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T₁ relaxation and inhomogeneous magnetization transfer in brain : physics and applications Manning, Alan Patrick 2018

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T1 Relaxation and Inhomogeneous MagnetizationTransfer in Brain: Physics and ApplicationsbyAlan Patrick ManningB.Sc., Carleton University, 2011M.Sc., The University of British Columbia, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2018© Alan Patrick Manning, 2018The following individuals certify that they have read, and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, the dissertation entitled:T1 Relaxation and Inhomogeneous Magnetization Transfer in Brain: Physics and Applicationssubmitted by Alan Manning in partial fulfillment of the requirements forthe degree of Doctor of Philosophyin PhysicsExamining Committee:Carl Michal, Department of Physics & AstronomyResearch SupervisorAlex MacKay, Department of Physics & Astronomy and Department of RadiologyCo-SupervisorSteven Plotkin, Department of Physics & AstronomySupervisory Committee MemberPiotr Kozlowski, Department of Radiology and Department of Urological SciencesSupervisory Committee MemberElliot Burnell, Department of Chemistry (Emeritus)University ExaminerLawrence McIntosh, Department of Chemistry and Department of Biochemistry & Molecular Bi-ologyUniversity ExaminerHarald E. Möller, Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig; andFelix Bloch Institute for Solid State Physics, University of LeipzigExternal ExamineriiAbstractA major goal of the Magnetic Resonance Imaging (MRI) community is quantifying myelin inwhite matter. MRI contrast depends on tissue microstructure, so quantitative models requiredetailed understanding of Nuclear Magnetic Resonance (NMR) physics in white matter’scomplex, heterogeneous environment. In this thesis, we study the underlying physics behindtwo different 1H contrast mechanisms in white and grey matter tissue: T1 relaxation and therecently developed inhomogeneous Magnetization Transfer (ihMT).Using ex-vivo white and grey matter samples of bovine brain, we performed a comprehensivesolid-state NMR study of T1 relaxation under six diverse initial conditions. For the firsttime, we used lineshape fitting to quantify the non-aqueous magnetization during relaxation.A four pool model describes our data well, matching with earlier studies. We also showexamples of how the observed T1 relaxation behaviour depends upon the initial conditions.ihMT’s sensitivity to lipid bilayers, like those in myelin, was originally thought to rely uponhole-burning in the supposedly inhomogeneously-broadened lipid lineshape. Our work showsthat this is incorrect and that ihMT only requires the presence of dipolar couplings, not aspecific kind of line broadening. We developed a simple explanation of ihMT using a spin-1system. Using solid-state NMR, we then performed measurements of ihMT and T1D (dipolarorder relaxation time) on four samples: a multilamellar lipid system (Prolipid-161), wood,hair, and bovine tendon. ihMT was observed in all samples, even those with homogeneousbroadening (wood and hair). Moreover, we saw no evidence of hole-burning.Lastly, we present results from ihMT experiments with CPMG acquisition on the bovinebrain samples. We show that myelin water has a higher ihMT signal than water outside themyelin. It was determined that this was due to the unique thermal motion in myelin lipids.In doing so, we developed a useful metric for determining the relative contributions frommagnetization transfer and dipolar coupling to ihMT. Also, we applied a qualitative fourpool model with dipolar reservoirs. Together, our results are consistent with myelin lipidshaving a T1D which is appreciably longer than the T1D of non-myelin lipids, despite recentmeasurements to the contrary.iiiLay SummaryIn diseases like Multiple Sclerosis (MS), a material in the brain called “myelin” is damaged.If nerve cells are like wires, then myelin is like their insulation: when myelin is damaged,nerve signals can’t travel properly. My research focuses on methods for measuring myelinusing an MRI scanner. This is important for more accurate diagnoses and for deeper studyof diseases such as MS.An MRI scanner is like an X-ray machine that’s really good at taking pictures of brain tissue.Instead of X-rays, the pictures taken by MRI scanners are made using magnets and radiowaves. How to distinguish myelin’s unique radio waves, and thereby be able to quickly andaccurately measure myelin using an MRI scanner, is the topic of my thesis.ivPrefaceThe experiments performed in Chapter 5 were suggested by my two co-supervisors, CarlMichal and Alex MacKay. The pulse sequences were written by Carl Michal and modified byme. I built the sample holders, modified the spectrometer, and carried out the experiments.I wrote the data analysis and simulation scripts with guidance from Alex and Carl.Chapter 6 of this thesis is based on the published paper:AP Manning, KL Chang, AL MacKay, CA Michal, Journal of Magnetic Resonance274, 125–136 (2017) of Section 4.4.3 were also taken from this paper. Carl Michal conceived of the initialexperiments. Kimberley Chang made the Prolipid-161 sample, wrote the pulse programs,and performed these experiments, under guidance from Carl and me. Carl and I developedthe spin-1 model together. I developed the Provotorov model (building off of previouslypublished work) and wrote the code to perform the simulations. Apart from Prolipid-161,I obtained all the other samples and ran the experiments on them. The majority of themanuscript was written by me based on Kimberley’s undergraduate thesis. Alex and Carlalso helped with the manuscript preparation.The ihMT-CPMG experiments in Chapter 7 were originally envisioned by Carl. They werecarried out by Patricia Angkiriwang and me. She wrote the final pulse program under guid-ance from Carl and I. This was based on an earlier version of the pulse program written byEsther Lin. Patricia also performed the preliminary analysis of the data for her undergradu-ate thesis. I performed the majority of the analysis and developed the four pool model withdipolar couplings independently.I developed the circuit analogies in Appendix C and the derivation of the Provotorov Equa-tions in Appendix A independently.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Symbols and Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 MRI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Motivation: quantitative MRI of myelin . . . . . . . . . . . . . . . . . . . . . 31.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 NMR theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 The Zeeman interaction and its implications . . . . . . . . . . . . . . . . . . 52.2 Classical treatment of NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.1 Precession and the rotating frame . . . . . . . . . . . . . . . . . . . . 72.2.2 The Bloch equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.3 A simple NMR experiment and the Lorentzian lineshape . . . . . . . 122.3 Quantum mechanical treatment of NMR . . . . . . . . . . . . . . . . . . . . 142.3.1 NMR in Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.2 NMR in Liouville space: density matrices . . . . . . . . . . . . . . . . 18vi2.3.3 A simple NMR experiment using density matrices . . . . . . . . . . . 202.4 Other spin interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.1 Chemical shielding and quadrupolar interactions . . . . . . . . . . . . 222.4.2 The dipolar Hamiltonian for two nuclei . . . . . . . . . . . . . . . . . 232.4.3 Dipolar line broadening in many-spin systems . . . . . . . . . . . . . 252.5 Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5.1 The problem statement and the local field . . . . . . . . . . . . . . . 262.5.2 Pulsed rf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.5.3 BPP theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.5.4 Spin temperature and Redfield theory . . . . . . . . . . . . . . . . . . 282.5.5 ADRF/ARRF: An application of Redfield theory . . . . . . . . . . . 312.5.6 Provotorov theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.6 Relaxation in homogeneous systems . . . . . . . . . . . . . . . . . . . . . . 342.6.1 What drives relaxation? . . . . . . . . . . . . . . . . . . . . . . . . . 342.6.2 BPP relaxation theory . . . . . . . . . . . . . . . . . . . . . . . . . . 342.7 Some experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.7.1 T1 measurement with inversion-recovery . . . . . . . . . . . . . . . . 362.7.2 T2 measurements: the spin echo and CPMG acquisition . . . . . . . . 372.7.3 Determining FID deadtime . . . . . . . . . . . . . . . . . . . . . . . . 393 White matter, grey matter, and myelin . . . . . . . . . . . . . . . . . . . 413.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 The nervous system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3 Myelin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3.1 Myelin structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3.2 Myelin function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.3.3 Multiple Sclerosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 Relaxation and spectra in brain: properties and applications . . . . . . 504.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.2 Spectral properties of white and grey matter . . . . . . . . . . . . . . . . . . 504.3 T1 relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.3.1 A common, simple model . . . . . . . . . . . . . . . . . . . . . . . . . 544.3.2 The controversy of quantitative T1 measurements . . . . . . . . . . . 544.4 Magnetization transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.4.1 The magnetization transfer ratio . . . . . . . . . . . . . . . . . . . . 554.4.2 qMT and the two pool model . . . . . . . . . . . . . . . . . . . . . . 56vii4.4.3 Inhomogeneous magnetization transfer . . . . . . . . . . . . . . . . . 574.5 T2 relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.5.1 Myelin water and intra/extra-cellular water . . . . . . . . . . . . . . 594.5.2 CPMG exchange correction . . . . . . . . . . . . . . . . . . . . . . . 605 Aqueous and non-aqueous T1 relaxation in brain under six different initialconditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2 Theory: the four pool model . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.3.1 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.3.2 NMR experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.3.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.3.3.1 FID fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.3.3.2 CPMG fitting . . . . . . . . . . . . . . . . . . . . . . . . . . 735.3.3.3 Combining CPMG and FID fits . . . . . . . . . . . . . . . . 745.3.3.4 Four pool model analysis . . . . . . . . . . . . . . . . . . . . 755.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.4.1 Spectra and FIDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.4.2 CPMG multi-exponential fitting . . . . . . . . . . . . . . . . . . . . . 815.4.3 White matter four pool and bulk water fitting . . . . . . . . . . . . . 825.4.4 White matter fitting variations . . . . . . . . . . . . . . . . . . . . . 865.4.5 Grey matter two pool fitting . . . . . . . . . . . . . . . . . . . . . . . 915.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.5.1 Comparison with other studies . . . . . . . . . . . . . . . . . . . . . . 935.5.2 Imaging applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.5.3 Is the four pool model necessary to understand T1 relaxation? . . . . 985.5.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986 Is “inhomogeneous” MT mis-named? . . . . . . . . . . . . . . . . . . . . 1006.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.2.1 ihMT model 1: a simple spin-1 system . . . . . . . . . . . . . . . . . 1036.2.1.1 Selective and non-selective pulses in a spin-1 system . . . . 1036.2.1.2 Application to ihMT . . . . . . . . . . . . . . . . . . . . . 1056.2.1.3 Spectral asymmetry from dipolar order . . . . . . . . . . . . 107viii6.2.2 ihMT model 2: a homogeneously-broadened system using Provotorovtheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.2.2.1 The Provotorov equations for continuous-wave ihMT prepulses 1086.2.2.2 Model details . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.2.2.3 Spectral asymmetry from dipolar order . . . . . . . . . . . . 1136.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.4.1 PL161 spectral asymmetry from dipolar order . . . . . . . . . . . . . 1156.4.2 Flip-angle dependence of spectral asymmetry . . . . . . . . . . . . . . 1176.4.3 PL161 dipolar order relaxation . . . . . . . . . . . . . . . . . . . . . 1206.4.4 Dipolar order of homogeneously-broadened spin systems . . . . . . . 1226.4.5 ihMT in lipids and homogeneously-broadened systems . . . . . . . . . 1246.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1267 Pool-specific ihMT in white matter . . . . . . . . . . . . . . . . . . . . . 1297.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1307.2.1 The four pool model with dipolar reservoirs . . . . . . . . . . . . . . 1307.2.2 The grey matter analogue . . . . . . . . . . . . . . . . . . . . . . . . 1337.3 Methods and materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347.3.1 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347.3.2 NMR experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347.3.3 CPMG fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1357.3.4 Four pool model fitting . . . . . . . . . . . . . . . . . . . . . . . . . . 1367.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1438 Conclusions and future work . . . . . . . . . . . . . . . . . . . . . . . . . 1478.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1478.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1488.2.1 The non-aqueous lineshape and the effect of soft pulses . . . . . . . . 1488.2.2 Improved quantification of T1 relaxation . . . . . . . . . . . . . . . . 1498.2.3 T1D measurements in brain . . . . . . . . . . . . . . . . . . . . . . . . 150Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151Appendix A Derivation of the Provotorov equations . . . . . . . . . . . . . 169ixAppendix B CPMG exchange correction . . . . . . . . . . . . . . . . . . . . 175B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175B.2 Equations from a two pool model . . . . . . . . . . . . . . . . . . . . . . . . 175B.3 Algorithm 1: Actual to observed values . . . . . . . . . . . . . . . . . . . . . 178B.4 Algorithm 2: Observed to actual values . . . . . . . . . . . . . . . . . . . . . 178Appendix C Circuit analogies in NMR relaxation . . . . . . . . . . . . . . 180C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180C.2 Four pool model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181C.3 Provotorov equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182Appendix D Model of ihMT using pulse-train prepulses . . . . . . . . . . 184xList of Tables3.1 The composition of human myelin, white matter, and grey matter . . . . . . 455.1 FID fit functions at equilibrium and the aqueous T ∗2 s. . . . . . . . . . . . . . 805.2 The fitted four pool model parameters, eigenvectors, and eigenvalues (whitematter) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.3 Comparison of chi-squares for three pool, weighted four pool, and unweightedfour pool model in white matter . . . . . . . . . . . . . . . . . . . . . . . . . 875.4 The fitted two pool model parameters, eigenvectors, and eigenvalues (greymatter) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.1 T1D values and measurement techniques . . . . . . . . . . . . . . . . . . . . 1227.1 Literature T1D measurements in white and grey matter. . . . . . . . . . . . . 1307.2 The dipolar reservoir fit parameters . . . . . . . . . . . . . . . . . . . . . . . 140xiList of Figures1.1 MR images of a patient with MS . . . . . . . . . . . . . . . . . . . . . . . . 22.1 Dynamics in a frame rotating at rf pulse frequency. . . . . . . . . . . . . . . 102.2 The relative orientations of the sample, B0, and B1(t) . . . . . . . . . . . . . 122.3 Dipolar broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.4 T1 and T2 as functions of correlation time τc . . . . . . . . . . . . . . . . . . 362.5 The inversion-recovery sequence . . . . . . . . . . . . . . . . . . . . . . . . . 362.6 The spin echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.7 The CPMG acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.1 A cartoon of a neuron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2 The physical and chemical composition of myelin . . . . . . . . . . . . . . . 433.3 Electron micrographs of white matter showing the myelin sheath . . . . . . . 443.4 The structure of the major lipid types in myelin . . . . . . . . . . . . . . . . 454.1 Dipolar couplings in lipid molecules and lipid bilayers . . . . . . . . . . . . . 514.2 Dipolar couplings in lipid acyl chains causes the super-Lorentzian lineshape . 524.3 How magnetization transfer works . . . . . . . . . . . . . . . . . . . . . . . . 564.4 The two pool model of tissue . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.5 ihMT compared to MTR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.6 Examples of the CPMG exchange correction under different conditions . . . 625.1 The four pool model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.2 The NMR pulse sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.3 The initial conditions after the preparation pulses . . . . . . . . . . . . . . . 695.4 The four pool model analysis and fitting flowcharts . . . . . . . . . . . . . . 715.5 The 1H NMR equilibrium spectra of white and grey matter . . . . . . . . . . 775.6 Fits to the equilibrium and non-equilibrium spectra . . . . . . . . . . . . . . 785.7 Sample fits to white matter FIDs . . . . . . . . . . . . . . . . . . . . . . . . 815.8 The regularized NNLS and sparse T2 distributions . . . . . . . . . . . . . . . 825.9 Four pool model fits to white matter data . . . . . . . . . . . . . . . . . . . 83xii5.10 Residuals from four pool model fits to short-TI MW data . . . . . . . . . . . 875.11 The effect of varying the MW weights . . . . . . . . . . . . . . . . . . . . . . 885.12 Comparison between three and four pool model fits . . . . . . . . . . . . . . 895.13 Two pool fits to grey matter data . . . . . . . . . . . . . . . . . . . . . . . . 925.14 Fitting the total aqueous IR-soft and IR-hard signals . . . . . . . . . . . . . 965.15 Eigenvector coefficients for each experiment . . . . . . . . . . . . . . . . . . 976.1 The original hole-burning explanation of ihMT . . . . . . . . . . . . . . . . . 1016.2 Simulation of non-aqueous ihMT in an isolated spin system using CW pre-pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.3 ihMT pulse sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.4 Flip angle dependence in PL161 non-aqueous spectra . . . . . . . . . . . . . 1166.5 Manifestation of dipolar order in ihMT. . . . . . . . . . . . . . . . . . . . . . 1176.6 Measured Zeeman and dipolar order . . . . . . . . . . . . . . . . . . . . . . . 1186.7 Beef tendon, human hair, and Western Red Cedar sapwood spectra followingCW ihMT prepulses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.8 Flip-angle dependence of spectral asymmetry and saturation method mea-surement of T1D in PL161 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.9 An ADRF/ARRF spectrum of PL161/D2O . . . . . . . . . . . . . . . . . . . 1206.10 Inversion-recovery measurements in PL161/D2O. . . . . . . . . . . . . . . . . 1216.11 T1D of beef tendon as measured by the saturation method. . . . . . . . . . . 1236.12 Aqueous and non-aqueous ihMTRs vs. offset frequency . . . . . . . . . . . . 1246.13 ihMTR vs. prepulse power . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.1 The four pool model with dipolar reservoirs . . . . . . . . . . . . . . . . . . 1317.2 The ihMT-CPMG pulse sequences . . . . . . . . . . . . . . . . . . . . . . . . 1357.3 Regularized NNLS distributions after the four different ihMT prepulses . . . 1377.4 MTR after single and dual prepulse irradiation . . . . . . . . . . . . . . . . . 1397.5 The ihMTR under varying prepulse times and prepulse strengths . . . . . . . 1417.6 Fits from the four pool model with dipolar reservoirs . . . . . . . . . . . . . 1427.7 ihMT in white matter samples and their grey matter analogues . . . . . . . 1437.8 The relative contribution of MT to ihMT . . . . . . . . . . . . . . . . . . . . 145C.1 The equivalent circuit of the four pool model . . . . . . . . . . . . . . . . . . 180C.2 The equivalent circuit of the Provotorov equations (Ω = 1) . . . . . . . . . . 182D.1 A model of ihMT prepulses of the pulse-train variety . . . . . . . . . . . . . 185D.2 Simulation of non-aqueous ihMT using pulse-train prepulses . . . . . . . . . 187xiiiList of Symbols and AcronymsβD Inverse spin temperature of the Dipolar reservoir, page 30βZ Inverse spin temperature of the Zeeman reservoir, page 30∆ Offset from resonance frequency, page 33η Weighting factor for MW residuals where TI < 37 ms, page 83γ nuclear gyromagnetic ratio, page 5HˆD Secular dipolar coupling Hamiltonian, page 24HˆZ Zeeman Hamiltonian, page 5B0 NMR/MRI main magnetic field, page 5B1 NMR/MRI oscillating, transverse magnetic field applied with coils., page 9Beff The total effective field in a rotating frame., page 9M0 Equilibrium magnetization, page 7Ω Fractional offset relative to ωD, page 33ω0 Larmor frequency, page 5ω1 RF (B1) field amplitude in rad/s, page 10ωD Residual (RMS) dipolar coupling strength, page 25ρ Density matrix, page 18ρ0 Equilibrium density matrix, page 20BL The local field, page 27xivm Magnetization in reduced units, page 63M(∞) Pool size, page 63Maq The total aqueous magnetization (MW+IEW in the four pool model), page 71Mnon-aq The total non-aqueous magnetization (M+NM in the four pool model), page 71S+ ihMT experiment with prepulse at an offset +∆, page 56S− ihMT experiment with prepulse at an offset −∆, page 56S0 ihMT experiment with no prepulse, page 56Sdual ihMT experiment with prepulse at offsets ±∆ simultaneously, page 56T1 Longitudinal relaxation time / spin-lattice relaxation time, page 11T2 Transverse relaxation time / spin-spin relaxation time, page 11T ∗2 Observed T2 relaxation time (T ∗2 < T2 because of field inhomogeneities), page 37T1D Dipolar order (magnetization) spin-lattice relaxation time, page 32T ∗1 The four pool model effective T1 relaxation times, page 61Tcr Cross-relaxation time for adjacent pools, page 63TI Cross-relaxation delay in T1 relaxation experiments, page 65W Saturation rate, page 33ADRF/ARRF Adiabatic Demagnetization/Remagnetization in the Rotating Frame, page 31BPP Bloembergen, Purcell, and Pound saturation/relaxation theory, page 27BW bulk water (isolated) pool, page 72CNS Central Nervous System, page 41CPMG Carr-Purcell-Meiboom-Gill acquisition, page 37CSF Cerebrospinal Fluid, page 2E/R Exchange/relaxation factor for four pool model eigenvectors, page 84FID Free Induction Decay, page 14xvGS Goldman-Shen NMR experiment, page 67IEW Intra/Extra-cellular Water, page 58ihMT inhomogeneous Magnetization Transfer, page 56ihMTR ihMT Ratio, page 57IR Inversion-Recovery experiment, page 36M four pool model Myelin: the non-aqueous protons in the myelin sheath, page 62MAS Magic Angle Spinning NMR experiments, page 61MRI Magnetic Resonance Imaging, page 1MS Multiple Sclerosis, page 1MT Magnetization Transfer, page 54MTR Magnetization Transfer Ratio, page 54MW Myelin Water, page 44MWF Myelin Water Fraction, page 59NM four pool model pool “Non-Myelin”: the non-aqueous protons outside the myelinsheath, page 62NMR Nuclear Magnetic Resonance, page 1NNLS Non-Negative Least Squares, page 59PNS Peripheral Nervous System, page 41qMT quantitative Magnetization Transfer, page 56rf magnetic fields at radio frequencies, page 7SNR Signal to Noise Ratio, page 6TR Tilted Rotating frame, page 30xviAcknowledgementsFirst and foremost, thank you to my parents for giving me the rare privilege of choosing thepath I wanted, the ability to take it, and the support to keep going.I also have been extremely fortunate by having an immensely patient, knowledgeable, andcaring supervisor. Carl, it’s truly been a pleasure working with and learning from youduring these past years. I know I’ll be frequently drawing upon your lessons in my futureendeavours, whatever they may be. My services as a laboratory interior designer/decoratorand ergonomics consultant will always be available to Room 100.Speaking of supervisors, Alex, thank you for guiding me with wisdom and patience into theworld of MRI. I would not have got very far into the journey without you.By the same token, thank you also to my other committee members, Piotr Kozlowski andSteve Plotkin, and the university/external reviewers, Elliot Burnell, Lawrence McIntosh,and Harald Möller. Their careful reading, pointed questions, and valuable suggestions havemade this work significantly stronger.Next, my colleagues and friends have been a constant source of support, humour, and hap-piness. Yael, I’ll simply say you brought sunshine into a windowless basement every day.Similarly, Nathan, I am grateful to the innumerable times you coaxed me out of the basementinto the sunshine, metaphorically and literally. Mohammad, thank you for listening and con-tributing to my constant ramblings. Our conversations always made me think with differentparts of my brain. Jimmy, I appreciated how you always brought the entrepreneurial spiritto work, I think it rubbed off on me. Francesco, our discussions were always enjoyable, thankyou for teaching me so much about Italy beyond, and including, the obvious subject of pasta.Kimberley and Patricia, this thesis would not have been possible without the exceptionalresearch from both of you.My list of colleagues and friends grew significantly when I joined the UBC MRI community.All of you are extremely welcoming, inclusive, and friendly people. The ISMRM conferenceswould not have been nearly as memorable had I not been part of your group.xviiOutside of academia, the sponsorship of Vancouver’s Proximis Digital has been tremendouslyappreciated. I frequently relied upon its consulting and implementation expertise, utilizing(pro-bono) its IT, communication, transportation, and fashion service lines. Moreover, thereal Dr. Manning, and her growing family, were always a constant source of encouragementand support as well. Likewise, Uncle John, I always looked forward to a beer and a chatwith you, and I am thankful for your hospitality when I needed a place to stay. And I can’tforget Harry and Rosie, whom I frequently relied upon for their superhuman listening skillsand calming presence.To the friends and relatives of Vesuvius Hammerbottom, our quests, imagined and real,stand among my best memories.Finally, Asha, thank you for your love and encouragement. I’m a better person because ofyour patience—and, through much practice, a better cook.xviiiDedicationThis work is dedicated to the memory of Evan MacRaexixChapter 1Introduction1.1 MRIMagnetic Resonance Imaging (MRI) uses the physical phenomenon of Nuclear MagneticResonance (NMR) to create images. Nowadays, MRI is a ubiquitous medical imaging tech-nology, given its excellent performance in soft tissue and its absence of ionizing radiation. Itis especially useful for brain and spinal cord imaging and is the only modality suitable fordiagnosing Multiple Sclerosis (MS) [1].Ultimately, the contrast in MRI scans depends only on the NMR properties of the nucleusunder study within distinct microstructural environments. In the majority of clinical andresearch contexts, the scanner detects the NMR signal from aqueous protons (1H nuclei).That said, this signal’s properties are dictated largely by interactions between aqueous andnon-aqueous protons. Hence, these aqueous/non-aqueous interactions can allow indirectimaging of the non–aqueous protons.Using different pulse sequences, an MRI scanner produces images which emphasize the varia-tion within one or more of the NMR properties of different tissue. For example, two commonways of distinguishing between white and grey matter is by their T1 relaxation times andaqueous proton concentrations. When a single aqueous T1 time is assumed, white and greymatter have values of ~0.7 s and ~1.2 s respectively at a field strength of 1.5 T [3]. And greymatter has a higher water content than white matter, so the two are distinguishable on animage showing only differences in aqueous proton density.To see these concepts in practice, Fig. 1.1 shows four images from different kinds of 1H MRIscans of the same MS patient with obvious lesions. The image in panel A is weighted by(that is, its contrast reflects) the T2 relaxation times. Aqueous proton density is emphasized1Figure 1.1: MR images of a 30 year old patient with MS. Different types of MRI techniqueswere used to generate contrast in different ways. Lesions (areas with myelin damage) canbe seen in all of the images. (A) A T2 weighted image. Short T2s appear dark. (B) Aproton density weighted image. High density appears light. (C) A fluid-attenuated inversionrecovery image, which attenuates signals from CSF. (D) T1 weighted image after injectionof a gadolinium contrast agent. Short T1s appear dark. Reproduced from ref. [2] [Journal ofNeurology, Neurosurgery & Psychiatry, Trip & Miller, Volume 76, iii11–iii18], © 2005, withpermission from BMJ Publishing Group the pulse sequence used to acquire the image in panel B. The image in panel C wasacquired after inverting the magnetization in the aqueous protons and waiting for a delaybefore acquisition. This reduces the signal from the cerebrospinal fluid (CSF), isolating thesignal from aqueous protons within the tissue. Finally, the patient has had a contrast agentinjected prior to acquiring the image in panel D, which is T1-weighted. The contrast agentselectively reduces the T1 relaxation time of specific tissues.This thesis will cover the physics behind these images extensively except for contrast agents.These are administered intravenously or orally and selectively change the relaxation times inspecific tissues, compartments, or organs. Currently, contrast agents are used in about 25%of all MRI exams [4,5]. Compounds with high specificity to certain tissues (such as myelin [6])or pathologies (such as amyloid plaques [7]) have been developed. However, because of safetyand regulatory considerations, their use is often limited to animal models [8]. In this thesis,2we are concerned with ways of improving the specificity of MRI without the use of contrastagents, so they won’t be discussed further.1.2 Motivation: quantitative MRI of myelinIn clinical practice, non-quantitative MRI scans are used most of the time; apart fromobserving the presence and morphology of lesions, the end result rarely contains precisemeasures of the tissue microstructure properties. Quantitative MRI is a highly desirable goalfor both clinicians and scientists, however: quantifying microscopic disease pathologies wouldlead to better diagnosis and management, more advanced research into disease mechanisms,and more precise metrics for judging treatment effectiveness. For these reasons, there is nowa push by the MRI research community towards developing and implementing quantitativetechniques.Quantitative MRI sequences and models seek to develop biomarkers for specific tissue com-ponents or morphologies. This requires a fundamental understanding of the physical originsof the NMR signal and its properties. In this thesis, we are concerned with the developmentof biomarkers for myelin. Myelin is a substance surrounding axons and is essential for propernervous signal transmission. It is a major component of white matter tissue.The research in this thesis covers two approaches to myelin quantification: T1 relaxation andinhomogeneous Magnetization Transfer (ihMT). We try to understand both of these on a fun-damental, physical level. Currently, measurements of T1 relaxation in white matter disagree.The values obtained by different groups show unexplained variation and even the numberof T1 components present is unclear. Our work emphasizes that there are indeed multiplecomponents, but they cannot be cleanly associated with specific compartments. ihMT is anew technique which allows one to calculate a simple ratio whose value may be a biomarkerfor myelin. In this thesis we argue that the original hypothesis explaining it—inhomogeneousbroadening of the non-aqueous lineshape—is incorrect. More generally, we explore its fun-damental physics, from its origin in the non-aqueous protons to its manifestation in separatecompartments of aqueous protons.Our tool to study white and grey matter is solid-state NMR spectroscopy, allowing straight-forward observation of both the aqueous and non-aqueous proton signals. The samples weuse are biological materials or phantoms of tissue. In particular, the grey and white matterwe investigate is from ex-vivo bovine brain. Because we are concerned with fundamentalNMR properties, there is no imaging performed in this thesis, but we expect that the workhere will be useful in guiding future quantitative MRI development.31.3 OutlineThe next three chapters are background material. In Chapter 2 we give an overview of theclassical and quantum physics of NMR. Topics which are relevant to the rest of the thesis areemphasized, including the dipolar interaction and saturation theories. After this, Chapter3 is a short introduction to the structure and function of myelin. Diseases of the myelinare briefly covered, focusing on multiple sclerosis (MS). Moving to Chapter 4, we give somenecessary information about the relaxation and spectral properties of NMR in white andgrey matter. The spectral lineshape of the non-aqueous 1H nuclei—the super-Lorentzian—isintroduced. Then, we explore some of the controversy surrounding T1 relaxation in whitematter. We show how Magnetization Transfer (MT) and the MT ratio (MTR) is a usefulconsequence of aqueous/non-aqueous magnetization exchange. T2 relaxation, which revealsdistinct aqueous compartments in white matter, is the final topic.With the background material out of the way, Chapter 5 is the first chapter with originalwork: an exhaustive study of T1 relaxation in bovine white and grey matter. There, the fourpool model is introduced and is used to analyze the results.Chapter 6 discusses a suite of experiments investigating the fundamental physics of ihMT.We performed ihMT experiments and measured dipolar order relaxation in a multilamellarlipid system (a phantom for myelin), hair, wood, and bovine tendon. Based on ihMT’sconnection with dipolar couplings, we also introduce a spin-1 model of ihMT. Our resultssuggest that ihMT does not rely on inhomogeneous broadening.Our last results are in Chapter 7, where we unite concepts from the previous two chapters.We carried out ihMT experiments with Carr-Purcell-Meiboom-Gill (CPMG) acquisition inthe same bovine white and grey matter. This allowed observation of the separate ihMTsignals from the myelin water and intra/extra-cellular water. We apply the four pool model,now modified with the addition of dipolar reservoirs, to qualitatively model our results.Finally, in Chapter 8 we review the results and suggest future experiments.The appendices contain additional calculations which are not integral to the main thesis.These include a derivation of the Provotorov equations, an outline of how to calculate thecorrection factor for exchange during CPMG acquisition, circuit analogies of the four poolmodel and ihMT, and modeling of pulse-train ihMT prepulses.4Chapter 2NMR theory2.1 The Zeeman interaction and its implicationsNuclei with unpaired protons or neutrons have nonzero nuclear spins. This leads to a nuclearmagnetic moment, µ = γ(~I), where γ is the gyromagnetic ratio and I the nuclear spin. (Wewill use the convention of unitless spin operators, hence the explicit factor of ~.) Nuclei likethis interact with magnetic fields via the Zeeman interaction. The Zeeman Hamiltonian is1~HˆZ = −µ ·B0.The units of HˆZ are rad/s and B0 is the main spectrometer field, which is taken to be in thez direction: B0 = B0zˆ. Hence, the Zeeman Hamiltonian (in rad/s) isHˆZ = −γB0Iˆz (2.1)= ω0Iˆz,where ω0 = −γB0 is the Larmor frequency of precession. This identification anticipates theconnection to the classical theory of precession.With an expression for the Zeeman energy, it is illuminating to calculate the thermal polar-ization in a typical NMR spectrometer or MRI scanner B0 field. Consider protons2, which,as spin-12 particles, have I =12 and m = ±12 . The energies of these two eigenstates are1In this chapter we will cite references only when necessary, since the content here parallels most introduc-tory textbooks. Sources which the author relied on for this content were Duer [9], Slichter [10], Schmidt-Rohr& Spiess [11], and a useful report by Goldman [12].2When NMR and MRI physicists say “protons”, they are always referring to a 1H nucleus. By the sametoken, “spins” almost always refer to nuclear spins.5〈±12∣∣∣ HˆZ ∣∣∣±12〉 = ±12ω0 and the energy