gradient pulse. An echo formed thereafter and was recorded by the spectrometer. In the case of pure water (or truly pure glycerol with no water contamination), it would suffice to use the echo amplitude for the purposes of calculating the diffusion coefficient. The echo amplitude is indicative of all the spins in the sample- since there is only one spin species, the echo accurately represents the diffusion rate of all spins. However, in the case of the water-glycerol samples used in this thesis, there are two species of diffusing spins, each with their own diffusion coefficient. By Fourier transforming the echo, we can distinguish the contributions of the OH and

CHx protons. To calculate the magnitude of the signal, one integrates the area under the OH or CHx peaks in the Fourier transformed dataset. These magnitudes are then plotted vs b-values, and curvefit to an exponential decay function (Eq. (2.13)) to obtain a value for the diffusion coefficient.

4.5.1 Pure solvents: water and glycerol

In order to verify that the methods used to analyze binary mixtures of water and glycerol are accurate, we began by making diffusion measurements of pure water and 64 glycerol. While it is possible due to measure DW using a standard pulsed gradient spin echo (PGSE) experiment, this is not easily done with glycerol. This is because glycerol diffuses very slowly: in order to obtain accurate results, it is necessary to allow the spins to diffuse a very long time. Unfortunately, due to the viscosity of glycerol and its larger size compared to water (and corresponding slower tumbling rate), the

correlation time for the molecule is relatively long, making it have a very short T2. This relaxation time is on the order of the diffusion time in a PGSE experiment (or even shorter). Signal attenuation due to relaxation can be greater than that due to diffusion, and cause an incorrect diffusion coefficient to be determined (too high). Therefore it is necessary to use the pulsed gradient stimulated echo (PGStE) experiment to measure the diffusion coefficient. While T2 is short, T1 for glycerol is considerably longer, more so than the typical diffusion time utilized in an pulsed gradient experiment. By storing the magnetization along the z-axis, T2 effects are mitigated completely, and accurate diffusion coefficients can be determined. For consistency, PGStE methods were used to obtain the diffusion coefficient of pure water as well.

Pure solvent diﬀusion data

We ﬁrst present the results of pulsed gradient experiments on pure glycerol and pure water in Fig. 4.3 and 4.4. These data were acquired at ambient temperatures using PGStE pulse sequences. Stimulated echoes were employed to mitigate the short T2 of the glycerol sample, as well as water samples doped with ferritin. Short

T2 values make standard PGSE experiments difficult because T2 is on the order of the diffusion time, ∆: the NMR signal attenuates due to diffusion, but also due to spin-spin relaxation, leading to improperly determined diffusion coefficients. In the case of glycerol, as well as ferritin doped water, T1 is considerably longer, as will

65 be seen in Chapter 5. This can be taken advantage by storing the magnetization along the z-axis with stimulated echoes. The data show excellent agreement with the theory of signal attenuation presented by Stejskal and Tanner [50], as well as with literature values of the diﬀusion coeﬃcient provided by Tomlinson [56] for glycerol, and various sources [57, 58] for water. In order to ensure that the presence of ferritin

and the resulting shorter T2 did not have an adverse effect on pulsed gradient diffusion measurements, data was taken for pure water with no ferritin, and with ferritin at an iron concentration of 4mM. As seen in Fig. 4.4, this has no appreciable effect on the determination of diffusion coefficients.

Literature Values

It is clear that pulsed gradient methods can determine diffusion coefficients. We compare the pure solvent data to the literature values as a function of temperature. Diffusion is highly temperature-dependent: a commonly quoted value for the diffusion coefficient of pure water is 2.299 × 10−9m2/s. However, this is at 25◦C, a temperature not used in this thesis. Therefore, to validate the results shown in Fig. 4.4, we compare the measured diffusion coefficient (D = 1.95×10−9m2/s) at 20◦C to previous researchers’ work in Fig. 4.5. The data acquired is in good agreement with previous results. The pure glycerol data was acquired at about 23◦C, and as seen in Fig. 4.6 agrees with data previous researchers acquired.

4.5.2 Diﬀusion in binary mixtures

The techniques that we have just examined were used to measure the diffusion coefficients of water and glycerol mixed together. PGStE data was acquired by varying the gradient strength, and leaving the pulse duration δ and diffusion time ∆ constant. The stimulated echo was acquired for several milliseconds, and the resulting data was

66 1 Glycerol 9 -12 -14 2 DG = 1.82 x 10 – 1.2 x 10 m /s 8

4 Normalized Echo Amplitude

3 9 0 200 400 600x10 2 2 2 (g G) d (D-d/3) [s/m ] Figure 4.3: PGStE results for pure glycerol diffusion at room temperature. The horizontal axis shows the b-value, which depends on the gradient strength G, the gradient pulse length δ, and the diffusion time ∆, while the vertical axis shows the normalized signal amplitude. b values were increased by increasing the applied magnetic field gradient. In this experiment, the maximum gradient strength employed was approximately 1.77 T/m, with a gradient pulse duration of 5 ms, and a diffusion time of 125.54 ms.

67 1 8 Water, 0mM [Fe] -9 -11 2 6 DW=1.95 x 10 – 3.0 x 10 m /s Water, 4mM [Fe] -9 -11 2 4 DW=1.92 x 10 – 3.2 x 10 m /s

0.1 8 6 4

2 Normalized Echo Amplitude 0.01 9 0.0 0.5 1.0 1.5 2.0x10

2 2 2 (g G) d (D-d/3) [s/m ] Figure 4.4: PGStE diffusion data for samples of pure water at room temperature. The two samples plotted differ only in that one sample contained ferritin at an iron concentration of 4mM (red dots), while the other did not (blue squares). The gradient pulse length used was 2 ms for the undoped water sample with a maximum gradient of .19 T/m, and 4ms for the 4mM [Fe] sample with a maximum gradient of .09 T/m; the diffusion time was 200 ms in both cases. The data was curve fitted to the exponential e−bD to determine the diffusion coefficient. The data shows that despite the presence of ferritin in the doped sample, the diffusion coefficient can be determined accurately, and is in agreement with the undoped sample.

68 -8 10

/s] 8 g

2 Holz et al, D0[T/Ts-1] 6 1/2 g Price et al, D0T [T/Ts-1] 4 Water, 0mM [Fe]

2.4 -9 10 8 2.2

-9 2.0

4 x10 1.8

2 1.6 288 292 296 300 -10 Water Diffusion Coefficient [m 10 240 280 320 360 Temperature [K] Figure 4.5: Water diffusion as a function of temperature. Data are from Ref. [57] and [58], as are the curvefits. The diffusion coefficient obtained for water using PGStE is in good agreement with these literature values at ambient temperatures. This means that the PGStE methodology as implemented works well: from gradient calibration, to pulse sequence design and data analysis. The inset shows how much the diffusion coefficient can be expected to change over the range of ambient temperatures seen in the lab, typically between 19 and 23 ◦C. This implies a maximum variation of about 0.2 × 10−9 m2/s.