difference is ∆E = ~ω0. If N+ + N− is the totalnumber of spins in a sample, with N± representing the number in the | ± 12〉 state, then therelative polarization isN+ −N−N+ +N−≈ 12(1− N−N+)= 12 (1− exp(−∆E/kBT ))≈ ∆E2kBT ,where kB is Boltzmann’s constant and T is the temperature in Kelvin. In a 100 MHz (2.3T) field3 at 300 K, this is only 8 ppm for protons—a very small polarization indeed! Wecan compare this to the typical polarizations seen in Electron Spin Resonance (ESR). Forprotons, γ/2pi = 42.577 MHz/T, and for an unpaired electron, γe/2pi = 28.025 GHz /T.The thermal polarization of a sample with unpaired electrons is about γe/γ ≈ 658× higher.The relative weakness of the nuclear Zeeman effect dictates the experimental constraints andfeatures of NMR and MRI. Without B0 & 0.1 T, performing NMR and MRI experiments isdifficult.4 For MRI, the range of nuclei which can be imaged in clinically reasonable timesis limited to 1H and a few others (eg. 23Na and 31P). All of these are abundant enough inthe human body to be imaged directly. However, protons are the most common nuclei toimage due to their superior signal to noise ratio (SNR) in biological samples. There are threereasons for this. First, with tissue being ~70% water, they are ubiquitous in biochemicalsystems. Second, NMR SNR is approximately proportional to ω20 [15, 16], so it makes senseto use nuclei with high Larmor frequencies.5 1H also leads in this category, for only theextremely rare 3H has a higher γ. Finally, naturally occurring Hydrogen is isotopically pure,with ~99.98% of all H atoms containing 1H nuclei [19].On a more fundamental level, the weak nuclear Zeeman effect sets NMR apart from manyother forms of spectroscopy because of its unique method of relaxation. Unlike optical spec-troscopy, excited states in NMR do not relax via stimulated and/or spontaneous emission(see Hoult [20] and references therein). Rather, relaxation is driven by environmental fluc-3The B0 and B1 field strengths of NMR spectrometers is typically stated in the Larmor frequency (inHz) of protons in that field. Conversely, in MRI these are usually given in Tesla.4But not impossible. In fact, NMR and MRI has been performed in the Earth’s magnetic field (eg. seereferences [13,14]). Although weak, the Earth’s magnetic field is extremely homogeneous. However, Earth’sField NMR and MRI are of limited practical utility, and typically can’t observe any nuclei except 1H.5In MRI, the SNR is more complicated. It is very sensitive to the sample geometry and composition.Also, because repeated scans are required to cover all of k-space, it doesn’t make sense to speak of the SNRfor a single acquisition. Furthermore, the image is typically made from the magnitude of the NMR signal,and this has a Rician noise profile. See Macovski [17] and Ocali & Atalar [18] for more details.6tuations, i.e. coupling to the lattice. Therefore, the relaxation processes in NMR and MRIcan be a useful window into the microscopic structure and motion of a sample. We will havemuch more to say on this in the following chapters.Because of the nuclear Zeeman effect’s low energy, NMR frequencies are in the 1 MHz –1 GHz range, corresponding to radiation wavelengths of 300–0.3 m. How then are MRIscanners capable of sub-millimeter resolution? Even though precessing spins emit radiowaves in the far-field limit, NMR and MRI operate in the near-field limit (i.e. the distanceto the emitters is small or comparable to the wavelength) [15, 20, 21]. In this regime, theemission from an ensemble of precessing spins manifests as an oscillating magnetic field which,unlike electromagnetic radiation, does not impose limits on resolution based on wavelength[20–22]. Correspondingly, the spins are manipulated with magnetic fields oscillating at radiofrequencies, and, in MRI, the spatial resolution is instead limited by the strength of themagnetic field gradients [23]. Obviously, this is the reason for the “M” in NMR! In theliterature, the transmitted and detected magnetic fields are commonly referred to as “rf”.Magnetization is the macroscopic result of the Zeeman interaction. The equilibrium magne-tization in a sample inside a field B0 isM0 = χ0B0, (2.2)where χ0 is the static magnetic susceptibility.2.2 Classical treatment of NMR2.2.1 Precession and the rotating frameThe classical picture of NMR is a straightforward way to introduce basic NMR dynamicsand experiments. The equations of motion for a magnetization M in a field B aredMdt= M× (γB), (2.3)which describes the precession of M around B at a frequency ω = −γB.Excluding relaxation, this compact equation contains all the classical dynamics of NMR. Itapplies whether M and/or B are constant or varying. Nonetheless, it isn’t terribly straight-forward to use in practice. One of the problems with Eq. 2.3 is that it describes the dynamics7of precession in the laboratory frame. Most calculations in NMR are simplified by workingin the rotating frame, a coordinate system which rotates around the z axis.Slichter [10] has a nice derivation of how vectors are transformed between the lab and rotatingframes. Consider an arbitrary vector function F(t) = ∑i=x,y,z Fiiˆ, where the iˆ unit vectorsare aligned with the lab frame. Now, assume that this coordinate frame rotates with anangular velocity ω. In this case, the lab frame time rate of change for iˆ isdˆidt= ω × iˆ.The time derivative of F(t) in the lab frame is now more complicated, but gives a usefulresult:dF(t)dt∣∣∣∣∣lab=∑i=x,y,z(dFidtiˆ+ Fidˆidt)=∑i=x,y,zdFidtiˆ+∑i=x,y,zFidˆidt=∑i=x,y,zdFidtiˆ+ ω × F(t)= dF(t)dt∣∣∣∣∣rot+ ω × F(t), (2.4)where dF(t)dt∣∣∣rotis the time rate of change in the rotating frame. If we apply this to M, thenfrom Eq. 2.3 we havedMdt∣∣∣∣∣lab=M× (γB)= dMdt∣∣∣∣∣rot−M× ω⇒ dMdt∣∣∣∣∣rot=M× (γB+ ω).And so, in the rotating frame Eq. 2.3 still applies, provided that we replace B0 with aneffective fieldBeff = B+ωγ. (2.5)When only B0 is present, by convention the precession axis is taken to be zˆ. Hence, in the8zˆ direction we haveBeff = B0 +ωγ(2.6)⇒ γBeff = γB0 + ωωeff = ω0 − ω.Where ωeff = −γBeff is the effective precession frequency in the frame rotating at ω. Ifω = ω0, then ωeff and Beff are both zero: the effect of the the main spectrometer field B0 hasbeen completely removed.NMR uses a B1 field to manipulate the magnetization. The B1 field is applied in thetransverse plane and oscillates at ω:B1(t) = B1 cos(ωt+ φ)xˆ+B1 sin(ωt+ φ)yˆ, (2.7)where φ is some arbitrary phase factor. We make no assumptions about ω: it could be onor off resonance. It’s easiest to analyze the problem in a frame rotating at ω. The timederivative is thendB1(t)dt∣∣∣∣∣rot= dB1(t)dt∣∣∣∣∣lab− (ωzˆ)×B1(t)= B1(ω − ω) sin(ωt+ φ)xˆ+B1(ω − ω) cos(ωt+ φ)yˆ,= 0.In this frame it appears as a constant magnetic field perpendicular to zˆ, around which Mrotates according to Eq. 2.3. This rotation around B1 is called nutation. Combining thisresult for B1 with the result for B0 in Eq. 2.6, the total effective field in the frame rotatingat the frequency of the rf pulse isBeff =(B0 +ωγ)zˆ +B1xˆ, (2.8)where we have chosen the phase of the rf pulse φ such that B1 is along xˆ. Another way towrite this isωeff = (ω0 − ω)zˆ + ω1xˆ (2.9)withω1 = |γ|B1 (2.10)9ω1 x^z(ω0−ω) z^yωeffzyxωeff M(A) (B)xFigure 2.1: Dynamics in a frame rotating at the B1 frequency ω. Field strengths are indicatedin units of rad/s (see Eqs. 2.8 and 2.9). (A) The effective field, ωeff, composed of the effectivemain field ω0 − ω and the amplitude of the rf field, ω1. (B) The magnetization M precessesaround the effective field. After Fig. 2.4 in Slichter [10].as the amplitude of the rf pulse in rad/s—which, by convention, is always positive. Thedynamics of M in this frame are shown in Fig. 2.1. If the rf pulse is on resonance (ω = ω0),the effect of B0 is completely removed.In the preceding discussion, we have glossed over the issue of the sign of γ and the directionof M’s precession and nutation. Most NMR-active nuclei, including 1H and 13C, have γ > 0.Referring to Eq. 2.3 and applying the right-hand rule, this implies a clockwise precession ofM about B, which is a negative angular frequency. Indeed, if B = B0, then the definitionω0 = −γB0 gives a negative Larmor frequency, as required. If γ < 0 (eg. 15N), then theprecession of M is in the counter-clockwise direction and ω0 > 0. Regarding nutation inthe B1 field, the definition of ω1 (Eq. 2.10) means it will always be positive—independentof the sign of γ—implying a counter-clockwise nutation of M about the effective field in therotating frame. Experimentally, we could ensure ω1 > 0 by appropriately selecting φ, the B1rf pulse phase.In practice, it is rarely necessary to keep track of the correct precession and nutation direc-tions in the analysis of single-nuclei experiments, so long as one is consistent. For a deeperdiscussion, the reader is referred to Levitt’s papers on the subject [24, 25]. For the sake ofclarity, we will assume counter-clockwise nutation of M about the B1 field for the remainderof this thesis.2.2.2 The Bloch equationsBefore getting too far into the discussion of how rf pulses (the B1 field) affect the spins,the Bloch equations should be introduced. These phenomenological equations incorporate10relaxation into the dynamics described by Eq. 2.3. The Bloch equations are approximateand do not accurately describe the dynamics of NMR in all situations—only in systemsof isolated spin-12 nuclei in low-viscosity solutions are they exact. Still, they provide anexcellent framework for intuitive understanding. And, even where they aren’t rigorouslyaccurate, they can often be modified to model the situation anyway.The Bloch equations in the lab frame aredMzdt= M0 −MzT1+ γ(M×B)zdMx,ydt= γ(M×B)x,y − Mx,yT2.(2.11)T1 is the spin-lattice or longitudinal relaxation time, T2 is the spin-spin or transverse re-laxation time, and T2 ≤ T1. T1 is a result of the spin-lattice coupling that returns themagnetization to thermal equilibrium M0 = M0zˆ viaMz(t) = M0(1−(1− Mz(0)M0)exp(− tT1)). (2.12)There can be no transverse components of magnetization in equilibrium: Mx and My mustdecay. They do so exponentially with time constant T2 viaMx,y(t) = Mx,y(0) exp(− tT2). (2.13)Dephasing of the precessing spins causes T2 relaxation. In practice, we often refer to spin-lattice and spin-spin relaxation as T1 and T2 relaxation, even in cases where there is not asingle, well-defined value for either.Now, consider the Bloch equations for a sample in an NMR spectrometer under the influenceof rf pulses on resonance. In this situation, Blab = B0 +B1(t), where B1(t) is rotating in thetransverse plane at ω0 (Eq. 2.7). Following the last section, in the rotating frame this becomesBrot = B1xˆ, where we have chosen the x-axis to be the direction of B1(0). Correspondingly,M×Brot = B1(Mzyˆ −Myzˆ), so the Bloch equations in the rotating frame aredMzdt= M0 −MzT1− ω1MydMydt= ω1Mz − MyT2dMxdt= 0.(2.14)11sampleB1(t)B0rf transceivercoilzxylab frameFigure 2.2: The relative orientation of the sample, B0, and B1(t).2.2.3 A simple NMR experiment and the Lorentzian lineshapeThe relative orientation of B0 and B1(t) to the sample and the transceiver coil is shown inFig. 2.2. The solenoidal coil style in this figure was used to complete all NMR experimentsin this thesis, though many other coil styles exist.In thermal equilibrium, the sample has a magnetization M0 = M0zˆ. In a simple NMRexperiment, a B1 pulse is applied to tilt M away from the z-axis. Experimentally, this isachieved by applying an oscillating voltage V (t) across the coil resonance circuit,V (t) ∼ exp(−i(ωt+ φ))where φ is phase under the experimenter’s control. This causes an oscillating, linearly-polarized B1 field,B1(t) = 2B1 cos(ωt+ φ′)xˆ,where 2B1 is an amplitude under the control of the experimenter. Regarding the phasefactors, φ and φ′ vary only by an additive constant for a given sample and experimentalset-up. Therefore, the relative phases of multiple B1 pulses (either in the same experimentor in repeated experiments) can be carefully controlled. In other words, the rf pulses usedin NMR and MRI pulse sequences are coherent.The oscillating B1 field along the x-axis may be mathematically decomposed into two12counter-rotating fields in the x-y plane:B1(t) = 2B1 cos(ωt+ φ′)xˆ= B1 (cos(ωt+ φ′)xˆ+ sin(ωt+ φ′)yˆ) +B1 (cos(ωt+ φ′)xˆ− sin(ωt+ φ′)yˆ) . (2.15)Under most circumstances, the field rotating opposite to the precession direction of M hasnegligible effect on the spins’ dynamics, and can be safely ignored in the analysis of mostexperiments. As such, with a linearly-polarized B1 field, half of the rf power is wasted. InNMR, this rarely poses a large enough problem to address, since doing so requires non-standard coil designs and resonance circuits. In these configurations, changing the sampleis often difficult. On the other hand, in MRI the wasted power is absorbed by the patient,posing limitations on B1 strength and duration. In modern MRI scanners, this is addressedby using quadrature coils [26, 27]. In the most simple configuration, two perpendicularlinearly-polarized B1 coils are used. When transmitting, the sinusoidal current in the twocoils has a relative phase shift of 90◦, producing circular B1 polarization. Due to the largebore of MRI scanners, coils with geometries designed for imaging specific regions can beplaced directly on the patient.Returning to the simple experiment, when B1 is turned on with an amplitude ω1  T−11and ω1  T−12 , relaxation effects may be temporarily ignored. If the total duration of theB1 pulse is τ , then in the rotating frameM(τ) = M0 cos(ω1τ)zˆ −M0 sin(ω1τ)yˆ.Let’s consider the case of ω1τ = pi2 , which is a “90-degree” pulse. Immediately following thispulse, M(τ) = −M0yˆ, which is a 90◦ rotation from equilibrium. With components in thetransverse plane, M precesses around zˆ at ω0.The precessing magnetization induces a voltage in the coil, which is now used as a receiver.The voltage is converted into a complex signal, S(t). Up to a constant, this is given byS(t) = Mtransverse(t) exp (i(ω0 − ωref )t) (2.16)=√Mx(t)2 +My(t)2 exp (i(ω0 − ωref )t) ,where Mtransverse(t) is the magnitude of the magnetization in the transverse (x-y) planeprecessing at ω0 and ωref is a reference frequency corresponding to the center of the spectrum.13Taking into account T2 relaxation, the signal after the 90◦ pulse isS(t) = M0 exp (−t/T2) exp (i(ω0 − ωref )t) . (2.17)The 90◦ pulse gives the maximum signal intensity. If a 45◦ pulse was used instead,Mtransverse(0)and S(0) would both be reduced by 1√2 .The NMR signal acquired with a single pulse is called the Free Induction Decay (FID). TheFourier transform of the FID gives the NMR spectrum, which in this case is [28]S(ω) = F{S(t)u(t)eiφ0}= eiφ0[T21 + (ω −∆ω)2 T 22− i (ω −∆ω)T221 + (ω −∆ω)2 T 22], (2.18)where ∆ω = ω0 − ωref . The first and second terms in the brackets represent the absorptionand dispersion parts of the Lorentzian lineshape. Two factors have been explicitly insertedprior to the Fourier transform: the zeroth-order phase factor, eiφ0 , and the unit step orHeaviside function u(t), given byu(t) =0 t < 01 t > 0.The first is necessary to complete the Fourier transform properly since S(t < 0) = 0. Thesecond arises from the NMR receiver chain. In NMR, spectra are represented using theabsorption lineshapes, so phase correcting S(ω) (multiplying by e−iφ0 for a zeroth ordercorrection) is necessary to isolate the pure absorption part. Exponential decay of the FIDfrom the T2 time corresponds to a Lorentzian lineshape in the frequency domain. Evidently,the Bloch equations naturally lead to a Lorentzian lineshape.2.3 Quantum mechanical treatment of NMR2.3.1 NMR in Hilbert spaceHaving seen the classical approach, the next step is some basic calculations using state vectorsevolving under the Schrödinger equation. This “Hilbert Space” approach is actually not usedvery often, since the ensemble average of ~1023 magnetic moments in a sample lends itself14to classical calculations (preceding sections) or density matrices (following section). Still, ita useful bridge into the quantum mechanics of NMR.The Schrödinger equation in the lab frame isi∂∂t|ψ〉 = Hˆ |ψ〉 (2.19)where Hˆ has units of rad/s. For the time being we shall only deal with cases where theHamiltonian is constant, leading to the formal solution|ψ(t)〉 = exp(−iHˆt) |ψ(0)〉 . (2.20)The operator exp(−iHˆt) is called the propagator. Two important properties of the propa-gator are i) Aˆ exp(Bˆ) = exp(Bˆ)Aˆ only if [Aˆ, Bˆ] = 0; and ii) if |α〉 is an eigenstate of Aˆ, thenexp(Aˆ)|α〉 = exp(α)|α〉.Let’s consider the expectation values of two different states for a spin-1/2 nucleus underHˆ = ω0Iˆz (no B1 field). The first is the eigenstate∣∣∣12〉. Unsurprisingly,〈Iz〉 = 〈12 | exp(iω0Iˆzt)Iˆz exp(−iω0Iˆzt)|12〉= 〈12 |Iˆz|12〉= 12In the same way we can also show that 〈Iˆx〉 = 〈Iˆy〉 = 0 for this state. For a more interestingexample, if the spin state is now |x; +12〉 = 1√2(|12〉+ | − 12〉), then 〈Iz〉 is〈Iz〉 = 12(〈12 |+ 〈−12 |)exp(iω0Iˆzt)Iˆz exp(−iω0Iˆzt)(|12〉+ | − 12〉)= 12(〈12 |+ 〈−12 |) (12 |12〉 − 12 | − 12〉)= 0.15And 〈Iˆx〉 is〈Iˆx〉 = 12(〈12 |+ 〈−12 |)exp(iω0Iˆzt)Iˆx exp(−iω0Iˆzt)(|12〉+ | − 12〉)= 12(〈12 |+ 〈−12 |)exp(iω0Iˆzt)Iˆx(exp(−iω02 t)|12〉+ exp(iω02 t)| − 12〉)= 12(〈12 |+ 〈−12 |)exp(iω0Iˆzt)(12 exp(−iω02 t)| − 12〉+ 12 exp(iω02 t)|12〉)= 12(〈12 |+ 〈−12 |) (12 exp(−iω0t)| − 12〉+ 12 exp(iω0t)|12〉)= 12 cos(ω0t).Similar calculations also show that 〈Iˆy〉 = 12 sin(ω0t). In the lab frame, 〈Iˆy〉xˆ+〈Iˆx〉yˆ representsthe precession of the spin’s magnetic moment around B0 at the Larmor frequency. Oneisolated spin’s expectation values behaves like the macroscopic magnetization.Now let’s apply a B1 pulse rotating around zˆ at a frequency ω with an amplitude ω1. In thelab frame,Hˆ = ω0Iˆz + ω1 cos(ωt)Iˆx + ω1 sin(ωt)Iˆy (2.21)= ω0Iˆz + ω1 exp(−iωIˆzt)Iˆx exp(iωIˆzt).On the second line we have applied a useful property of the angular momentum operators,exp(−iφIˆl)Iˆm exp(iφIˆl) = Iˆm cosφ+ Iˆn sinφ (2.22)for cyclic permutations of l,m, n = {x, y, z},which arises from their well-known commutation relations, [Iˆl, Iˆm] = iIˆn (and cyclic permu-tations thereof). In other words, exp(iφIˆl) is a generator of rotations around axis l. We willuse this property extensively when working with density matrices under rf pulses.The lab frame Hamiltonian in Eq. 2.21 is unsuitable to solve with the propagator in Eq. 2.20because of its time dependence. As in the classical case, a transformation into the rotatingframe simplifies the problem. We do this by moving the time-dependent part of the Hamil-tonian into the state kets. This technique can be applied to any Hamiltonian in the formHˆ(t) = Hˆ0 + Vˆ (t) and moves the problem into what is known as the interaction represen-tation. Consider an arbitrary state ket in the lab and rotating frame, indicated by |ψ〉 and|ψ′〉 respectively. They are related by|ψ′〉 = exp(iωtIˆz)|ψ〉⇒ |ψ〉 = exp(−iωtIˆz)|ψ′〉.16The procedure to find the dynamics of the transformed state is similar to the one above forthe transformation of a classical vector into the rotating frame (Eq. 2.4). The time derivativeis∂∂t|ψ〉 = −iωIˆz exp(−iωtIˆz)|ψ′〉+ exp(−iωtIˆz) ∂∂t|ψ′〉,which we equate to Hˆ|ψ〉 (via Schrödinger’s equation, Eq. 2.19). After some algebra, we findSchrödinger’s equation in the rotating frame,i∂∂t|ψ′〉 =((ω0 − ω)Iˆz + ω1Iˆx)|ψ′〉. (2.23)i∂∂t|ψ′〉 = Hˆ ′|ψ′〉.The effective Hamiltonian in the rotating frame, Hˆ ′, is now static. The B1 field is constantalong the x-axis and the effective B0 field isBeff = B0 +ωγ,as in the classical case (Eq. 2.5).Now, we apply this rotating frame representation to calculate the effect of a B1 pulse ofduration τ . If we start with a |12〉 state, the expectation values change in time:〈Iˆz〉 = 〈12 | exp(iHˆ ′τ)Iˆx exp(−iHˆ ′τ)|12〉= 〈12 | exp(i((ω0 − ω)Iˆz + ω1Iˆx)τ)Iˆx exp(−i((ω0 − ω)Iˆz + ω1Iˆx)τ)|12〉.This is tedious to calculate when ω 6= ω0, since eAˆ+Bˆ = eAˆeBˆ only if [Aˆ, Bˆ] = 0, which isn’tthe case for Iˆz and Iˆx. One possible approach would be to use the matrix form of Hˆ ′ anddiagonalize it. This would be written as H ′ = PDP−1, where D is a diagonal matrix withentries λ1, λ2, . . . , λn, the eigenvalues of H ′. Then, the properties of the matrix exponentialare such thatexp(−iH ′τ) = exp(−iPDP−1τ)= P exp(−iDτ)P−1= Pe−iλ1τ · · · 0... . . . ...0 · · · e−iλnτP−1.Another approach is to use a different frame transformation, such as the double-rotated17frame or the tilted frame, which we will meet in Section 2.5.4 below. In this example, wewill proceed assuming the B1 pulse is on resonance.With ω = ω0, 〈Iˆz〉 becomes〈Iˆz〉 = 〈12 | exp(iω1Iˆxt)Iˆz exp(−iω1Iˆxt)|12〉= 〈12 | cos(ω1t)Iˆz − sin(ω1t)Iˆy|12〉= 12 cos(ω1t),where we used the spin operators as generators of rotation (Eq. 2.22). By the same token,〈Iˆy〉 = 12 sin(ω1t).B1 rf pulses allow transitions between eigenstates of Iˆx, Iˆy, and Iˆz. After a 90◦ pulse (ω1t =pi2 ), the system has moved from an eigenstate of Iˆz to one of Iˆy, corresponding classically tonutation of the magnetization around the B1 field by 90◦.2.3.2 NMR in Liouville space: density matricesDensity matrices are used extensively in NMR since they provide a concise way of dealingwith ensembles of many spins. We will briefly review their properties and motivation, fol-lowing Lynden-Bell [22]. Consider a spin with a wavefunction |Ψ〉 written in terms of theeigenstates |ψn〉:|Ψ〉 = c1|ψ1〉+ c2|ψ2〉+ · · ·+ cn|ψ3〉.The density matrix ρ has elements given byρij = cic∗j ,which are the elements of the Cartesian product |Ψ〉 〈Ψ| of the wavefunctions. One of thedensity matrix’s strengths is that expectation value calculations are straightforward. For an18arbitrary operator Qˆ with a matrix representation Q in the {|ψn〉} basis,〈Qˆ〉 = 〈Ψ|Qˆ|Ψ〉=∑i,jc∗i cj〈ψi|Qˆ|ψj〉=∑i,jρjiQij= Tr{ρQ} = Tr{Qρ}.Moreover, it can be shown that the trace is independent of the basis set for ρ and Q, allowingone to choose the simplest representation for the calculation at hand.The density matrix formalism provides a method of dealing with ensembles of spins. Considera system of N spin-12 nuclei which are spatially localized and therefore distinguishable. Toa good approximation, this is the case for protons in a sample of liquid water. The systemcould be described by the set of wavefunctions for each spin, {|ψ1〉 , |ψ2〉 , . . . , |ψN〉}. Thesewould have forms like|ψ1〉 =∣∣∣12〉|ψ2〉 = −∣∣∣−12〉|ψ3〉 = 1√2∣∣∣12〉+ 1√2 ∣∣∣−12〉. . . etc.,for example. Keeping track of these wavefunctions is impossible in a sample with ~1020water molecules. Moreover, it isn’t useful, since in NMR we can only ever measure expec-tation values from all spins at once. In other words, we don’t care about each individualwavefunction; it is the ensemble average which is the useful quantity. Finding this is sim-plified by the density matrix. Formally, the density matrix for this system is given byρ = 1N(|ψ1〉 〈ψ1|+ |ψ2〉 〈ψ2|+ · · ·+ |ψN〉 〈ψN |), which is a 2x2 matrix—a significant reduc-tion in the number of terms to keep track of! And since the equilibrium density matrix isknown (see below), there is never any need to deal with the wavefunctions corresponding toindividual spins.The time evolution of the density matrix under a Hamiltonian Hˆ (with units of rad/s) isgiven by the Liouville-Von Neumann equation,∂ρ∂t= −i[Hˆ, ρ]. (2.24)19When Hˆ is constant, the formal solution isρ(t) = exp(−iHˆt)ρ(0) exp(iHˆt), (2.25)where exp(−iHˆt) is the propagator we met before in the formal solution to Schrödinger’sequation (Eq. 2.20).Density matrices exist in Liouville space, where they are the analogue of Hilbert space statevectors. In Liouville space the trace is the analogue of the inner product and “superoperators”act on state density matrices via commutators. While a comprehensive overview of theLiouville formalism would be useful for some topics in NMR, it isn’t necessary for thisthesis.2.3.3 A simple NMR experiment using density matricesWe’ll calculate again the evolution of a simple NMR experiment consisting of a 90◦ pulsefollowed by signal acquisition, this time using density matrices. Our model system now is acollection of isolated protons coupled loosely to the lattice. Together, they form a canonicalensemble. At equilibrium in the spectrometer field B0, the density matrix isρ0 =exp(−~HˆZ/kBT )Tr{exp(−~HˆZ/kBT )}≈ 1− ~HˆZ/kBTTr {1}= 1N− ~HˆZNkBT= 1N− ~ω0NkBTIˆz.(2.26)with 1 as the unit operator and N as the number of spins. A density matrix of 1N1 describesan ensemble of spins oriented completely randomly, which isn’t measurable. Rather, it is thedeviation from the random orientation which is detectable, so the constant term is dropped.Also, the constant prefactor in front of Iˆz is usually set to 1 since it doesn’t typically changeover the course of an experiment. We are left with the simple result,ρo = Iˆz. (2.27)20The equilibrium magnetization is proportional to 〈Iˆz〉. If the spins have I = 12 ,M0 ∝ 〈Iˆz〉= Tr{ρ0Iˆz}= Tr{Iˆz Iˆz}= 12 .Up to now, the calculations have been valid for the rotating or lab frame, since only Iˆzoperators have been involved. From now on, we will assume that ρ is in the rotating frame.With the system in equilibrium, we now apply a pulse of the B1 field at frequency ω. Thefirst step in determining the evolution of ρ is to calculate the propagator of the Hamiltonianin the rotating frame (Eq. 2.23):exp(−iHˆ ′t) = exp(−i((ω0 − ω)Iˆz + ω1Iˆx)τ).Again, we’ll assume that ω = ω0. Hence,ρ(τ) = exp(−iω1τ Iˆx)ρ(0) exp(iω1τ Iˆx)= exp(−iω1τ Iˆx)Iˆz exp(iω1τ Iˆx)= cos(ω1τ)Iˆz − sin(ω1τ)Iˆy,where we have used the rotation relations for the spin operators given in Eq. 2.22. We areconsidering a 90◦ pulse, so ω1τ = pi2 and nowρ = −Iˆy.This corresponds to magnetization precessing in the transverse plane. The signal measuredby the spectrometer after quadrature detection, S(t), is easily found using density matrices.In Section 2.2.3, we found S(t) classically from the precessing magnetization. The signalfrom the FID detected in the coil is always mixed with a reference frequency, ωref . So, wefirst need to determine the evolution of the density matrix in a frame rotating at ω0 − ωrefrelative to the stationary lab frame. Since ρ = −Iˆz is in a frame rotating at ω0, the new21frame has a relative rotation of −(ω0 − ωref ). The density matrix in this new frame, ρ′, is:ρ′ = exp(i (ω0 − ωref ) tIˆz)ρ exp(−i (ω0 − ωref ) tIˆz)= exp(i (ω0 − ωref ) tIˆz)(−Iˆy) exp(−i (ω0 − ωref ) tIˆz)= − cos((ω0 − ωref )τ)Iˆy − sin((ω0 − ωref )τ)Iˆx.In the rotating frame, S(t) is given by [9, 11]S(t) ∝ Tr{Iˆ+ρ′} ,where Iˆ+ = Iˆx + iIˆy. Up to a constant we haveS(t) = Tr{(− cos((ω0 − ωref )τ)Iˆy − sin((ω0 − ωref )τ)Iˆx) (Iˆx + iIˆy)}= − sin((ω0 − ωref )τ)Tr{IˆxIˆx}− i cos((ω0 − ωref )τ)Tr{Iˆy Iˆy}= −12 sin((ω0 − ωref )τ)− i12 cos((ω0 − ωref )τ)= −i12 (cos((ω0 − ωref )τ) + i sin((ω0 − ωref )τ))= −i12 exp(i(ω0 − ωref )τ).Ignoring the exponential decay from T2 relaxation, this is the same signal we calculatedclassically (Eq. 2.17). The presence of the phase factor −i here is irrelevant and arises onlyfrom how we have defined our pulse phases. We are free to rotate the coordinate systemwithin the rotating frame around zˆ without changing the physics.2.4 Other spin interactions2.4.1 Chemical shielding and quadrupolar interactionsSo far we have not explicitly considered any interactions apart from the nuclear Zeemaneffect. This thesis focuses mostly on the effects of the dipolar interaction, a topic we willsoon cover in detail. First, however, we briefly discuss two other ways a nuclear spin mayinteract with its environment.Chemical shielding is the interaction of the electrons surrounding a nucleus in a fieldB0. Thisresults in a slightly higher or lower field at the nucleus, changing the Larmor frequency in ameasurable way. Because of this effect, the same nucleus in a molecule often has site-specificspectral frequencies. In rapidly-tumbling molecules in solution, an isotropic part of the22chemical shielding interaction remains. This is one of the main reasons NMR spectroscopyis so useful in analytical chemistry.Nuclei with I > 12 (~74% of all NMR-active nuclei [9]) experience quadrupolar couplings.This is an interaction of the nucleus with electric field gradients. Quadrupolar coupling isusually quite strong relative to other interactions. Because its interaction strength dependson its orientation relative to B0, it can be a useful tool for studying the structure and motionof certain molecules, like liquid crystals. To first order, the quadrupolar interaction has noisotropic part, so it is averaged away in small molecules in solution.2.4.2 The dipolar Hamiltonian for two nucleiDipolar coupling6 is the interaction of one nuclear spin’s magnetic moment with one or moremagnetic moments from neighbouring nuclear spins. Imagine a system with two spins, I andS, separated by a vector r. Their dipolar Hamiltonian (in rad/s) isHˆD,tot =(µ04pi)γIγS~(Iˆ · Sˆr3− 3(Iˆ · r)(Sˆ · r)r3)= −d [A+B + C +D + E + F ]in units of rad/s. Here, γI,S are the gyromagnetic ratios, Iˆ = Iˆxxˆ+ Iˆyyˆ + Iˆz zˆ (and similarlyfor Sˆ), andd =(µ04pi)γIγSr3~A = IˆzSˆz(3 cos2 θ − 1)B = −14(Iˆ+Sˆ− + Iˆ−Sˆ+)(3 cos2 θ − 1)C = 32(IˆzSˆ+ + Iˆ+Sˆz)sin θ cos θ exp(−iφ)D = 32(IˆzSˆ− + Iˆ−Sˆz)sin θ cos θ exp(+iφ)E = 34(Iˆ+Sˆ+)sin2 θ exp(−2iφ)F = 34(Iˆ−Sˆ−)sin2 θ exp(+2iφ),(2.28)where Iˆ± = Iˆx± iIˆy. In these terms θ and φ are the polar and azimuthal angles and zˆ is thedirection of B0. The angles refer to the orientation of the inter-nuclear vector with respect6In this thesis we are dealing exclusively with magnetic dipoles and dipolar coupling.23to B0. Proton-proton dipolar couplings in organic solids have a maximum strength of 20–30kHz. This is small compared to the Zeeman interaction (~100 MHz), so HˆD,tot may be treatedas a perturbation. In light of this, the dipolar Hamiltonian simplifies immensely [10]. TermsC, D, E, F are off-diagonal—they connect non-degenerate states—so their contributions tothe spectrum are small (although they are responsible for relaxation). Conversely, terms Aand B are diagonal. In any Hamiltonian, keeping terms like A and B is called the secularapproximation. The secular, homonuclear (γI = γS) dipolar Hamiltonian isHˆD = −d(3 cos2 θ − 1)(IˆzSˆz − 14(Iˆ+Sˆ− + Iˆ−Sˆ+))= −d(3 cos2 θ − 1)(IˆzSˆz − 12(IˆxSˆx + IˆySˆy))= −d2(3 cos2 θ − 1)(3IˆzSˆz − Iˆ · Sˆ).(2.29)We will refer to the secular approximation as the dipolar Hamiltonian from now on, usingthe three forms given in Eq. 2.29. As usual, the Hamiltonian has been written in units ofrad/s. Iˆ+Sˆ−+ Iˆ−Sˆ+ is often called the “flip-flop” term since it swaps the z-component of thespins in a spin pair. The heteronuclear Hamiltonian (γI 6= γS) doesn’t have these terms—thestates it connects are non-degenerate—and so retains only IˆzSˆz.The Zeeman Hamiltonian for these two spins, HˆZ ∼ (Iˆz + Sˆz), commutes with HˆD: [Iˆz +Sˆz, IˆzSˆz] = 0 trivially, and by using the identity 2Iˆ · Sˆ = (Iˆ + Sˆ)2 − Iˆ2 − Sˆ2, we can seethat [Iˆz + Sˆz, Iˆ · Sˆ] = 0 as well (this is only true under the assumption that Iˆz and Sˆz haveidentical prefactors). This implies that HˆD is the same in the lab frame and in any framerotating around zˆ. Also, it means that HˆD and HˆZ have simultaneous eigenstates.As an example application, consider the special case of two identical dipolar-coupled spin-12nuclei, such as protons in a methylene group. For this system,HˆD = −d2(3 cos2 θ − 1)(3Iˆ2z − Iˆ2).Since the spins are identical, we can only measure the total spin, not the spin of any onenucleus. The total spin eigenstates are separable into the singlet state,|00〉 = 1√2 (| ↑↓〉 − | ↑↓〉) ,24and the triplet states,|11〉 = | ↑↑〉|10〉 = 1√2 (| ↑↓〉+ | ↑↓〉)|1− 1〉 = | ↓↓〉,where the arrows indicate the spin of the two protons and |Is〉 indicates the total spin statehas quantum numbers I and s. The singlet state is of no interest here since no transitionsare possible and it is therefore unobservable. Using only the triplet state, we have Iˆ2 =I(I + 1)1 = 21, where 1 is the unit operator. And so, in the B0 field the total HamiltonianisHˆZ + HˆD = ω0Iˆz − 13ωD(θ)(3Iˆ2z − 21)(2.30)with 13ωD(θ) =d2(3 cos2 θ − 1). (2.31)This causes transitions at frequencies ω0 ± ωD(θ) (Fig. 2.3A and B). The splitting 2ωD(θ)depends on the relative orientation of the two nuclei (as above, θ is the angle the inter-nuclearvector makes with B0). At the “magic angle” ωD(θMA) = 0. This angle isθMA = cos−1(1/√3)≈ 54.7◦ (2.32)We will use this Hamiltonian again in Chapter 6 to describe ihMT.2.4.3 Dipolar line broadening in many-spin systemsThe two-nucleus dipolar Hamiltonian above (Eq. 2.29) is simple enough to solve exactly.However, in naturally-occuring organic solids or soft matter, proton-proton dipolar couplingsare rarely limited to two spins. These systems have a many-spin dipolar Hamiltonian,HˆD = −∑i 6=jdij(3 cos2 θij − 1)(Iˆz,iIˆz,j − 14(Iˆ+,iIˆ−,j + Iˆ−,iIˆ+,j)).This still has simultaneous eigenstates with the Zeeman Hamiltonian for the system,HˆZ =∑iω0,iIˆz,i,25one isolated spinfreqamptwo coupled identical spinsmany coupled spins(A) (B) (C)2ωD(θ)Figure 2.3: The effects of dipolar broadening in two identical spins and in many spins. (A)The spectrum of an isolated nucleus, with slight broadening from T2 relaxation. (B) Whentwo of these spins are coupled via the dipolar interaction, the spectrum is a doublet, whereωD(θ) is given by Eq. 2.30. (C) When many spins are coupled together, individual linescannot be distinguished and the spectrum is broad. The spectral intensities are not to scale.but we have no easy way of finding what they are. As more spins are added to the system,the Zeeman energy levels are smeared out into a continuum. As a result, the spectrum isbroadened, as illustrated in Fig. 2.3C. In organic solids, 1H spectral broadening from thedipolar interaction can be up to ~50 kHz.Experimentally, it is found that the spectral broadening from dipolar couplings in manysystems is approximately Gaussian. This cannot easily be derived directly from the Hamilto-nian. However, it can be motivated using other models, like assuming a randomly-fluctuatingfield [29]. Also, we can approximate the lineshape of a system under the influence of themany-spin dipolar Hamiltonian without explicitly knowing the energy eigenstates. Thistechnique is called the Van Vleck expansion [10, 30]. This expands the lineshape in termsof its moments (e.g. the second moment of the Gaussian is the variance, σ2). Using thistechnique, many systems, such as cubic lattices, are only non-Gaussian in negligible higherorder moments [30].2.5 Saturation2.5.1 The problem statement and the local fieldImagine performing an NMR experiment on a dipolar-coupled system of many spins. Whenrf is applied via the B1 field, the lab frame Hamiltonian is composed of the Zeeman, dipolar,and rf parts:Hˆlab = HˆZ,lab + HˆD + ω1 cos(ωt)Iˆx,lab.26We can pose what may seem to be a simple question: what is the evolution of the densitymatrix ρ under this Hamiltonian? In fact, this is a difficult problem to solve because Iˆx,labdoes not commute with HˆZ,lab or HˆD, so only in simple systems are there exact solutions.In practice we must use approximations, each valid in a different regime. (Note that we areassuming HˆD is the secular dipolar Hamiltonian, which is the same in the lab and rotatingframes.)In order to determine the regime, we need a relative measure of the energies of each interac-tion. The Zeeman and rf terms have the energy scales ~ω0 and ~ω1 respectively. The dipolarinteraction energy is less straightforward. What we need for the dipolar Hamiltonian is ameasure for the typical field strengths at one spin due to its neighbouring spin. When thereis only once spin species, this is given by [31]ωD =√〈Hˆ2D〉=√13〈∆B2〉. (2.33)Here, 〈∆B2〉 is the second moment of the absorption lineshape, and we call ωD the local fieldstrength or RMS average dipolar interaction strength. It has units rad/s. In the literature,it is often called the local field BL, defined asBL = ωD/γ. (2.34)Note that ωD applies to one coupled network of spins, not to all spins of a specific species.With an energy scale at hand for all parts of the total Hamiltonian, we can now exploresome fundamental theories which apply in various regimes. The following is the programfor the remainder of Section 2.5. After describing the trivial case of pulsed rf, we willintroduce the work of Bloembergen, Purcell, and Pound (BPP), which uses perturbationtheory to model saturation under extremely small rf irradiation (we will make explicit what“extremely small” means later). However, BPP theory fails to explain much of the behaviourseen in solids. This was the motivating factor for development of the next topic, Redfieldtheory, which introduces the concept of spin temperature. Redfield theory applies understrong rf irradiation. Provotorov Theory, the last topic discussed, extends the concept ofspin temperature to explain the saturation of solids under weak rf irradiation. For a moredetailed picture of how these theories fit together, the reader is referred to the introductionof Janzen’s paper [32].What is meant by saturation? If the system starts in thermal equilibrium, then a saturationrf pulse slowly equalizes the populations of the states, reducing the magnetization, 〈Iˆz〉.27Depending on the specific regime, 〈Iˆz〉 → 0 (the sample is completely saturated) or 〈Iˆz〉 →const. Note that saturation also implies randomly distributed phases among the spins. A90◦ pulse may tip the magnetization into the transverse plane, causing 〈Iˆz〉 = 0, but thephases of the spins are coherent, so this is not saturation.On a historical note, many of the theories describing saturation were developed in the earlydays of NMR. At that time, continuous-wave NMR spectroscopy was used. In this technique,one applies a constant oscillating B1 field to the sample as the B0 field is increased, allowingfor measurements of the absorption and dispersion of the sample. Obviously, quantifyingthe effects (eg. saturation narrowing) on the spectrum from the continuous B1 field wasimportant. Nowadays, Fourier transform spectroscopy is used, where the B0 field is constantand the spectrum is determined from the FID after a series of intense B1 pulses. However,saturation theories are useful in MRI, where certain techniques we discuss in the next chapteruse long, low-power B1 pulses to generate contrast.2.5.2 Pulsed rfThe simplest regime: ω1  ωD, ω0  ω1, and B1 duration τ is much less than the lifetimeof the FID. Hence, we can safely ignore the effects of dipolar couplings during the pulse.Working in the rotating frame allows us to directly calculate solutions, as exemplified inSection 2.3.3. Saturation would occur when ω1 . ωD and τ & T2.2.5.3 BPP theoryBloembergen, Purcell, and Pound were the first to treat the rf pulse as a perturbation to themain Hamiltonian, consisting of the Zeeman and dipolar terms [33]. Their calculation usedtwo coupled spin-1/2 nuclei under a weak rf field. However, the BPP saturation theory is onlyuseful in the regime ω21T1T2  1. Higher rf powers will cause population equilibration, whichchanges the wavefunction of the system, invalidating the perturbation theory approach [34].While BPP saturation theory predicts the right behaviour in liquids, in solids it deviatesfrom observations at long times.2.5.4 Spin temperature and Redfield theoryRedfield theory is applicable to strong rf pulses, where ω21T1T2  1 [10, 12, 30]. It uses theconcept of spin temperature. This emerges because nuclear spins form a canonical ensemble,28describable by a Boltzmann distribution with a well-defined temperature. This is true undermost circumstances in solids, where there is weak coupling to the lattice and spin-spincouplings redistribute the populations of the energy levels in a time T2  T1.Let us first consider the system at equilibrium with no rf applied. The total Hamiltonianconsists of the Zeeman and dipolar parts,Hˆ = HˆZ + HˆD.Then, following the derivation of Eq. 2.27, the density matrix isρ = −βHˆ.