69 -[K/(T-T0)] Tomlinson, D=D0e Glycerol /s] 2 -10 10

-11 2 10

-12 10 9 Glycerol Diffusion [m -12 8 10 288 292 296 300 320 360 400 440 Temperature [K] Figure 4.6: Glycerol diffusion as a function of temperature. Data is from Ref. [56]; the curvefit is to an empirical equation described in the same reference. The diffusion coefficient obtained for glycerol at a temperature of approximately 23◦C using PGStE is in good agreement with the values listed in this article. The inset shows how much the diffusion coefficient can be expected to change over the range of ambient temperatures seen in the lab, typically between 19 and 23◦C. This implies a maximum variation of about 0.5 × 10−12m2/s. However, pure glycerol was not used for any ferritin experiments.

70 Fourier transformed to reveal the NMR spectrum of the sample for diﬀerent b-values.

The areas under the OH and CHx peaks were integrated and plotted vs b-values as demonstrated with pure water and glycerol samples, and the diffusion coefficients obtained by a curvefit to e−bD. Of interest was whether or not the presence of ferritin would adversely affect this process. Indeed, as will be seen in Chapter 5, T2 becomes shorter in the presence of ferritin, reducing the signal amplitude. However, this was expected, and was one reason why the PGStE sequence was chosen to determine D instead of the PGSE method. Figs. 4.7 and 4.8 show the results for both OH and CHx protons in a system of 92.5%-7.5% water-glycerol by OH fraction with 0 and 4mM [Fe] respectively. Within experimental error, the results are in agreement, demonstrating that the presence of ferritin, while affecting relaxation rates, does not noticeably affect the diffusion rates of water and glycerol molecules.

Diﬀusion as a function of composition

Diffusion data was acquired for samples of varying water and glycerol content. The diffusion coefficient of water as a function of glycerol OH fraction for the 0mM and 4mM [Fe] datasets is plotted in Fig. 4.9. It is compared to literature values acquired by D’errico et al. in Ref. [59]. The diffusion coefficient is in good agreement with the literature values, demonstrating the validity of the PGStE data and the Fourier analysis used to obtain the diffusion coefficients of water and glycerol separately from the same dataset.

4.6 Summary of Experimental Results

The addition of glycerol to water slows the diﬀusion of water molecules down. This eﬀect is independent of ferritin concentration. The use of Pulsed Gradient Stimulated

71 1 6 92.5-7.5% by OH fraction Water-Glycerol, 0mM [Fe] 4

2 0.1 6 4

OH Peak 0.01 -9 -12 2 DOH = 1.17 x 10 ± 3.4 x 10 m/s 6 CHx Peak -10 -12 2 D = 5.01 x 10 ± 2.5 x 10 m/s 4 CHx

Echo Amplitude [Arb. Units] [Arb. Amplitude Echo

2 9 0 1 2 3 4 5x10 2 2 2 (γ G) δ (Δ−δ/3) [s/m ] Figure 4.7: Diffusion in a binary mixture of water and glycerol, which was mixed according to OH fraction. The OH peak contains information about both the glycerol and water diffusion, as both molecules have hydroxyl protons. In contrast, the CHx peak shows the diffusion behavior of only the glycerol molecules. This allows for the water diffusion rate to be extracted from the OH data. Unsurprisingly, glycerol diffuses more slowly than water: it is a much larger molecule.

72 1 92.5-7.5% by OH fraction 6 Water-Glycerol, 4mM [Fe] 4

0.1 6 4

2 OH Peak -9 -12 2 DOH = 1.12 x 10 ± 1.4 x 10 m/s 0.01 CHx Peak -10 -12 2 D = 4.41 x 10 ± 4.1 x 10 m/s 6 CHx

Normalized Echo Amplitude NormalizedEcho 4

9 0 1 2 3 4 5x10 2 2 2 (γ G) δ (Δ−δ/3) [s/m ] Figure 4.8: Diffusion in a binary mixture of water and glycerol containing ferritin. The experiment was performed identically to that used to acquire the data in Fig. 4.7; the results are the same within experimental error. This indicates that the presence of ferritin does not affect the measuremetn of diffusion coefficients.

73 DW, 4mM 2.0 DW, 0mM DW, D'errico DW, D'errico model

]

2 1.5

-9 1.0

[10

D 0.5

0.0 0.2 0.4 0.6 0.8 1.0

FG, OH Figure 4.9: Water diffusion as a function of composition in water-glycerol systems. Water and glycerol were mixed according to OH fraction coming from each. Displayed is the diffusion coefficient of water molecules obtained from such mixtures, both with and without ferritin, demonstrated the repeatability of the data. Also plotted are literature values from D’errico [59] for the water diffusion in such systems at 25◦C. The unfilled squares represent a curvefit presented in the D’errico paper. The final data point from the D’errico data is extrapolated. Unseen are the glycerol data for clarity. In both water and glycerol at very dilute glycerol concentrations, the predicted D’errico diffusion rates are slightly higher than my data, possibly explained by the slightly higher temperature of the samples.

74 Echo pulse sequences in combination with Fourier analysis of the water-glycerol samples yielded diffusion coefficients in excellent agreement with literature values. Data for diffusion in water-glycerol systems at very dilute glycerol concentrations was acquired, representing a new contribution to the literature, and extending the range of concentrations over which such measurements have been made. Finally, it was shown definitively that diffusion can be controlled in a predictable manner in binary mixtures of water and glycerol. This sets the stage for being able to see the diffusion dependence of proton relaxation in the presence of ferritin.

75 Chapter 5 Relaxation Data

5.1 Introduction and Premise

In the previous chapter, we examined how to characterize binary mixtures of water and glycerol by their diffusion coefficients, even in the presence of ferritin. This provides a means of testing the effects of diffusion on the relaxation enhancement due to ferritin. However, we still need to quantify how ferritin affects the relaxation rate. In this chapter, we will examine both longitudinal and traverse relaxation rates for binary mixtures of water and glycerol with varying concentrations of ferritin. The data will show two main effects on relaxation rate: that increasing amounts of glycerol lead to a faster relaxation of water protons, both longitudinally and transversely, while the addition of ferritin also leads to enhanced relaxation. The hydroxyl protons of water and glycerol chemically exchange: data will show that this effect can be minimized by choosing appropriate pulse sequences. Further analysis and discussion of relaxation data will take place in Chapter 6, in particular how relaxation changes as a function of ferritin concentration and mixture composition, and by association, diffusion rates. This will show the importance of the outer sphere mechanism in the proton relaxation rate enhancement due to ferritin.