= −βHˆZ − βHˆD (2.35)where β is the inverse lattice temperature. The spin temperature in each reservoir is thesame, but the order (or magnetization/polarization) is not. To see this, we writeρ = −βω0[Iˆz]− βωD[HˆDωD].The operators in [· · · ] are written this way to make them both unitless. The order p in eachreservoir ispD =〈HˆDωD〉= −βωDpZ = 〈Iˆz〉 = −βω0.(2.36)Because ωD  ω0, pD  pZ . Since the energy of the system is bounded, the order may benegative or positive in general.The partition function isZ = Tr{exp(−βHˆ)}= Tr{1− βHˆ + 12β2Hˆ2 + · · ·}≈ (2I + 1)N + 12β2Tr{Hˆ2}.We have made use of the fact that Tr{Hˆ} = 0 since Tr{HˆZ} = Tr{HˆD} = 0. Tr{1} =(2I+1)N is the dimensionality of a system with N particles of spin I. We may now calculate29expectation values. The magnetization is [10,12]M =Tr{Mˆ exp(−βHˆ)}Z≈ Tr{Mˆ(1− βHˆ)}(2I + 1)N (2.37)= CβB, (2.38)where Mˆ = (Iˆxxˆ+ Iˆyyˆ+ Iˆz zˆ) and C is the Curie constant. This result is Curie’s law. It saysthat the local field has no effect on the magnetization, which is either parallel or anti-parallelto B. In the same way we can find the entropy of the system, s, a quantity we will makeuse of later. This is [10,12]s = sZ + sD (2.39)= −12β2Tr{Hˆ2Z}− 12β2Tr{Hˆ2D}+ const (2.40)=(−12Cβ2B20)+(−12Cβ2B2L)+ const. (2.41)Now, imagine suddenly turning on an rf field ω  ωD, so Beff = (B0 − ω/γ)zˆ + (ω1/γ)xˆ(Eq. 2.5). According to Curie’s law (Eq. 2.38),Mmust eventually point along Beff. How longdoes this take? When the rf field is turned on, M precesses around Beff until its “transverse”components (the components perpendicular to Beff) dephase. This happens in a time ~T2(~1/ωD). Only the projection of M along Beff is retained.Say B1 is rapidly turned on with an amplitude and frequency such that Beff is 45◦ to B0(realized when ω = ω0 − ω1). If the equilibrium magnetization in the lab frame was M0 =M0zˆ, then in a time ~T2 after the rf is turned on it will be parallel to Beff and have amagnitude of M0/√2 . The tilted rotating frame (henceforth the TR frame) is a rotatingframe where zˆ is along Beff. The rapid application of B1 means that once M0||Beff, thesystem is in quasi-equilibrium: both the Zeeman and dipolar reservoirs in this tilted-rotatingframe are describable by different inverse spin temperatures β′Z and β′D, but neither is inequilibrium with the lattice.The new Zeeman inverse spin temperature β′Z in the tilted frame may be found from the30Curie law in Eq. 2.38:β′ZβZ= B0Beff√2M0M0=√2 B0Beff 1. (2.42)Thus, β′Z  βZ and the Zeeman reservoir has a significantly colder spin temperature in theTR frame than in the lab frame. We shall not attempt a similar calculation for the dipolarorder—there is no analogue of the simple Curie law and the dipolar Hamiltonian in the tiltedframe, Hˆ ′D, is complicated.We may also calculate the new entropy:s′ = s′Z + s′D=(−12CB2effβ′2Z)+(−12CB′2Lβ′2D)+ const, (2.43)where primes indicate the TR frame.In the above discussion, the rapid application of the B1 field with ω1  ωD led to a quasi-equilibrium state after time ~T2. In the TR frame, the Zeeman reservoir has energy spacingsof γBeff. In the dipolar reservoir, the energy spacings are γBL. When ω1  ωD, γBeff  γBLand “heat” (magnetization) cannot flow between the two reservoirs. Such transitions wouldbe energy non-conserving. Yet when ω1 ∼ ωD these transitions can take place: the reservoirsare coupled together, and their spin temperatures equilibrate on timescales of ~1/ωD.As for equilibrium with the lattice, in the TR frame this has timescales of ∼ T1ρ for theZeeman reservoir (the spin-lattice relaxation time in the rotating frame, ~0.1–1 s) and T1Dfor the dipolar reservoir (the dipolar relaxation time, ~0.1–10 ms for the samples consideredin Chapter 6). In the following discussion, we assume the rf duration is much shorter thanT1ρ or T1D, so we may ignore their effects while the rf is on.2.5.5 ADRF/ARRF: An application of Redfield theoryWe will now give an example of an experiment which can be quantified using Redfieldtheory. This is called Adiabatic Demagnetization/Remagnetization in the Rotating Frame(ADRF/ARRF) [10, 30]. We will use this experiment in Chapter 6 to measure the dipolarrelaxation time, T1D.From an equilibrium state with a spin temperature of β, a 90◦y on-resonance pulse rotates31the magnetization: M = M0xˆ. Immediately after (taken to be t = 0), the rf is phase-shifted to the rotating frame’s x-axis. At this point, the Zeeman reservoir is already inquasi-equilibrium since M0||Beff. An analysis like the one leading to Eq. 2.42 shows that theZeeman reservoir’s inverse spin temperature at t = 0 is βZ = B0B1β.The amplitude of the rf pulse, which starts off at ω1  ωD, is now ramped down adiabaticallyto zero, leaving a final inverse spin temperature β’. The entropy is constant, so we can usethe expression in Eq. 2.43. Dropping the primes for convenience, we equate the entropy atthe start and end of the ADRF ramp (state 1 and 2 respectively):s1 = s2sZ,1 + sD,1 = sZ,2 + sD,2(−12CB21βZ,12)+(−12CB2Lβ2D,1)=(−12CB2eff,2β2Z,2)+(−12CB2Lβ2D,2)B21(B0B1β)2≈ B2L,2β2D,2ω20β2 = ω2Dβ2D,2→ βD,2 = β ω0ωD.where we have used the fact that ω1  ωD and that Beff,2 = 0 in the third line. At thispoint, Beff = 0 henceM = 0, meaning that there is no magnetization in the Zeeman reservoir.Instead, it has been moved to the dipolar reservoir, which now has an inverse temperatureof β ω0ωDand a polarization (Eq. 2.36) of −βω0. Note that because we are on resonance, ωDis the same as in the lab frame. During demagnetization, the Zeeman and dipolar reservoirsremain uncoupled until ω1 ∼ ωD.Since we reached this using an adiabatic process, reversing it will transfer observable magne-tization back to the Zeeman reservoir. This is the ARRF part of the sequence. While in thedipolar reservoir, the dipolar order decays with spin-lattice relaxation time T1D. Therefore,the ADRF/ARRF sequence can be used to measure T1D.2.5.6 Provotorov theoryProvotorov Theory deals with the case where ω1 is weak so that we can’t assume the Zeemanand dipolar reservoirs have the same spin temperature [10, 30, 34, 35]. Experimentally, thisis usually the case in solids when ω1  ωD. It was first introduced by Provotorov [36] butthe canonical reference is Goldman’s book [30].In Appendix A we derive the Provotorov equations. Here, we simply state the results and32put them into a useful form for later. We will use the form of the equations introduced byLee et al. [37, 38]. The density matrix in a rotating frame isρ = −(ω0 − ω)βZ Iˆz − βDHˆD= −2pi∆βZHˆZ − βDHˆD= −2pi∆βZ Iˆz − ωDβD(HˆDωD). (2.44)In the above, 2pi∆ = ω0 − ω, where ω is the frequency of the rf (not yet applied). ∆ is theoffset from the center of the spectrum and is stated in Hz. We can now express ρ as a vectorwith{Iz, HˆD/ωD}as the basisρ = −(2pi∆)βZ−ωDβD . (2.45)The components of ρ are the magnetizations or orders in each reservoir〈Iz〉 = (ρ)1 = −(2pi∆)βZ〈HˆDωD〉= (ρ)2 = −ωDβD.(2.46)When weak rf is applied with amplitude ω1  ωD, the Provotorov equations aredρ±dt=W −1− 1WT1 ΩΩ −Ω2 − 1WT1Dρ± + 〈Iz〉0T10 , (2.47)withW = piω21g(∆) (2.48)Ω = 2pi∆ωD. (2.49)Here, g(2pi∆) the symmetric, normalized lineshape (in units of s). In Appendix A we alsoshow how rf applied at offsets ±∆ simultaneously decouples the Zeeman and dipolar reser-voirs, leading to [30]dρdualdt= W −1− 1WT1 00 −Ω2 − 1WT1Dρdual + 〈Iz〉0T10 . (2.50)33Eq. 2.47 and 2.50 are at the heart of ihMT, discussed in Chapter 6.2.6 Relaxation in homogeneous systems2.6.1 What drives relaxation?Environmental fluctuations in magnetic fields felt by the nuclei are responsible for spin-spin(T2) and spin-lattice (T1, T1D) relaxation. Generally, molecular motion causes these fluctua-tions. Consider a proton on a tumbling molecule in a B0 field. Depending on the molecularorientation, there may be different electron screening around the nucleus. Also, there willinevitably be fluctuating fields from the magnetic moments of other protons and nuclei.Time-independent couplings also cause precessing nuclei to dephase, so these contribute toT2 as well. Finally, the fluctuations of paramagnetic centers also play a role.In MRI, it isn’t typically necessary to perform exhaustive calculations of the quantum ori-gins of relaxation. In fact, given the complexity of most biochemical environments, likelipid bilayers, this would be impossible without molecular dynamics simulations (e.g. refer-ence [39]). Instead, it is usually adequate to either measure the relaxation rate, predict itsmagnitude from the fundamental physics, or predict its value based on knowledge of similarsystems. For example, we can measure the T1 and T2 times of aqueous protons in white andgrey matter. We can predict that the non-aqueous protons will have a T2 ∼10–100 µs dueto their slow tumbling (see next section). This short T2 time can be confirmed experimen-tally using NMR spectroscopy and computationally using molecular dynamics simulations.Similarly, in Chapter 7 there is an extensive discussion on predicting the relative T1D timesin different types of white matter lipids.We shall now present some general results from BPP relaxation theory (the same theorythat describes saturation in the limit ω21T1T2  1). This provides a suitable framework forunderstanding the quantum origin of relaxation in tissue.2.6.2 BPP relaxation theoryHere we provide a flavour of BPP theory without going into the details. Relaxation is theredistribution of populations in the density matrix. So for an arbitrary matrix elementρnm = 〈n|ρ|m〉, how does this change in time? It can be shown (e.g. see Slichter [10]) thatddtρnm =1~2Jmn(ωn − ωk)34where Jmn(ω) is the is the spectral density and ωn−ωk is the energy difference between |m〉and |n〉. The spectral density is a measure of which frequency components are present inthe random fluctuations felt by a nucleus. Intuitively this makes sense: when the randomfluctuations are on resonance for a transition, the populations of the corresponding levels willchange. The spectral density is given by the Fourier transform of the correlation functionGmn(τ):Jmn(ω) = F {Gmn(τ)}andGmn(τ) = 〈m|Hˆr(t− τ)|n〉〈n|Hˆr(t)|m〉.Here, Hˆr is some Hamiltonian that is responsible for the fluctuations, such as the dipolarHamiltonian. The overbar indicates an ensemble average. We often make the assumptionGmn(τ) ∼ exp(−τ/τc),where τc is called the correlation time. This is very nearly exact for small molecules likewater. With this assumption, Jmn(ω) has a Lorentzian profile around ω = 0 with width ~τc.The above is the starting point for deriving expressions for the relaxation times. By consider-ing the case of two rapidly-tumbling dipolar-coupled spins (e.g. protons in a water molecule),it can be shown that [33,40,41]1T1= C(τc1 + (ωoτc)2+ 4τc1 + (2ω0τc)2)1T2= C(32τc +(5/2)τc1 + (ω0τc)2+ τc1 + (2ω0τc)2).(2.51)Where C is a constant and ω0 the Larmor frequency. Fig. 2.4 plots these expressions undervarious conditions, we see that they give the right qualitative behaviour. In solids, T2  T1and in liquids T1 ∼ T2. In tissues, the molecules are often restricted by compartment walls,so T2 < T1.The same sort of approach has also been applied to T1D for a dipolar-coupled proton pair [43].Using a more general relaxation theory (Redfield theory [10]), the value is found to be [43–45]1T1D= 278 γ4~J1(ωD). (2.52)35-12 -10 -8 -6 -4 log10(rotational correlation time [s]) 10-1-2-3-4-5log 10(T1, T2 [s]) Higher B0 Lower B0SmallmoleculesLarger molecules,polymers, proteinsSolidsT1T2increasing temperaturedecreases correlation timeFigure 2.4: T1 and T2 as functions of correlation time τc. Modified from reference [42] withpermission from Hans Reich. Figure is based on work in reference [33].180 90tAcquisitionFigure 2.5: The inversion-recovery sequence used to measure T1.Here, J1(ω) is the spectral density of the spherical tensor function F (1), andF (1) = sin θ cos θ exp(−iφ)r3,where r is the vector connecting the two nuclei. Eq. 2.52 says that slow fluctuations aroundthe frequency of the local field strength ωD drive T1D relaxation.2.7 Some experimental methods2.7.1 T1 measurement with inversion-recoveryT1 is commonly measured in NMR studies by using an inversion-recovery (IR) experiment.Its pulse sequence is shown in Fig. 2.5. In equilibrium the magnetization is M0zˆ. If a 180◦B1 rf pulse is applied, now the magnetization is −M0zˆ. According to Eq. 2.12, the system36will now return to equilibrium viaMz(t) = M0 (1− 2 exp (−t/T1)) .Mz(t) can be observed by a 90◦ pulse and the T1 time extracted. The IR experiment isconsidered the “gold standard” for T1 measurements. But in MRI scanners, it requiresprohibitively long scan times [46–48]. Instead, Look-Locker methods (where a 180◦ pulse isfollowed by a train of low flip-angle pulses) or variable flip angle methods (where a sequenceof images are acquired with varying flip angles) are used [48].2.7.2 T2 measurements: the spin echo and CPMG acquisitionFrom the Bloch equations, it seems that the envelope of the FID will decay with exp(−t/T2).We assumed this in our example above (Eq. 2.17). In reality, we often find that the FID foraqueous protons decays with exp(−t/T ∗2 ), where T ∗2 < T2. T ∗2 takes into account inhomo-geneities in the B0 field which are static on timescales of T2:1T ∗2= 1T2+ 1T2,inhomo.T2,inhomo is the contribution from inhomogeneities in the static field. These are causedby varying magnetic susceptibilities throughout the sample, limitations on shimming, andparamagnetic impurities. However, there is a simple method to reverse the effects of T2,inhomoand measure the true T2 time. To explain what this is, we first need to introduce the conceptof the spin echo.Fig. 2.6 gives a pictorial description of the spin echo, which is a sequence consisting of a 90◦pulse, then a delay τ , then another 180◦. Following the 90◦ pulse, the spins will precess atslightly different frequencies due to static field inhomogeneities. Correspondingly, they losephase coherence and the net magnetization decays with a time constant T ∗2 . However, afterthe 180◦ pulse, the spin with the lowest precession frequency now has the most advancedphase, and vice versa. Therefore, at a time 2τ , we are only left with the effects of T2relaxation.The Carr-Purcell Meiboom-Gill (CPMG) acquisition is a series of repeated spin echoes. Itspulse sequence is shown in Fig. 2.7. The amplitude of each echo, plotted as a function oftime, decays with time constant T2. The experiments in Chapters 5 and 7 use this acquisitionmethod.In MRI, B1 and B0 inhomogeneities mean that the 180◦ refocusing pulses are imperfect,37"slow" spin,ω0 - Δω "fast" spin,ω0 + Δωnetmagnetization along yspinphasesτ 180oy pulse τM0 M0 exp-2τT2M0 exp -τ T2 T2,inhomo1 1+Figure 2.6: The spin echo in the rotating frame. Because of static field inhomogeneities,each spin has a slightly different speed of precession. As they lose phase coherence, the netmagnetization is reduced. After evolving for a time τ , a 180◦ pulse flips the phases: now theslow spin is in front and the fast spin is behind. At 2τ , an echo is observed when the spinsrefocus. While the effects of T2,inhomo can be removed, T2 relaxation from field fluctuationsis unavoidable.τ90x 180ynacquireτ τFigure 2.7: The CPMG acquisition sequence. A few points around the echo are acquired oneach echo. The 90◦ phase shift between the 90x and 180y pulses prevents small errors fromimperfect 180◦ pulses from accumulating.38leaving some magnetization in the longitudinal direction after each pulse. Ultimately, thisproduces stimulated echoes, which arise after two or more pulses, and causes errors whenfitting the CPMG decay curve. The effect of these stimulated echoes can be corrected post-acquisition and is necessary when analyzing MRI CPMG curves [49]. In NMR, because thesample is significantly smaller and the pulse widths shorter, the B1 and B0 inhomogeneitiesare relatively insignificant and this correction is not usually necessary.2.7.3 Determining FID deadtimeAfter an intense B1 pulse, there is a period where the resonant circuit in the probe ringsdown. Following an observation pulse, this prevents immediate detection of the FID, forcingthe experimenter to wait before turning on the receiver. This delay, td, is known as thereceiver deadtime and is typically a few µs.The deadtime must be accounted for when different frequency components of the signaldephase or decay appreciably during the deadtime. For example, the FID of tissue containsa signal from non-aqueous protons that decays in 10–100 µs, whereas the signal from theaqueous protons can last up to about 1 s. If one desires to precisely model the non-aqueoussignal, a sizable portion of it may be missing due to the deadtime. It is important, therefore,to know the true t = 0 point, otherwise the modeling could over or under-estimate the non-aqueous signal’s amplitude. If only the aqueous protons are of interest, then the deadtime iscomparatively insignificant and the start of the acquired FID is taken to be the t = 0 point.With some equipment, the receiver clock is easily synchronized with the pulse sequence suchthat the true t = 0 time is known. However, signal propagation delays depend on filters inthe receiver chain, which may vary between experiment setups. For this reason, it is ofteneasier to measure td. For the work in this thesis, the following measurement technique wasused.Let f(t) be the envelope of an FID that has a resonance with a lineshape g at a frequencyω0 in a field B0. The spectrum S(ω) with zero deadtime isS(ω) = F {f(t) exp(−iω0t)}= exp(−iφ0)g(ω) ∗ δ(ω − ω0)= exp(−iφ0)g(ω0)Here, φ0 is the the zeroth-order phase correction chosen to make a pure absorption spectralline at ω0: S(ω) exp(iφ0) = g(ω0). Its value is determined by timing in the spectrometer39receiver chain and can be measured by visually phasing the spectrum.To find td, imagine two spectra are acquired on the same sample in slightly stronger andweaker B0 fields, B0 + ∆B and B0 −∆B, where ∆B is a small offset achieved through thez0 shim. Including the deadtime, their FIDs are f(t− td) exp(−iω0,±t), leading to spectraS(ω±) = exp(−iφ0) exp(−iωtd)g(ω) ∗ δ(ω − ω0,±)= exp(−i(ω0,±td + φ0))g(ω0,±)= exp(−i(θ± + φ0))g(ω0,±),where θ± = ω0,±td Now, θ± + φ0 is the phase correction term, where φ0 is approximatelyconstant over the small changes in frequency at hand. Finally, the difference (θ+ + φ0) −(θ− + φ0) = td(ω0,+ − ω0,−) leads totd =1360(θ+ − θ−)f0,+ − f0,− (2.53)where θ± is in degrees and f0,± = ω0,±/2pi is in Hz.In practice, the procedure is as follows:1. Set the spectrometer frequency to be at the center of the spectrum.2. Adjust the z0 shim to shift the spectrum about +10 kHz off of the spectrometer fre-quency and acquire. Phase the spectrum using the zero-order correction and recordthe value (θ+). Also record the frequency of the spectrum’s central line (f0,+).3. Adjust the z0 shim the other way to shift the spectrum about -10 kHz off of thespectrometer frequency and acquire. Record the corresponding values for θ− and f0,−.4. Use Eq. 2.53 to calculate td.Note that defining ∆θ = θ+ − θ− and ∆f = f0,− − f0,+, Eq. 2.53 implies ∆θ ∝ ∆f , whichis a first-order phase correction (i.e. the phase correction is a linear function of frequency).So finding td is the same as finding the first-order phase correction. In a spectrum withmultiple, well-defined lines over a reasonably broad frequency range, this correction can bedone by eye from a single spectrum.40Chapter 3White matter, grey matter, andmyelin3.1 IntroductionImproving the sensitivity and specificity of MRI to myelin is the ultimate goal of this thesis.This chapter’s aim is to briefly explore the biology and physics of myelin. We first introducethe nervous system and the role played by myelin in neuron signal propagation. Myelin’sunique structure, which is the source of its MR properties, is discussed in detail. We alsodiscuss Multiple Sclerosis (MS), a disease in which MRI plays a central role in diagnosis andstudy.3.2 The nervous systemThe nervous system in vertebrates is separated into the Peripheral Nervous System (PNS)and the Central Nervous System (CNS). Neurons in the PNS relay sensory information fromexternal and internal sources. They also relay signals to control muscles and other organs.The CNS receives and processes information from the PNS and coordinates responses.The CNS, comprising the spinal cord and the brain, receives a great deal of study withMRI. However, the function of the CNS, from a cognitive to a genetic level, does not muchconcern us here. Instead, our focus is on the CNS microstructure, since this determines theproperties of the NMR signals within an MR image.The CNS tissue is separated into grey matter and white matter. White matter tissue containsmyelinated axons, glial cells, and capillaries. Its pale white appearance is from the high lipid41nucleussomadendritemyelin internodeNode of Ranvier axon terminalaxonFigure 3.1: A cartoon of a neuron. Image modified from the original created by “QuasarJarosz” on English Wikipedia [50] with permission under the CC-BY-SA.content in the myelin sheaths. White matter connects different parts of the grey mattertissue, which is composed of neuronal dendrites and cell bodies, glial cells, and capillaries.Its relatively low myelin content is responsible for its darker appearance. In the brain, greymatter is mostly found on the surface whereas white matter is mostly found underneath inthe bulk tissue. In the spinal cord, the opposite is true.On a microscopic level, the vertebrate nervous system is composed of neurons and glial (sup-port) cells. Fig. 3.1 shows an idealized neuron, the fundamental unit of the nervous system.Signals, called action potentials, are received on dendrites. Outgoing signals propagate downthe axon. At the axon terminals, neurotransmitters are released into the synapse, reachingthe next neuron’s dendrite.3.3 Myelin3.3.1 Myelin structureMyelin forms a multi-layered sheath around nerve cell axons (also called compact myelin).The myelinated regions are called internodes and alternate periodically with short, unmyeli-nated regions called the Nodes of Ranvier. These nodes are typically about 1–2 µm long andare spaced at intervals about 100× the axon diameter [52,53]. Myelin is a plasma membraneextension of a specialized glial cell. In the PNS, Schwann cells form the myelin with one cellper sheath. In the CNS, the myelin sheaths are extensions of oligodendrocyte cells, and oneoligodendrocyte can form up to about 30 nodes [53]. At the end of the internode, each layerof the cytoplasmic membrane is attached to an invagination in the axon called the perinodalloop. While these nodes occur at regular intervals, the relative total length of unmyelinatedaxon is quite short at <0.5% of the surface length [54].42Figure 3.2: The physical and chemical composition of myelin. The proteins are myelinbasic protein (MBP), proteolipid protein (PLP), cyclic nucleotide phosphodiesterase (CNP),and myelin-associated glycoprotein (MAG). Note that the extracellular water and cytoplasmwater indicated on this diagram together form the myelin water pool which will covered indetail in Section 4.5. The intra/extra-cellular water we also introduce in that section is foundareas outside the myelin sheath, such as the axon shown here. Reprinted by permission fromSpringer Neurotherapeutics, Laule et al. [51], © 2007.43Figure 3.3: Electron micrographs of human brain white matter with increasing levels ofmagnification in panels A–C. In (C) the multi-layered structured of the myelin sheath isobvious, showing the alternating major dense and intraperiod lines. Between the myelin’sbilayers a pool of water called the Myelin Water (MW) is trapped. Figure modified fromLiu & Schumann [59] with permission under CC-BY-4.0.Fig. 3.2 shows a diagram of the myelin sheath’s physical and chemical composition. Ateach internode, the myelin sheath wraps around the axon like a toilet paper roll, forminga system of alternating lipid bilayers, where the bilayers are the plasma membrane of theoligodendrocyte. Cytoplasmic fluid is contained between apposed internal surfaces of themembrane, and extra-cellular fluid is contained between apposed external surfaces. Theseare called the major dense lines and intra-period lines respectively, due to their alternatingappearance on electron micrographs (Fig. 3.3). Both the extra-cellular fluid in the intra-period line and the cytoplasmic fluid in the major dense line share a unique NMR relaxationproperty: their T2 time (~10 ms) is measurably shorter than the T2 time (~50 ms) of waterelsewhere in the tissue [55–58]. Because of this collective behaviour, fluids in both theintraperiod lines and the major dense lines are known as Myelin Water (MW) [51].The composition of myelin (Table 3.1) is what enables its remarkable structure, where themembrane surfaces in compact myelin are “zippered” together [60]. Its lipid content isunusually high, comprising around 70% of the dry weight [53, 61, 62], with the remainingweight from proteins. This is in contrast to typical biomembranes, which have much higherprotein to lipid ratios, typically somewhere between 1:1 and 4:1 [61]. Also, a high proportionof myelin lipids are saturated (94%) and/or have very long hydrocarbon chains (20% havemore than 18 carbons). This is significant compared to the grey matter average, where only80% of lipids are saturated and just 1% have chains longer than 18 carbons [61]. The reducedmembrane fluidity from the tight packing of the saturated chains is offset by myelin’s high(~30%) cholesterol content, which increases fluidity [63]. The structure of the major lipid44Myelin White matter Grey matterProteins 30% 39% 55%......myelin basic protein (MBP) .....30% - -......proteolipid protein (PLP) .....50% - -......cyclic nucleotide phosphodiesterase (CNP) .....4% - -......myelin-associated glycoprotein (MAG) .....1% - -Lipids 70% 55% 33%......cholesterol .....28% .....28% .....22%......phospholipids .....43% .....46% .....70%......glycosphingolipids .....28% .....26% .....7%Table 3.1: The composition of human myelin, white matter, and grey matter. Bold numbersare percent weight in wet tissue, all others are percent dry weight of total lipid or total proteincontent. Myelin protein values are from Laule et al. [51], white and grey matter values arefrom Norton & Cammer [62].fatty acidfatty acidphosphateRglycerolsphingosinefatty acidgalactoseglycosphingolipidphospholipidFigure 3.4: The structure of the major lipid types in myelin. After van der Knaap [64].types seen in myelin is shown in Fig. 3.4.Myelin’s major proteins are myelin basic protein (MBP) and proteolipid protein (PLP). MBPhelps to stabilize the membrane structure by neutralizing the charge on the phospholipid headgroup. PLP is often referred to as a “spacer” and has domains in both the intraperiod andmajor dense lines. It maintains a constant spacing between the plasma membranes [64].Other less-abundant proteins include cyclic nucleotide phosphodiesterase (CNP, an enzyme)and myelin-associated glycoprotein (MAG, which plays a role in cellular recognition andintra-cellular interactions) [51,64].The values for myelin composition in Table 3.1 are averages across a number of individualsand structures and variation from these values is expected. Variation is also seen on themicroscopic level. For example, the corpus callosum (which enables communication betweenthe cerebral hemispheres) has some fibre tracts with myelin sheaths on only 30% of the axons,despite being highly myelinated in general. [65]. There may also be significant differences inmyelination between individuals, especially those of different age. Children are born withrelatively few CNS structures fully myelinated and myelination isn’t completed until earlyadulthood [64].453.3.2 Myelin functionNeurons have resting potentials around -70 mV. The net negative charge inside the cytosol isestablished by K+ and Na+ ion pumps. At rest, the Na+ concentration is higher outside butK+ is higher inside. Signal transmission is a temporal and spatial change in this membranepotential. It happens in two main ways: by graded potentials and by action potentials. Weshall describe graded potentials first, then action potentials, and then finally bring the twoconcepts together by discussing the role of myelin.Graded potentials are changes in membrane potential which are variable in size, are additive,and decay spatially from a source. These can be observed in unmyelinated axons withvoltage clamp experiments and naturally occur at the postsynaptic dendrites in response toneurotransmission. There, neurotransmitters activate ion channels (distinct from the Na+and K+ channels already mentioned), causing membrane depolarization. The depolarizationis localized at the postsynaptic dendrite and causes a decaying membrane potential—thegraded potential—away from the synapse. This potential spreads via the attraction andrepulsions of ions inside the cytosol [66]. The length constant of this decay is a function ofthe axial resistance and the membrane resistance. In a cylindrical axon the axial resistanceis [67, 68]ra =ρcpia2,where ρc is the resistivity of the cytosol and a the axon radius, so ra has units of Ω/m.The membrane resistance, rm, is a function of the specific resistance of an area of plasmamembrane, Rm:rm =Rm2pia.Rm has units of Ωm2, so rm has units of Ω m. Graded potentials spread away from thesource, decaying exponentially in strength with a length constant λ given by [67,68]λ =√rmra=√Rmρca2 . (3.1)Hence, increasing the membrane resistance and the axon diameter increases the length con-stant, allowing graded potentials to spread over longer distances. The duration of the graded46potential is given by the time constant τ via [67,68]τ = rmcmwhere cm is the membrane capacitance in F/m. An estimation of the velocity of the gradedpotential down the nerve cell is then [67]v ≈ λτ= 1√rmracm∝√√√√ a3ρRmc2m=√aρRmC2m(3.2)where Cm is the capacitance per unit area of the membrane. The constants which weredropped are given explicitly in Tasaki’s work [69].Action potentials are the second type of signals which are transmitted in neurons, where theyoccur in the axons. In contrast to graded potentials, action potentials are capable of propa-gating over long distances without decaying in strength. They are caused by voltage-gatedNa+ and K+ ion channels in the axon membrane. When an action potential approaches,voltage-gated Na+ ion channels open, allowing external Na+ ions to diffuse down their con-centration gradient into the axon. This causes a positive voltage across the membrane,eventually triggering the K+ ion channels. The K+ ions then start to move out of the cy-tosol across the membrane, causing a negative membrane potential. Finally, there is a smalldelay when the section of axon is unable to transmit any signals. During this refractoryperiod, K+ and Na+ ion pumps re-establish the resting potential. (The number of ions thatmove across the membrane is actually quite small; for K+, this is less than <0.03% of thetotal number of ions within the axon [60].)Both graded potentials and action potentials can propagate in unmyelinated axons. Ofthe two, graded potentials have a much faster signal velocity, relying on short-range chargereorganization within the cytosol instead of diffusion through ion channels, as for actionpotentials. However, action potentials are able to propagate over long distances withoutdiminishing in strength. Nature’s ingenious solution to these problems is the myelin sheath,which allows the best of both worlds.In a myelinated axon, action potentials occur only at the nodes of Ranvier. These cause a47graded potential, which then spreads underneath the internode at greater speed and over alonger distance compared to what it could in an unmyelinated axon. To see why the speed isgreater, consider that myelin increases the membrane resistance, Rm. Referring to Eq. 3.1,this increases the length constant λ. Myelin also increases the membrane capacitance Cm.With a myelin thickness of b, Rm ∝ b and Cm ∝ b−1, if the approximation of a parallel platecapacitor is used. Substituting these into Eq. 3.2 gives [67]v ∝√ab.The myelin thickness and axon diameter are key parameters in determining the conductionvelocity.When the graded potential reaches the next node of Ranvier, it triggers an action potential,and the process repeats. Because of the refractory period, the graded potential cannot causean action potential at the previous node. Thus, the signal appears to “hop” from node tonode, hence its name: saltatory conduction, from the latin word for hopping, saltare. Myelinincreases the speed of action potential transmission 10–100× compared to an unmyelinatedaxon and also requires less energy, since ion pumping is required only at the nodes. [51].Also, saltatory conduction ensures signals can travel in one direction and do not diminish instrength.3.3.3 Multiple SclerosisChanges in myelin have been associated with many cognitive disorders, including Alzheimers[70] and schizophrenia [71]. Even mild trauma can affect CNS myelin [72]. There are alsodiseases which directly damage the myelin, known as demyelinating diseases. Of these,Multiple Sclerosis (MS) is perhaps the most well-known. MS is of particular concern inCanada, which has one of the highest rates of in the world [73]. The cause of the diseaseis unknown, and while it is likely autoimmune in nature, vitamin D deficiency (prevalenceis higher is northern countries), viral (e.g. Epstein-Barr), and genetic factors have all beenimplicated [74–76]. Onset of MS is typically in early adulthood and ultimately leads todecreased motor and cognitive abilities, although progression is generally slow—on average,patients live 30 years after diagnosis [76]. There is no cure, but symptomatic treatmentsexist.In MS, localized areas of demyelination occur, which can be followed by axonal loss or remyeli-nation (in earlier stages of the disease). These significant changes to the tissue microstructure48are often (although not always) visible using MRI. For this reason, MS diagnostic criteriarelies upon localization of MRI-visible lesions in time and space [77].49Chapter 4Relaxation and spectra in brain:properties and applications4.1 IntroductionMR imaging of white and grey matter tissue makes use of their relaxation and spectral prop-erties, which is the concern of this chapter. We first emphasize the theory behind the NMRsignal from non-aqueous protons, which is measured directly in Chapter 5’s experiments.At the heart of this is the super-Lorentzian spectral