76 5.2 Methodology and Experimental Results

Longitudinal relaxation data were acquired ﬁrst. The primary reason T1 data were acquired was to be able to maximize the SNR per unit time in experiments. CPMG pulse trains were used to acquire transverse relaxation data- these pulse trains allow

∗ for reversal of dephasing due to T2 effects caused by inhomogeneities in the external field, made important by diffusion of molecules. In addition, these CPMG pulse sequences allow for a clear measure of the inherent relaxation of OH protons: OH protons from water and glycerol chemically exchange: that is that hydrogens from water interchange with hydrogens bonded with oxygen atoms in the glycerol molecule. This process leads to additional dephasing. We will describe this in further detail

when presenting the transverse relaxation data. Rather than list T2 values, we will

present transverse relaxation data in terms of the relaxation rate, R2, which will later facilitate discussion of the results.

5.2.1 Longitudinal relaxation and saturation recovery

Saturation recovery was the method of choice for acquiring T1 data. Primarily, this is because saturation recovery experiments took less time than inversion recovery experiments, and are also less sensitive to imperfections in the duration of the 180◦

pulse. Furthermore, in the case of pure glycerol or solutions containing ferritin, T2 can be so short as to make it difficult to create echoes of sufficient amplitude for detection in an inversion recovery experiment. The results of the saturation recovery experiments were used to help set the repetition time between scans, allowing for an optimization of the SNR of the data per unit time in the CPMG experiments. Because water and glycerol molecules have different longitudinal relaxation rates in binary solution, the longer of the two relaxation rates was used to determine the repetition time between scans. 77 T1 is affected both by the composition of the solution in terms of water and glycerol content, as well as by the concentration of iron in the form of ferritin. Most research has focused primarily on T2; the presentation of this data for systems of water and glycerol, and for those systems with ferritin is a new contribution to the literature. In terms of correlation time, as the fraction of glycerol increases, and water molecules move more slowly, the correlation time increases. This will cause the spectral density function to change, such that there are more fluctuations at the Larmor frequency, leading to faster longitudinal relaxation. First we shall examine typical saturation recovery data sets, first for pure water and glycerol, and then for 92.5-7.5 water-glycerol by hydroxyl fraction. For small

◦ recovery time, τR, the spins remain saturated, and a 90 pulse has no eﬀect as there is not net magnetization on which to work. For long recovery time, τR T1, the magnetization is allowed to relax to its equilibrium value. The magnetization depends on τR as:

−τR T ) M(τR) = M0(1 − e 1 (5.1)

It is most useful to plot the data as 1 − M(τR)/M0. For short recovery times, the

value of this function will be close to unity, as M(τR) will be near 0, and for very long recovery times, the value of the function will be 0. Assuming an exponential

recovery of the magnetization with time constant T1, Eq. (5.1) will look exactly like

an exponential decay, one whose time constant is also T1. Fig. 5.1 plots the results from a typical saturation recovery experiment, performed on a sample of distilled water with 0.15M NaCl. As expected, the relaxation time is relatively long (2.82 s) due to the relatively short correlation time on the order of

picoseconds. Because τC << 1/ωL, the density of ﬁeld ﬂuctuations that are respon-

78 1 6 Water, 0.15M NaCl T = 2.82 ± 0.01 s 4 1

2 0.1

0 6

)/M 4

τ 2

1-M( 0.01 6 4

2 0.001 0 4 8 12 16

Recovery Time, τR [s] Figure 5.1: Saturation recovery data for 0.15M NaCl aqueous solution. In a saturation recovery experiment, the magnetization of the spins is ﬁrst saturated by repeated application of RF pulses with slight delays in between them. The combination of relaxation and spin manipulation scrambles the magnetization so that there is zero net magnetization, i.e. no signal. The signal is then allowed to recover for a time ◦ period, τR, at which point a 90 pulse is applied, and the resulting free induction decay is recorded. As τR grows longer, the magnetization has more time to return to thermal equilibrium, a process that occurs on the timescale of T1. By plotting 1 − M(t)/M0, where M0 is the equilibrium polarization, against τR, we can determine the value of T1.

79 1 8 6 Pure Glycerol T = 316.4 ± 2.3 ms 4 1

)/M 0.1

R 8 τ 6 4

1-M(

0.01 8 Magnetization fully recovered 6 0 1 2 3 4 5

Recovery Time, τR [s] Figure 5.2: Saturation recovery data for pure glycerol. In comparison with the results for water, glycerol has a shorter T1 by an order of magnitude. This is expected as the glycerol molecule is larger than water, leading to a slower tumbling rate, and the intermolecular interactions are stronger, giving it a greater viscosity. This has a net eﬀect of increasing the correlation time, leading to more eﬀective longitudinal relaxation. The later data points represent when the magnetization has fully recovered, causing the plotted function to approach zero. Deviations from zero are caused by experimental noise.

80 sible for causing longitudinal relaxation are weak. It is also agreement with results from the literature, of 2.86 s for a phantom sample of water at 7 tesla.[60]. Fig. 5.2

shows similar data for pure glycerol, with a much shorter T1 of 316 ms, a diﬀerence of one order of magnitude. This can be explained by a diﬀerence in the correlation times for water and glycerol: the longer correlation time of glycerol molecules causes the longitudinal relaxation rate to be higher, or inversely, the long τC leads to a short longitudinal relaxation time.

With saturation recovery established as a valid method of measuring T1, we performed the same experiments on solutions of water and glycerol, both with and without ferritin. In order to distinguish between the longitudinal relaxation of water and glycerol molecules, all the data acquired was Fourier transformed from the time domain into the frequency domain so that analysis could be performed on a peak by peak basis. Example data is shown in Fig. 5.3 for a system of 92.5-7.5 water-glycerol by hydroxyl fraction. Data were acquired for all samples of water and glycerol with ferritin. The results

are presented in Tables 5.1 and 5.2. A general trend towards shorter T1 for the hydroxyl protons (mostly water) can be seen with increasing glycerol content, as well as with higher iron concentrations.