lineshape. Then, the section on T1relaxation highlights its inconsistent experimental results. We offer an explanation for theseinconsistencies later in our work in Chapter 5. A section on the physics and uses of magne-tization transfer follows which includes a detailed overview of inhomogeneous magnetizationtransfer (ihMT). This will be useful for Chapters 6 and 7, where we explore the fundamentalphysics of ihMT. Finally, we end this chapter with an overview of T2 relaxation, which canbe used to separate signals from water inside and outside the myelin sheath.4.2 Spectral properties of white and grey matterThe molecules in tissues can be broadly separated into two types: non-aqueous and aqueous.The non-aqueous molecules are restricted in their motion in some way, whether by their size(such as large proteins) or by their environment (such as molecules in lipid membranes).11We will use the terms “aqueous” and “non-aqueous” throughout this work. However, the nomenclaturein the literature is inconsistent. Some of the synonyms encountered are:aqueous protons = free protons, water protons, unbound protonsnon-aqueous protons = macromolecular protons, tissue-associated protons, bound protons, semi-solid protons50B0 θωD (θ)∝|(3cos2θ−1)/2|(A) (B)Figure 4.1: Dipolar couplings in lipid molecules and lipid bilayers. (A) A generic lipid acylchain. There is strong 1H intra-methylene coupling. Weaker inter-methylene couplings leadto spectral broadening. (B) In a lipid bilayer, rapid spinning averages away inter-moleculardipolar couplings. The residual strength of the dipolar couplings is now a function of bilayerorientation.The aqueous protons are mostly on water molecules, which have very similar 1H spectralproperties to unconfined water. Tissue water in a homogeneous B0 field also has an FID thatdecays exponentially, leading to a Lorentzian lineshape with a precisely defined T2 (Eq. 2.18).There is significantly more to say regarding the 1H spectral properties of non-aqueousmolecules. Their motion is restricted, so the dipolar interaction is not averaged to zeroas it is for aqueous molecules. Lipid-rich tissues like white matter have a super-Lorentzian2non-aqueous lineshape [79–82]. Fundamentally, this is caused by strong intra-methylenedipolar couplings in the lipid acyl chains (Fig. 4.1A). At physiological temperature, lipid bi-layers in brain are in the fluid lamellar (Lα) phase [83,84], which has three kinds of motionsthat average the proton-proton dipolar couplings in the acyl chains. First, lateral diffusionaverages away the effects of intra-molecular dipolar couplings on the spectrum (althoughthese can still produce relaxation) [85, 86]. Second, rapid spinning of the lipid moleculesabout their long axes creates a P2(cos θ) dependence on the average strength of the dipolarcouplings [87], where P2 is the second Legendre polynomial and θ is the angle between B0and the bilayer normal, as shown in Fig. 4.1B. Third, the lipid tails fluctuate via trans-gauche isomerisation, with motion increasing towards the tail ends in the middle of thebilayer [87,88].Collectively, these motions average the inter-methylene proton couplings more than the intra-2The Super-lorentzian is named because its ratio of the width of the line at half max to the width at theinflection points is greater than a Lorentzian lineshape [78]. It is not actually a special case of a Lorentzianline.51Figure 4.2: Dipolar couplings in acyl chains causes the super-Lorentzian lineshape. (A) The1H NMR spectrum of lamellar-phase potassium palmitate - (β–ω) - d29 in D2O. In thesedeteurated molecules, only the α-methylene group retains its protons. The top trace is thesignal from all protons, which is a sum of the spectrum just from the α-CH2 methylenes(middle trace) plus contamination from the protons in the headgroup and in water (bottomtrace). The middle trace is the angular average of two Gaussian-broadened doublets witha splitting determined by the dipolar coupling ∼ 12(3 cos2 θ − 1). This is also called aPake pattern. Because the acyl chain is deuterated, inter-methlyene dipolar couplings areminimized. (B) A super-Lorentzian lineshape with σ0=20 kHz and σmin=40 Hz and examplecontributions from specific angles. One would expect undeteurated potassium palmitate tohave such a lineshape since it has significant inter-methylene coupling. Hence, its spectrafrom bilayers at specific orientations would be well described as single Gaussians rather thanthe doublets seen in the deuterated case in panel A. Panel A is modified slightly to improvereadability from ref. [85] [Chemistry and Physics of Lipids, Volume 20, Higgs & MacKay,Determination of the complete order parameter tensor for a lipid methylene group from 1H-and 2H-NMR spin labels, 105–114], © 1977, with permission from Elsevier.52methylene couplings. (Methyl groups can also produce super-Lorentzians [89], but theircontribution is small compared to the large number of methylenes.) Consequently, the lipidtail spin system could well be called a system of strongly-coupled spin-12 pairs weakly coupledtogether [88,90]. Such systems nominally have spectra of superimposed Gaussian-broadeneddoublets [87, 91]. Indeed, in a sample of lamellar-phase potassium palmitate - (β–ω) - d29in D2O where the protons have been retained only in the α-methylene group, the spectrum(Fig. 4.2A, from ref. [85]) shows this doublet behaviour clearly. In this sample, the inter-methylene interaction responsible for more significant Gaussian broadening is not present.In naturally-ocurring lipids in the lamellar phase, however, the 1H spectra are substantiallybroadened by these inter-methylene interactions. Therefore, the most common approachto modeling the super-Lorentzian is as a spectrum of superimposed Gaussians with widthsmodulated by P2(cos θ) [78,79,81,87,91,92]. The Gaussian standard deviation for a bilayerwhose normal makes an angle θ to B0 isσ(θ, σ0, σmin) =√14(3 cos2 θ − 1)2σ20 + σ2min. (4.1)Here, 3σ0/2 is the maximum linewidth at θ = 0◦. Bilayer fluctuations and field inhomo-geneities are responsible for the minimum linewidth σmin of molecules oriented at the magicangle θMA ≈ 54.7◦ [81]. When all bilayer orientations are possible, integrating over them(powder averaging) gives the super-Lorentzian FID and spectral lineshape: [79–81,89,93,94]S(t;σ0, σmin) =∫ pi/20dθ sin θ exp(−12σ(θ, σ0, σmin)2t2)(4.2)s(ω;σ0, σmin) =∫ pi/20dθ sin θg [ω, σ(θ, σ0, σmin)]=∫ pi/20dθ sin θσ(θ, σ0, σmin)exp(− ω22σ(θ, σ0, σmin)2). (4.3)Here, g is the Gaussian lineshape at one orientation. An example of the super-Lorentzianalong with g from different orientations is displayed Fig. 4.2B. Note that because of the nervefibre tracts in white matter, its in vivo lineshape in certain regions may more accuratelybe described as a partially-averaged super-Lorentzian [81]. As we will discuss in the nextchapter, the ex vivo samples used in this thesis do not retain fibre tract orientation andisotropic powder averaging (i.e. a super-Lorentzian lineshape as described by Eq. 4.3) isseen.The super-Lorentzian is typically discussed in the context of lipids, but it also arises from53other macromolecules (like proteins) which undergo similar averaging. As an example, super-Lorentzian lineshapes have been observed from suspensions of whole cells [80], and havesuccessfully modeled quantitative Magnetization Transfer (qMT, Section 4.4.2) in muscle [82]and hydrated durum wheat and gluten [95]. In light of these varied applications, it seemsthat a super-Lorentzian may arise in systems where there are strongly-coupled proton groups(e.g.methylene or methyl groups) undergoing rotational thermal averaging, creating angular-dependent dipolar couplings.We will return to the super-Lorentzian in Chapter 5 when it is used to fit FID data fromwhite matter.4.3 T1 relaxation4.3.1 A common, simple modelT1 relaxation is a key source of contrast in brain and spinal cord imaging, but a comprehensiveunderstanding of its physics has been elusive. One reason for this is simplicity—the goalof clinical MRI is to obtain adequate contrast in the shortest amount of time possible.Complicated models that involve many parameters are less useful in a clinical context thansimple empirical or semi-empirical ones. Indeed, the assumption of a single T1 in aqueousprotons is suitable for understanding and developing many forms of MRI contrast.The simplest approach to T1 in tissue is the solvation layer model [96, 97]. Briefly, this as-sumes three distinct populations of protons in tissue: free water, a solvation layer surroundingnon-aqueous molecules, and non-aqueous protons. The assumption of fast exchange leads tothe well-known empirical relation,1T1∝ 1WC + const, (4.4)where T1 is the single value measured for aqueous protons, and WC is the water content (theweight fraction of water) in the tissue. This model suffices in many cases.4.3.2 The controversy of quantitative T1 measurementsFor quantitative imaging, the specifics of T1 relaxation become important. And in brain andspinal cord, the details of the observed aqueous T1 relaxation unfortunately remain unclear.54For example, when a single T1 component is assumed, the reported values in white mattertissue at 3 T vary from 690–1735 ms depending on the site and technique used [48,98].Many recent MRI studies have measured both a short (~100 ms) and a long (~1 s) T1component [99–103]. These have been associated with two separate aqueous pools andsome have suggested that myelin water is responsible for the faster T1 component [101,103].However, still more studies suggest that axonal water (water inside myelinated axons) mayhave its own unique intrinsic T1 time and should be accounted for explicitly [104–106].Finally, confounding effects from T1’s sensitivity to iron content may also play a significantrole (see MacKay et al. [3] and references therein).Our work in Chapter 5 is an attempt to show how some of these disagreements arise andin what situations simple models are suitable. We emphasize there that cleanly associatingany one T1 component with a specific aqueous compartment is generally not possible. Thisnaturally arises from multiple-compartment models, such as the two pool model explainedbelow and the four pool model used in Chapter 5.4.4 Magnetization transfer4.4.1 The magnetization transfer ratioIn tissue, magnetization can exchange between the aqueous and non-aqueous protons. Un-der physiological conditions, this happens primarily via proton exchange between water andthe hydroxyl protons in lipid headgroups on macroscopic timescales of 0.1–1 s [83,107,108].Microscopically, the residence time of protons in a macromolecular group is often as lowas 10−11 s [107]. If the magnetization in the non-aqueous pool is reduced, magnetizationexchange causes a subsequent reduction in the aqueous pool’s magnetization. We can re-alize such a situation by making use of the 1H spectral properties of tissue, exemplified inFig. 4.3A. Magnetization Transfer (MT) is depicted in panel B. Low-amplitude rf irradia-tion (a prepulse) is applied far off resonance from the water peak prior to acquisition. Thissaturates the non-aqueous protons, reducing their magnetization. Magnetization exchangethen leads to a net magnetization reduction in both the non-aqueous and aqueous pools.To easily see the effects of MT in MRI, we need to compare our image with (S) and without(S0) the weak, off-resonance MT pulse. The Magnetization Transfer Ratio (MTR) is [109]MTR = S0 − SS0.55FrequencyIntensity macromolecular protons~10 kHzwater protons~10 Hz tissue spectrum+ =ΔRFsaturation(A) (B)magnetizationexchangeFigure 4.3: How magnetization transfer works. (A) A cartoon of the 1H NMR spectrum intissue, consisting of a broad line from the non-aqueous protons and a narrow line from theaqueous protons. (B) Magnetization transfer. Weak rf irradiation at an offset ∆ from thewater peak reduces the non-aqueous magnetization. Because of magnetization exchange, theaqueous magnetization decreases as well.Because MT originates in the non-aqueous pool, MTR can reveal changes in tissue structure.For example, in CNS tissue, MTR is lower in MS lesions and in areas with inflammation [110].Also, MTR is sensitive to changes that may be invisible to other techniques. For example,non-lesion white matter in MS patients (the normal-appearing white matter) was observedto have a measurably different MTR than white matter in healthy controls [110,111].4.4.2 qMT and the two pool modelMT is often modeled using a two pool model, also called the binary spin bath model (Fig. 4.4)[35, 112, 113]. Under rf irradiation ωrf at an offset ∆ from resonance, the coupled Blochequations are [113,114]:dMx,1(t)dt= −Mx,1(t)T2,1− 2pi∆My,1(t)− ω1 sin(2pi∆t)Mz,1dMy,1(t)dt= −My,1(t)T2,1+ 2pi∆Mx,1(t) + ω1 cos(2pi∆t)Mz,1dMz,1(t)dt= M1(∞)−Mz,1(t)T1,1− k12Mz,1(t) + k21Mz,2(t)+ωrf sin(2pi∆t)Mx,1(t)− ω1 cos(2pi∆t)My,1dMz,2(t)dt= M2(∞)−Mz,2(t)T1,2− k21Mz,2(t) + k12Mz,1(t)−WMz,2.(4.5)Here, kij is the exchange rate for magnetization from pool i to pool j and the M(∞) termsare the pool sizes. Because of the short T2 in the non-aqueous pool, it may be assumed thatits transverse magnetization is zero at all times. The saturation rate W is the same as for56T1,2T1,1Non-aqueous poolAqueous poolk12T2,2~10-100 μsT2,1~10-100 msM2(∞)M1(∞)LATTICEk21g(Δ)Figure 4.4: The two pool model of tissue.Provotorov theory (introduced in Section 2.5.6),W = piω21g(2pi∆),where g(2pi∆) is the lineshape for the non-aqueous pool. Note that these are modifiedBloch equations, because g(2pi∆) is usually assumed to be Gaussian or super-Lorentzian,not Lorentzian. This is not rigorous, but works well enough in practice [35,79,112–114].Using these equations, “quantitative MT” (qMT) imaging can be performed by fitting dif-ferent MT experiments to the model. Typically, one fixes T1,2 = 1 s, assumes a functionalform for g(2pi∆), and measures T2,1 and T1,1 in separate experiments. Then, the remainingparameters are fit to repeated MT experiments at different prepulse offsets.Because of the need for multiple values of ∆, qMT MRI takes a relatively long time toacquire in practice, about 20–30 minutes [115]. Moreover, with only one aqueous pool,qMT is sensitive to any changes in the tissue microstructure, not only those associated withmyelin [109,115,116]. Hence, while qMT can detect changes in the myelin, it is not specificto them [117].4.4.3 Inhomogeneous magnetization transferRecently, a modification of MT, called inhomogeneous Magnetization Transfer (ihMT), hasbeen developed which appears to be selective for lipid bilayers, such as those found in myelin[84, 110, 118–131]. ihMT was originally thought to arise from inhomogeneous broadeningof the non-aqueous lineshape, a hypothesis we explore at length in Chapters 6 and 7. Ourresults indicate that ihMT’s specificity arises from dipolar couplings alone (a connection alsomade by others), so the “inhomogeneous” name is perhaps unfortunate. In this section, weshall describe the technique and some applications of ihMT, leaving the details of the physicsfor later chapters.In ihMT experiments, a series of NMR spectra or MRI images are acquired: first, one with57a prepulse at a single offset frequency +∆ (spectrum S+); next, one with an offset −∆(spectrum S−); then, one with the same power split between both +∆ and −∆ (spectrumSdual); and finally, one with no prepulse (spectrum S0). Experiments have shown that manynon-aqueous non-lipids (such as heat-denatured ovalbumin, agarose, and gelatin [84, 118])result in an equal attenuation of the aqueous signals in S+, S−, and Sdual. However, materialscontaining a substantial proportion of lipid bilayers (such as brain and spinal cord WM, hairconditioner, Prolipid-161, and DPPC:Cholesterol phospholipids) show larger attenuation inSdual [84, 118]. A quantitative measure of this difference, the ihMT ratio, has been definedas [118]ihMTR = S+ + S− − 2SdualS0. (4.6)The sum S+ +S− provides a first-order correction for MT asymmetry. In this thesis, we usea definition that also includes a 2 in the denominator:ihMTR = S+ + S− − 2Sdual2S0 . (4.7)This is to maintain consistency with our publication on ihMT [131].An example of ihMTR in various phantoms is shown in Fig 4.5. MTR shows similar signalsfrom the two samples with non-aqueous protons (heat-denatured ovalbumin in row 1 andhair conditioner in row 2). Hair conditioner is a phantom for myelin, containing a highconcentration of lipid bilayers. For this reason, ihMTR’s selectivity to hair conditioner is ofsubstantial interest.Since its introduction, work on ihMT has rapidly advanced. Various improvements to thetechnique have been made, such as enabling measurements of T1D [127] and generating T1D-dependent contrast [127,132,133]. Moreover, by modifying the concentration of power duringthe prepulse, the ihMT signal can be boosted considerably [123,134].Comparison between ihMT and other techniques in vivo have shown promise of myelinsensitivity. Ercan et al. performed ihMT, MTR, diffusion tensor imaging, and myelin waterfraction imaging (MWF imaging; see Section 4.5 below) in different white matter tracts [121].Those authors found a strong correlation between ihMTR and MWF, the latter which isknown to be a biomarker for myelin. MTR was only weakly correlated with MWF. In a studyby Geeraert et al., they found strong correlation between the myelin volume fraction andihMTR [129]. Finally, recent work by Prevost et al. used mice that were genetically modifiedto produce green fluorescent protein in their myelin [135]. They observed a linear correlationbetween the fluorescence intensity and the ihMTR values, albeit with a considerable offset.ihMT also holds considerable promise for rapid adoption in clinical settings given the sim-58Figure 4.5: ihMTR compared to in different phantoms. Row 1 contains heat-denaturedovalbumin, row 2 contains hair conditioner, rows 3–10 are aqueous solutions with varyingT2 times. (A) The measured T2 times of the phantoms. (B) The MTR shows similar signalsfrom two samples (hair conditioner and ovalbumin) with a non-aqueous proton pool. (C) TheihMTR is selective to the sample with lipid bilayers (hair conditioner). Figure reproducedfrom Varma et al. [118], © 2015, with permission from John Wiley & Sons, Inc.plicity of calculating ihMTR. Like MTR, ihMTR is just a ratio and isn’t dependent onmodel fitting. Two studies have shown its potential for use in patients. Rasoanandrianina etal. used ihMT to image spinal cord in Amyotrophic Lateral Sclerosis (ALS) patients [128].That work found correlation between ihMTR and the clinical disability scores. In a similarstudy, ihMTR in lesions and in normal-appearing white matter of MS patients had a strongercorrelation than MTR with their clinical disability scores. [110].One of the possible confounding factors with ihMT is its orientation dependence, which hasbeen shown in-vivo in brain [121, 129] and ex-vivo in sections of spinal cord [136]. This islikely because ihMT is sensitive to the strength of dipolar couplings, and dipolar couplingsin lipid bilayers are orientation-dependent.4.5 T2 relaxation4.5.1 Myelin water and intra/extra-cellular waterIn contrast to T1 relaxation, there is a consensus on T2 relaxation in white matter [3,51,55,58,137–139]. T2 distributions calculated from CPMG decay curves show distinct peaks. MWtypically has T2 ∼ 10 ms, and the water in intra/extra-cellular compartments (the IE wateror IEW) has T2 values in the range 40–90 ms [3,51,57,139]. Often, a third peak at ~1 s from59CSF is also present. It is somewhat remarkable that the IEW appears as one pool givenits complex environment. However, glial cells are about 80% water and their membraneshave a high concentration of aquaporin, allowing the free diffusion of water molecules on thetimescale of one CPMG echo [54].Regularized non-negative least squares (NNLS) is used to obtain the T2 distribution fromCPMG decay curves. Let A be a matrix whose elements areAij = exp(−ti/T2,j).Then, the T2 distribution f(T2) is found from the CPMG echo amplitudes y(ti) by solvingarg minf ||Af − y||22 + λ2||f ||22 subject to f > 0.In this expression, λ is the regularization parameter chosen so the regularized NNLS fitchi-squared is 1%–2% larger than in the non-regularized fit [58]. Regularization preventsover-fitting by forcing f to be smooth. Examples of these types of distributions are shownin the next chapter. Once f(T2) has been calculated, a useful quantity called the myelinwater fraction (MWF) can be found. If AMW, AIEW, and Aother are the areas of the peakscorresponding to the MW, IEW, and other water, thenMWF = AMWAMW + AIEW + Aother.The MWF has been validated as a marker for myelin using histology [140] and has foundextensive use in research. However, as a myelin-sensitive MRI technique, it does have somedrawbacks. Acquisition can be lengthy and processing can be technically demanding. Also,MWF imaging has difficulty distinguishing between intact myelin and myelin debris. Finally,MW/IEW and MW/non-aqueous exchange can introduce errors. For in-depth discussion ofthe applications and challenges of MWF imaging, the reader is referred to the reviews byMacKay & Laule (and references therein) [55,141].4.5.2 CPMG exchange correctionMagnetization exchange between the MW and IEW pools can lead to slight shifts in the mea-sured T2 and pool sizes. Exchange during CPMG acquisition will lead to an underestimationof the T2 times and the MW size, and slight overestimation of the IEW size [142]. Previouswork by Bjarnason et al. in bovine brain showed that these corrections were approximately605–10% of the MW size [142]. Kalantari et al. used a slightly different procedure to estimatecorrections in human brain. They fit the four pool model (discussed in the next chapter) torelaxation data with and without MW/IEW exchange, finding corrections up to 15% of theMW size [143]. With this approach, the corrections were sensitive to the initial conditions.In this thesis, we will use the correction method introduced by Bjarnason et al., which usesa two-pool model of exchange between the MW and IEW to calculate the correction factorsfor both the pool sizes and the T2 times. How to do so was introduced by Bjarnason etal. in ref [142]. In Appendix B, their derivation is repeated using slightly different notation.Fig. 4.6 gives some examples of this correction on the T2 times and the MW pool size. Inthat plot, Tcr is a measure of the MW/IEW exchange time, where lower values mean fasterexchange. With a typical Tcr time of about 1 s, corrections are usually <20%. With thisapproach, the correction factor is independent of the initial conditions.61Figure 4.6: Examples of the CPMG exchange correction to pool sizes and T2 under differentconditions. Tcr (defined in Section 5.2) is a measure of the MW/IEW exchange time, wherelower values mean faster exchange. (A) T2/T˜2 (corrected T2 / observed T2) for MW and IEWfor different observed MWFs and Tcr,MW/IEW. The tilde indicates the observed value prior tohaving the correction applied. (B) The actual MWFs under the same conditions. Here weare assuming MW and IEW are the only aqueous pools and have fixed T˜2,MW = 6 ms andT˜2,IEW = 55 ms. These values were typical of our measurements in bovine brain (Chapter 5).62Chapter 5Aqueous and non-aqueous T1relaxation in brain under six differentinitial conditions5.1 IntroductionEver since Edzes and Samulski’s pioneering work on cross-relaxation in hydrated tendon[144], it has been known that in general, the intrinsic spin-lattice relaxation time of aqueousprotons in tissue is not directly observable. Instead, the observed aqueous T1s (called T ∗1 s inthis work) convolve intrinsic spin-lattice relaxation with cross-relaxation to the non-aqueousprotons and pools of other aqueous protons.Some recent studies have emphasized the fact that only T ∗1 s are directly observable [100,102, 145, 146]. They have also highlighted how a system’s relaxation depends on the initialconditions [102, 142, 145–147]. This may help explain the disagreement in the number of T1components present in white matter as highlighted in Section 4.3.2. As was noted, repro-ducibility is difficult in T1 measurements, where pulse sequence, field strength, and site seemto be confounding variables [48,98,147].One significant limitation to studying T1 relaxation with MRI scanners is their difficulty inobserving the rapidly-decaying signal from the non-aqueous protons. Yet this is straight-forward with NMR spectrometers, and there has been some work on solid-state NMR spec-troscopy in white matter. For example, some groups have looked at magic angle spin-ning (MAS) spectra of white matter [148, 149], while others have investigated the short T ∗2times [89,148,150]. Wilhelm et al.’s recent publication on high-resolution spectra of extracted63myelin lipids and rat thoracic spinal cord in D2O clearly showed their super-Lorentzian line-shapes [89]. However, the author is only aware of one previous paper by Bjarnason et al. [142]that followed the aqueous and non-aqueous magnetization during T1 relaxation.This chapter discusses T1 relaxation in bovine white matter brain tissue under six differentinitial conditions. We analyzed the data in the context of a four pool model, where the twoaqueous pools are MW and IEW, and the two non-aqueous pools are in myelin and non-myelin. As in Bjarnason’s work, an NMR spectrometer was used to acquire both FIDs andCPMGs, allowing the magnetization in different pools (MW, IEW, and total non-aqueous)to be found. The data and analysis here are significant improvements on this previousstudy, however. Because of improved sample holders and new equipment, we were able toacquire more FID data points and perform lineshape fitting instead of relying on momentanalysis. This new approach gives deeper insights into the physico-chemical compositionof the non-aqueous signal fraction. Furthermore, our CPMG analysis technique allowed formodeling of negative amplitude components, whereas Bjarnason et al. required subtractionof two complimentary experiments to ensure a positive signal. Also, their four pool modelparameters were estimated from relaxation analysis on the data, whereas we fit the datadirectly. Finally, we also present FID experiments on grey matter for the first time.Our analysis is performed with an eigenvector solution to the four pool model, introduced inBarta et al.’s recent work [145]. We emphasize how the eigenvector coefficients provide richinformation about the effects the initial conditions have on relaxation. Lastly, we discusshow our results may be used guide pulse sequence design.5.2 Theory: the four pool modelIn white matter, there are at least two distinct pools of aqueous protons, separable bytheir T2 times: the myelin water (MW) and the intra/extra-cellular water (IEW). Also, asdiscussed in Section 4.3.2, measurements of T1 using MRI typically reveal two additionaltime constants, one short (~100 ms) and one long (~1 s). With the assumption that thesefour time constants are distinct, then in a simple model with first-order exchange betweenpools, four pools are required. The Four Pool Model [142,143,145,151–153] assumes that theMW and IEW pools exchange with each other via self-diffusion. Physiological considerationsalso require that each aqueous pool also exchanges with an adjacent non-aqueous pool. ForMW, this is the non-aqueous myelin (pool M), composed of lipids and proteins in the myelinbilayers. IEW exchanges with the non-aqueous non-myelin (pool NM), which is mostly lipidsand proteins in the glial cell membranes. The aqueous/non-aqueous exchange is mediated64primarily by proton exchange (see Section 4.4).Fig. 5.1 is the schematic of the four pool model. The dynamics of the longitudinal magneti-zation in the four pools are described by a system of coupled differential equations:dMMdt=− MM −MM(∞)T1,M− kM,MWMM + kMW,MMMWdMMWdt=− MMW −MMW(∞)T1,MW+ kM,MWMM − kMW,MMMW−kMW,IEWMMW + kIEW,MWMIEWdMIEWdt=− MIEW −MIEW(∞)T1,IEW+ kMW,IEWMMW − kIEW,MWMIEW−kIEW,NMMIEW + kNM,IEWMNMdMNMdt=− MNM −MNM(∞)T1,NM+ kIEW,NMMIEW − kIEWMNM.(5.1)Here, Mi(∞) and T1,i are the size and intrinsic spin-lattice relaxation time of pool i, respec-tively; and kij is the magnetization exchange rate between pools i and j. kij and kji arerelated throughkijMi(∞) = kjiMj(∞) (5.2)and1Tcr,ij= 1kij+ 1kji. (5.3)Tcr,ij is the cross-relaxation time between pools i and j, which depends on the pool sizethrough the kijs.It is convenient to work in reduced magnetization units m, which are defined for pool ias [144]mi = −Mi −Mi(∞)2Mi(∞) . (5.4)In these units, m = 0 corresponds to equilibrium magnetization, m = 1 is inverted magneti-zation, and m = 1/2 is zero magnetization (in the longitudinal direction). This transformsEq. 5.1 into a homogeneous system of differential equations,dmdt= Rm, (5.5)65where R isR =− 1T1,M− kM,MW +kM,MW+kMW,M − 1T1,MW − kMW,M − kMW,IEW0 +kIEW,MW0 0· · ·· · ·− 1T1,M− kM,MW +kM,MW+kMW,M − 1T1,MW − kMW,M − kMW,IEW0 +kIEW,MW0 0(5.6)and m is a vector of the pools’ reduced magnetizations. The solution to Eq. 5.5 is [154]m(t) =4∑p=1cpvp exp(λpt)=4∑p=1cpvp exp(−t/T ∗1,p), (5.7)where vp and λp are the pth eigenvector and eigenvalue of R, cp is a coefficient that dependson the initial conditions, andT ∗1,p = −1/λp (5.8)is the pth apparent T1 relaxation time. T ∗1,p is associated with eigenvector p, not with justone specific pool.1Finally, in regular magnetization units, the solution for pool i isMi(t) = Mi(∞)1− 2 4∑p=1cpvp,i exp(λpt) , (5.9)where vp,i is the ith component of the pth eigenvector.In Appendix C, we show how there is an analogous electronic circuit for the four pool model.1A few words on notation: despite sharing the “*” superscript with T ∗2 , T ∗1 has nothing to do with B0homogeneity. Both are apparent relaxation times, but T ∗1 is intrinsic to the sample microstructure andreflects the inability to distinguish between changes in longitudinal magnetization due to exchange and dueto spin-lattice relaxation. On the other hand, T ∗2 is a function of sample geometry and shimming and canbe distinguished from T2 by means of a CPMG acquisition. Also, our definition of “apparent T1 relaxationtime” is different than Rioux et al.’s [147]. They define T ∗1 (same symbol) to be the single T1 time one wouldmeasure if mono-exponential behaviour is assumed in a system relaxing with multiple exponentials. In ourwork, we take it to mean the time constants of those multiple exponentials.66T1,M T1,MW T1,IEW Tcr,IEW/NM T1,NMPool 1: non-aqueousmyelin (M)Pool 2: myelin water (MW)Pool 3: intra/extra-cellular water (IEW)Pool 4: non-aqueousnon-myelin (NaNM)Tcr,MW/IEWTcr,M/MWT2,M~10 μs T2,MW~6 ms T2,IEW~60 ms T2,NM~10 μsMM(∞) MMW(∞) MIEW(∞) MNM(∞)LATTICEFigure 5.1: The Four Pool model.5.3 Methods5.3.1 Sample preparationA chilled, unfrozen bovine brain was ordered from Innovative Research (Novi, MI, USA) andreceived about 30 hours after harvesting. The age of the cow at the time of slaughter wasless than 30 months old. Tissue samples were immediately excised and stored at 5 ◦C untiluse. All experiments were completed within 72 hours of receiving the brain.Four tissue samples were extracted: 11.6 mg and 36.7 mg of white matter from two differentlocations in the splenium of the corpus callosum (samples WM-sp1 and WM-sp2), 52.2 mgof frontal white matter (sample WM-fr), and 28.6 mg of basal ganglia grey matter (sampleGM-bg). Each sample was sandwiched between two cylindrical spacers made from proton-free Kel-F (the flourinated polymer PCTFE) inside a 3.5 mm NMR tube. This improved B0homogeneity over previous experiments which did not use spacers [145]. Proton-free o-ringson the spacers minimized water loss. During white matter sample preparation, the tissuewas folded several times, ensuring that the nerve fiber tracts were oriented isotropically.5.3.2 NMR experimentsThe pulse sequences, shown in Fig. 5.2, consisted of three parts. First, the preparationpulses put the longitudinal magnetization into a non-equilibrium state. Second, the lon-gitudinal magnetization relaxed (via intrinsic spin-lattice relaxation and exchange betweenpools) during the variable cross-relaxation delay, TI. Finally, an FID or CPMG echo trainwas acquired. The preparation pulses are the only variation between the sequences. Fig.5.3 shows how the preparation pulses of each experiment gives the four pools unique ini-tial magnetization values. The size and direction of the arrows qualitatively represent themagnitude and direction of the longitudinal magnetization.The hard inversion-recovery (IR-hard, Fig. 5.2A) and soft inversion-recovery (IR-soft, Fig.67τf=1 or 50 ms(D) Goldman-Shen (4 kinds: GS-1ms-up, GS-1ms-down, GS-50ms-up, GS-50-ms-down)(C) Soft inversion-Recovery (IR-Soft)(B) Hard Inversion-Recovery (IR-Hard)TI=0.7 ms – 10 s90Preparation Cross Relaxation Acquisition(CPMG or FID)or(A) Pulse sequence scheme180 90TI180 90TI~6 μs rectangular (broadband)inversion pulse3 ms 3-lobe sinc(narrowband) inversion pulse180ϕ90ϕ90-ϕ ("down", magnetization along +z)AcqAcq90TI90ϕ ("up", magnetization along -z)Acq7 s recycle delay7 s recycle delay7 s recycle delay7 s recycle delayFigure 5.2: The NMR pulse sequences. All sequences have the general form shown in (A) con-sisting of three periods: preparation, cross relaxation, and acquisition. The hard inversion-recovery sequence (IR-hard) shown in (B) uses a broadband inversion pulse, inverting theaqueous and non-aqueous magnetization. The soft inversion-recovery sequence (IR-soft) in(C) uses a narrowband inversion pulse (1.1 kHz bandwidth), completely inverting only theaqueous magnetization. (D) shows the Goldman-Shen sequences, consisting of a spin-echoT2 filter of 1 or 50 ms followed by a broadband pulse that either rotates the magnetizationin the +z (“up”) or −z (“down”) direction.68~0GS-50ms-upGS-50ms-down GS-1ms-down GS-1ms-upIR-softIR-hardNon-aqueousmyelin (M)Myelinwater (MW)intra/extra-cellularwater (IEW)Non-aqeousnon-myelin (NM)~0 ~0~0 ~0~0~0 ~0~0 ~0Figure 5.3: The initial conditions on the four pools immediately after the preparation periodin the various pulse sequences. The arrows’ sizes and directions represent the longitudinalmagnetization’s direction and magnitude. Ideally, no two experiments have similar initialconditions, ensuring that the behaviour during cross-relaxation depends on a wide range ofthe four pool model parameters.5.2B) experiments use different types of inversion pulses. The “hard” inversion pulse is ashort (∼6 µs) rectangular pulse whose broadband excitation profile completely inverts themagnetization in all the pools. The “soft” inversion pulse is a 3 ms three-lobe sinc pulse thathas a narrow (1.1 kHz) excitation bandwidth. This pulse is designed to invert the magnetiza-tion of the aqueous protons (due to their narrow linewidth) while only marginally decreasingthe non-aqueous protons’ magnetization. This is similar to the soft pulses typically used inMRI sequences.Goldman-Shen (GS) experiments separate proton populations with distinct T2 times [155,156]. Our implementation (Fig. 5.2C) uses a spin-echo as a “T2 filter” followed by a pulsethat puts the magnetization back in the +z (“up”, parallel to B0) or -z (“down”, antiparallelto B0) direction. The spin echo in the middle of the T2 filter is necessary due to the short T ∗2time, which is ~10 ms for the aqueous signals from all samples. The echo or filter time, τf , iseither 1 ms or 50 ms. τf = 1 ms separates the non-aqueous and aqueous magnetization due tothe latter’s short, ∼10–500 µs decay time. τf = 50 ms separates the myelin water (T2∼6 ms)and the IE water (T2∼60 ms). The two τf times, combined with the two directions possiblefor the magnetization after the spin echo, give a total of four Goldman-Shen experiments:GS-1ms-up/down and GS-50ms-up/down.Experiments were performed using a Bruker solenoidal probe (HP WB73ASOL10) in a 200MHz (4.7 T) magnet with a home-built NMR spectrometer. The temperature was regulatedat 37 ◦C. This setup allows acquisition of both FID and CPMG signals. During the FID,69131072 points were acquired with a dwell time of 1 µs (106 samples/sec).The CPMG acquisition collected 300 echoes spaced 2 ms apart. 