5.2.2 Transverse relaxation (R2) and fast CPMG sequence

Perhaps the most important data acquired in this thesis is the transverse relaxation data of water-glycerol systems of various compositions with different concentrations of iron in the form of ferritin. This data proved more difficult to obtain than either T1 or diffusion coefficient for two main reasons: the increased relaxation rate of water protons due to changes in the correlation time induced by the addition of the highly viscous glycerol, and also due to the presence of ferritin, which serves to

81 1 92.5-7.5 by OH Fraction Water-Glycerol OH Peak T1 = 1.963 ± .003 s 0.1 CHx Peak T1 = 1.084 ± .005 s

)/M

τ 0.01

1-M(

0.001

0.0001 0 5 10 15 20 25 30

Recovery Time, τR [s] Figure 5.3: Saturation recovery data for a mixture of water-glycerol, 92.5%-7.5% by hydroxyl fraction. Data were acquired on a peak by peak basis by Fourier transforming the time-domain free induction decay of the spins after a given recovery time following their initial saturation. In comparison with the saturation recovery data for pure water and pure glycerol, it can be seen that the water molecules have a shorter T1, while that of the glycerol molecules (solely responsible for the CHx peak) is considerably longer. This can again be attributed to a change in the correlation time (longer for water, shorter for glycerol) caused by mixing the two liquids together, with the resulting change in viscosity as seen by the diﬀusion data presented in Chapter 4.

82 Table 5.1: OH T1 as a function of composition and iron concentration, in seconds

Iron Concentration Composition 0mM 1mM 4mM 100/0 2.82 2.88 2.48 97.5/2.5 2.75 - 2.06 95/5 2.22 2.23 1.82 92.5/7.5 1.96 - 1.56 90/10 1.87 1.79 1.41 85/15 1.46 1.35 1.07 80/20 1.14 1.13 0.90

Table 5.2: CHx T1 as a function of composition and iron concentration, in seconds

Composition 0mM 1mM 4mM 97.5/2.5 1.74 - 1.41 95/5 1.23 1.29 1.27 92.5/7.5 1.08 - 1.13 90/10 .998 1.05 .993 85/15 .829 .846 .779 80/20 .666 .664 .641

enhance proton relaxation via mechanisms outlined earlier in Chapter 1. These two eﬀects serve to create much higher relaxation rates than those seen in pure water. Further complicating matters is chemical exchange: hydroxyl protons of water and glycerol exchange with one another, separate from proton exchange between these same protons and hydroxyl groups on the ferritin core surface. Because the Larmor frequency of the hydroxyl protons in water and glycerol diﬀers ever so slightly, this will lead to an enhanced dephasing which depends on the time between pulses in an

echo train, the Carr-Purcell time τCP . This chemical exchange typically takes place on the order of milliseconds (compared to tens of nanoseconds for proton exchange at

83 the ferritin core surface). To mitigate the eﬀects of chemical exchange, we employed rapid pulse sequences so that the additional dephasing caused by exchange events would not be allowed to accumulate and present itself in the form of artiﬁcially high

R2-values. Specifically, a fast CPMG sequence prevented this chemical exchange- induced dephasing from interfering from a proper determination of the relaxation rate of protons in the presence of ferritin. We will demonstrate how chemical exchange was characterized later in this chapter, in section 5.2.4, as well as provide a physical model for the chemical exchange process. For the time being, it will suffice to be qualitative: when there are many chemical exchange events occurring in one τCP , the dephasing caused by this process will be irreversible by the next 180◦ pulse, leading to higher relaxation rate. The first step in characterizing water-glycerol solutions is to know what their relaxation rates are in a pure state, i.e. not mixed together, and without any ferritin. This also allows for verification of technique, as exemplified previously by comparison to literature values. In the case of glycerol, it is sufficient to use a standard spin echo experiment to determine the transverse relaxation time, T2. However, this is not true for water in the apparatus used for this thesis because the diffusion of water is sufficiently rapid at room temperature (D ≈ 2.0 × 10−9m/s2) that the inhomogeneity in the external magnetic field over the sample, on the order of 1ppm (300 Hz), is enough to lead to significant and irreversible dephasing. For this reason, a Carr-Purcell-Meiboom-Gill pulse train was used to characterize the relaxation rates of both water and glycerol, as seen in Figs. 5.4 and 5.5 respectively. As is expected, the relaxation time for glycerol (27.1 ms) is much shorter than that of water (2.49 s), by almost two orders of magnitude. This can be explained in terms of a much longer correlation time for the glycerol molecule: it is larger than than a water molecule, and interactions with its neighbors are stronger such that glycerol is many times more

84 viscous than water, as evidenced by the diffusion data presented in the previous chapter. This longer correlation time makes the glycerol protons much more susceptible to transverse dephasing as they will experience a given field strength for a much longer time than a water molecule, leading the population of glycerol molecules as a whole to experience slightly different Larmor frequencies for longer periods of time, leading to a greater spread in the phases across the entire ensemble. This manifests itself as a smaller NMR signal as the spins are no longer as coherent as they would be with a much shorter correlation time.

In order to measure T2 (or R2) in binary mixtures of water and glycerol, very fast CPMG pulse trains must be used. In other words, the time between 180◦ pulses must be as short as possible in order to minimize the effects of not only diffusion, but also of chemical exchange. Because these pulses are so frequent, it is not possible to obtain an echo between each pair of pulses in the train, as seen in Fig. 2.12. Instead a variable number of CPMG pulses are used to sample the relaxation at increasing time values. Furthermore, simply looking at the amplitude of the echo does not suffice to determine the relaxation rates. There is no reason by which water and glycerol molecules should have the same relaxation rate. As seen in Chapter 4, the diffusion rates for water and glycerol molecules are very different: the larger glycerol molecule translates, and in related fashion, tumbles more slowly than water molecules. This means that its correlation time is longer, and accordingly (as explained in Chapter 2) the relaxation rate of glycerol is higher. Therefore it is necessary to Fourier transform the echo so

that the relaxation rates of OH and CHx protons may be determined separately. By integrating the area under reach peak and plotting this as a function of experiment time, the R2 for each species can be obtained by curve ﬁtting. Since only glycerol has a CHx peak, the relaxation rate for this peak is synonymous with the relaxation rate for glycerol.

85 1 Water, 0.15M NaCl t m 9 CP=28 s, T2=2.48 s

4 Normalized Echo Amplitude

0.0 0.5 1.0 1.5 2.0 2.5 Time[s] Figure 5.4: CPMG results of 0.15M NaCl water. A CPMG pulse train was used to constantly excite spins, with τCP =28 µs. This type of excitation reverses any dephasing induced by inhomogeneities in the external magnetic ﬁeld. Therefore, any decrease in signal amplitude as a function of time is strictly due to spin-spin interactions. The signal amplitude was sampled once every 125 ms.

86 0 10 Pure Glycerol -1 τCP = 1.142 ms 10 T2 = 27.1 ± .2 ms

-2 10

-3 10 Signal indistinguishable from noise -4 10

-5 10

Normalized Echo Amplitude NormalizedEcho -6 10 0.0 0.1 0.2 0.3 0.4 0.5 Time [s] Figure 5.5: CPMG results of pure glycerol. A CPMG pulse train was used to constantly excite spins, with τCP =1.142 ms. Because pure glycerol has a much slower diﬀusion rate, it is less susceptible to additional transverse dephasing due to ﬁeld inhomogeneities, allowing the use of a longer τCP . The noise level in these data is on the order of 10−4.

87 First we will examine the eﬀects of τCP . In many experiments in the literature,

it is common for relatively long τCP (in comparison with those used in this thesis) to be employed. This may be satisfactory given the simple aqueous solutions used to analyze the eﬀects of ferritin on proton relaxation. However, it is not acceptable to have the time between RF pulses be on the same order of magnitude as the chemical

exchange time, τCE, which is deﬁned as:

1 1 1 = + (5.2) τCE τW τG

where τW and τG are the times spent on average by a hydroxyl proton on either a water or glycerol molecule. These times are related to the fraction of water or glycerol hydroxyl sites (as deﬁned in Eq. (4.2)):

τ F W = W (5.3) τG FG

◦ When τCP ≥ τCE, there can be many exchange events between 180 pulses. The dephasing caused by these exchange events is therefore irreversible by the 180◦ pulse, and will show up as an apparently higher transverse relaxation rate. In eﬀect, the apparent relaxation rate, R2,app is given by:

R2,app = R2,0 + R2,CE (5.4)

where R2,0 is the inherent relaxation rate of the spins in the absence of chemical

exchange, and R2,CE is the additional relaxation caused by exchange events. Using the 92.5-7.5 water-glycerol system to demonstrate the effect, two Carr-Purcell times were used: one was made as short as possible, 32 µs, and the other significantly longer at approximately 2 ms. The effects can be seen in Fig. 5.6. Chemical exchange can cause a striking change in the measured relaxation rate. This can hide the effects of ferritin on proton relaxation enhancement in the case that the chemical exchange

88 eﬀects causes more relaxation than the presence of ferritin. To ensure this does not

happen, it is imperative to use values of τCP that are short enough to reverse the dephasing caused by any chemical exchange.

5.2.3 Distinguishing water and glycerol relaxation

A natural question to ask is whether the CHx protons of glycerol also chemically exchange. Such exchange would make it more diﬃcult to ascertain the true transverse

relaxation rate of glycerol. The proof is in the pudding: Fig. 5.7 shows the CHx

proton relaxation rates using the same two τCP as in Fig. 5.6. There is virtually no diﬀerence in the data, indicating that chemical exchange is not playing a role in the dephasing of CHx protons. This means that these same protons are constantly residing on glycerol molecules alone, so that their relaxation rate is synonymous with the relaxation of glycerol. This is important because it allows for the relaxation of water protons to be extracted from the OH proton data in a similar fashion to how diﬀusion data was obtained using Eq. (4.3).

R2,OH − FGR2,G R2,W = (5.5) FW

Tables 5.3 and 5.4 summarize the relaxation data acquired using the methods described previously, as a function of iron concentration (in the form of ferritin) as solution composition.

5.2.4 Chemical exchange and T2 dispersion

The hydroxyl protons of water and glycerol chemically exchange with one another, characterized by the chemical exchange time, τCE. This process can lead to enhanced dephasing, as protons experience the diﬀerent Larmor frequencies of water

89 1

92.5-7.5 by OH fraction 0.1 Water-Glycerol

OH Peak, τCP=32 µs, -1 R2=.685 ± 0.004 s

OH Peak, τCP=2.032 ms, -1 R2=11.75 ± 0.1 s 0.01

Normalized Amplitude Normalized

0.001 0.0 0.5 1.0 1.5 2.0 Time [s] Figure 5.6: Transverse relaxation data for OH protons in a 92.5-7.5 water-glycerol mixture. The two datasets were acquired with different τCP , with noticeably different results in the relaxation rate determined from an exponential curve fit. For the short Carr-Purcell time, dephasing caused by exchange events is reversed by the application of a 180◦ pulse, and the relaxation rate extracted from an exponential curve fit reflects the inherent relaxation rate. The 2.032 ms time between RF pulses allows for many exchange events to occur, making their resulting dephasing irreversible by the pulses. This is reflected in the much higher calculated relaxation rate. Dephasing due to diffusion does not have an effect in this dataset, since the diffusion coefficient of 1.17× 10−9 m2/s, as determined in Fig. 4.7, is insufficient to cause significant decoherence given the slight inhomogeneity in the magnetic field and the small diffusion distance in 2 ms (on the order of a micron). Thus the different measured relaxation rates are purely a result of chemical exchange.

90 1 92.5-7.5 by OH fraction 9 Water-Glycerol CHx Peak, τCP=32 µs, 8 -1 R =0.874 ± 0.002 s 7 2 CHx Peak, τCP=2.032 ms, -1 6 R2=0.869± 0.005 s 5

Normalized Amplitude Normalized 2

0.0 0.5 1.0 1.5 2.0 Time [s]

Figure 5.7: Transverse relaxation data for CHx protons in a 92.5-7.5 water-glycerol mixture. The two datasets were acquired with diﬀerent τCP , with nearly identical relaxation rates determined from exponential curve ﬁts. This is indicative of a lack of chemical exchange, especially when compared with Fig. 5.6.

91 0 10 92.5-7.5% by OH fraction Water Glycerol, 4mM [Fe] -1 OH Peak, R2 = 14.51 ± .02 s -1 CH Peak, R = 16.86 ± .13 s -1 x 2 10

-2 10

-3 10

Normalized Echo Amplitude NormalizedEcho -4 10 0.0 0.5 1.0 1.5 2.0 Time [s]

Figure 5.8: Transverse relaxation data for OH and CHx protons in a 92.5-7.5 water- glycerol mixture with ferritin at an iron concentration of 4 mM. The two datasets were acquired with the same τCP .

Table 5.3: Water R2 as a function of composition and iron concentration, in units of s−1.

Composition 0mM 1mM 4mM 100/0 0.40 2.6 7.1 97.5/2.5 .48 - 8.7 95/5 .62 2.5 12.2 92.5/7.5 .68 - 13.6 90/10 .85 3.1 11.3 85/15 1.1 5.1 17.6 80/20 1.5 5.4 25.7

92 Table 5.4: Glycerolx R2 as a function of composition and iron concentration, in units of s−1.

Composition 0mM 1mM 4mM 97.5/2.5 .63 - 9.7 95/5 .80 2.75 14 92.5/7.5 .87 - 16.1 90/10 .98 3.6 12.3 85/15 1.3 5.9 20.4 80/20 1.6 6 27.9

and glycerol molecules. In order to use water-glycerol solutions as a means to control diﬀusion, the eﬀects of this chemical exchange must be measured, and if possible, mitigated by pulses which are more frequent than exchange events. A CPMG pulse

sequence was used to measure R2 as a function of τCP , which is deﬁned as half the time between 180◦ pulses.