100 data points spaced 10µs apart were collected around the center of each echo and averaged. For all samples, the90◦ pulse width was within the range of 3.1–3.3 µs (a B1 amplitude of 18–19 mT). Therecycle delay was 7 s and 8 acquisitions were averaged in all experiments. During the cross-relaxation delay, 23 TI times were used, arrayed logarithmically from 0.77 ms to 10 s. WithTI = 10 s, the system had fully recovered to equilibrium.5.3.3 Analysis70all input datasignals isolated from all aqueous poolstotal non-aqueous signalBW size, T1, T2fit equilibrium CPMG using regularized NNLSfit CPMG curves with sparse exponentialslineshape fittingtotal aqueous signalM Aq (TI )=M MW (TI )+M IEW (TI )+MBW (TI )~M MW (TI )=~pMW(TI)MAq (∞)~M IEW(TI )=~p IEW(TI )MAq (∞)M BW(TI )=pBW(TI )M Aq (∞)aqueous pool signal fractions~pMW (TI ) ,  ~p IEW(TI) ,pBW (TI )fit all BW data with single T1T 1 ,BWT 2 , BWM BW(∞)CPMG ExperimentsFID ExperimentsM Non-Aq (TI )=MNM(TI)+MM (TI )observed T2 times of aqueous pools~T 2, MW ,  ~T 2, IEW ,  T 2 ,BWM Non-Aq (TI )=MNM(TI)+MM (TI )~M MW (TI ),~M IEW(TI)initial conditions for each experiment~M MW (0) ,  ~M IEW(0)M Non-Aq(0)MW, IEW, and total non-aqueous pool sizes~M MW (∞) ,  ~M IEW(∞)M Non-Aq(∞)=MNM(∞)+MM (∞)assumed non-aqueous fraction in myelinαM=0.5Four pool model fittingNM size, T1T 1 ,NMM NM(∞)M size,T1T 1 ,MM M(∞)MW size, T2, T1T 1 ,MWT 2 , MWM MW (∞)IEW size, T2, T1T 1 ,IEWT 2 , IEWM IEW(∞)Tcr timesT cr,M / MWT cr,MW / IEWT cr,IEW / NMtotal aqueous sizeM Aq (∞)=M MW (∞)+M IEW(∞)+MBW (∞)Solver iterationChoose test parametersT cr,M / MW ,  T cr,MW / IEW ,  T cr,IEW / NMT 1 , M=T 1 , NM ,T 1 ,Non-AqT 1 ,MW ,  T 1 , IEWApply CPMG exchange correction Observed MW and IEW signals~M MW (TI ),~M IEW(TI)~M MW (0) ,~M IEW(0)~M MW (∞) ,~M IEW (∞)True MW and IEW signalsM MW (TI ),  M IEW (TI )True MW and IEW pool sizesM MW (∞) ,  M IEW(∞)True MW and IEW initial conditionsM MW (0) ,  M IEW(0)non-aqueous initial conditionseigenvectors and eigenvaluesv i ,  T i*calculate coefficients from initial conditions for each experimentcalculate residualsGlobal minimum?NOYESObserved MW and IEW T2s~T 2 ,MW ,~T 2, IEWOutputs:● All parameters● Corrected MW and IEW pool sizes● Corrected MW and IEW T2 timesTrue MW and IEW T2sT 2 , MW , T 2,IEW(A) FID/CPMG fitting (B) Four pool model fitting routineMW, IEW, BW signals~AMW (TI) ,  ~AIEW(TI ) ,ABW (TI )total non-aqueous pool sizeM Non-Aq(∞)total non-aqueous initial conditionsM Non-Aq (0)M M(0)=αM MNon-Aq (0)M NM(0)=(1−αM)MNon-Aq (0)non-aqueous pool sizesM M(∞)=αM MNon-Aq (∞)M NM(∞)=(1−αM )MNon-Aq (∞)combining CPMGand FID dataFID fittingCPMG fittingcore simulation stepsinput data modificationinput data to fitting routineoutputFigure 5.4: (A) The analysis flowchart for fitting and combining the FID and CPMG data and (B) for fitting the four poolmodel parameters. BW is an isolated bulk water pool, discussed below.71The analysis pipeline, drawn in Fig. 5.4, broadly consists of two parts. In part A, the FIDand CPMG data are fitted and then combined to extract the signals of different separablepools as functions of TI. Ultimately, separate signals from MW, IEW, and the sum of allnon-aqueous protons are extracted. We remind the reader that in the context of the fourpool model introduced in Section 5.2, the non-aqueous protons are composed of separatepools of non-aqueous myelin (pool M) and non-aqueous non-myelin (pool NM). In part B,these separate signals (MW, IEW, and M+NM) are fit to the four pool model. These stepswill be covered in detail below.Except when stated otherwise, for each parameter pi its errors σ+i and σ−i were determined inthe following way. Using a least squares minimizer (regardless of the minimizer used to findpi), 50 repeated fits were performed with synthetic Gaussian noise whose standard deviationwas equal to the standard deviation of the best fit residuals. The initial guess for the ithparameter in these fits was chosen randomly from [0.8pi, 1.2pi] each time. Then, σ±i weredetermined from the average positive and negative deviation from pi over the repeated fits.All analysis was performed in Python with the Scipy/Numpy library [157] and with theLMFIT library [158]. FID fittingThe FID signals were modeled using one or more super-Lorentzians for the broad, non-aqueous component and a combination of Voigtians, Lorentzians, and Gaussians for theaqueous components (yellow boxes on the flow chart in Fig. 5.4A). In a perfectly-shimmedB0 field, the aqueous lineshapes are nominally Lorentzian. However, one Lorentzian couldnot adequately model the aqueous component in our samples due to field inhomogeneities.In such cases, the Voigtian lineshape, which is a Gaussian-broadened Lorentzian, is oftenappropriate [159, 160]. In the time domain, the functions of the Lorentzian, Gaussian, andVoigtian positioned at f0 aregLorentzian(w, f0; t) = exp(−piwt) exp (−i(2pif0)t) (5.10)gGaussian(σ, f0; t) = exp(−(2piσ)2t2)exp (−i(2pif0)t) (5.11)gVoigtian(σ, f0, w; t) = exp(−piwt) exp(−(2piσ)2t2)exp (−i(2pif0)t) . (5.12)We use widths to characterize all the lines. For the Lorentzian, w is the FWHM, andT ∗2 = (piw)−1. For the Gaussian, the width σ is the standard deviation. The Voigtian’s widthscorrespond to its Gaussian and Lorentzian parts. In the frequency domain, the Voigtian is72a convolution of Gaussian and Lorentzian lines, which is a complicated function [160]. Forcompleteness, the super-Lorentzian function is repeated from Eq. 4.2:gSL(σ, σ0, f0; t) =∫ pi/20dθ sin θ exp(−12σ(θ, σ0, σmin)2t2)=∫ pi/20dθ sin θ exp(−12(14(3 cos2 θ − 1)2σ20 + σ2min)t2),where 3σ0/2 is the maximum linewidth at θ = 0◦ and σmin is the minimum linewidth at themagic angle.Fitting was performed directly on both the real and imaginary parts of the phased time-domain data; there are two reasons why this was done instead of fitting in the frequencydomain. First, the non-aqueous component in the frequency domain is low amplitude andspread out over multiple data points. Second, because of probe ring-down, the FIDs donot start immediately after the end of the 90◦ acquisition pulse. With our equipment, thisdeadtime had to be determined by calculating the first-order phase correction, the procedurefor which is outlined in Section 2.7.3. Without knowledge of this, the amplitude of therapidly-decaying non-aqueous signals may be over- or under-estimated by fitting. With thetime of the first FID datapoint known, the data were phased to achieve a zero imaginarycomponent at t = 0 when extrapolated backwards. Accounting for the deadtime in thefrequency domain is not straightforward.Fits to all FIDs give the total aqueous and total non-aqueous magnetizations at each TI:Maq(TI) = [MMW(TI) +MIEW(TI) +MBW(TI)] (5.13)Mnon-aq(TI) = [MNM(TI) +MM(TI)] . (5.14)The terms in [...] cannot be individually determined by FID data alone. But, by fittingthe CPMG, one can determine the relative sizes of the isolated MW, IEW, and BW terms.Note that this is not so for the NM and M terms: they cannot be separated by any of theexperiments in this work. CPMG fittingMulti-exponential regularized NNLS distributions, introduced in Section 4.5, are the usualway of fitting CPMG data when one is interested in separating the MW and IEW [51,55,57,58]. However, this approach can’t account for signals where different components may haveopposite signs, as may occur in our experiments. Moreover, in regularized NNLS, separate73peaks in the T2 spectrum often coalesce when dealing with low-amplitude signals. In light ofthese issues, we used a sparse exponential distribution instead to fit the CPMG data (blueboxes in flowchart A). With this method, there is one exponential for the low-amplitude MWpeak, two exponentials for the IEW peak (separated by 10 ms), and a final exponential forthe low-amplitude ~200 ms peak.Two comments in anticipation of the results are necessary here. First, we found the use oftwo exponentials instead of one for the IEW signal gave superior fits to the CPMG. This wasespecially true when the longitudinal magnetization was in a non-equilibrium state. Second,we have identified this last ~200 ms peak as bulk water (BW). This will be justified later.With this sparse exponential distribution, we can measure the magnetization, positive ornegative, of each aqueous component as a function of TI. The signal fraction p(TI) of eachaqueous pool isp˜MW(TI) =A˜MW(TI)A˜MW(∞) + A˜IEW(∞) + ABW(∞)p˜IEW(TI) =A˜IEW(TI)A˜MW(∞) + A˜IEW(∞) + ABW(∞)pBW(TI) =ABW(TI)A˜MW(∞) + A˜IEW(∞) + ABW(∞),(5.15)where Ai(TI) is the total intensity of the signal from pool i and Ai(∞) is the amplitude ofthe pool’s exponential fits to the CPMG at TI = 10 s. The tilde on the MW and IEW termsindicates observed values—as covered in Section 4.5.2 and Appendix B, MW/IEW exchangeduring the CPMG acquisition means the actual pool amplitudes and T2 times cannot bedirectly measured. The actual values are determined later in the analysis. In contrast, theBW pool is relatively isolated from exchange, so its actual value can be directly observed(indicated by the absence of a tilde). Combining CPMG and FID fitsSteps in combining the CPMG and FID fits are shown in grey in part A of the analysispipeline in Fig. 5.4. Using Eqs. 5.15 and 5.17 together gives the magnetization in eachaqueous pool,M˜MW(TI) = p˜MW(TI)Maq(∞)M˜IEW(TI) = p˜IEW(TI)Maq(∞)MBW(TI) = pBW(TI)Maq(∞).(5.16)74M˜MW(TI), M˜IEW(TI), and, from Eq. 5.14,Mnon-aq(TI) are the inputs to the four pool modelfitting routines. For each of the six experiments, we take the initial conditions in pool i fromextrapolation back to TI = 0. Similarly, at the longest TI time the magnetization will beclose to equilibrium, giving a measure of the pool sizes:Mi(∞) ≈Mi(10 s). (5.17)The median value from Mi(10 s) in each of the six experiments is used. MBW(TI) is fitseparately to the standard equation for T1 relaxation,MBW (TI) = MBW (∞) (1− f exp(−TI/T1,BW )) , (5.18)where f is the inversion efficiency, which depends on the initial magnetization (f = 2 forcomplete inversion) and T1,BW is the single T1 time of the BW pool. When fitting, f isallowed to vary in each experiment while MBW (∞) and T1,BW vary as global values. Four pool model analysisIn Fig. 5.4’s flowchart B, the input data to the fitting routine is in green boxes and cal-culations to modify those data are in red. One such modification is the CPMG exchangecorrection, applied at each iteration to find the “actual” magnetization in MW and IEW(assuming the trial parameters are correct in that iteration). This is done first by correctingthe pool size (finding MMW(∞) and MIEW(∞) as described in Section 4.5.2). Then, eachdata point in these pools is multiplied by M˜MW (∞)/MMW(∞) or M˜IEW (∞)/MIEW(∞) topropagate the correction across all TI times.Another modification is splitting the total non-aqueous data, Mnon-aq(TI) into its approx-imated constituent contributions from MM(TI), and MNM(TI). To do so, we define a pa-rameter αM = MM(∞)/(MNM(∞) +MM(∞)) such thatMM(TI) = αMMnon-aq(TI)MNM(TI) = (1− αM)Mnon-aq(TI).This also gives values for the initial conditions in these pools,MM(0) andMNM(0). Followingprevious work [142,143,145,152], we set αM = 0.5, which assumes equal proton amounts inboth non-aqueous pools.Fig. 5.4B’s white boxes are the core steps to simulate the magnetization evolution. We usedScipy’s implementation of the Differential Evolution algorithm, which is a global minimizer75[161]. On each iteration, the algorithm first introduces new four pool model test parameters(4 T1s and 3 Tcrs). It also could vary the pool sizes by ±5% (not shown on the flowchart).The penalty function is found using the model’s estimation for the MW, IEW, and M+NMdata. Once the global minimum is found, the fitting terminates. The penalty function isnominally the sum of the residuals squared, and we use this calculation with one slightadjustment—as we explain later, some of the MW early-time data data points are weightedslightly higher.5.4 Results5.4.1 Spectra and FIDsThe 1H equilibrium spectrum of white matter (Fig. 5.5A) has a narrow, intense line from theaqueous protons sitting on a low-amplitude, broad ~10–15 kHz line from the non-aqueousprotons. This non-aqueous lineshape was fit well by a super-Lorentzian, characterized bybroad wings and a sharp central peak. The grey matter spectrum (Fig. 5.5B) is similar, butwith a much smaller non-aqueous amplitude. In both the grey and white matter spectra,some additional structure is also visible, likely from metabolites, lipid headgroups, and/ormethyl groups [80,89,148,149]. The B0 shimming was imperfect due to sample geometry, sointerpreting small spectral details is difficult.While the fitting was ultimately performed in the time domain, we will discuss the frequency-domain spectra in detail since some aspects are easier to interpret. Fig 5.6A and B shows twoexamples of fitting WM-sp2’s spectrum at a short and long time after a soft inversion pulse,and Table 5.1 lists all of the lineshape functions used for fitting each sample’s spectra. Whilenumerous functions are involved, the end result is simply an overall amplitude for the aqueousand non-aqueous signals. Through trial and error, we found that the aqueous line in eachsample was fit well by one main Lorentzian (for WM-fr, WM-sp2, and GM-bg) or Voigtian(for WM-sp1). The need for the Voigtian indicates the B0 field was less homogeneous inWM-sp1, likely due to its small size (11.6 mg of tissue compared to WM-sp2’s 36.7 mg).Lower-intensity lines were also necessary to supplement this main aqueous line. These helpedto isolate the wings of the main water peak, which due to its intensity, had a width of about2 kHz near the baseline. In the case of IR-soft (Fig 5.6A), the central portion of the aqueouspeak was inverted by the 1.1 kHz-wide inversion pulse, so the extremities of the wings werelargely untouched. Even though the maximum amplitude of these wings was <1% of theaqueous line, we found they must be accounted for in order to distinguish their signal from76Figure 5.5: The equlibrium NMR spectra from WM-fr and GM-bg. The narrow peak isfrom aqueous protons: IEW, MW (in white matter), and BW. The broad super-Lorentzianis from non-aqueous protons: NM and, in white matter, M. “SL” indicates the peak of thesuper-Lorentzian, clearly visible on the white matter spectra in (A) but not in the greymatter spectrum in (B), where the non-aqueous pool is smaller and composed of relativelyfewer lipids.77Figure 5.6: Fits to the equilibrium and non-equilibrium spectra for sample WM-sp2. (A)When the center of the aqueous peak is inverted, multiple low-amplitude lines are necessaryto fit the non-inverted wings of the aqueous signal. (B) In equilibrium, the aqueous peak inthis sample is largely accounted for by a Lorentzian, although a small, negative correctionis required for better definition near the baseline. In both cases, two super-Lorentzians (onebroad, one narrow) at the same frequency account for the non-aqueous magnetization. (C)When the second, narrower super-Lorentzian is not present, the fit to the same data as (B)is comparatively worse, as indicated by the arrows.the super-Lorentzians from the non-aqueous protons. As seen in the example in Fig 5.6A, aseries of Gaussians modeled these wings. These Gaussians were fixed at zero amplitudes atall TI times after the entire aqueous line relaxed enough to have a positive amplitude.Two super-Lorentzians, one broad and one relatively narrow, with both constrained to beat the same frequency, were required in all white matter samples to fit the non-aqueoussignal. Fig 5.6B and C demonstrates why two were needed by comparing the same equi-librium spectrum fit with two and one super-Lorentzians. Arrows in plot C indicate wherethe single super-Lorentzian’s fit was inferior. Moreover, when single super-Lorentzian fitswere attempted on non-equilibrium spectra like the one in plot A, the results were oftenunphysical estimations for the non-aqueous magnetization (i.e. Mnon-aq(TI) > Mnon-aq(∞)).Specifically, using only one broad super-Lorentzian alone did not adequately fit the intensitynear the center. The additional ~2 kHz super-Lorentzian ensured the central singularity fitwell at all TI times. This extra intensity in the second, narrower super-Lorentzian may befrom other, distinct proton groups. For example, in Wilhelm et al.’s high-resolution spec-tral fits of purified myelin [89], they included lower-intensity super-Lorentzians from CholineCH2s and acyl CH3s at approximately the same frequency as the broad, intense methy-lene super-Lorentzian. Whether this is the reason the spectra here required an additional,78narrow super-Lorentzian is difficult to judge. Our B0 shimming was comparatively worsethan in the Wilhelm study (their sample was a solution of myelin extract, allowing a high-resolution probe to be used), so the lines from these other protons, if present, could not bedefinitively resolved. Another potential explanation for the necessity of including the nar-rower super-Lorentzian is inadequate numerical integration of the broader super-Lorentzian.From Eq. 4.3, one can see that around the magic angle, the super-Lorentzian’s constituentGaussians rapidly decrease in width, which has has a nearly P2(cos θ) dependence. How-ever, doubling the resolution from the 400 evenly-spaced steps in θ used here to 800 did notchange the results appreciably. Beyond this, the fitting routine takes too long, though amore efficient discretization scheme could be used.Since FID fitting was performed mainly through trial and error, there was a risk of over-fitting. Yet, for our purposes, any over-fitting introduced negligible error, since we were onlyinterested in the sum of all aqueous and non-aqueous lineshape function amplitudes. Withthe non-aqueous lineshape functions consisting solely of super-Lorentzians, the aqueous/non-aqueous contributions were easily distinguishable. On the other hand, if the goal was todistinguish contributions from specific metabolites, then the inclusion of multiple lineshapefunctions must be carefully mapped to each species (which would probably be impossiblewith the B0 field homogeneity seen here). As it stands, we added additional lineshapefunctions until plots of the total aqueous and total non-aqueous amplitudes as a function ofTI were seen to vary negligibly (not shown).For all white matter samples, the spectral frequency of the super-Lorentzians were fixedat the position of the visible central peak of the non-aqueous spectrum. This peak wasnot visible in GM-bg so the position was approximated from the white matter samples.Correspondingly, without the obvious central peak, it wasn’t necessary to include the morenarrow super-Lorentzian when fitting the data as in the white matter samples. The centralpeaks were found to be about 3 ppm upfield from the water line. This is consistent with thesignal originating mostly from acyl methylene groups [87, 89, 94, 162]. The minimum widthwas set to σmin=40 Hz, which is close to the narrowest width of the aqueous line.Examples of time domain (FID) data are given in Fig. 5.7, along with the total aqueous andtotal non-aqueous fits (the sum of the functions in Table 5.1). Here, the broad, non-aqueoussuper-Lorentzians correspond to signals which have mostly decayed by ∼100 to 500 µs. Thismeans that the number of FID data points with non-aqueous signals was relatively small.Consequently, we found it essential to have appropriate lineshape functions for the beginningof the FID. Also given in this table are the approximate T ∗2 times, when the aqueous part ofthe FID decayed to about 1/e of its initial amplitude. These times are much shorter than79Sample Non-aqueous equilibrium lines Aqueous equilibrium lines AqueousT ∗2 (ms)WM-fr 54 · SL(σ=16.2 kHz, σmin=40 Hz, f0=−673 Hz)19 · SL(σ=2.4 kHz, σmin=40 Hz, f0=−673 Hz)158 · L(w=44 Hz, f0=0 Hz) 10.3WM-sp1 27 · SL(σ=14.9 kHz, σmin=40 Hz, f0=−679 Hz)15 · SL(σ=4.5 kHz, σmin=40 Hz, f0=−679 Hz)161 · V (σl=41 Hz, σg=33 Hz, f0=4 Hz)5 ·G(σ=11 Hz, f0=7 Hz)8.6WM-sp2 29 · SL(σ = 13.6 kHz, σmin = 40 Hz, f0 = −679 Hz)1 · SL(σ = 549 Hz, σmin = 40 Hz, f0 = −679 Hz)−5 · V (σl = 224 Hz, σg = 176 Hz, f0 = 35 Hz)5 · L(w = 9 Hz, f0 = 3 Hz)126 · L(w = 45 Hz, f0 = 0 Hz)6.6GM-bg 12 · SL(σ=18.8 kHz, σmin=40 Hz, f0=−679 Hz) −35 · V (σl=118 Hz, σg=795 Hz, f0=39 Hz)33 ·G(σ=1.0 kHz, f0=21 Hz)99 · L(w=96 Hz, f0=−3 Hz)13 · L(w=25 Hz, f0=10 Hz)6.3Table 5.1: The functional forms of the equilibrium FID fits (TI = 10 s) for all samples andtheir aqueous T ∗2 s. The functions’ amplitudes are adjusted at every TI and then summedtogether to determine the total aqueous and total non-aqueous signal. All lineshape functionsare normalized to 1. The amplitudes are in relative units. SL=super-Lorentzian function(Eq. 4.2; 3σ/2 is the standard deviation of the widest component Gaussian, σmin is standarddeviation the narrowest component Gaussian), G=Gaussian function (Eq. 5.11; σ is standarddeviation), L=Lorentzian function (Eq. 5.10; w = (piT2)−1 is FWHM) V=Voigtian function(Eq 5.12; σ and w correspond to its Gaussian and Lorentzian widths). The widths varyfrom sample to sample to account for differences in B0 homogeneity. The centers of the non-aqueous lines were fixed at the center of the super-Lorentzian, visible in the white mattersamples. In GM-bg, the super-Lorentzian center not visible on the spectrum and so wasapproximated from the white matter samples. The aqueous T ∗2 s are approximately the timeit took for the aqueous portion of the FIDs to decay by 1/e.80Figure 5.7: Examples of fits to FIDs in an IR-Soft experiment on WM-sp1 at different TItimes. (A) The FID shortly after the soft inversion pulse. Here, the non-aqueous signal’smagnetization is positive, whereas the aqueous magnetization is inverted. (B) The same TIafter a hard inversion pulse. The non-aqueous magnetization is now inverted as well. Theequilibrium FID in (C) shows the long-lasting signal from the aqueous protons as well as theshort-lived signal from the non-aqueous protons.the IEW T2 (~60 ms) because of B0 inhomogeneity.5.4.2 CPMG multi-exponential fittingFig. 5.8 shows the difference between the regularized NNLS and sparse exponential distribu-tions of WM-fr. The distributions of all white matter samples were similar. In GM-bg, noMW contribution was detected, even in the sparse exponential distribution. Possible reasonsfor this are explored in the discussion. Another difference was that GM-bg’s BW T2 timewas closer to 300 ms (instead of 200 ms as for the white matter samples). In the sparsedistribution, two exponentials were necessary to fully account for the IEW’s potential forhigh positive and negative signals across the whole range of experiments. The sparse distri-butions were fairly consistent with Barta et al.’s work in bovine brain [145], although theirBW T2 was much higher (650 ms) than ours. This was likely due to the confined volume ofour sample, where the BW could only form a thin film between the sample and the walls ofthe NMR tubes and plugs. This restriction would have reduced the T2. Barta et al. useda small piece of tissue inside an NMR tube without any spacers, so the BW pool was lessrestricted.81Figure 5.8: The regularized NNLS distribution compared to the sparse exponential distri-bution. The distributions were calculated from WM-fr’s equilibrium CPMG. The sparsedistribution allowed for negative amplitudes. Two exponentials were used for the IEW peak,which has the highest intensity.5.4.3 White matter four pool and bulk water fittingUsing the four pool model, MMW(TI), MIEW(TI), and Mnon-aq(TI) (=MM(TI) +MNM(TI))data series were fit for the three white matter samples. Given its lack of a measurableMW pool, sample GM-bg was fit to a two pool model, discussed later. These results arepresented in Fig. 5.9, which also includes the BW fits using Eq. 5.18. The MW and IEWdata in this plot have been corrected for MW/IEW exchange during the CPMG and sorepresent the actual magnetization in these pools. In the case of MW, we found data pointsnear MMW(TI) ≈ 0 tended to be unreliable and were omitted (indicated by “×” markers).The sparse exponential distribution could not adequately detect the MW when its absoluteintensity is very small, leading to values which tended to correlate (erroneously) with the IEWsignal. Because the omitted data includes the start of the GS-50ms-up/down experiments,in these cases MMW(0) was fixed at 0. The MW plots also show the observed magnetization,calculated by performing the CPMG exchange correction in reverse. The same correctionwas applied to the IEW data but the difference is not visible on the graph.82Figure 5.9: Four pool model fits to WM-fr. All MW and IEW data with opaque markers havebeen corrected for exchange during the CPMG and are the actual magnetizations in thosepools. For MW, the observed magnetization, M˜MW(TI), is plotted (translucent markers)along with its fit (dashed line). These are not plotted for IEW because the difference betweenits observed and actual magnetization is very slight. “×” markers indicate data which wereomitted (data points where MMW(TI) ≈ 0, as described in the text). The BW was fitseparately using Eq. 5.18; for this pool, each experiment was well characterized using aunique initial magnetization and a single, global T1 time. This is evidence that the BW poolcan be considered isolated from the the other four pools.83 (A)​​ White matter four pool model fit parameters  Parameter Units WM-fr WM-sp1 WM-sp2 T​cr​,M/MW s 0.13 (+0.01/-0.01) 0.16 (+0.01/-0.02) 0.154 (+0.007/-0.016) T​cr​,MW/IEW 0.82 (+0.05/-0.04) 0.84 (+0.07/-0.07) 0.73 (+0.05/-0.07) T​cr​,IEW/NM 0.67 (+0.04/-0.03) 0.86 (+0.16/-0.06) 0.86 (+0.19/-0.05) T​1,M s 0.23 (+0.01/-0.01) 0.19 (+0.05/-0.01) 0.149 (+0.066/-0.004) T​1,NM  0.63 (+0.06/-0.06) 1.0 (+0.5/-0.2) 2.5 ​b T​1,IEW ​a 3.0 (+0.2/-0.1) 2.4 (+0.2/-0.1) 2.24 (+0.10/-0.08) T​1,MW ​a 3.0 (+0.2/-0.1) 2.4 (+0.2/-0.1) 2.24 (+0.10/-0.08) M​M​(∞) rel. ​c 14.92 (+0.06/-0.07) 9.26 (+0.07/-0.07) 9.11 (+0.04/-0.04) M​NM​(∞) 14.92 (+0.06/-0.07) 9.26 (+0.07/-0.07) 9.11 (+0.04/-0.04) M​IEW​(∞) 85.9 (+0.1/-0.2) 91.65 (+0.14/-0.10) 84.2 (+0.2/-0.2) M​MW​(∞) 8.92 (+0.08/-0.10) 5.46 (+0.03/-0.02) 6.92 (+0.08/-0.04) M​non-aq​(∞)=​M​NM​(∞)+​M​M​(∞) 29.8 (+0.1/-0.1) 18.5 (+0.1/-0.1) 18.22 (+0.09/-0.08) T​2,MW ms 8.6 7.9 8.3 T​2,IEW 63 66 66  - 0.9 0.86 0.87  0.93 0.93 0.92  0.78 0.7 0.73  1 1.00 1 M​BW​(∞) rel. ​c 5.19 (+0.03/-0.03) 2.892 (+0.02/-0.02) 8.88 (+0.04/-0.04) T​1,BW s 2.20 (+0.02/-0.02) 1.60 (+0.02/-0.02) 2.19 (+0.02/-0.02) T​2,BW ms 211 (±5%) 211 (±5%) 184 (±5%) MW residual weight​, 𝜂 - 8 10 5  a) Parameters were constrained to be the same value b) Parameters are at the limit of allowed range c) Relative pool size units are scaled so that aqueous pools add to 100  (B)​​ Fit eigenvectors and eigenvalues  WM-fr T1* 22.9 (+1.1/-0.8) ms 74 (+3/-4) ​​ms 130 (+4/-4) ​​ms 1.268 (+0.004/-0.005) ​​s M 6.16 2.19 -11.71 5.25 MW -8.11 0.32 -5.09 4.05 IEW 2.67 -14.97 7.65 51.55 NM -0.15 14.52 3.32 8.29 Size 17.09 32 27.78 69.15 E/R 0.03 0.06 0.21 1  WM-sp1 T1* 22 (+1/-2) ​​ms 67 (+5/-4) ​​ms 102 (+6/-5) ​​ms 1.453 (+0.006/-0.005)​​ s M 2.94 1.25 -8.01 3.01 MW -5.17 0.01 -2.55 2.59 IEW 2.66 -9.6 4.92 53.71 NM -0.11 9.13 1.62 5.29 Size 10.88 19.98 17.1 64.6 E/R 0.03 0.04 0.24 1  WM-sp2 T1* 24 (+1/-2) ​​ms 72 (+3/-3) ​​ms 91 (+6/-2) ​​ms 1.376 (+0.005/-0.004) ​​s M 3.99 2.73 -7.22 2.74 MW -6.21 0.78 -3.45 3.17 IEW 2.96 -10.81 3.19 49.1 NM -0.13 8.55 3.16 5.46 Size 13.3 22.87 17.02 60.47 E/R 0.05 0.05 0.25 1          Table 5.2: The results of the four pool model fits on all white matter samples. (A) The fit parameters. Errors on the T2sare estimated to be 5%, which is close to the FWHM of the peaks on the regularized NNLS distributions (Fig. 5.8). Errorson other parameters are the standard deviations of repeated fits with noise, as described in the Methods section. (B) Theeigenvectors and eigenvalues derived from the fit parameters. To show the amount of magnetization flow represented eacheigenvector, the components listed are v′i = viMi(∞). The size (∑4i=1 |v′i|) is a measure of how much magnetization flow isassociated with each eigenvector. Eigenvectors with larger sizes generally have more easily observable T ∗1 values. E/R is theexchange/relaxation factor from Eq. 5.21 (E/R=0 means the T ∗1 corresponds to pure exchange, whereas E/R=1 indicates purespin-lattice relaxation).84The data show that the expected initial conditions in the six experiments (Fig. 5.3) wereachieved, leading to unique relaxation behaviour in each case. The fits from the modeldescribed the magnetization in each pool well, with the largest relative deviation occurringin MW during the GS-50ms-up experiment around TI ∼ 100 ms. However, this largerelative error was consistent with MW being the smallest pool. In fact, when the standardpenalty function was used in the fitting routine (the sum of residuals squared), the solver wasrelatively insensitive to the MW pool due to its comparatively lower amplitude. This led tothe solver omitting a short, ~30 ms T ∗1 component, present most obviously in MW. To ensurethis component was accounted for, the fitting results shown here have had the residuals fromthe start (TI<37 ms) of the MW data up-weighted by a factor η when calculating the fittingpenalty function:MW Residuals =MMW,data(TI)−MMW,fit(TI) TI > 37 msη (MMW,data(TI)−MMW,fit(TI)) TI < 37 ms. (5.19)We will deal with choosing the value of η and the consequences of this choice in the nextsection.Turning now to the best fit parameters listed in Table 5.2A, the size of the MW pool,MMW(∞), is immediately understandable. Because the pool sizes in the table have beennormalized so that MMW(∞) + MIEW(∞) + MBW(∞) = 100, this is equal to the myelinwater fraction (MWF). WM-fr had the highest MMW(∞), likely due to the presence of moremyelin. This was reflected in WM-fr’s total non-aqueous pool size, Mnon-aq(TI), which waslarger than in the other samples. In all cases T1,MW and T1,IEW were constrained to have thesame value, which could vary. Without this constraint, T1,MW had large (~1 s) variability.The ratios of the observed to true values for IEW and MW pool sizes and T2s are also givenin the table. The largest correction was for M˜MW(∞) of WM-sp2, where the true pool size,MMW(∞), was about 40% larger. In other words, the observed value only underestimatedthe pool size by less than 2% of the total aqueous signal (the size of MMW(∞) in WM-sp2is 6.9% of the total aqueous signal). The effect on the IEW pool size was negligible.The T ∗1 s and eigenvectors in Table 5.2B offer a simple way to consider the meaning of all ofthe fitted parameters at once. In the table, the values listed are scaled eigenvectors v′ whosecomponents arev′i = Mi(∞)vi. (5.20)v′i can be interpreted as amount of physical magnetization entering (positive values) orleaving (negative values) the pool, keeping in mind that only the relative sign differences85in the eigenvector components matter—multiplying and eigenvector by -1 does not changethe physics. Each eigenvector describes a varying amount of magnetization exchanging orrelaxing, so it is convenient to define their “size” as ∑4i=1 |v′i|. Generally, the larger the size,the easier it is to observe the T ∗1 associated with this eigenvector—although this ultimatelydepends on the initial conditions. Another related metric is the relaxation/exchange factor[145],E/R = |∑4i=1 v′i|∑4i=1 |v′i|, (5.21)where E/R=0 indicates the corresponding T ∗1 arises from inter-pool exchange only, andE/R=1 indicates the T ∗1 is from pure spin-lattice relaxation. All the white matter samples’eigenvectors had a similar structure and matched well with those found in a previous study inbovine brain [145]. Looking at WM-fr, the most rapid relaxation time was T ∗1 = 23 ms froman eigenvector that primarily described M/MW exchange. This matches with the theory thatMW’s short T2 time is caused by exchange with the myelin lipids [57, 145]. There was alsosome MW/IEW exchange on this timescale as well, which was more significant with the WM-sp1/sp2 samples. This is consistent with their smaller non-aqueous pool sizes. A smallermyelin sheath would be more permeable, increasing the MW/IEW exchange. The nexteigenvector (T ∗1≈60–80 ms) was almost pure IEW/NM exchange. The last two eigenvectorscan be associated with MW/IEW or (M+MW)/(IEW+NM) exchange (T ∗1≈90–130 ms) andspin-lattice relaxation (T ∗1 > 1 s). Because this last one had the largest size and the longestT ∗1 , it is therefore the easiest to observe using MRI. T1 measurements of brain often onlyreport this relaxation time.Fig. 5.9 and Table 5.2 also show the results of fitting the BW data to Eq. 5.18, the singleinversion-recovery equation. BW was well-described using a single, global T1 and MBW(∞)values, but with unique initial magnetization for each experiment. The majority of itsrelaxation was single-exponential, with only small fluctuations indicating some exchangewith other pools. In light of this, treating BW as an isolated pool outside of the four poolmodel seems justified.5.4.4 White matter fitting variationsAs mentioned in the last section, when all MW, IEW, and M+NM residuals were treatedwithout weighting in the fitting penalty function, the four pool model tended to ignorethe ~30 ms T ∗1 component. Its associated eigenvector in Table 5.2 shows this componentrepresents M/MW exchange. Fig. 5.10 summarizes this issue by plotting the residuals fromfits to the first 10 points of the MW magnetization on a linear scale. The GS-50ms-up/down86Figure 5.10: Residuals from four pool model fits to short-TI MW data in sample WM-fr.In the weighted fit, the residuals for the 10 MW data points across all experiments whereTI < 37 ms were multiplied by η (defined in Eq. 5.19), emphasizing the importance of thesepoints. This forced the rapid, 20–30 ms decay most visible in the MW pool to be accountedfor. This is particularly clear on this graph’s linear scale in the case of IR-Soft, where theweighted fit was superior. The GS-50ms-up/down experiments were not included since theirinitial MW data points were excluded from the four pool model fit.χ2totWM-fr WM-sp1 WM-sp2three pool 6.83 7.61 3.67four pool, η = 1 (unweighted) 2.98 4.16 2.27four pool, η > 1 (MW start weighted) 5.86 6.79 3.05Table 5.3: Comparison of total chi-squares for three pool, weighted four pool, and unweightedfour pool model in white matter. The three pool model consisted only of MW, IEW, and onenon-aqueous pool. The unweighted four pool model treated the residuals of all data pointsthe same. η, the weighting factor, is given in Table 5.2. Larger values force recognition ofthe T ∗1 ∼ 30 ms time. Weighting was removed prior to calculation of χ2tot (see Eq. 5.24). Thevalues of η when η > 1 are listed for each sample in Table 5.2.87Figure 5.11: The effect of varying the weighting of the start of the MW residuals (η, Eq. 5.19)for sample WM-fr. (A) Chi-square from each data series across all experiments (Eq. 5.23)and the total chi-square (Eq. 5.24). (B) The change in the four T ∗1 times as η is increased.Only at larger values were the times from MW/M exchange and IEW/NM distinguishable.The longest T ∗1 time, associated with spin-lattice relaxation, is minimally influenced by η.For this sample, the optimal factor was η=8, determined by the start of the plateau in mostof the chi-square and T ∗1 values.88Figure 5.12: A comparison of three different models applied to WM-sp2: two four poolmodels (one weighted, one not) and one unweighted three-pool model (consisting only ofMW, IEW, and one combined non-aqueous pool). The chi-square plotted, χ2(xs, xe), is thesum over all data points for each data series xs and experiment xe (Eq. 5.22). Overall, thefour pool models provided a better fit, particularly in the IEW pool. However, weightingwas necessary in order to fit the MW pool properly.MW data was not plotted since it was excluded from the fits (as explained above).The four pool model fits shown were performed in two different ways. The first fit methodtreated all data points in the residuals with the same weight (η = 1, where η was definedin Eq. 5.19). With no weighting, the rapid decay seen most clearly at the beginning ofthe IR-soft and GS-1ms-up/down experiments was ignored. This was also visible in IR-softnon-aqueous magnetization (not shown). In the second method, the residuals from the firstten MW data points (TI < 37 ms) in each experiment were up-weighted by the factor, ηdefined in Eq. 5.19. As evident by the weighted fits, η > 1 forced recognition of this shorttime constant. However, forcing the fit to more closely match the beginning of the MWcurve came at the price of a worse overall fit.Fig. 5.11 motivates the ultimate choices of η, which are reported for each sample in Table5.2. The total chi-square for fits to the MW, IEW, and total non-aqueous (M+NM) data areshown in panel A, and T ∗1 times are in panel B. Both are plotted as functions of η. There isan inflection point in the plotted quantities around a similar value of η. For WM-fr, this wasaround η = 8. Note that without a large η, the T ∗1 times associated with MW/M exchangeand IEW/NM exchange were unacceptably close (the corresponding eigenvalues were also89equal).Table 5.3 gives the total chi-square values, χ2tot, for different fitting methods, includingweighted and unweighted four pool models.2 The chi-square χ2(xs, xe) for data series xs ∈{MW,IEW, M+NM} and experiment xe ∈{IR-Hard, IR-Soft, GS-50ms-up, GS-50ms-down, GS-1ms-up, GS-1ms-down} isχ2(xs, xe) =NTI∑i=1(data(xs, xe;TIi)− fit(xs, xe;TIi))2 , (5.22)where NTI is the number of TI times in that series (which depends on xs and xe since datapoints were removed in certain series). From this, the total chi-square for a particular seriesxs across all experiments isχ2(xs) =∑xe∈experimentsχ2(xs, xe), (5.23)and the total chi-square for all series and experiments has the formχ2tot =∑xs∈seriesχ2(xs). (5.24)Importantly, these values are calculated independent of the value of η.In the three white matter samples, χ2tot was ~1.2–1.8× worse in the weighted fits than inthe unweighted fit. To investigate why, the chi-square values for each experiment and dataseries, χ2(xs, xe), are shown in Fig. 5.12. Unsurprisingly the weighted four pool model fitgave superior MW modeling, particularly for the IR-soft experiment where the ~30 ms T ∗1component is most obvious. However, the trade-offs were worse IEW and total non-aqueousfits.To confirm the four pool model is necessary to model white matter, we also tried fitting thesame data to an unweighted (η = 1) three pool model. In the three pool model, the twonon-aqueous pools in the four pool model (M and NM) are combined into a general non-aqueous pool. Exchange happens between MW/IEW, IEW/non-aqueous, and MW/non-aqueous. This was motivated by the fact that we were unable to observe the M and NMmagnetization separately. When this model was used, the result was a worse overall chi-square than either of the four pool fitting methods (Table 5.3). Looking at the chi-square2The χ2as defined here is actually the Residual Sum of Squares (RSS). If the errors on each data pointwere independently and identically distributed with a variance σ2 (which is not the case for our data), thenRSS/σ2 has a chi-square distribution.90   (A)​​ Grey matter two pool model fit parameter  Parameter Units GM-bg T​cr​,non-aq/aq s 1.42 (+0.12/-0.11) T​1,non-aq s 0.23 (+0.02/-0.01) T​1,aq 3.63 (+0.25/-0.18) M​non-aq​(∞) rel. 8.26 (+0.12/-0.04) M​aq​(∞) 98.52 (+0.20/-0.16) T​2,IEW s 1.482 (+0.016/-0.014) M​BW​(∞) rel. 1.480 (+0.016/-0.011) T​1,BW s 1.823 (+0.036/-0.026) T​2,BW ms 288 (±5%)  (B)​​ Fit eigenvectors and eigenvalues  T​1​* 73 (+1/-2) ​​ms 2.000 (+0.003/-0.008) ​​s Aqueous -5.83 80.5 Non-Aq 8.25 4.7 Size 13.08 85.1 E/R 0.16 1   Table 5.4: The fitted two pool model parameters (A), and eigenvectors and T ∗1 s (B) for thegrey matter sample GM-bg. Values listed in the eigenvectors are viMi(∞). Error values aredescribed in the caption of Fig. 5.2.values of the individual experiments, the three pool model performed better in MW thanthe unweighted four pool model. This may hint that the assumption of equal non-aqueousprotons in the M and NM pools was incorrect. Yet, the three pool fit was significantly worsein both the IEW and the total non-aqueous than either of the four pool models.5.4.5 Grey matter two pool fittingWith no detectable MW, the GM-bg sample was fit to a two pool model representing aqueousand non-aqueous protons. The fit results are summarized in Fig. 5.13 and Table 5.4. Thenon-aqueous data are comparatively noisier than in the white matter samples, due to GM-bg’s smaller non-aqueous pool size (8% of the total aqueous pool size compared to about18% in WM-sp1/sp2 and 30% in WM-fr). The IR-soft non-aqueous data was especiallynoisy; as in the white matter samples, this was the most difficult experiment for FID fittingbecause of its complicated lineshape. This was exacerbated by the relatively low non-aqueoussignal in the grey matter. Looking at the eigenvectors and T ∗1 times, the rapid, 73 ms T ∗1 isassociated with aqueous/non-aqueous exchange, whereas the longer 2.0 s time is associatedwith spin-lattice relaxation.91Figure 5.13: Two pool fits to grey matter sample GM-bg. Since there was no detectable MWin this sample, the two pool model was more appropriate. Bulk water was fit separately usingEq. 