We will discuss the behavior of R2 in the limits of very short and very long τCP .

Initially, τCP τCE, and the relaxation rate is given by the population-weighted average of the intrinsic relaxation rates of water and glycerol, R2,CE = FW RW +FGRG.

◦ As τCP increases, the number of exchange events occurring between successive 180 pulses rises, leading to enhanced dephasing. Eventually, this enhancement reaches a plateau when τCP τCE, and the time an individual spin spends on each site approaches a population-weighted average. The chemical exchange process for a given spin in this regime can be envisioned as a telegraph process, where the Larmor frequency the spin experiences oscillates between two diﬀerent values, ωW and ωG, on

2 2 a timescale τCE. The rate of dephasing is then given by ∆ω τCE, where ∆ω is the

93 mean squared deviation from the average frequency, ω, experienced by the spin.

2 2 2 ∆ω = (ωW − ω) FW + (ωG − ω) FG (5.6)

ω = FW ωW + FGωG (5.7)

2 R2,CE = FW FG (ωW − ωG) τCE describes the relaxation rate due to chemical exchange in the long τCP limit. To characterize chemical exchange experimentally, CPMG experiments were conducted with diﬀerent values of τCP . In order to demonstrate the validity of this technique, we performed such a τCP -dispersion experiment on a control sample of distilled water with 0.15M NaCl concentration. The Carr-Purcell time was varied across two orders of magnitude, starting at 28 µs and going all the way up to 10 ms. The data is shown in Fig. 5.9. There is no signiﬁcant change in the measured transversed relaxation rate until τCP approaches several ms. This is in the crossover

∗ regime between where T2 effects due to field inhomogeneity and the relatively fast diffusion of protons in pure water solutions can be ignored, and where they become experimentally important. The data indicates that there is no chemical exchange in this system, which is expected since the hydroxyl sites are identical: all are on water molecules.

Fig. 5.10 demonstrates the result of a τCP -dispersion experiment on a solution of .925/.075 water-glycerol by hydroxyl fraction. For very long τCP , one can see the increase in relaxation rate begin to level oﬀ. This is when many chemical exchange events are occurring, and the Larmor frequency experienced by a spin behaves like a telegraph process. It is clear that in the fast pulse (short τCP regime), the eﬀects of chemical exchange between water and glycerol hydroxyl protons are mitigated. This allows for an accurate measurement of the inherent relaxation rate of water and glycerol when mixed together, enabling the determination of the relaxation rate

94 0.15M NaCl Water 0.55

0.50 ] -1 [s 2 R 0.45

0.40

4 6 8 2 4 6 8 2 4 6 8 -4 -3 -2 10 10 10 t CP [s]

Figure 5.9: T2 dispersion of pure water. CPMG data were collected with a changing Carr-Purcell time, τCP . The measured relaxation rate, R2, shows little variation till τCP is on the order of several milliseconds. This is the crossover point where the inhomogeneity in the external ﬁeld (about 1ppm, or roughly 300 Hz/7 µT over the sample diameter of 750 µm)causes additional dephasing due to diﬀusion of the water molecules.

95 enhancement for a given composition of solution and concentration of ferritin. The curveﬁt used in Fig. 5.10 was obtained by using the Carver-Richards model of chem-

ical exchange, details of which can be found in Ref. [61]. The short and long τCP limits have been described above in terms of a physical picture. In the limit of short

τCP , the eﬀects of chemical exchange are eliminated, and inherent relaxation rates are measured. This validates the R2 acquired for water-glycerol systems with and without ferritin, where the Carr-Purcell time used was on the order of 30 µs.

96 14

R2, OH R 10 2, CHx

]

-1 8 FW/FG=12.33 Δω =1830 ± 40 rads [s -5 -7 2 τb=5.21 x 10 ± 5.9 x 10 s 6 -1

R R2, OH=0.559 ± 0.006 s -1 R =0.865 s 4 2, CHx

0 2 4 6 2 4 6 2 4 6 -5 -4 -3 -2 10 10 10 10 -1 τCP [s ] Figure 5.10: Chemical exchange in a water-glycerol system. The sample composition by molar fraction of hydroxyl protons was 92.5% water and 7.5% glycerol. A CPMG T2 dispersion experiment was used to characterize the process of chemical exchange between the hydroxyl protons of the two molecular species. When τCP is of short duration, there are very few, if any, chemical exchange events occurring between RF pulses. As τCP approaches τCE, the CPMG sequence is no longer able to reverse the effects of chemical exchange, and dephasing begins to accumulate. This causes the measured relaxation rate, R2 to increase significantly over its true value. These dispersion graphs are useful for identifying the range of τCP for which chemical exchange is negligible, and in which the inherent relaxation rate of each chemical special can be accurately determined. The OH data has been curvefit to the Carver-Richards chemical exchange model.

97 Chapter 6 Discussion

In the previous two chapters, we showed the diffusion and relaxation data acquired in our experiments, designed to reveal information about the relaxation mechanisms of protons in the presence of ferritin. In this chapter, we will combine these data to describe what is physically happening in water-glycerol systems doped with ferritin molecules. Specifically, we will examine the diffusion dependence of ferritin-induced relaxation, and conclude that the diffusion mechanism (OS) in the systems studied is dominant. By extension, this mechanism likely plays a role in vivo, and must be accounted for when quantifying brain iron via MR methods.

6.1 Relaxation Enhancement, ∆R2

In Chapter 5 we presented the relaxation data for protons in the presence of ferritin. However, unlike in previous experiments,[35, 36, 55, 62, 63] the ferritin proteins were not merely in aqueous solution, but in a binary mixture of water and glycerol. Higher glycerol content means longer correlation times, and therefore increased relaxation rates, separate from the eﬀect caused by adding ferritin. Therefore it is important to subtract out the inherent relaxation rate of protons, determined in mixtures without any ferritin using a fast CPMG sequence. The total relaxation rate has several diﬀerent contributions: the inherent relaxation rate for the binary mix-

98 ture, the OS and PE contributions, and any dephasing caused by diﬀusion and the inhomogeneity of the external ﬁeld.