5.18.925.5 Discussion5.5.1 Comparison with other studiesOverall, the values measured in the four pool model fits were reasonable and in-line withprevious ex-vivo and in-vivo studies. Comparing the size of the MW pool (i.e. the MWF) inwhite matter, we find it is on average lower than in previous human and bovine studies (seeMinty et al. and references therein [163]). For example, our white matter values were about2–10% smaller than those from two previous studies in bovine brain [142, 145]. That said,our data was internally consistent: the fittedMMW(∞) in WM-sp1 and WM-sp2 were similar(within 2 units of amplitude, where 100 units comprises the entire aqueous magnetization).This is expected given that they are from the same brain and same area. Differences in thesample preparation and between individual cows (such as their age) likely account for thediscrepancies between our work and the other studies.Between the WM-sp1/sp2 and WM-fr samples, WM-fr had a ~2× larger non-aqueous frac-tion. Confirmation of this in the literature is mixed: one quantitative MT (qMT) in-vivohuman study also showed a 10-15% higher non-aqueous fraction in frontal white matter thanin splenium [116, 164], although a different study found little difference [165]. Still, becausewe are able to directly compare the aqueous and non-aqueous FIDs, our accuracy in thisregard is likely better than in previous work, with one caveat: with the ex-vivo sample usedhere, there is the potential for water loss.The intrinsic T1s of the aqueous and non-aqueous pools are also of interest. In all our samples(acquired at 4.7 T and 37 ◦C), T1,M ≈ 200 ms, which is similar to both the value of 171±22 msmeasured in human brain in-vivo at 1.5 T [166]; and the values of 250 ms at 3 T and 500ms at 7 T for human brain in-vivo [102]. Both of these studies used a two-pool model, sotheir values would include protons in what we consider the NM pool as well. However, ourvalue is shorter than another study’s 0.5–1 s measured in bovine optic nerve at 20 ◦C and 1.5T [151]. Evidently, the T1 of the non-aqueous pool is temperature and field dependent. Theformer is primarily a result of correlation time changes from increased or decreased thermalmotion away from physiological temperatures [83, 84, 167]. The field-dependence, discussedin Section 2.6, arises from T1 relaxation’s sensitivity to fluctuations at ω0 and 2ω0 (ω0 isthe Larmor frequency) [3, 10, 102, 168]. In contrast, the intrinsic aqueous protons alreadyexperience significant averaging, so their T1s (T1,MW and T1,IEW) don’t depend as stronglyon the field [102]. Our data suggest a large, >1 s T1 for both MW and IEW. However, thevalues obtained from the data here may be imprecise.Another important quantity to compare with previous work is the MW/IEW “exchange93time”. Using the T ∗1 associated with the MW/IEW exchange eigenvector, our samples are inthe range of about 90–130 ms. These are perhaps slightly faster than with previous reportedvalues of similar quantities, including ~200 ms (for human white matter in-vivo [169]), 127ms (for human white matter in-vivo [170]) ~220 ms (bovine splenium ex-vivo [145]), and~140 ms (bovine white matter ex-vivo [142]). Still, this confirms previous conclusions thatMW/IEW exchange has only a slight effect on the MWF measurements. Indeed, the error inthe MW amplitude due to exchange during the CPMG, (MMW(∞)− M˜MW(∞))/M˜MW(∞),is about 20% for WM-fr and for WM-sp1/sp2.Turning now to the FID results, there are relatively few studies which have directly observedthe non-aqueous proton FIDs in CNS tissues (e.g. references [89, 142, 148]). In fact, mostmeasurements of the super-Lorentzian lineshape in white and grey matter have been pre-sented in qMT studies [79, 81, 93, 94, 116, 171, 172]. In those experiments, the non-aqueousprotons are indirectly observed via their influence on the aqueous protons’ magnetization.Grey matter is less frequently studied this way, given its smaller non-aqueous pool [171].Still, qMT studies have generally shown that the effective T2—the T2 that is measured ifone assumes a Lorentzian lineshape—for the non-aqueous signals is similar in both whiteand grey matter (see Sled [171] and references therein). Although assigning an effective T2to the distinctly non-Lorentzian non-aqueous NMR signal is crude, it is consistent with ourresults: The wide super-Lorentzians in our white (13.5–16.2 kHz) and grey matter (18.8kHz) samples have similar spectral widths. Techniques to perform in-vivo observations ofthe non-aqueous signal also support this: in T ∗2 measurements in ovine (sheep) brain, Fan etal. found white and grey matter values of 209±9 µs and 258±4 µs respectively [173].The similarity between the non-aqueous lineshapes in white and grey matter may hint at theinability of FID fitting to distinguish between M and NM protons (those non-aqueous protonswithin the myelin bilayers and those outside it). If NM had a very different lineshape, thiswould dominate in grey matter where there is little myelin. However, this doesn’t appear tobe the case. There is an analogous observation in the aqueous protons: nothing in our resultssuggests MW and IEW have different lineshapes, a possibility previously raised [169]. Whilethe broad linewidths from field inhomogeneities may have been a limitation in this regard,the same result was present in a high-resolution study of rat thoracic spinal cord [89]. Thatsaid, fitting the non-aqueous component is intrinsically difficult compared to the aqueouscomponent; in the frequency domain its highest amplitude is ~50× lower than the aqueouspeak (Fig. 5.5), and in the time domain its longest signal lasts for ~1/300 as long as theaqueous FID.The lack of a MW pool in the GM-bg sample is surprising. Indeed, grey matter has signif-94icantly less myelin, yet in-vivo human MWF measurements typically show non-zero values(eg. Laule et al. [140]). It may be that the little myelin present in our GM-bg was damagedduring sample preparation and loading. Due to its higher water content, grey matter tissueis significantly more delicate than white matter tissue. Still, with only one grey matter sam-ple, it is hard to draw any definitive conclusions, and more measurements would be required.This is especially true given that GM-bg is sub-cortical grey matter; hence, future workshould also include samples of cortical grey matter as well.5.5.2 Imaging applicationsIt is unlikely the four pool model could ever be completely characterized in-vivo, given thenumber of parameters involved. Moreover, acquiring the FIDs of both the aqueous and non-aqueous protons is probably unfeasible on clinical MRI systems due to the short, intense,broadband pulses required. Instead, this work may prove useful in experimental design andanalysis, particularly when there is an interest in the MW pool signal.The total aqueous signal, MMW(TI) +MIEW(TI), is approximately what is measured in animaging experiment (BW is excluded since it is external to our samples). As an example, wecan analyze this quantity in detail for IR-soft and IR-hard. Following standard procedurein IR experiments, each recovery curve is fit to a multiple-component inversion-recoveryequation,MMW(TI) +MIEW(TI) =n∑i=1ai(1− fi exp(−TI/τi)). (5.25)Here ai is the amplitude of the component with an apparent relaxation time of τi and fi itsinversion efficiency. We use the generic notation ai instead of Mi(∞) to emphasize that eachcomponent is most likely associated with multiple pools—and using this type of analysisalone, we can’t know which pools these are, only what their combined magnetization is.Fig. 5.14A shows how a single component fit is clearly inadequate in the soft-IR experiment:it requires two components (τ1 = 1.4 s and τ2 = 63 ms), whereas in plot B the IR-hard fitswell with just one (τ1 = 1.4 s). These relaxation times are comfortably similar to two of theT ∗1 s (71 ms and 1.27 s) from the four pool model fitting (Table 5.2B). Taken together, theresults strongly suggest that broadband inversion and selective inversion will generally resultin measurably different T1 relaxation values. Note that broadband inversion is possible inclinical MRI using adiabatic pulses [174].This behaviour—where the value and number of components measured depends on the initialconditions—has been discussed by some recent publications [102, 142, 145–147]. Both twoand four pool models can explain this result. However, the eigenvector formalism that we95Figure 5.14: One and two-component fits to IR-soft and IR-hard for sample WM-fr. Thesewere fit to Eq. 5.25. The IR-soft curve in (A) appears to be bi-exponential (τ1 = 1.4 s andτ2 = 63 ms), whereas the IR-hard curve in (B) is mono-exponential (τ1 = 1.4 s).used (following Barta et al. [145]) can give a particularly clear depiction of how the initialconditions affect T1 relaxation. This is made explicit by Fig. 5.15 where the eigenvectorcoefficients are plotted. Plot A are these coefficients (the cjs in Eq. 5.7), showing which T ∗1components are excited by the preparation pulses. Plot B shows cp(∑4i=1 |v′pi|), where eachcoefficient is multiplied by the sum of the absolute values of the eigenvector components. Thisroughly corresponds to the magnitude of the perturbation from equilibrium correspondingto that eigenvector.Turning again to the contrast between IR-hard and IR-soft, the coefficient plots make afew things clear. First, the most prominent appearance of the shortest T ∗1 time (23 ms) isin the IR-soft experiment; in IR-hard, it is negligible. In fact, IR-hard primarily excitedthe T ∗1 = 1.3 s eigenvector, consistent with Fig. 5.14. The IR-soft behaviour is remarkablydifferent, despite having almost exactly the same total aqueous signal amplitude. Here, thesoft inversion pulse excited all components except T ∗1 = 130 ms. The nonzero signal strengths(plot B) from the other eigenvectors hints they should all be observable. Indeed, even inthe bi-exponential T ∗1 fit of IR-soft data above (Fig. 5.14A), there appears to be a smallcomponent in the <100 ms range, likely corresponding to the 23 ms T ∗1 . Still, it is not verywell-defined, since the lowest two T ∗1 s were separated only by a few tens of ms (23 ms vs. 74ms). The difficulty separating these components was one of the reasons for up-weighting thestart of MW residuals during the four pool fitting.This type of eigenvector analysis may be useful when interpreting the T ∗1 components mea-sured in a particular experiment. Importantly, it can be applied to two or three pool modelsas well, since—as we discuss next—the four pool model may not always be necessary.96Figure 5.15: The eigenvector excitation and aqueous pool magnetizations of the initial con-dition for sample WM-fr. (A) The eigenvector coefficients cjs, showing which T ∗1 componentsare excited by the different preparation pulses. (B) The same coefficients weighted by thescaled eigenvector sizes. This is approximately a measure of the expected signal size fromeach component. In IR-hard and IR-soft, there are distinctly different sets of eigenvectorswhich will relax. This is despite the aqueous magnetization being essentially the same, whichis shown in (C).975.5.3 Is the four pool model necessary to understand T1 relax-ation?The comparison of the MW weighted and unweighted four pool model fits with the threepool model fits (Table 5.3 and Fig. 5.12) makes it clear that while the four pool model’sfits were superior, none of the three performed best in all pools simultaneously. What’smore, different models performed better in different experiments. The fact that we had toup-weight the start of the MW data in order to fit the ~30 ms T ∗1 component indicates thatthe experiments did not equally expose all time constants. In Barta et al.’s recent work,they fit only the MW and IEW signals from IR-soft and IR-hard experiments similar to theones performed here. Yet, their fit picked up the ~30 ms T ∗1 component without any need toweight the MW residuals as we have done [145]. This is an indication that our inclusion offour more experiments washed out the importance of the ~30 ms component’s appearancein the IR-soft experiment.To summarize, the four pool model provides the most comprehensive description of relaxationin all the protons at once, and future refinements will probably help improve its accuracy.However, this information may not always be required. For example, the commonly-usedtwo pool model (Section 4.4.2) ignores the distinction between MW and IEW on the onehand, and M and NM on the other. This is reasonable in many applications given the MW’ssmall size compared to the IEW. The eigenvector analysis just discussed could be a usefultool in determining which model is adequate for quantitative and/or qualitative modeling.5.5.4 LimitationsWhile the present study has shown the applicability of the four pool model to white matter,there were a number of limitations. Perhaps the most significant of these was the assumptionof equivalent M and NM pool sizes. Myelination varies in different brain structures, andpresumably this results in different proportions of M and NM pool sizes [65]. At present,there is no clear way to separate the FIDs from these two non-aqueous pools, although someideas are given in the last chapter. When this proportion was allowed to vary, it was poorlyconstrained by the data from these experiments and tended to put all the protons in one ofM or NM and none in the other. Histology may yield better estimation in specific tissues,and one recent study applied this to quantify the myelin volume fraction in mice [117].On the experimental front, the choice of initial conditions could be improved in similar futurestudies. The need for additional weighting in order to fit the short ~30 ms T ∗1 component98means that most of the experiments used here did not sufficiently excite this correspondingeigenvector. Eigenvector analysis (the plots in Fig. 5.15) may be a useful tool in choosinggroups of experiments in the future. Combinations with MT and ihMT (see the next twochapters) could also be explored. Because we can directly measure the magnetization afterthe preparation pulses, these can be arbitrarily complicated without concern about modelingthem. Some suggestions for future experiments are given in the last chapter.Regarding the samples, experimental constraints meant that we were limited in the numberof samples we could study. With only three white matter samples, one grey matter sample,and all of them coming from the same brain, more work will be needed before the resultshere can be conclusively generalized. Sample preparation could also be improved, perhapsby soaking the tissue in D2O to reduce the intensity of the aqueous peak. This approach wassuccessfully used by Willhelm et al. in spinal cord and by Fan et al. in ovine brain [89,173].99Chapter 6Is “inhomogeneous” MT mis-named?6.1 IntroductionInhomogeneous Magnetization Transfer (ihMT) seems to show enhanced contrast in ma-terials containing lipid bilayers, such as myelin.1 We introduced the ihMT technique inSection 4.4.3, and in the work below we investigate its physical origins in detail. The origi-nal ihMT paper by Varma et al. suggested that it relied upon the non-aqueous spectrum oflipids being inhomogeneously-broadened [118]. They claimed it would be possible to “burn ahole” in such a spectrum, thereby causing sensitivity to the prepulse frequency [118,120,175].Fig. 6.1 is reproduced from their paper and its caption outlines this hypothesis, which wasthe origin of ihMT’s name.Portis was the first to define homogeneous and inhomogeneous broadening of magnetic res-onance spectra (the publication focused on electron spin resonance, but the same physicsapplies to NMR) [176]. That paper defined a homogeneously-broadened spectrum as onewhich spreads any absorbed energy equally throughout the spin system. In other words,an rf saturation pulse will attenuate the magnetization in a homogeneously-broadened spec-trum equally. Additionally, Portis called spectra consisting of overlapping narrow lines fromisochromats inhomogeneously broadened. In such a spectrum, a low power rf pulse will sat-urate a localized frequency range only, corresponding to the spins whose resonance conditionis met. Thus, one will “burn a hole” in the spectrum.Maricq and Waugh introduced a slightly different definition of homogeneous and inhomo-geneous in their paper on magic angle spinning (MAS) [177]. If a spectrum is broadened1This chapter is modified from the following publication:AP Manning, KL Chang, AL MacKay, CA Michal, Journal of Magnetic Resonance 274, 125–136 (2017) (Hz)Intensity (a.u.)Figure 6.1: The original explanation of ihMT, which suggested that the prepulses were burn-ing holes in the non-aqueous spectra of lipids. This would necessarily require the spectrumto be inhomogeneously-broadened under the definition of Portis (see text). Under theseconditions, the S− (A) and S+ (B) prepulses would burn a hole at single offsets, whereas theSdual prepulse (C) would burn a hole at both offsets. The difference (D) would lead to anobservable ihMT signal. Figure modified slightly for readability from Varma et al. [118], ©2015, with permission from John Wiley & Sons, Inc.101by a Hamiltonian Hˆ (such as the many-spin dipolar Hamiltonian) and [Hˆ(t1), Hˆ(t2)] = 0 attimes t1 6= t2, then by their definition the spectrum is inhomogeneously-broadened. DuringMAS experiments, one would see spinning sidebands as a result. These are NMR signalsthat appear in the spectrum separated by the NMR spinning frequency. Conversely, if[Hˆ(t1), Hˆ(t2)] 6= 0, then under this definition it is homogeneous and no spinning sidebandsare seen.Portis’s definition relies only on the spectrum, whereas Maricq and Waugh’s definition arisesfrom the properties of the Hamiltonian. Hence, the nomenclature in the literature is incon-sistent. Schmidt-Rohr and Spiess discuss this point in Section 3.13.4 of their book [11]. Infact, in certain situations, an inhomogeneous Hamiltonian may give rise to a homogeneously-broadened spectrum [11,177]!To connect this to the lipid systems of interest to this thesis, we discussed in Section 4.2how dipolar couplings within methylene groups on lipid acyl chains lead to their super-Lorentzian 1H lineshape. The rapid translational diffusion and spinning of the lipid moleculesresults in an effective homonuclear dipolar Hamiltonian that commutes with itself at alltimes while spinning [178]. High-resolution MAS spectra of lamellar lipids can be obtained,where spinning sidebands are evident [178,179]. Therefore, in these systems the Hamiltonianis inhomogeneous by the Maricq and Waugh definition. However, as we explore below, theoverlapping nature of the orientation-dependent subspectra in the super-Lorentzian lineshapecharacteristic of lipids does not permit asymmetric hole burning—so the spectra are notinhomogeneous in the Portis sense.In any case, in addition to the explanation based on inhomogeneous broadening, more recentwork by Varma et al. used Provotorov Theory to describe the fundamental physics of ihMT[30,119]. When rf irradiation is applied at one offset, magnetization is able to flow betweenthe Zeeman and dipolar reservoirs, but when rf is applied at both offsets simultaneously, thecoupling between the reservoirs is severed. In this framework, the dipolar relaxation time T1Dis a key parameter in determining the ihMT signal magnitude. Notably, the application ofProvotorov Theory does not require any assumptions about the type of spectral broadening,only that there be dipolar couplings present.To summarize, to date there are two possible explanations for ihMT: one based on hole-burning, which assumes an inhomogeneously-broadened spectrum in the Portis sense for lipidmembranes’ methylenes; and one based on Provotorov theory, which makes no assumptionson the type of broadening. In this chapter, we test both explanations by focusing on the NMRbehavior of the non-aqueous protons. In the Theory section, we describe the fundamentalphysics of ihMT rigorously, first through a minimal model of an isolated methylene group102using density matrices, and then for any system using Provotorov Theory. In doing so wecharacterize the fundamental timescales of the non-aqueous protons required for a non-zeroihMTR. Our experimental results show that the model lipid system, Prolipid-161 (PL161),does not exhibit hole-burning and that its non-aqueous spectrum behaves like that of aweakly-coupled ensemble of strongly-coupled spin pairs. Moreover, our results make clearthat some samples with homogeneously-broadened lineshapes do exhibit ihMT, refuting theexplanation based upon inhomogeneous broadening. Taken together, we show that ihMTarises simply from the dipolar interaction, not from a specific broadening mechanism.6.2 TheoryThe physics of both MT and ihMT may be considered as two separate processes: 1) theirradiation of non-aqueous protons, and 2) magnetization exchange between the non-aqueousand aqueous protons. A complete model of ihMT using Provotorov Theory that considersboth processes (by including aqueous and non-aqueous protons) has already been publishedby Varma et al. [119], and we do not seek to replicate it here. What sets ihMT apart fromMT is only a change in prepulse irradiation (comparing single vs dual-sided irradiation).Therefore, fundamental understanding why ihMT appears more selective to lipids requiresmodeling the behavior of the non-aqueous protons only, which we do in the models below.Both of our models describe calculation of a “non-aqueous ihMTR” in analogy to the “aque-ous ihMTR” typically used in ihMT. Experimentally, since NMR spectroscopy can detectboth the non-aqueous and aqueous parts of the proton spectrum, the two ihMTRs are foundby integrating the corresponding parts of the spectrum. In a sample that exhibits MT sig-nals, a non-aqueous ihMT (where the non-aqueous ihMTR 6= 0) will cause an aqueous ihMTbecause of magnetization exchange.Our first model of ihMT is based on the simplest system in which ihMT can occur: a spin-1system. Our second model uses Provotorov theory to describe ihMT. It predicts ihMT canarise in spectra with either inhomogeneous or homogeneous broadening.6.2.1 ihMT model 1: a simple spin-1 system6.2.1.1 Selective and non-selective pulses in a spin-1 systemWe now consider the behavior of a simple spin-1 system, which is motivated by the behaviourof coupled protons in the methylene groups in lipid acyl chains (Section 4.2). We will103use this to both semi-quantitatively model the non-aqueous protons in the lipids and as astraightforward means of understanding the general mechanism of ihMT.The Hamiltonian of a dipolar-coupled spin-12 pair or a single spin-1 particle was given inEq. 2.30. Repeating it here:Hˆ = HˆZ + HˆD= −ω0Iˆz + ωD3 (3Iˆ2z − 21).(6.1)The first and second terms are the Zeeman and dipolar Hamiltonians, ω0 is the Larmor fre-quency, and ωD is the dipolar interaction strength (where ωD  ω0). In thermal equilibriumthe density matrix isρ0 = M0 diag(1, 0,−1) = M0Iˆz. (6.2)The spectrum of this system, g(ω), following a broadband on-resonance rf pulse isg(ω) ∝ δ(ω0 + ωD) + δ(ω0 − ωD). (6.3)This is a doublet centered at ω0 with a splitting of 2ωD.Now, a non-selective pulse is applied at ω0 with an amplitude ω1  ωD such that bothtransitions are affected. If the pulse is applied on the y-axis in the rotating frame, then viaEq. 2.22,ρ = M0 cos(ω1τ)Iˆz − sin(ω1τ)Iˆx,where τ is the duration of the non-selective pulse. In this spin-1 system, this is exactly thecase of the dual ihMT prepulse, so we may writeMdual = 〈Iˆz〉= M0 cos(ω1τ). (6.4)Alternatively, we may also use a selective pulse with ω1  ωD such that it is on resonancefor only one transition. If the pulse frequency is ω0 +ωD, then we can analyze just the {1, 0}subspace of ρ [9, 180]. In this subspace, there is one transition, so it is identical to a spin-12104particle. In equilibrium, the subspace isρ0,1/2 = 1 00 0= M02 1 00 1+ M02 1 00 −1= M02 11/2 +M0Iˆz,1/2,where we have used spin-12 operators. The same subspace in the rf Hamiltonian isHˆrf = 0 −iω1iω1 0= 2ω1Iˆy,1/2.Outside of this subspace, Hˆrf is zero, since the pulse is selective to the one transition. Again,using Eq. 2.22, we find that the density matrix evolves under this Hamiltonian toρ1/2 = M02 11/2 +M0 cos(2ω1τ)Iˆz,1/2 −M0 sin(2ω1τ)Iˆx,1/2.The nutation frequency is twice as fast than in the dual case. This is a well-known effectand is seen in the nutation of the central transition in quadrupolar couplings as well [9,180].Now, substituting this subspace back into the complete spin-1 density matrix, we find for aselective pulse on a single transitionMsingle = M0(34 +14 cos(2ω1τ)). (6.5)Selectively irradiating the other transition at ω0 − ωD yields the same result. Application to ihMTWe now show how the spin-1 system is sensitive to the frequency of selective pulses explicitlythrough one simplified example. The selective pulses are used to saturate or invert one orboth transition populations. This has a corresponding impact on the spectral line amplitudes.Consider a selective pulse applied at ω0 + ωD, with an amplitude ω1 and pulse length τcalibrated to invert the transition (a pi pulse), yielding105ρ+ = M0 diag(0, 1,−1) = M02 Iz − M02 (3I2z − 21), (6.6)in which the subscript + on ρ+ indicates a pulse applied at a positive frequency offset. The(3I2z − 21) term indicates the presence of dipolar order. Similarly, applying the selective pipulse at ω0 − ωD to invert the other transition yieldsρ− = M0 diag(0, 1,−1) = M02 Iz + M02 (3I2z − 21), (6.7)In either case, the coefficient of the Iˆz term is reduced from the thermal equilibrium valueof M0 to M0/2. Next, we consider applying the same pulse power to both transitions simul-taneously. We know from Eqs. 6.4 and 6.5 that if 2ω1τ = pi in the selective case, the flipangle in the dual case will be pi/2 (half as much). Since we don’t care about the off-diagonalelements, we will call this a saturation pulse. Thus,ρdual = M0 diag(0, 0, 0), (6.8)where now clearly there is no magnetization or dipolar order.The difference from equilibrium for the single prepulse cases areM0−M0/2 = M0/2, whereasin the dual case it is M0 − 0 = M0. Therefore in his example, irradiating both transitionssimultaneously provides twice as much difference from the equilibrium value of 〈Iz〉 for thesame rf power. Calculating the non-aqueous ihMTR for this experiment, we haveihMTR = 〈Iz〉+ + 〈Iz〉− − 2〈Iz〉dual2〈Iz〉0= M0/2 +M0/2− 2(0)2M0= 1/2(6.9)Using Eqs. 6.4 and 6.5, we can determine this generally for any prepulse power, showing that106the spin-1 system always has a nonzero ihMTR:ihMTR = 〈Iz〉+ + 〈Iz〉− − 2〈Iz〉dual2〈Iz〉0=2M0(34 +14 cos(2ω1τ))− 2M0 cos(ω1τ)2M0= 3 + cos(2ω1τ)− 4 cos(ω1τ)4 > 0.(6.10)In the absence of spin-lattice relaxation, the limit τ →∞ averages the cosine terms to zeroand ihMTR approaches 3/4.If our spin-1 system here represents a typical methylene group in a lipid bilayer, then magne-tization exchange would take place with aqueous protons on the timescale of 10–100 ms. Thiswould decrease the aqueous magnetization by an amount proportional to the non-aqueous〈Iz〉, thereby causing a non-zero aqueous ihMTR. This shows that a non-zero aqueous ihMTsignal is expected from a system consisting of water in contact with strongly-coupled spinpairs. Moreover, it shows that ihMT does not arise from a specific type of spectral broad-ening, but from dipolar couplings alone. Spectral asymmetry from dipolar orderThe population differences in the density matrices of Eqns. 6.6, 6.7, and 6.8, suggest thatinverting (or saturating) the transition of one of the lines in the doublet spectrum shouldresult in an enhancement in the amplitude of the other line. Perhaps surprisingly, a non-selective pi2 (or any (2n + 1)pi2 ) pulse applied to this system produces a spectrum with bothpeaks of the doublet having identical amplitudes. However, if the flip angle is 6= npi2 , anintensity difference between the two lines is observed. This is a manifestation of the “flip-angle effect” [181,182].The diagonal of the density matrix following an arbitrary prepulse can be written asρ = diag(ρ11, ρ22, ρ33) = aIz + b3(3I2z − 21) + c1, (6.11)where a = 12(ρ11 − ρ33), b = 12(ρ11 + ρ33)− ρ22, and c = 13(ρ11 + ρ22 + ρ33). The off-diagonalcomponents are set to zero, which may be accomplished experimentally by appropriate phasecycling. Next, we calculate the effect of a hard observe pulse of flip angle α along the +xaxis using product operators (Eq. 2.22). Keeping only the observable terms, we findρ = −a sinαIy − b cosα sinα(IzIy + IyIz). (6.12)107The Iy term is from the initial Zeeman magnetization (given by a), and the IzIy+IyIz term isantiphase magnetization from the initial dipolar order (given by b). Evolving this expressionunder the Hamiltonian in Eq. 6.1 gives the amplitudes of the two transitions:A± = a sinα± b cosα sinα, (6.13)where A± are the amplitudes of the transitions at ±ωD. It is clear that when the flip angleis pi2 , A+ and A− are identical (A±(pi2 ) = a =12(ρ11 − ρ33)), and both lines have identicalamplitudes regardless of the amount of dipolar order. However, for flip angles 6= npi2 , theamplitudes of the lines will differ by 2b cosα sinα. For small flip angles, where cosα ≈ 1,A± = (a± b) sinα, so that A+ = (ρ11−ρ22) sinα and A− = (ρ22−ρ33) sinα as expected. Forall flip angles, the amplitudes of the two lines are correlated. The amplitude of one cannot bereduced independently of the other. This is in contrast to hole burning, where the amplitudesof the overlapping lines that make up the spectrum can be changed independently.Experiments like the ones considered here, demonstrating the interplay between dipolarcoupled spectral lines, have been carried out experimentally on the ensemble of dipolarcoupled 1H spin pairs in oriented 5CB (which has a spectrum of two lines) by Lee et al. [37].When one of the spectral lines of the dipolar coupled pairs was irradiated, the intensitiesbehave as predicted by the model described above. Nakashima et al. have also shown similareffects in the behavior of spin-3/2 systems in 23Na NMR of NaNO3 crystals [182].Spin-1 behavior is not immediately obvious from the proton NMR spectra of lamellar lipids,where the chain-position dependent coupling strength, along with residual inter-molecularand inter-methylene couplings broaden the doublets to the point where the lipid lineshapeis well described as a superposition of Gaussian singlets having widths and intensities mod-ulated by P2(cos θ) [80, 87,91]. This fact is explored more deeply in Section ihMT model 2: a homogeneously-broadened system usingProvotorov theory6.2.2.1 The Provotorov equations for continuous-wave ihMT prepulsesWhile the spin-1 theory just described forms a minimal model demonstrating the origin ofthe ihMT effect, a general approach for all systems is based on Provotorov Theory. Thisdescribes the evolution of dipolar-coupled spins under weak rf irradiation (i.e. 2piν1  ωD),such as ihMT prepulses [30, 183, 184]. We introduced Provotorov Theory in Section 2.5.6and derive it in Appendix A.108Previous ihMT studies have used two varieties of prepulses. The “Continuous-Wave” (CW)type applies a single rectangular prepulse, which is sine-modulated in the case of the Sdualexperiment [118]. The “pulse-train” type uses prepulses consisting of a train of shaped pulses(typically Hann or Gaussian) [84,119,120]. We analyze the behavior of a coupled spin systemunder both varieties. Because our experiments use CW prepulses, their analysis is presentedbelow. Appendix D contains a similar treatment of pulse-train prepulses, which are commonin imaging applications. The derivation below follows the approach by Lee et al. [37, 38].We briefly review the Provotorov equations. The density matrix for a dipolar-coupled systemin a rotating frame at angular frequency ω0 + 2pi∆ is [30, 38]ρ = 1− (2pi∆)βZIz − ωDβD(HˆDωD). (6.14)Or, as a vector with {Iz, HˆDωD } as the basis (the 1 term is dropped):ρ = −(2pi∆)βZ−ωDβD . (6.15)Here, βZ,D are the inverse spin temperatures for the Zeeman and dipolar reservoirs, and thevector basis is {Iˆz, HˆDωD }, with HˆD as the dipolar Hamiltonian. When weak rf is applied at asingle offset ∆, the Provotorov equations including spin-lattice relaxation are [30]dρ±dt= W −1− 1WT1 ΩΩ −Ω2 − 1WT1Dρ± + 〈Iz〉0T10 , (6.16)withW = pi(2piν1)2g(2pi∆) (6.17)Ω = 2pi∆ωD. (6.18)Here, ν1 is the prepulse amplitude (in Hz), and g(2pi∆) the symmetric, normalized lineshape.ωD is the RMS average dipolar interaction strength (the residual dipolar couplings).Eq. 6.16 describes the evolution during the S+ or S− prepulse, the only difference betweenthe two being the sign of Ω and consequently the sign of dipolar magnetization. However,qualitatively different behavior occurs during the Sdual prepulse, where rf irradiation withamplitude ν1/√2 is applied to dual offsets ±∆ simultaneously. This causes the Zeeman and109dipolar reservoirs to decouple, leading to [30]dρdualdt= W −1− 1WT1 00 −Ω2 − 1WT1Dρdual + 〈Iz〉0T10 . (6.19)Applying standard differential equation techniques (e.g. see reference [154]) to solve Eqs.6.16 and 6.19 under a prepulse of duration τ yieldsρ(t) = c1v1eλ1Wτ + c2v2eλ2Wτ + v0 (6.20)whereλ1,2 = −12 1WT1+ 1WT1D+ 1 + Ω2 ±√( 1WT1− 1WT1D+ 1− Ω2)2+ 4Ω2 (6.21)v0 =〈Iz〉0WT1(λ1 − λ2) λ2+Ω2+(WT1D)−1λ2 − λ1+Ω2+(WT1D)−1λ1Ωλ2− Ωλ1 . (6.22)Lee et al. also give the steady-state solution vector v0 explicitly, assuming the system startsfrom thermal equilibrium [37]. The non-aqueous ihMTR as a function of prepulse durationτ isihMTR(τ) = 〈Iz〉+(τ) + 〈Iz〉−(τ)− 2〈Iz〉dual(τ)2〈Iz〉0 . (6.23)In Appendix C, we present an analogous electronic circuit of the above equations. Model detailsWe now consider the behavior of the non-aqueous protons under the three different typesof prepulses, assuming the system starts at equilibrium, ρ(0) = 〈Iz〉0Iz. In the S+ and S−cases, 〈Iz〉 decays bi-exponentially since the eigenvalues Wλ1 and Wλ2 are always negative.This is in contrast to the mono-exponential decay behavior in the Sdual case.Under the Sdual prepulse, the two reservoirs are decoupled. In the Zeeman reservoir, relax-ation and saturation are responsible for a loss of magnetization at the rate (W + T−11 )〈Iz〉.The solution to Eq. 6.19 is〈Iz〉dual(τ) = 〈Iz〉0 1 +WT1e−(W+T−11 )τ1 +WT1,〈HˆDωD〉(τ) = 0.(6.24)110The magnetization in the Zeeman reservoir decays exponentially toward the steady-statevalue (1 +WT1)−1 at a rate that is independent of T1D, because no magnetization enters thedipolar reservoir.During the S+ prepulse (or with Ω → −Ω, the S− prepulse), the dipolar reservoir mustalso be considered, where magnetization saturates and relaxes at a rate (WΩ2 + T−11D )〈 HˆDωD 〉.Magnetization flows into the dipolar reservoir from the Zeeman reservoir at a rate WΩ〈Iz〉.If T1DW  1, rapid relaxation means very little magnetization is left to flow back to the Zee-man reservoir. The net result is that 〈Iz〉+(τ) (and 〈Iz〉−(τ)) behaves similarly to 〈Iz〉dual(τ)and ihMT is negligible. Alternatively, if T1DW & 1, then non-negligible magnetization flowsback into the Zeeman reservoir at a rate WΩ〈 HˆDωD〉. This leads to a significant deviation inbehavior of the Zeeman magnetization from the Sdual case, so ihMT is measurable.From this, we see that WT1D is a key parameter in controlling ihMT. If W is held constant,then, as others have pointed out, ihMT generates T1D-dependent contrast in MRI [84, 119,133]. However, since W is, within limits, under control of the experimenter (by controllingν1), it may afford detection of ihMT in systems with short T1D.Fig. 6.2 explores the dependence of non-aqueous ihMTR on prepulse duration, WT1D andoffset frequency. Fig. 6.2A shows the ihMTR dependence on prepulse length, τ . Generally,there is a peak followed by a falloff to a lower steady-state value. At short times (τ ∼ 1ms), the bi-exponential decay of 〈Iz〉+(τ) (and 〈Iz〉−(τ)) and the mono-exponential decayof 〈Iz〉dual are similar, so ihMTR is small. At times τ ∼10–100 ms, the difference betweenthe two behaviors is at a maximum, leading to a maximum ihMTR. At longer times, thedifference decreases as the system achieves steady state, yielding constant ihMTR values.Also shown in this plot is that ihMTR increases with WT1D.Fig. 6.2B isolates the WT1D dependence more explicitly for a prepulse duration of τ = 500ms. This plot shows similar behavior in ihMTR as in Fig. 6.2A, where ihMTR is plottedas a function of τ . Analysis of Eqs. 6.21 and 6.24 shows that only Wτ appears, not τalone. Therefore, holding τ and T1D constant and varying W produces the same behaviouras holding T1D constant and varying Wτ . This plot also shows the highly-sensitive T1D-dependence, which is approximately linear unless T1 ∼ T1D.Fig. 6.2C shows the dependence on the offset frequency of the prepulse. The general shapeof these curves is similar to plots of aqueous ihMTR vs. |∆| in previous studies [84,118–120].At short prepulse lengths, the maximum ihMTR occurs near the resonance condition in thelocal field (Ω = 1). However, at long prepulse lengths, the maximum shifts to higher offsetfrequencies where saturation effects are suppressed.This model considers a system of isolated, non-aqueous protons only. However, including an111Figure 6.2: Simulation of non-aqueous ihMT in an isolated spin system using CW prepulses(Eq. 6.23). No coupling to aqueous protons is included. (A) The dependence of ihMTR onprepulse length. If couplings to aqueous protons were included, the long-time behavior woulddeviate from what is shown here. (B) The effect of WT1D and T1D. The T1D-dependenceis approximately linear and for WT1D  1, ihMT is unobservable. (C) The dependence onoffset for different prepulse lengths. The resonance condition in the local field is Ω = 1.Parameters unless otherwise indicated: T1 = 1 s, Ω = 1, ν1 = 400 Hz, with g(2pi∆) as aGaussian with standard deviation ωD = 10 kHz.112exchangeable aqueous proton pool would not change the qualitative behavior significantly,except in the case of ihMTR as a function of prepulse length (Fig. 6.2A). When ihMT isperformed in the presence of an aqueous proton pool, a longer prepulse produces a greaterchange in the aqueous magnetization due to the slow exchange. In this case, there will notbe a maximum in the aqueous ihMTR at a prepulse length of 10–100 ms.Appendix D contains simulations of the pulse-train model (Fig. D.2) similar to those shownfor the CW model. The term WeffT1D is shown to play the role of WT1D, where the effectiveW (Weff) is scaled by the duty cycle. Spectral asymmetry from dipolar orderAs in the spin-1 model, dipolar order in large spin systems gives rise to spectral asymmetry.Starting with the rotating-frame density matrix in Eq. 6.14, a hard pulse of flip angle α isapplied along y. After a time t, the components of magnetization are [30]〈Ix〉(t) ∝ −(βZ(2pi∆) sinα)f(t)〈Iy〉(t) ∝ (βD sinα cosα)df(t)dt .(6.25)Here, f(t) is the envelope of the FID. The resonance frequency at the center of the spectrumis ω0, so the spectrum (up to a constant) isA(ω) = [−βZ(2pi∆) sinα− (ω − ω0)βD sinα cosα] g(ω − ω0). (6.26)Here, g(ω−ω0) is the Fourier transform of f(t) and describes a symmetric spectrum centeredat ω0. The factor of ω − ω0 in the second bracketed term causes spectral asymmetry, whichis only visible when α 6= npi/2.We can use this equation to re-derive the spin-1 model amplitudes. Substituting the spin-1spectrum (Eq. 6.3) into the expression for A(ω), we calculateA± ∝∫ ±∞0A(ω)dω= −βZ(2pi∆) sinα∓ βDω0 sinα cosα,(6.27)which has the same form as Eq. 6.13, showing that it is applicable to any dipolar-coupledlineshape.113modulated @   (oset=±  )ooset =190o 0.70711ADRF/ARRFihMT: SdualihMT: S+, S-, S0 (1=0)Dipolar ordercreationoooset =Figure 6.3: Pulse sequences used in this work. The dipolar order creation sequence uses aGaussian prepulse. The two ihMT sequences were used to measure ihMTR. During theirprepulses the rf power was the same. The ADRF/ARRF sequence was used to measure T1D.6.3 MethodsPL161 samples were prepared by melting ca. 50 mg of prolipid-161 (Ashland Specialty Ingre-dients, DE, USA) at 80-90◦ C in distilled water and/or deuterated water (Cambridge IsotopeLaboratories, Inc. MA, USA). Samples of PL161/D2O (10%/90% w/w), PL161/D2O/H2O(10%/88%/2% w/w), and PL161/H2O (10%/90% w/w) were made. PL161 forms a lamellarliquid crystal with MT properties similar to those of myelin [118,185–187]. A 63 mg sampleof curly, black human hair was obtained 10–20 cm from the scalp. This was thoroughlywashed in water and soap then air-dried prior to measurement. A 61 mg sample of DouglasFir (Pseudotsuga menziesi) sapwood was obtained from a branch with a ∼3 cm diameterand was air-dried for 2 weeks prior to measurement. A 57 mg sample of Western Red Cedar(Thuja plicata) sapwood was obtained from a branch with a ∼1 cm diameter and dried thesame way. Beef (Bos taurus) tendon was obtained frozen from a local butcher. A 70 mgsample was extracted from the tendon sheath, patted dry, then sealed inside an NMR tube.Experiments on tendon were completed within 48 hours of thawing.The four styles of pulse sequences used are shown in Fig. 6.3. The dipolar order creationsequence was used for spectral asymmetry experiments. It features a short, intense Gaus-sian prepulse three standard deviations wide (typically, τ =1–3 ms, ν1 = 2.5 kHz, ∆ = 8kHz, δ variable). For ihMT-related experiments, the S+ and S− spectra were producedby a rectangular prepulse at offsets +∆ and −∆ respectively, whereas the Sdual spectrawere produced with a rectangular prepulse modulated by sin(2pi∆), thereby irradiating ±∆simultaneously. The S0 spectra had no prepulse. When observing the differences in thenon-aqueous portions of S0, S+, S−, and Sdual spectra, typically τ =2–50 ms and ν1 = 10kHz. When measuring aqueous ihMTR, typically τ = 500 ms and ν1 = 460 kHz. In bothcases, δ ≤ 0.5 ms. Lastly, the Adiabatic Demagnetization/Remagnetization in the RotatingFrame (ADRF/ARRF) sequence was used for measuring the dipolar order decay constantT1D, discussed in Section 2.5.5. It had a ramp time τ = 1 ms and a variable relaxation delay114δ. In all experiments, the observe pulse flip angle α was either 33◦ (pulse width 2 µs) or 90◦(pulse width 5.5 µs).All spectra were acquired at 21±1◦C using 64 acquisitions, a 10 s recycle delay, and wereprocessed with 500 Hz of Gaussian line broadening. ∆ = 0 kHz is approximately the centerof the non-aqueous lineshape. Experiments were carried out using a horizontal solenoid coilprobe on a 200 MHz home-built NMR spectrometer incorporating a digital receiver, basedon an Oxford Instruments 4.7 T, 89 mm bore superconducting magnet [188]. Curve fittingwas performed with SciPy’s least squares package [157]. Errors given on fitted parametersare one standard deviation found with the bootstrap method using 1000 permutations.6.4 Results6.4.1 PL161 spectral asymmetry from dipolar order1H NMR spectra of a PL161/D2O sample showing the effects of an off-resonance prepulse for33◦ and 90◦ observe pulses are shown in Fig. 6.4. The spectra consist of a super-Lorentzianarising from the non-aqueous lipid protons and a residual HDO line. Because τ = 1 msand δ = 0.5 ms, magnetization transfer to the aqueous protons is insignificant. Therefore,plotting S+ − S0 removes the HDO line and highlights differences between the non-aqueousS+ and S0. Spectra acquired with an observe flip angle of α = 33◦ (Figs. 6.4A and 6.4C)are asymmetric: the prepulses with ∆ = 8 kHz attenuate the spectrum near +8 kHz andenhance it near -8 kHz, indicating the presence of dipolar order. When α = 90◦ (Figs. 6.4Band 6.4D), the prepulse appears to attenuate the entire non-aqueous spectrum uniformly. InFig. 6.4, a single prepulse at positive offset was used. In work by Swanson et al. [84], plotsof S+ − S− for PL161 also show asymmetry from dipolar order, but remove the effects ofZeeman order, obscuring any potential hole-burning. Plotting S+ − S0 as done here showsboth the effects of both Zeeman order and dipolar order on the spectrum.These experiments show no evidence of hole-burning in the non-aqueous parts of the PL161spectrum as would be expected based on the explanation of the ihMT effect in Ref [118].The behavior is consistent with the presence of strong dipolar couplings within the lipidmethylenes. Two discrete spectral peaks are not observed for the non-aqueous componentdue to the orientation and chain-position dependence of the dipolar coupling strengths andresidual dipolar couplings to neighboring methylene groups.S0, S+, S−, and Sdual for PL161/D2O using CW ihMT sequences are plotted together inFig. 6.5. The purpose of these experiments is to highlight the response of the non-aqueous115−40−2002040−0.4− (C)Intensitydifference(a.u.)Frequency, f (kHz)S+ − S0−40−2002040(D)(B)Observe pulseflip angle α = 90◦01234(A)1HIntensity(a.u.)Observe pulseflip angle α = 33◦S0S+Figure 6.4: Effect of the observation flip angle α on the PL161 non-aqueous proton spectrum.For S+, a Gaussian prepulse (τ = 1 ms, ν1 = 2.5 kHz, ∆ = 8 kHz, δ = 0.5 ms) was appliedwith α = 33◦ (A) or α = 90◦ (B). (C) and (D) are difference spectra which highlight theasymmetry caused by dipolar order observed in α = 33◦ case.protons to the various ihMT prepulses. Again, enhancement of the non-irradiated side of thespectrum can be seen in S+ and S−. In contrast, the non-aqueous component of the Sdualspectrum is symmetric and strongly suppressed.We can approximate the degree of Zeeman and dipolar magnetizations in these four spectra.Integrals of the positive frequency (I>) and negative frequency (I<) sides of the non-aqueouscomponents of a spectrum S(f) are calculated byI> =∑fi=+80 kHzfi=+3 kHz S(fi),I< =∑fi=−3 kHzfi=−80 kHzS(fi).(6.28)Following the description of the spectral lineshape in the presence of dipolar order (Eq. 6.26),the sum and difference are approximate measures of the Zeeman and dipolar magnetizations,respectively:〈Iz〉 ≈ I> + I< (6.29)∣∣∣〈 HˆDωD〉∣∣∣ ≈ |I> − I<|. (6.30)Fig. 6.6 compiles these results for our samples, allowing comparison of the different behaviors.The spectra used in the calculations are shown in Figs. 6.5 and 6.7. The prepulse length τvaries and was chosen to give the maximum dipolar order in the S+ and S− spectra for each116−60−40−200204060Frequency, f (kHz)0123Douglas Fir0123PL161/D2O1HIntensity(a.u.)S0S+S−SdualFigure 6.5: Manifestation of dipolar order in ihMT. Spectral asymmetry from dipolar orderis present in S+ and S− but not S0 or Sdual. Sequence parameters are listed in the captionof Fig. 6.6.sample. Also, the recovery time δ was decreased for samples with smaller T1D values. Incontrast to the short duration used here, in ihMT-MRI experiments reported to date, τ + δis typically ∼ 1 s [84, 118–120]. However, a prepulse this long would push the non-aqueousprotons toward steady-state conditions, decreasing the visible differences in their portion ofthe spectra, observation of which is our goal for these experiments.In all of these samples, we see that non-aqueous ihMT occurs since 〈Iz〉dual < 〈Iz〉±. InPL161/D2O, these results are consistent with the behavior of the spin-1 view of the lipidspin system. The non-aqueous Zeeman magnetization 〈Iz〉 in the S+ and S− cases are 0.71and 0.70, respectively, which is a reduction of about 0.3 from the S0 case where 〈Iz〉 = 1.In the Sdual case the reduction is about twice as much, i.e. 〈Iz〉 = 0.34 ≈ 1− 2× 0.3. Thistwo-fold reduction in the case of Sdual is consistent with the predictions of the spin-1 modelabove.6.4.2 Flip-angle dependence of spectral asymmetryFig. 6.8A shows the observe pulse flip-angle (α) dependence of the non-aqueous spectrum ofPL161/D2O. I> and I< of S+−S0 as functions of α are fit to the A+ and A− line intensitiesof the spin-1 model (Eqs. 6.13 and 6.27). In order to account for B1 inhomogeneity effects,A+ and A− were multiplied by exp(−pi δB1 α/2piB1), where δB1 is the full-width at half-maxof a Lorentzian distribution of B1 field strengths.The best fit was found with a Zeeman order of a = 6.4±0.7, a dipolar order of b = 4.4±0.6,117Figure 6.6: Summary of the positive/negative frequency integrals I>/I<, approximatedipolar magnetization∣∣∣〈 HˆDωD〉∣∣∣, and Zeeman magnetization 〈Iz〉 for the non-aqueous protonportions of the S+, S−, and Sdual spectra from Figs. 6.5 and 6.7. I< and I> are normalized tothe S0 values. The ihMT sequences were used with ν1 = 1 kHz, |∆| = 10 kHz, and α = 33◦.τ was chosen to maximize the amount of dipolar order, and δ decreased for samples with ashorter T1D. PL161/D2O: τ = 50 ms, δ = 0.5 ms, Douglas fir: τ = 50 ms, δ = 0.1 ms, hair:τ = 2 ms, δ = 0.1 ms, tendon: τ = 2 ms, δ = 0.01 ms.118−60−40−200204060Frequency f (kHz) Cedar0.00.51.01HIntensity(a.u.)Hair0.00.51.0 TendonS0S+S−SdualFigure 6.7: Beef tendon, human hair, and Western Red Cedar sapwood spectra followingCW ihMT prepulses. These are used to calculate the Zeeman and dipolar magnetizationsshown in the chart of Fig. 6.6, which also lists the pulse sequence parameters. Spectralasymmetry from dipolar order is evident in all three samples. The hair and tendon spectrahad more line broadening applied due to their lower signal-to-noise ratio.0 90 180 270 360 450Observe pulse flip angle, α (degrees)−505S+−S0integratedintensitydifference(a.u.) (A)I>I<A± fit10−3 10−2 10−1 100 101Recovery time, δ (s)−20−10010S+−S0integratedintensitydifference(a.u.) (B)I>I<T1D , T1 fitFigure 6.8: Flip-angle dependence of spectral asymmetry and saturation method measure-ment of T1D in PL161. (A) The I> and I< integrals of S+−S0 as functions of α closely followEq. 6.13. A single Gaussian prepulse was used with τ = 3 ms, ν1 = 2.5 kHz, ∆ = +10 kHz,and δ = 0.5 ms. (B) Saturation method data, showing T1 and T1D relaxation in PL161 fromthe recovery of the S+ − S0 difference spectrum. I> and I< are fit to Eq. 6.32. A Gaussianprepulse with τ = 1 ms, ν1 = 2.5 kHz, ∆ = +8 kHz, and α = 33◦ was used. In both plots,deviations apparent in nearby data points provide estimates of the measurement error.119−60−40−200204060Frequency, f (kHz)01231HIntensity(a.u.)PL161/D2OS0ADRF/ARRFδ = 10 msFigure 6.9: An ADRF/ARRF spectrum of PL161/D2O. There are no contribution fromaqueous protons, since their dipolar couplings are averaged away. This also applies for lipidsoriented near the magic angle, hence the dip at 0 kHz.and a B1 inhomogeneity of γδB1/2pi = 11± 1 kHz (about 20% of ν1).6.4.3 PL161 dipolar order relaxationMeasurements of the dipolar order relaxation time T1D in PL161 were made with two differentmethods. The ADRF/ARRF sequence converts Zeeman magnetization to dipolar order andallows it to relax for time δ before reconverting to an observable signal. An example ofan ADRF/ARRF spectrum of PL161/D2O is shown in Fig. 6.9. In comparison to the S0spectrum, the residual HDO peak is absent and the peak of the super-Lorentzian is replacedwith a dip. The peak of the super-Lorentzian corresponds to lipids in bilayers whose normalpoints along the magic angle. When aligned with the magic angle, the intra-methyleneresidual dipolar coupling strength is averaged to near zero. The signal from these lipids andfrom the water are eliminated by the pulse sequence phase cycle.Measurements of T1D made with the ADRF/ARRF sequence were not well fit with singleexponential decays, likely due to the distribution of bilayer orientations and chain positions.Stretched exponentials of the form〈Iz〉(t) = C exp(−(t/T1D)s), (6.31)did adequately describe the data. Best fit parameters are given in Table 6.1. Our value of~60 ms for PL161/D2O is similar, but somewhat greater than, the 48.8 ± 2.5 ms measuredby Swanson et al. [84]. The discrepancy is probably due to their slightly higher sample tem-perature (25◦C) and their measurement technique. They used a Jeener-Broekaert sequence,which weights T1D distributions differently than ADRF/ARRF sequences [189].We have also measured T1D for PL161 in 88%/2% D2O/H2O and 90% H2O. Increasing the1200.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Recovery time, δ (s)−30−20−100102030Integratedspectrumintensity(a.u.)Entire spectrumAqueous line onlyFitFigure 6.10: Inversion-recovery T1 measurements in PL161/D2O. The aqueous proton in-tensity is integrated in a 1 kHz-wide window around the aqueous peak. Integrals ofthe entire spectrum and the aqueous protons only were fit to a stretched exponential ofthe form 〈Iz〉(t) = C [1− γ exp (−(δ/T1)s)] . The fit values are as follows. Entire spec-trum: T1 = 1.245 ± 0.007 s, γ = 1.831 ± 0.004, s = 0.818 ± 0.005. Aqueous peak only:T1 = 1.594± 0.005 s, γ = 1.961± 0.003, with s set to 1. Here, γ is the inversion efficiency, δis the recovery time, and C is a constant.aqueous proton concentration decreases the T1D, from 61 ms in D2O to 23 ms in H2O. Thissuggests that both spin diffusion from within the bilayer to the surface and exchange withaqueous protons at the surface, which destroys the dipolar order, are important contributorsto the rate.Another way to measure the T1D is by using the “saturation method” [190–192]. A weak, off-resonance pulse first creates dipolar order. Then, the sample’s T1D is extracted by observingthe decay of spectral asymmetry (assuming α 6= pi2 ) as a function of δ. We have performed thisexperiment on PL161/D2O. In the difference spectrum S+ − S0, the non-aqueous integralsI> and I< relax toward zero with time constant T1D. The sum I> + I< decays toward zerowith time constant T1. The data were fit simultaneously toI< = C1 exp(−δ/T1) + C2 exp(−δ/T1D)I> = C1 exp(−δ/T1)− C2 exp(−δ/T1D),(6.32)where C1,2 are constants. The results are shown in Fig. 6.8B. The best fit is T1D = 58±4 msand T1 = 390± 60 ms. This T1D value agrees with the value found from the ADRF/ARRFsequence, but the T1 value disagrees with measurements made with an inversion-recovery (IR)sequence. These data, which are shown in Fig. 6.10, were also fit to a stretched-exponential,yielding T1 = 1.245± 0.008 s. However, the IR non-aqueous signal is likely contaminated bythe residual HDO, causing the apparent T1 to shift toward the longer aqueous T1. This issupported by a fit to the aqueous peak intensities only, which was found to be longer (about1.6 s).121Sample T1D (ms) s Measurement TechniquePL161/D2O (10%/90% w/w) 61± 1 0.74± 0.01 ADRF/ARRF58± 4 1 (fixed) saturation methodPL161/D2O/H2O (10%/88%/2% w/w) 49.8± 0.8 0.70± 0.01 ADRF/ARRFPL161/H2O (10%/90% w/w) 23± 1 0.66± 0.02 ADRF/ARRFDouglas fir 2.6± 0.1 0.84± 0.05 ADRF/ARRFHuman hair 1.58± 0.03 0.84± 0.02 ADRF/ARRFBeef tendon 0.634± 0.010 1 (fixed) saturation methodTable 6.1: T1D values for our samples. T1D values greater than about 1 ms could be measuredusing the ADRF/ARRF technique. The saturation method had to be used to measure theshort tendon T1D. s is the stretched exponential parameter defined by Eq. 6.31. In the caseof Beef Tendon, a single exponential fit the data well, hence s was fixed at Dipolar order of homogeneously-broadened spin systemsHere, we show explicitly that homogeneously-broadened spin systems can behave similarlyto the PL161 spin system discussed so far, in agreement with the predictions from Provo-torov Theory. Measurements were made of three biological materials with homogeneously-broadened non-aqueous spectra: Douglas fir sapwood, human hair, and beef tendon. Fig.6.6 displays measures of their Zeeman and dipolar magnetizations following the ihMT pre-pulses, as well as their T1Ds. In contrast to PL161, these samples are not well-described bythe spin-1 model, due the large size of their coupled spin systems.The Douglas fir S0, S+, S−, and Sdual spectra are shown in Fig. 6.5. As in PL161, the S+and S− spectra display spectral asymmetry from dipolar order, as predicted by Eq. 6.26.The S0 lineshape is approximately Gaussian, due to the large, rigid spin system of woodconstituents including cellulose, xylan, and lignin [193]. Of these, cellulose makes up abouthalf the mass of wood. In Douglas fir, these exist as crystalline microfibrils with diametersof about 12–20 nm, interspersed with amorphous regions every 100–200 nm [194,195]. Fromthe aqueous and non-aqueous zero-time intercepts of the FID, the moisture content of theDouglas Fir sample was found to be about 10% [196]. At this level, most of the remainingwater is in the cell wall, where it hydrates the cellulose microfibrils [196]. Experiments on asample of Western Red Cedar sapwood (Fig. 6.7) gave similar behavior as the Douglas Fir.The spectra of Human hair (Fig. 6.7) also revealed the presence of dipolar order and asimilar ratio of aqueous to non-aqueous intensities. In hair, crystalline α-keratin filamentsabout 8 nm in diameter are embedded in an amorphous hydrophilic keratin matrix [197,198].The water signal originates from this matrix, where the water hydrates the keratin filamentexteriors.1220.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8Relaxation time, δ (ms)−3−2−10123S+−S0Integratedintensitydifference(a.u.)I< I> FitFigure 6.11: T1D of beef tendon as measured by the saturation method. The data are fit toa mono-exponential decay (Eq. 6.31 with s fixed at 1), yielding T1D = 634± 10 µs. For S+,a Gaussian prepulse was used with a width τ = 3σ = 0.5 ms and ν1 = 7 kHz. The observepulse flip angle was 33◦.The behavior of tendon (Fig. 6.7) is similar, although the short T1D hinders the creation andobservation of dipolar order under these experimental conditions. The non-aqueous spectrumis visually similar to the PL161 super-Lorentzian. Tendon is highly-ordered, consisting oftriple helices of collagen, which are organized into fibrils that run parallel to the tendon [199].As large, relatively immobile molecules, collagen proteins have substantial dipolar couplings.At the same time, mobile water molecules permeate the entire fibril and are partially orderedby its structure [200].The T1D values from the three non-lipid samples (Table 6.1) reflect the microstructure of thesamples. As with lipid bilayers, dipolar order in the non-aqueous protons of hair, wood, andtendon is destroyed at the aqueous/non-aqueous proton interface due to proton exchange.When there is no exchange, the dipolar order can evolve for much longer: crystalline celluloseprepared in D2O has T1D ∼ 50 ms, for example [201].The T1D of tendon was too short to be measured by the ADRF/ARRF sequence. Instead,the saturation method was used. As shown in Fig. 6.11, this gave a value of T1D = 634± 10µs. This is comparable to the value of 230 ± 20 µs measured by Swanson et al. for chickenhyaline cartilage at 25◦ using a Jeener-Broekaert sequence [84]. The unique samples probablyaccount for the difference between these values. Tendon is mostly type I collagen (about86% of the dry weight), whereas hyaline cartilage is mostly type II collagen (about 60% ofthe dry weight) [202–204].The saturation method was used to measure the T1D of beef tendon. Fig. 6.11 shows thedecay of the spectral integrals of S+ − S0 and the fit to Eq. 6.31 with s was fixed at 1,describing a mono-exponential decay. In contrast to PL161, in tendon T1  T1D, so T1relaxation can be ignored during this analysis. A single exponential fit this data well so it1230 5 10 15 20 25 30 35 40Prepulse offset, |∆| (kHz)−202468Tendon02468 Hair02468ihMTR(%) Douglas Fir0102030 PL161/D2O/H2O0102030 PL161/D2ONon-aqueousAqueousFigure 6.12: Aqueous and non-aqueous ihMTRs as functions of offset frequency. CW ihMTsequences were used with τ = 500 ms, ν1 = 460 Hz, and δ = 0.5 ms. Samples withhomogeneously broadened spectra show non-zero ihMT, in contradiction to the hypothesisthat ihMT occurs due to inhomogeneous broadening.was not necessary to introduce the stretched exponential (i.e. s was fixed at 1).6.4.5 ihMT in lipids and homogeneously-broadened systemsFig. 6.12 shows the ihMTR for hair, Douglas fir sapwood, PL161, and tendon as functions ofoffset frequency |∆|. We have calculated two ihMTRs: one using aqueous proton intensities(integrated in a 1 kHz window around the aqueous peak), and one using non-aqueous protonintensities (= I< + I>, where I< and I> are defined in Eq. 6.32). The general shape thesecurves follow is consistent with previous studies and follows our model of non-aqueous ihMTRusing CW prepulses (Fig. 6.2C). At low offset frequencies, the ihMTR becomes unreliable,due to discretization of the sin(2pi∆t) modulation in the Sdual prepulse, the direct saturationof the aqueous protons, and smaller differences between the S+/S− and Sdual spectra.Another observation is that the ∼ 3× larger ihMTR seen in the non-aqueous protons of1240.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9ν21 (kHz2), ∝ WT1D02040PL161/D2O0510ihMTR(%)Tendon Non-aqueous AqueousFigure 6.13: ihMTR vs. prepulse power (which is ∝ WT1D) in the samples with the longestand shortest T1D values. These curves are qualitatively similar to those in our model of CWihMTR (Fig. 6.2B). A CW ihMT sequence was used with τ = 500 ms, δ = 0.5 ms, and withν1 calibrated by nutation of the water peak. |∆| is 11 kHZ in PL161 and 19 kHz in tendon,which are close to the values that maximize the ihMTR in Fig. 6.12.PL161/D2O than in the aqueous protons is due to diffusion-limited proton exchange to theaqueous pool. Even the slightly higher concentration of aqueous protons in the sample with2% H2O causes more similar non-aqueous and aqueous ihMTRs. This provides confirmationthat ihMT occurs due to the behavior of the non-aqueous protons’ dipolar reservoir only,and that it is observable via the aqueous protons because of proton exchange. In sampleswith abundant protons, under long prepulses the aqueous and non-aqueous ihMTRs will bevery similar.These results are in agreement with Varma et al.’s claim that ihMTR is highly sensitive toT1D [119]. However, the differences in T1D alone are not enough to account for the differencesin ihMTR between the samples. For example, despite having a T1D that is 20–40 times largerthan Douglas Fir or hair, PL161 has a maximum ihMTR only 4–5 times greater. This islikely due to differences in W contributing to WT1D. Different values of W arise due tothe different lineshapes g(2pi∆) and local field strengths ωD amongst the samples. Testingthe model’s predicted sensitivity to WT1D would involve quantifying these values accurately,and was not attempted here.We have not attempted to fit the results of Fig. 6.12 to the models described above, asour models do not include the effects of exchange with the aqueous proton pools. Varmaet al. did quantitative fits of a Provotorov-theory based model to experimental data andfound excellent agreement [119]. Our goal here is not to reproduce that work, but insteadto consider simplified models that provide greater understanding of the fundamental physicsunderlying the ihMT effect.125We have also measured ihMTR vs. prepulse power (ν21) for the samples with the highest(PL161/D2O) and lowest (beef tendon) T1Ds. These results are shown in Fig. 6.13. Theoffset frequencies were set to the values that maximized the non-aqueous ihMTR in Fig.6.12, these are 19 kHz for tendon and 11 kHz for PL161. As shown in Eq. 6.17, W ∝ ν21 .Furthermore, T1D is constant, so these are effectively plots of ihMTR vs. WT1D. Theseresults follow the behaviour predicted by our model of ihMT in Fig. 6.2B: at low values ofWT1D ihMTR increases linearly and at high values it saturates.6.5 DiscussionOur results have clearly shown that ihMT originates in the non-aqueous protons but isobserved in the aqueous protons following magnetization exchange. Moreover, ihMT does notarise because of inhomogeneous broadening but instead through dipolar couplings, and doesin fact occur in homogeneously-broadened spin systems. Finally, a hole cannot be burnedin the PL161 proton spectrum. Hole-burning is not a part of the mechanism responsible forthe ihMT signals observed here.These experimental findings are consistent with our theoretical models of the non-aqueousspin systems under ihMT prepulses. Spin-1 theory provides a minimal model allowing anintuitive understanding of the origin of the effect. The Provotorov theory-based approachallowed the identification of the timescale ratioWT1D for CW ihMT, andWeffT1D and τ2/T1Dfor pulse-train ihMT (see Appendix D), as key parameters controlling whether ihMT occurs.Moreover, the spin-1 model results can be derived from the Provotorov theory model, showingtheir generality beyond spin-1 systems. Neither of our models require assumptions aboutthe type of spectral broadening (homogeneous vs. inhomogeneous) present. All they requireis the presence of the dipolar interaction.We are now in a position to definitively answer the question, “When will a material havean ihMT response?” The short answer to this for CW ihMT sequences is that WT1D forthe non-aqueous protons must be “large enough”. A quantitative cutoff can be estimatedby estimating WT1D for tendon and PL161, which respectively have the lowest and highestihMTR among our samples. We assume a Gaussian lineshape with a standard deviation ωDequal to the offset that maximizes the ihMTR in Fig. 6.12 (about 11 kHz for PL161 and 19kHz for tendon). Then, using |∆| = ωD and ν1 = 460 Hz, we find WT1D ≈ 5.6 for PL161and WT1D ≈ 0.04 for tendon. From this, we may estimate a rule of thumb that ihMT willnot be easily observable unless WT1D > 0.01.Experimentally, for a given W , ihMT depends sensitively upon a material’s T1D. This value126reflects structure of the microstructure and its motion. Dipolar order relaxes from motionsthat occur on a timescale ∼ ω−1D , and is also destroyed by proton exchange. In most tissues,motions from exchange with water are probably the primary driver of T1D [201, 205]. Inproteins, reorientation of methyl groups also play a role, and in lipids the residual inter-and intra-molecular dipolar couplings weakly contribute as well [205, 206]. If a bottleneckexists for spin diffusion from a reservoir to these sites of relaxation, then T1D can be largeenough for ihMT to occur. This is exemplified using simple spin diffusion models of PL161and cellulose crystallites.Lamellar lipids tend to have very slow spin diffusion along the lipid tails due to the weakinter-methylene coupling. In contrast, dipolar order relaxation proceeds rapidly once mag-netization reaches the headgroup because of proton exchange. Taken together, this meansspin diffusion along the lipid tails in PL161 is the rate-limiting process for dipolar orderrelaxation. As shown in Table 6.1, T1D of PL161 seems sensitive to aqueous proton concen-tration only when the aqueous proton concentration is small. T1D decreases by ∼20% whenthe sample is changed slightly from pure 90% D2O (10% PL161) to a 88%/2% D2O/H2Omixture. Yet, the PL161 sample in neat H2O shows a further decrease of only 50% in T1D.Aqueous protons destroy non-aqueous dipolar order at the headgroups via proton exchange,but after a critical concentration of aqueous protons is reached, this is not the rate-limitingprocess, rather spin diffusion inside the lipid bilayer is.A simple one dimensional model of spin-diffusion in an infinite plane can be solved analyt-ically (e.g. Eq. 4.16 in Crank [207]). With parameters chosen to represent a single lipidbilayer (D ≈ 0.016 nm2/ms [208] and layer thickness l ≈ 3 nm and D ≈ 0.016 nm2/ms [208],corresponding to a 15–18 carbon chain), initial conditions of uniform dipolar order withinthe plane, and boundary conditions where the dipolar order is destroyed at the plane sur-faces, the decay of dipolar order predicted fits well to a stretched exponential (Eq. 6.31),with T1D ≈ 45 ms and s = 0.8. This T1D value is within a factor of 2 from the measuredvalue of 23 ms.A similar analysis can be performed for wood cellulose. We assume that H2O permeatesthe cellulose microfibrils and relaxes dipolar order at the surface of each cellulose crystal-lite. These crystallites have a diameter of about 5 nm and lengths 4–10× this [209]. Thecrystallites can be modelled as infinitely long cylinders with radius a ≈ 2.5 nm. In solidorganic polymers like these, the spin diffusion coefficient is ∼ 1 nm2/ms [11]. With theseparameters, and similar initial conditions and boundary conditions as before, the solution tothe diffusion equation in this geometry (e.g. Eq. 5.18 in Crank [207]) is again well describedas a stretched exponential, with T1D = 1.05 ms, s = 0.98. Here T1D is within a factor of127three from the measured value of 2.6 ms.The agreement between these models and our results show that in systems with magnetiza-tion exchange to abundant aqueous protons, T1D is largely dependent on the spin diffusionrate. It also suggests that even if spin diffusion is relatively fast, T1D may be long enoughfor ihMT to be observable if the physical size of the reservoir is large enough, as in the caseof cellulose crystallites in wood and keratin crystallites in hair.The observation of non-zero ihMT in tendon suggests that this technique may not be asmyelin-specific as previously thought. In brain, lipid membranes in myelin and glial cellsare likely the only structures with an ihMT response. Such experiments however may proveto be useful in imaging other areas of the body, Further work to rigorously identify tissuesproducing non-zero ihMT is required.Taken together, this work suggests that thinking of ihMT as resulting from a type of spec-tral line broadening is misleading. While ihMT may occur in inhomogeneously-broadenedsystems, it occurs in homogeneously-broadened ones as well. The presence of a dipolar termin the Hamiltonian of the non-aqueous protons is enough to ensure the presence of a dipolarreservoir, and if WT1D is large enough, then ihMT will be visible from the non-aqueous pro-ton intensities. If magnetization exchanges with aqueous protons, then ihMT will be visiblefrom their intensity too. Others have already shown the applicability of Provotorov theory,and our results have confirmed that ihMT is driven by the dipolar interaction alone and thatinhomogeneous broadening is not involved. For this reason, we suggest changing the nameihMT to dipolar magnetization transfer (dMT) to better reflect the underlying mechanism.128Chapter 7Pool-specific ihMT in white matter7.1 IntroductionThere is little doubt of ihMT’s sensitivity to T1D. Indeed, studies performed on phantoms[84, 125, 131] and in-vivo [127] show how ihMT may be considered a T1D-weighted imagingmodality. It is also clear that ihMT is sensitive to myelinated tissues such as white matter.Moreover, myelin bilayers are known to be unique, possessing on average fewer proteins, morelong-chained lipids, and a higher proportion of saturated lipids than in other biomembranes[51,61,63,64,84]. Accordingly, the prevailing theory for ihMT’s sensitivity to myelin is thatmyelin’s unique lipid bilayers possess a long T1D.Recent measurements of T1D in white and grey matter are inconsistent with this under-standing of ihMT. Using the sequence developed by Varma et al. to measure T1D in-vivowith ihMT prepulses [127], multiple studies have shown remarkably similar T1D values inwhite and grey matter. However, still other measurements by Swanson et al. give very dif-ferent T1D values in these two tissues. The state of T1D measurements is summarized inTable 7.1. Differences in the samples (fixed vs. in-vivo) may account for the large variation.Also, each technique for measuring T1D is biased towards certain values.These discrepancies show the need for closer examination of the ihMT signal from myelin.To this end, the present study combines ihMT with CPMG acquisition in order to measurethe distinct signals from myelin water (MW) and intra/extra-cellular water (IEW). This ispossible through multi-exponential fitting of the CPMG decay. Since MW is nominally thefirst aqueous pool in which the ihMT signal from myelin arises, observing its ihMT signaldirectly may highlight the differences between the non-aqueous protons inside and outsidethe myelin. To qualitatively model the results, we apply the four pool model of white matter129Study Sample Technique Whitematter T1D(ms)Grey matterT1D (ms)MuscleT1D (ms)Prevost etal. [125]Mousein-vivoT1D-ihMT 6.1±0.8 (IC) 5.6±1.2(cGM)2.2±0.6Swanson etal. [84]Formalin-fixed bovinespinal cordin D2OJeener-Broekaert11.1±1.8(SC)4.06±1.20(SC)-Varma etal. [127]Humanbrain in-vivoT1D-ihMT 6.2±0.4(avg)5.9±1.2(avg)-Carlvalho etal. [210]Mouse brainin-vivoT1D-ihMT 4.9±1.0 (IC) 4.8±0.7(cGM)1.9±0.3Rat spinalcord ex-vivo6.6±0.5 (SC) 6.6±0.9 (SC) -Table 7.1: Measurements from the literature of white and grey matter T1Ds. IC = internalcapsules, cGM = cortical grey matter, SC = spinal cord. T1D-ihMT refers to the sequencedeveloped by Varma et al. for measuring T1D using ihMT prepulses [127].from Chapter 5. The model is modified to include dipolar reservoirs in each non-aqueouspool.7.2 Theory7.2.1 The four pool model with dipolar reservoirsThe four pool model, used extensively in Chapter 5, models longitudinal relaxation in whitematter tissue. To model ihMT as well, a four pool model with dipolar reservoirs is required(Fig. 7.1). The dipolar couplings in the non-aqueous pools must now be taken into account.The protons in these pools are on large molecules like lipids, which either tumble slowly or arerestricted in some way. As a result, the proton-proton dipolar interactions are incompletelyaveraged. This causes broad non-aqueous NMR lineshapes, such as the super-Lorentzian seenextensively in previous chapters. What is relevant here is that the residual dipolar couplingsalso forms a thermodynamic reservoir. This reservoir can store dipolar magnetization (alsocalled dipolar order) [10,30].Provotorov Theory, derived in Section 2.5.6 and applied in the previous chapter, describes thecoupling of Zeeman and dipolar magnetization. We remind the reader of its key equations.In an isolated system of dipolar-coupled protons, the evolution of a vector ρ = [MD, M ]T130T1,M T1,MW T1,IEW Tcr,IEW/NM T1,NMPool 1: non-aqueousmyelin (M)Pool 2: myelin water (MW)Pool 3: intra/extra-cellular water (IEW)Pool 4: non-aqueousnon-myelin (NM)Tcr,MW/IEWTcr,M/MWMM(∞) MMW(∞) MIEW(∞) MNM(∞)non-aqueous myelin dipolar reservoirnon-aqueous non-myelin dipolar reservoirT1D,M T1D,NMRate depends ongNM, Δ, ωD,NM, B1Rate depends ongM, Δ, ωD,M, B1Figure 7.1: The four pool model with dipolar reservoirs. Pools 1 to 4 can hold Zeeman(longitudinal) magnetization. The dipolar reservoirs hold dipolar magnetization.