PE OS ∗ R2 = R2,0 + R2 + R2 + R2 (6.1)

∗ We can ignore R2 because the CPMG sequences used were designed to continuously undo dephasing caused by this eﬀect on a time scale much shorter than the time needed for appreciable and irreversible dephasing to have happened due to inhomogeneity in the external ﬁeld. This leaves us with the intrinsic rate R2,0, which is

PE OS measured in solutions without any ferritin, and ∆R2 = R2 + R2 , which can be determined by subtracting the intrinsic rate for a given solution composition from the overall transverse relaxation rate, i.e. ∆R2 = R2 − R2,0. In addition, it is important to distinguish between the OH relaxation and that of water. By assuming that the time between 180◦ pulses is short enough to completely mitigate the eﬀects of chemical exchange between water and glycerol, the overall OH relaxation rate will be a weighted average according to molar fraction of water and glycerol hydroxyl protons, as explained in Chapter 5.2.3: the water relaxation rate is given by Eq. (5.5). This method is justiﬁed because the CHx protons of glycerol are not subject to chemical or proton exchange. They remain solely on glycerol, so that their relaxation rate would be the relaxation of the glycerol hydroxyl protons as well in the absence of water. The inherent relaxation rates of OH and CHx protons in water- glycerol systems, and the enhancement in solutions containing ferritin, are listed in Tables 5.3 and 5.4. The relaxation rate enhancements for water and glycerol molecules are listed below in Tables 6.1 and 6.2. There is a clear trend in both datasets towards higher enhancement as a function of ferritin concentration and glycerol content.

99 Table 6.1: Water ∆R2 as a function of composition and iron concentration, in units of s−1.

Composition 1mM 4mM 100/0 2.2 6.7 97.5/2.5 - 8.2 95/5 2.0 11.6 92.5/7.5 - 12.9 90/10 2.5 10.4 85/15 4.4 16.5 80/20 4.5 24.2

Table 6.2: Glycerol ∆R2 as a function of composition and iron concentration, in units of s−1.

Composition 1mM 4mM 97.5/2.5 - 9.1 95/5 2.0 13.4 92.5/7.5 - 15.2 90/10 2.6 11.3 85/15 4.8 19.1 80/20 4.5 26.4

100 6.2 ∆R2 as a Function of OH concentration

The main focus of this dissertation is to determine the diffusion dependence of ferritin-induced relaxation enhancement. To this end, diffusion was controlled by adding glycerol to aqueous solutions, and examining the resulting effects on proton relaxation. Looking more closely at Eq. (1.1), the OS mechanism exhibits a 1/D dependence, while the PE mechanism, Eq. (1.4) is not dependent on diffusion. However, the PE mechanism is dependent on the concentration of hydroxyl protons, ∼ 1/COH . As it so happens, water and glycerol have similar proton concentrations, 111 M and 109 M at room temperature. Not all the protons of glycerol are hydroxyl protons: accounting for this makes COH for pure glycerol only 41 M. As glycerol is added to water, the concentration of hydroxyl protons in the solution will change. However, the relaxation enhancement evident from the data presented previously is not explainable by hydroxyl concentration. The range of hydroxyl concentrations in the samples used here ranges from 85 (80%-20% water-glycerol by OH fraction) to 111 M (water only), calculated using density data on water-glycerol systems provided by DOW[64] and the molar fractions of water and glycerol used in these experiments. Given the dependence of the PE mechanism on COH , this should result in no more than a 30% change in the relaxation rate enhancement over the range water-glycerol compositions. As clearly evidenced in the data presented in Tables 6.1 and 6.2, the enhancement increases by a factor of 4 over the range of compositions. This is inconsistent with solely a proton exchange mechanism. Even if the possibility of glycerol diffusing as close to the ferritin core as water does is removed (which will be discussed later), so that only the concentration of water hydroxyls is considered, the maximum resulting enhancement for the water protons over the range of compositions increases by a factor of 50%, still inconsistent with the proton exchange mechanism. This shows the need for another explanation. 101 6.3 ∆R2 as a Function of Diffusion

Because hydroxyl concentration is insuﬃcient to explain the results of the relaxation experimental results, we will examine the proton relaxation rate enhancement,

∆R2, as a function of diffusion coefficient, D. We will look at the results for water and glycerol molecules separately, given that each molecule diffuses at different rates. Finally, the different dependence of relaxation enhancement on diffusion rates will be explained in terms of the distance of closest approach to the ferrihydrite core, which is an outstanding question in the case of glycerol.[65]

6.3.1 Water enhancement

By combining the relaxation rate enhancements in Table 6.1 with the water diffusion data presented in Chapter 4, we can plot the enhancement vs 1/D, as seen in Fig. 6.1. By doing so, a clear dependence of the relaxation enhancement of water protons by ferritin as a function of diffusion coefficient is revealed. As the protons diffuse more slowly, and spend more time in the field generated by the ferrihydrite core, their relaxation rate is increased. The enhancement increases linearly with 1/D, in good agreement with the OS model.

6.3.2 Glycerol enhancement

Of key importance is examining the CHx proton data, which is associated only with glycerol. These protons do not experience chemical exchange with the hydroxyl protons; this assertion holds up experimentally as shown in Figs. 5.7 and 5.8. It stands to reason that the diﬀusion dependence of glycerol’s relaxation enhancement should be readily apparent since the PE mechanism is not available to it. Fig. 6.1 shows the eﬀect in 1mM and 4mM [Fe] solutions. There is a clear inverse dependence on DG. 102 30 Water, 1mM [Fe] Water, 4mM [Fe] Glycerol, 1mM [Fe] Glycerol, 4mM [Fe] 25

]

-1

2 15

Δ 10

0 0 1 2 3 4 5

9 2 1/D [10 s/m ]

Figure 6.1: ∆R2 of water and glycerol vs 1/DW and 1/DG, respectively, for 1 and 4mM [Fe] concentrations in water-glycerol systems. For water, the linear fits have slopes of 10.75 s−2 m−2 for the 4mM solutions, and 2.59 s−2 m−2 for the 1mM. These differ by a factor of 4.15, in rough agreement with the prediction of the OS model for linear concentration dependence. The glycerol linear fits have slopes of 4.59 s−2 m−2 for the 4mM solutions, and 0.88 s−2 m−2 for the 1mM.

103 6.3.3 Water vs glycerol- distance of closest approach

Water and glycerol molecules both exhibit relaxation dependence that is inversely proportional to their respective diffusion coefficients. However, this strength of this diffusion dependence differs for the two. Water molecules experience more dephasing for the same value of D than glycerol molecules. This can be explained in terms of the distance of closest approach to the ferrihydrite core. The OS relaxation enhancement is proportional to 1/RD. A relevant parameter to study the different dependence on diffusion for water and glycerol is therefore the product of the products of relaxation rate enhancements and D for water and glycerol molecules. These have been calculated by taking the average of this product for the data points shown previously:   ∆R D 2.61 ± .20, 1mM 2,W W = ∆R2,GDG  2.68 ± .28, 4mM

Until recently, the outer-sphere mechanism has been modeled with the ferritin core as a hard sphere, more or less ignoring the presence of the apoferritin shell. This gives the dependence seen in the OS equation, Eq. (1.1). In the case of aqueous solutions, this assumption is ﬁne because water is known to diﬀuse to the core surface, as it is a vital component of the ferrihydrite structure. However, this may not be the case for glycerol. Whether or not glycerol penetrates the apoferritin shell is somewhat of an outstanding question, with only one article being written regarding this issue,[65] and indicating only the possibility glycerol molecules bind to the inside of the apoferritin shell, or at least to sites along the 3-fold channels. Strictly looking at the 1/RD dependence seen in the OS model, we can examine the radius of closest approach for water compared to glycerol. For a typical ferrihydrite core radius of 3 nm, a radius of closest approach for glycerol would be approximately 8 nm. This exceeds the radius of the apoferritin shell (6.5 nm), so in of

104 itself cannot explain the results of the diffusion and relaxation experiments. Another explanation is needed. One possible explanation for the data in Fig. 6.1 is that water is entering the ferritin cavity, but that it diffuses more slowly while doing so, as the channels most likely restrict motion. Glycerol may not enter the pores at all, remaining outside the apoferritin shell. Because the water molecules get closer to the core, and spend a longer time doing it, they are relaxed more effectively. However, there is less quantitative justification for this given that to date, no one has measured the diffusion of water molecules in the 3-fold pores, and such a measurement would be difficult to obtain. However, a recent contribution by Zeiner provides a theoretical model for this type of behavior.[30] The OS relaxation model can be rewritten as follows[26, 27]:

4 ROS = v∆ω2 τ (6.2) 2 9 RMS D

2 For the hard sphere OS model, the RMS frequency shift is ∆ωRMS, proportional to 1/R6 since the dipolar ﬁeld responsible for the Larmor frequency shift falls oﬀ as R3.

2 The diﬀusion time is τD = R /D, and v is the volume fraction taken up by the ferritin cores, proportional to R3. This gives the 1/R dependence seen in Eq. (1.1). However, if we set aside the volume fraction having dependence on core size, something not invoked by Gueron in his work on outer sphere relaxation,[33] the dependence on distance of closest approach goes as 1/R4. Taking the fourth root of the average of the product exhibited in Tables 6.1 and 6.2 yields RG = 1.28RW . Horse spleen ferritin from Sigma has an average loading factor ranging from 1700- 2100 iron atoms, which translates into a ferrihydrite core radius of 2.9-3.1 nm. This means that glycerol approaches the core to within a distance of 3.7 to 4.0 nm, which corresponds to partial penetration of the apoferritin shell at best. This is consistent

105 with a physical picture where water enters the ferritin cavity and closely approaches the ferrihydrite core, but glycerol does not, or at least not easily. What is clear from these data, no matter which model is used to explain the distance of closest approach, is that glycerol diffuses more slowly and farther away from the ferritin core than water does. This is an interesting result, but not the main focus of this dissertation. The key concept to take from the data is that glycerol molecules are affected by an outer-sphere mechanism, and that water molecules exhibit the same behavior. In other words, both water and glycerol have relaxation enhancements in the presence of ferritin that are dependent on diffusion rates.

6.4 Conclusions

The diffusion dependence of ferritin-induced proton relaxation enhancement was explicitly measured. This was accomplished by the addition of glycerol to water, slowing the diffusive motion of water molecules. Due the slower diffusion, protons spent more time in the dipolar field of ferritin, leading to an enhanced transverse relaxation rate. This relaxation was found to be inversely proportional to the diffusion coefficient, in agreement with an outer sphere mechanism of proton relaxation. To date, no one had explicitly tested the diffusion dependence of ferritin-induced relaxation. Only the field dependence, iron concentration, and in one report the hydroxyl proton concentration had been investigated. It was with this in mind that we set out to perform experiments with control of diffusion. We chose to use glycerol to modify the diffusion rate of protons because of its miscibility in water, and a distinct chemical shift peak unshared with water protons. Because glycerol is much more viscous than water, we were able to successfully control the diffusion of water and slow it down considerably. Glycerol has protons that are bonded to carbon atoms; these have a slightly different Larmor frequency

106 than than the hydroxyl protons. Water only has hydroxyl protons, and the chemical shift between the protons next to carbon and the ones next to oxygen atoms was well within the spectral resolution of our NMR apparatus. This allowed the determination of data for strictly glycerol molecules, notably diffusion coefficients and relaxation rates. Knowing the composition of the solutions coupled with glycerol-only data permitted the calculation of diffusion and relaxation rates for water molecules, which share the hydroxyl peak with glycerol. The range of known diffusion coefficients in water-glycerol systems was extended on the side of very dilute glycerol solutions, a new contribution to the literature. An unexpected consequence of this choice of system was the presence of chemical exchange between the hydroxyl protons of water and glycerol, a process which under normal circumstances leads to additional relaxation of protons. However, this relax-

◦ ation enhancement is dependent on τCP , the time between 180 pulses in a CPMG pulse sequence. By using a rapid CPMG pulse sequence such that the time between pulses was much less than the characteristic exchange time for hydroxyl protons, we were able to eliminate the eﬀects of this dephasing, and measure the intrinsic relaxation rate for water and glycerol molecules, and then quantify the relaxation rate enhancement due to ferritin. The chemical exchange time was on the order of milliseconds; these data were acquired because the literature is severely lacking on chemical exchange data for water-glycerol systems. As shown previously in this chapter, there is a clear diﬀusion-dependence of the relaxation rate enhancement of protons of water and glycerol in the presence of ferritin proteins. The slight change in hydroxyl proton exchange that accompanies the addition of glycerol to water does not explain the change in relaxation rate enhancements, as previously discussed by Gilles et al. [28] The enhancement is too large and the variation in hydroxyl concentration too small to account for the behavior

107 measured. The data is best explained by a dominant outer sphere mechanism of relaxation for the protons by ferritin. Given the higher field strengths employed in this dissertation, and the slower diffusion rates that more closely mimic those seen in the brain of a living person, the outer sphere relaxation mechanism must be taken into consideration when examining in vivo iron. Ferritin can serve as one of the best biomarkers we have for a variety of diseases, many of them related to iron deficiency or overloading. Because invasive methods of measurement are expensive, risky, and sometimes impossible, it is paramount that non-invasive non-destructive methods for evaluating non-heme iron in the body be further developed. MRI provides a non-invasive imaging modality, and ferritin clearly affects the MRI signal. Developing the necessary background for interpreting those images requires a firm understanding of the NMR relaxation mechanisms by which ferritin affects the MRI signal-generating protons. We have done a series of experiments designed to probe the role of diffusion in ferritin-induced relaxation enhancement, and have shown that diffusion plays a dominant and key role.

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