(where MD and M are the dipolar and Zeeman magnetizations respectively) under weak rfirradiation is described bydρdt= −W − 1T1 WΩWΩ −WΩ2 − 1T1Dρ± + 〈Iz〉0T10 . (7.1)T1 and T1D are the Zeeman and dipolar spin-lattice relaxation times andΩ = 2pi∆/ωD,where ∆ is the offset frequency of the rf and ωD is the RMS dipolar interaction strength.Finally,W = pi(2piB1)2g(∆), (7.2)which is a function of the prepulse RMS amplitude B1 and the lineshape of the non-aqueousprotons g evaluated at the offset. The Sdual prepulse causes the off-diagonal elements of thematrix above to vanish.Combining this with the four pool model is straightforward. This combined model buildsoff of two pool models with a dipolar reservoir for modeling MT [35,93], and Varma et al.’sintroduction of this two pool model for simulating ihMT [119]. In the last chapter, we usedProvotorov Theory in a single non-aqueous proton system as a model for discussing ihMTphysics. In the combined model below, the dipolar reservoirs remain coupled to non-aqueouspools only, which in turn can exchange with aqueous pools. The combined model is describedby a coupled system of homogeneous differential equations,dMdt= RM. (7.3)131M is a vector of Zeeman (M) and dipolar (MD) magnetizationsM(t) = [MD,M(t), MM(t), MMW(t), MIEW(t), MNM(t), MD,NM(t), 1]T. (7.4)Starting from thermal equilibrium, the initial condition vector isM(0) = [0, MM(∞), MMW(∞), MIEW(∞), MNM(∞), 0, 1]T. (7.5)The last component of these two vectors is a constant equal to 1 whose purpose is to convertthe inhomogeneous differential equations in Eqs. 2.47 and 5.1 to homogeneous equations.This approach is possible because the inhomogeneous terms (the Mi(∞)/T1,is) are constant.This last dimension in M(t) has no physical interpretation and can be ignored. When thefour pool model is used without a dipolar reservoir, substituting reduced magnetization units,mi = −Mi−Mi(∞)2Mi(∞) , also makes the system homogeneous [144, 145]; this was our approach inChapter 5. That is not possible here because M(∞) in the dipolar reservoirs is close tozero [30].The matrix R containing the dynamics is given byR =− 1T1D,M−WMΩ2M ΦWMΩM 0ΦWMΩM − 1T1,M − kM,MW −WM kMW,M0 kM,MW − 1T1,MW − kMW,M − kMW,IEW0 0 kMW,IEW0 0 00 0 00 0 0...0 0 0 00 0 0 MM(∞)T1,MkIEW,MW 0 0 MMW(∞)T1,MW− 1T1,IEW− kIEW,MW − kIEW,NM kNM,IEW 0 MIEW(∞)T1,IEWkIEW,NM − 1T1,NM − kNM,IEW −WNM ΦWNMΩNMMNM(∞)T1,IEW0 ΦWNMΩNM − 1T1D,NM −WNMΩ2NM 00 0 0 0(7.6)132whereΦ =1, during S+ and S− prepulse0, during Sdual prepulse. (7.7)In experiment S0 there is no prepulse and WM = WNM = 0.One significant limitation of this combined model is that it assumes single values for physicalparameters. For the four pool model alone, its excellent fit to the relaxation experiments inChapter 5 shows little need to model distributions for pool sizes, Tcr times, and T1 times.This is likely because rapid diffusion of magnetization in aqueous compartments averages outlocal variations. However, the same simplification is not necessarily expected with regardsto the dipolar reservoir parameters.We showed that the non-aqueous lineshape from our white matter samples were super-Lorentzian in Chapter 5; in the notation here, that is the function gM(∆) + gNM(∆). Thesuper-Lorentzian itself is an integral of orientation-dependent lineshapes, primarily from theacyl chains in lipid bilayers. Therefore, the use of single values for gM and gNM (via Eq. 7.2)implicitly assumes the presence of angular averaging in the lipid bilayers, occurring muchfaster than magnetization transfer to the aqueous pool. Orientation dependence of ωD andT1D (see Eq. 2.52) means there is a distribution of these parameters as well. Nonetheless,accounting for distributions is beyond the scope of this work. Rather, the purpose of thissimple model is to lead to straightforward, qualitative conclusions.7.2.2 The grey matter analogueIn addition to modeling the MTR and ihMTR data for a white matter sample, the fourpool model with dipolar reservoirs can be used to model a “grey matter analogue”. Froma structural perspective, grey matter is very roughly like white matter with the myelinremoved. In both cases, the components (glial cells, unmyelinated axons, blood vessels, etc.)are similar, though there are some obvious structural differences: the presence of neuron cellbodies in actual grey matter, for example. Still, in a thought experiment where the myelin inwhite matter is removed and replaced with water, this reduces the quantity of non-aqueousprotons by about half, nicely matching with the fact that there are roughly half as manynon-aqueous protons in grey matter compared to white matter (when measured as a fractionof total proton number) [165,211]. And so, to simulate ihMTR in the white matter sample’sgrey matter analogue, we set Tcr,IEW/MW → ∞ and record the signal from the IEW poolonly.133This grey matter analogue is indeed a crude model, but it has been used before in biochemicalassays. Norton & Cammer note that despite being an oversimplification, this model doesyield accurate results when quantifying lipids in grey and white matter [62]. Its use here isjustified given the qualitative nature of our analysis.7.3 Methods and materials7.3.1 Sample preparationThe same four samples (bovine white matter samples WM-fr, WM-sp1, WM-sp2, and greymatter sample GM-bg) were used here as in the T1 relaxation experiments in Chapter 5. SeeSection 5.3.1 for details of the sample preparation.7.3.2 NMR experimentsThe pulse sequences used are shown in Fig. 7.2. This style of ihMT experiment, where theprepulse is continuous-wave rf, requires three experiments with different prepulses plus areference experiment for a total of four experiments. Experiments S+ and S− use prepulsesat offsets +∆ and −∆ relative to the center of the lipid super-Lorentzian. ExperimentSdual uses a prepulse which is sine-modulated at ∆ to irradiate ±∆ simultaneously. Themaximum prepulse amplitude in Sdual is√2B1, but in S+ and S− it is B1. This ensuresthat the prepulse RMS power is the same in all three cases. In the reference experiment S0,B1 = 0 during the prepulse.For sample WM-sp1, 20 ihMT experiments were acquired, varying the prepulse duration, τ ,and the relaxation delay, δ. In the first 14 experiments, τ was increased from 10 to 1000 ms(with δ held at 0.01 ms), and in the last 7 experiments, δ was increased from 0.01 to 500ms (with τ held at 1000 ms). In order to complete all experiments on all samples within 72hours, for the remaining samples only 12 experiments were acquired, where τ was increasedfrom 10 to 1000 ms in the first 7 (holding δ constant at 0.01 ms), and δ was increased from0.01 to 500 ms in the last 6 (holding τ constant at 1000 ms). More data was acquired at lowvalues of τ and δ in order to capture any short-time behaviour.|∆| was fixed at 7 kHz and B1 was either 141 or 283 Hz (3.32 or 6.64 µT). In a previousstudy by Varma et al., similar parameters gave a large ihMT in human WM [119]. This isalso close to the optimal parameters (∆=7–9 kHz and an RMS B1 of 4–5.5 µT) suggested134Figure 7.2: The four ihMT NMR experiments with CPMG acquisition. Sdual uses a sine-modulated prepulse to irradiate offsets at ±∆ simultaneously. In experiment S0 there is Mchinda et al., although their in-vivo study used a pulse-train prepulse instead of cosine-modulation [123]. Our CPMG acquisition used broadband rectangular pulses with a typical90◦ pulse duration of 3.1–3.3 µs (a B1 amplitude of 18–19 mT). The CPMG train used 300echoes with 2 ms spacing. 8 or 4 transients were acquired with a recycle delay of 7 s.Like the experiments on these samples reported on earlier, the data for this study wascollected using a Bruker solenoidal probe (HP WB73ASOL10) in a 200 MHz (4.7 T) magnetwith a home-built NMR spectrometer. The temperature was regulated at 37 ◦C.7.3.3 CPMG fittingAs in the T1 relaxation experiments in Chapter 5, the CPMG curves were analyzed usingsparse exponential distributions. Section outlined this approach, where the CPMGsignal was modeled as a sum of four exponentials: one corresponding to the MW peak(~6 ms), two corresponding to the IEW peak (constrained to be 10 ms apart, centerednear 60 ms), and a last one for a small ~200 ms component. Previously, we identified thislongest component as an external bulk water (BW) pool, based off of its mono-exponentialT1 relaxation behaviour. Our magnetization transfer results below confirms this association.The amplitudes of these exponentials give the relative amount of magnetization in each pool.From the four experiments, the standard Magnetization Transfer Ratio (MTR) can be cal-135culated separately for single and dual-sided ihMT prepulses. For each pool, the MTR isMTRsingle =2A0 − A+ − A−2A0(7.8)MTRdual =A0 − AdualA0, (7.9)and the ihMTR calculated throughihMTR = MTRdual −MTRsingle= A− + A+ − 2Adual2A0 .(7.10)As in the last chapter, we caution the reader that this definition of ihMTR has a two in thedenominator, which most imaging studies now omit. We keep it here for internal consistencywithin the thesis. When we later quote results from other papers, we will convert the valuesappropriately.Because the quantities of interest in this chapter are ratios, we can ignore the effects ofMW/IEW exchange during the CPMG in this chapter. MW/IEW exchange causes the themeasured amplitudes (and T2s) of the MW and IEW pools to deviate slightly from their truevalues (see Section 4.5.2). In Chapter 5, we calculated a multiplicative correction factor forthe pool amplitudes which was applied to each of the observed A0, Adual, A+, and A− termsat all times. However, this cancels out in MTR and ihMTR.Fitting was performed using the least squares solver in the SciPy package [157]. Errors on T2component amplitudes are the standard deviations of 50 repeated fittings to the CPMG withsynthetic Gaussian noise. The noise standard deviation was equal to the standard deviationof the best fit residuals.7.3.4 Four pool model fittingWhen the four pool model with dipolar reservoirs is fit to the data from the white mattersamples, the following constraints are imposed on physiological grounds:T1D,NM ≤ T1D,MωD,NM ≤ ωD,M. (7.11)These are justified by the properties of the myelin lipid bilayers, which Chapter 3 mentionedare unique in three ways. When compared to other membranes, myelin has, on average, i)136longer lipid acyl chains, ii) a higher degree of saturated lipids, and iii) a smaller protein:lipidratio [51, 61, 63, 64, 84]. Taken together, compact myelin is more closely-packed and rigidthan other biomembranes [60], meaning the membrane fluctuation amplitudes which driveT1D relaxation are expected to be smaller [84,210], whereas ωD is similar or slightly higher.This last point is supported by the similar linewidths seen in our measurements of the non-aqueous spectra of WM-fr, WM-sp1, WM-sp2, and GM-bg (Table 5.1).Fitting used SciPy’s implementation of the Differential Evolution algorithm [161] via theLMFIT package [158]. The four pool parameters from the results in Chapter 5 were used,so only the dipolar reservoir parameters need to be fit. These include T1D, ωD, and g(∆) inboth non-aqueous pools, which is six parameters in total.7.4 ResultsFigure 7.3: Comparison of regularized NNLS distributions in WM-fr after the four ihMTprepulse conditions. The equilibrium distribution (experiment S0, no prepulse) shows thethree peaks from distinct populations of aqueous protons. The integrated intensity is dis-played beside each peak. Immediately after a prepulse (B1 = 283 Hz, τ = 215 ms, δ = 0.01ms), the Sdual experiment showed a significantly larger reduction of the MW and IEW peakscompared to the S+ or S− experiments. The BW pool appears to increase due to regularizedNNLS fitting artifacts.To illustrate how the different prepulses uniquely affect each aqueous pool, Fig. 7.3 shows theregularized NNLS distributions in WM-fr for all four experiments under a 215 ms prepulse.Each peak is labeled with its integrated intensity. The MW and IEW peaks decreased morewhen a dual-sided prepulse was used (Sdual) than in the case of single-sided prepulses (S+and S−). Conversely, the BW pool appeared to increase in intensity. However, this is an137artifact of the NNLS fitting. As the total signal decreases, so too does the SNR. Smaller SNRvalues are known to introduce errors into the regularized NNLS distribution [212]. However,the more robust sparse exponential fits below showed only a decreasing BW amplitude forthe white matter samples.The MTRs (Eqs. 7.8 and 7.9) are plotted in Fig. 7.4 for the white matter samples (MW,IEW, and BW pools) and for GM-bg (IEW and BW pools). The corresponding ihMTRs(Eq. 7.10) are given in Fig. 7.5. We first focus on the white matter MTR and ihMTR, leavingthe GM-bg results for later.The ihMTR and MTR values are plotted as functions of increasing prepulse duration τ andrecovery time δ. The results in all white matter samples are visually similar. The MT re-sponse from MW was the largest, followed closely by IEW. This matches with the resultsfound in previous MT-CPMG experiments [169]. The BW MT response was significantlylower, confirming that this pool is relatively isolated. Across all pools, B1 = 283 Hz causeda higher MT response. There are striking differences in MW and IEW MTRs during therelaxation period, when δ > 0. These can be explained by comparing the size of their corre-sponding non-aqueous pools (given by the four pool fits in Table 5.2). MW relaxes quickly,since it is ~0.5× the size of pool M and these two pools are in close contact. Conversely, theIEW pool size is about 5–7× the NM pool, so the magnetization exchange rate between thetwo is much smaller. For example, when B1 = 141 Hz, the IEW MTR appears to plateauuntil about δ = 100 ms, whereas MW’s MTR starts decreasing immediately.Focusing now on the ihMTR plots (Fig. 7.5), given the relatively small difference betweenMTRdual and MTRsingle, these ihMTR data were much noisier than the MTR data. Allwhite matter samples showed gradual growth in MW and IEW ihMTR as τ increases whenB1 = 141 Hz. However, whenB1 = 283 Hz, the MW ihMTR peaks and then decays. Previousin vivo studies [125] and modeling [119,131] that looked at the total aqueous response havealso shown similar behaviour. The same feature would be expected in the IEW pool at higherB1 values. This occurs because the MW magnetization became saturated more rapidly thanthe IEW magnetization. When this saturation happened, the effect of the single and dualprepulses became similar, and ihMTR decreased.We also plot ihMTR for the combined MW and IEW pools in white matter, which is ap-proximately what would be observed for the total aqueous signal in-vivo (assuming TE, thetime between the MRI excitation pulse and acquisition, was short compared to the MWT2). Excluding the BW pool is justified due to its isolation. This MW+IEW ihMTR closelyfollowed the IEW ihMTR since that pool was 90–95% of the aqueous signal.Concerning the grey matter, the results indicate a relatively poor fit. Firstly, the MTR from138Figure 7.4: MTR for all samples after single and dual prepulse irradiation as a function ofprepulse duration τ and recovery time δ. The consistently lower MTR in the BW poolsconfirms its relative isolation. In general, MW and IEW showed a higher MTRdual thanMTRsingle, indicating ihMT occurs in these pools. Error bars are plotted but are smallerthan the data points for most series. In GM-bg, no MW signal was observed and the BWMTR was constrained to be positive. This was necessary due to the large fitting error on theBW pool. Note that in the interests of completing all experiments on the samples promptly,fewer τ and δ times were acquired on WM-fr, WM-sp2, and GM-bg compared to WM-sp1.139WM-fr WM-sp1 WM-sp2T1D,M (ms) 7.4 30* 8.9ωD,M/2pi (kHz) 4.1 8.4 6.0gM/10−6(s) 3.4 5.0 12T1D,NM (ms) 3.8 7.5 4.0ωD,NM/2pi (kHz) 3.8 7.4 5.3gNM/10−6(s) 3.0 9.6 7.5Table 7.2: The dipolar reservoir fit parameters. In these simulations, the four pool modelparameters in Table 5.2 were used. Errors on parameters were not included since this modelis qualitative only, as evident from the fits in Fig. 7.6. T1D,M for WM-sp1 was at the allowedlimit for that value (30 ms). Parameter constraints are listed in Eq. 7.11.GM-bg’s BW pool (Fig. 7.4D) had to be artificially constrained to be positive, otherwiseunphysical negative values appeared. This fitting difficulty is reflected in the large errorbars on GM-bg’s BW MTR data, which are about ±2%; in the white matter samples, theseare <0.5%. It is unsurprising that the lowest ihMTR signal did come from GM-bg’s IEWpool, which had a maximum signal about half that of the white matter samples. This isexpected, given the low quantity of myelin in grey matter. Yet, the negative ihMTR values,particularly in the B1=141 Hz data, are not expected—these are clearly unphysical. We willdelay an exploration of possible causes for this until the Discussion.The plots in Fig. 7.6 show the four pool model fits to the data, using the technique describedin the methods. It’s immediately clear that the model describes the MTR data well, butonly qualitatively describes the ihMTR. Still, the general trends are captured. For instance,the model matches how the MW ihMTR rose rapidly compared to the IEW ihMTR. Also,in the case of B1 = 283 Hz, it simulates the maximum in the MW ihMTR at short τ . Wherethe model falls short is the magnitude of the IEW ihMTR. This is the case even though wemake no assumptions about the functional form of gM and gNM or the relationship betweenthese factors and the respective ωDs. No fitting was performed on the GM-bg sample—itsnegative ihMTR values at small τ are unphysical.Table 7.2 gives the dipolar reservoir fit parameters (the four pool parameters were fit inChapter 5 and are listed in Table 5.2). The qualitative nature of this model means theprecision of these values is low. Indeed, two pool model fits to similar data tended to havelarge variations in the parameters [119]. Even so, the relative magnitudes are illuminating.For example, these parameters are consistent with the view that M has a significantly longerT1D than NM. Also, the gM and gNM values are similar, which is expected if these two poolshave similar lineshape widths.140Figure 7.5: The ihMT response for all samples under the two different prepulse amplitudes.The maximum MW ihMTR was always higher than IEW’s. The total aqueous ihMTR(MW+IEW) was very close to the IEW ihMTR, since that pool contains the majority of theaqueous protons. Error bars are not drawn for clarity on the line for MW+IEW, but theyare approximately the same size as IEW’s error bars. GM-bg sample had no measurableMW pool so no combined ihMTR is plotted. The negative ihMTR values in that sample arenon-physical. In all cases, the solid lines are plotted as guides to the eye.141Figure 7.6: The fits of the four pool model with dipolar reservoirs in all white matter data.The fits to the MTR data are shown in plots (A,C,E) and the fits to the ihMTR data areshown in plots (B,D,F). There is significantly higher error in the ihMTR fits due to ihMT’ssensitivity to subtle differences between MTRdual and MTRsingle. The model fails to capturethe details, but is qualitatively correct. The simulation used parameters from Tables 5.2 and7.2.142Figure 7.7: ihMT in white matter samples and in their grey matter analogues. The fourpool simulations are the same fits shown in Fig. 7.6. The grey matter analogues are asimulation of the same sample with the myelin removed. The maximum ihMTR in bothcases is indicated.Using these parameters, we also simulated the grey matter analogue. This is plotted inFig. 7.7. If we imagine the white matter sample’s myelin being removed, the behaviour inthe dashed lines is expected, which shows lower ihMTRs in every case.7.5 DiscussionThe work here has two main portions: ihMT-CPMG measurements, which allow the ihMT inMW and IEW to be observed separately; and qualitative simulations using a four pool modelwith dipolar reservoirs. While the precise values of ihMTR depend on sequence, sample, andparameters used, in general the total ihMTR (MW+IEW) we measured matches well with143previous in-vivo ihMT studies. For example, in a previous study also using cosine-modulatedSdual prepulses, doubling the RMS prepulse power also roughly doubled the ihMT signal [118].The same was seen with pulse-train prepulses [120]. Moreover, the grey matter sample had amaximum ihMTR that was roughly half the maximum ihMTR in the white matter samples,which was also seen previously [118,120].The measurements of ihMT in MW are one novel aspect of the work here. Its higher MTresponse reflects the close contact between this pool and the myelin lipids. Tracing theorigin of the higher ihMT response is less clear. On one hand, we expect the myelin lipidsto have more ihMT compared to non-myelin non-aqueous protons since their lipid bilayersare unique: as mentioned previously, they are more rigid, more compact, and—althoughmeasurements disagree on this—are thought to have a long T1D [84, 119,131]. On the otherhand, MW’s MTR is larger (especially at δ<500 ms, see Fig. 7.4), which, all things beingequal, would allow a higher ihMTR to be realized. The essential question is the following:is MW’s ihMTR higher than IEW’s because of i) a difference in the dipolar reservoirs of thenon-aqueous pools to which they are coupled, or ii) the higher MT in MW because of itsmore intimate contact with its non-aqueous pool?In an attempt to answer this, we define the ratioihMTRMTR= MTRdual −MTRsingle12 (MTRdual + MTRsingle), (7.12)which will be large when ihMTR is due mainly to the behaviour of the dipolar reservoirand small when it is mostly caused by MT. This ratio doesn’t contain much information onits own, but it is illuminating to compare its values for different pools like MW and IEW,as is plotted in Fig. 7.8 for the duration of the prepulses. Because of MW/IEW exchangeeffects, 50 ms <δ< 200 ms is the easiest period to interpret. (The eigenvectors from the fourpool model fits show that MW/IEW exchange in these samples operates on a timescale of100-150 ms, see Table 5.2.) While the MW data were noisy, a general trend emerges acrossthe samples and prepulse amplitudes: ihMTRMTR is higher in MW than in IEW. In short, thisis good evidence that MW’s higher ihMT is at least partially due to a distinction betweenthe dipolar reservoir in myelin non-aqueous protons and the dipolar reservoir in non-myelinnon-aqueous protons.The four pool model with dipolar reservoirs, though qualitative, provides a description ofihMT that is also consistent with the dipolar reservoir in myelin being unique. In the model,this uniqueness manifests as T1D,M > T1D,NM, which has indeed been the accepted theoryfor the sensitivity of ihMT to tissues with a high abundance of myelin [119]. However, as144Figure 7.8: The relative contribution of MT to ihMT during the prepulse. The ratio plottedis defined in Eq. 7.12. Higher values indicate ihMTR is from dipolar reservoir properties,not from MT behaviour. After about 200 ms, MW/IEW exchange effects start to becomesignificant, making interpretation difficult. Values of δ<50 ms are not plotted because inthat regime there is little distinction between MTRsingle and MTRdual.145mentioned, there have been conflicting measurements of T1D in white and grey matter, thelatter of which has similar structural and biochemical properties as the NM pool [62]. Somestudies report different values [84], and others have measured similar values [127,210]. Takentogether with our work, there is obviously a clear need for a careful measurement of T1D inwhite and grey matter. We explore how this could be accomplished in Chapter 8.Future studies like the work here could be improved in a number of ways. Firstly, we onlyvaried two parameters: the prepulse duration, δ, and the prepulse peak amplitude, B1.Varying the offset frequency, ∆, may allow the lineshapes of the M and NM pool to bedetermined. More generally, the SNR could be improved upon with more signal averaging.This is particularly important for the small MW pool. FID acquisitions may also be usefulin constraining model fitting by providing information about the non-aqueous amplitudes.Any continuation of this work should also ensure multiple grey matter samples are studied,for the negative ihMTR values at low prepulse times in the single grey matter sample (GM-bg) are difficult to interpret. If real, it would suggest that response to single-sided prepulsesis completely different in grey matter than in white matter. It is not an artifact of theanalysis: in the cases where GM-bg’s ihMTR is negative, the CPMG decay curves (notshown) do have a slightly larger amplitude in Sdual than in any other experiments. Onepossibility is the presence of paramagnetic ions (eg. iron in blood), which could introducespectral asymmetries. Still, such an effect should be mitigated by the inclusion of bothS+ and S− in the ihMTR calculation. Besides, the S+ and S− experiments show similarattenuation. In any case, it is a small enough effect to largely ignore for the purposes of thiswork. And notably, none of the other experiments discussed in this chapter or in Chapter 5show similar behaviour, so there is little concern of a systematic error.146Chapter 8Conclusions and future work8.1 ConclusionsThis thesis looked at the fundamental physics of T1 relaxation and ihMT in brain. T1relaxation is a key contrast mechanism which is highly-dependent on myelin, but quantitativestudies have so far disagreed on the value, number, and source of T1 components in whitematter. ihMT is a new technique that is sensitive to materials rich in lipid bilayers, likemyelin. However, the hypothesis that it requires inhomogeneous spectral broadening isunproven. Also, recent studies with conflicting T1D measurements in white and grey matterhave questioned how it is selective to myelin lipids in particular. Together, these factorsprovided the motivation for this work. More broadly, the research here is part of a largereffort to improve quantitative MRI of myelin.In Chapter 5 we reported on a suite of solid-state NMR spectroscopy experiments on ex-vivobovine grey and white matter brain tissue. We separately observed T1 relaxation of the MW,IEW, and total non-aqueous protons from six unique initial conditions. For the first time, weperformed non-equilibrium lineshape analysis on the non-aqueous signal from these samples.These data were fit to a four pool model, and the fit parameters in general matched wellwith the literature values. Our results also confirmed that MW/IEW exchange only causesminor errors in the accuracy of MWF measurements. We also explored why different initialconditions lead to different relaxation behaviour, showing this explicitly for hard and softinversion-recovery experiments. In doing so we exemplified how eigenvector analysis couldbe a useful tool for predicting the relaxation behaviour under different pulse sequences.Chapter 6 encompassed a close look at the physics of ihMT. We introduced a simple spin-1model, showing how ihMT arises from the dipolar interaction. Then, an analysis of the147Provotorov Theory model of ihMT showed how WT1D was a key parameter in determiningits strength. Our experiments showed how the off-resonance pulses created dipolar orderin PL161, tendon, wood, hair—the last two having homogeneously-broadened non-aqueousspectra. None of the samples showed evidence of hole burning, but all exhibited ihMT. T1Dmeasurements were carried out using ADRF/ARRF and saturation-recovery experiments,showing the validity of WT1D as a rough measure of the intensity of ihMT.The last study in Chapter 7 combined aspects of the four pool model and ihMT. ihMTexperiments with CPMG acquisition were used to observe MW and IEW separately in thesame bovine brain samples from Chapter 5. A higher ihMT signal from MW was observed.To separate out the relative contribution from MT and ihMT in this larger signal, the ratioIHMTR/MTR was compared for MW and IEW. This new metric showed evidence that thecomparatively higher MW ihMT is due to a distinction between the myelin and non-myelinlipids. A qualitative four pool model with dipolar reservoirs suggested that this was due toa difference in T1Ds, which matches with earlier explanations. Together, this suggests thatrecent observations of similar T1Ds in grey and white matter need to be carefully examined.8.2 Future workThe work performed in this thesis has shed new light on T1 relaxation and ihMT in brain.Inevitably, certain areas of this research could be refined, and it has also raised new questions.Below, we offer some suggestions on how research in this area could proceed.8.2.1 The non-aqueous lineshape and the effect of soft pulsesWhen fitting the non-aqueous lineshapes in Chapter 5, we used super-Lorentzians, whichare superpositions of Gaussians with orientation-dependent widths. When a low-amplitude,off-resonance pulse is applied, do these Gaussians get saturated individually, or does theentire lineshape get saturated as a whole? Typically, the latter behaviour is assumed whensuper-Lorentzians are used in qMT studies (e.g. see references [79,81,93,171]), but the formerseems correct on a fundamental level. We were unable to study this in the ex-vivo samplesbecause of the intense aqueous line. Varma et al. considered this in one of their models forihMT [119], but they did not directly observe the non-aqueous protons and could not makeany firm conclusions on this point.The ideal experiment might be to simply use the PL161/D2O from Chapter 6 with a sat-uration pulse of variable length and offset followed by a 90◦ pulse and FID acquisition. In148order to remove confounding effects from dipolar order [93], one could saturate at an offset±∆ simultaneous (decoupling the Zeeman and dipolar reservoirs), as in an Sdual ihMT pre-pulse. Experiments on ex-vivo brain samples could also be performed. Soaking the samplein D2O to reduce the intensity of the water line may help to isolate the non-aqueous signal.This would be similar to Wilhelm et al.’s recent high-resolution spectra of rat spinal cord inD2O [89].It may be worthwhile attempting these experiments in a high-resolution probe. This wouldrequire longer 90◦ pulse lengths, therefore causing more truncation at the start of the FID.Also, adiabatic pulses may be required to completely excite the broad, ~20 kHz non-aqueousline. However, these may be worthwhile trade-offs if the resolution is significantly better.8.2.2 Improved quantification of T1 relaxationThe six different experiments in Chapter 5 were chosen to establish a diverse set of initialmagnetization within the distinct pools of protons in the brain tissue sample. Future studiescould build upon this by using other experiments to better separate the two non-aqueouspools (M and NM). We used the assumption of an equal number of protons in M and NM,but this should be a well-constrained free parameter, or if possible, a measured value.One experiment which may help reveal this is saturation of the non-aqueous protons duringthe cross-relaxation period [100, 213, 214]. Low-amplitude, continuous rf would be appliedat offsets ±∆, saturating the non-aqueous protons (the dual offsets are to prevent dipolarorder creation). If the rf power was high enough, it may be possible to saturate the non-aqueous protons completely while leaving the aqueous protons relatively unaffected. Thiswould turn the non-aqueous pools into magnetization sinks. The relaxation dynamics wouldbe extremely different, and may help reveal differences in the M and NM pools. It may alsomore obviously show the contribution from M/MW exchange, which had to be manuallyemphasized when fitting our data in Chapter 5.Better separation of the relaxation associated with MW/M exchange (T ∗1 ≈ 30 ms) andIEW/NM exchange (T ∗1 ≈ 70 ms) is also desirable. One way to do this may be with a doubleinversion-recovery sequence [215]. In this sequence, one inverts the magnetization and thenwaits until the MW magnetization passes through zero (~30 ms). Another inversion pulsebrings the magnetization into the +z direction. Then, during the cross-relaxation period, itmay be easier to view the M/MW exchange behaviour. The inversion pulses could be hardor soft.Complimentary data could also be collected using two different B0 field strengths. We149discussed how the intrinsic spin-lattice relaxation time (T1 for each pool) is a function ofthe B0 field strength in Section 2.6. However, all other four pool model parameters—the Tcrand M(∞) values—will remain constant. One could run the same experiments on the samesample in two different spectrometers with different B0 fields. Then, when fitting the model,one could impose all parameters except the T1 times to be identical in the two experimentsets. Sample aging, shimming, and variation in T2 times may be confounding factors withthis approach.8.2.3 T1D measurements in brainIn the introduction to Chapter 7, it was mentioned that recent measurements of grey andwhite matter T1Ds are inconclusive. Some research has suggested that grey and white matterT1Ds are the same, whereas other papers have measured very different (~10 ms) values. Theoutcome of these measurements seems to depend on the technique and sample used. Still,the sensitivity of ihMT to myelin is thought to rely on it having a uniquely-long T1D time.Our results seemed to confirm this, although we did not measure T1D directly—somethingthat future research should focus on.Measuring T1D is difficult because most techniques are biased towards certain values. Forexample, the ADRF/ARRF sequences used in Chapter 6 could not measure T1Ds less than ~1ms. It would likely be necessary (and illuminating) to perform the measurements with mul-tiple techniques on both white and grey matter. These could include the saturation method,the ADRF/ARRF sequence, the Jeener-Broekaert sequence, and the ihMT sequence. Re-garding the analysis of this type of data, regularized NNLS may help identify distributionsin T1D.Because the presence of the intense water signal makes measurement difficult in tissue sam-ples, a sample soaked in D2O may prove useful here too. 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This Hamiltonian does not include terms leading to spin-lattice relaxation. Also, weassume [Hˆ1, HˆT ] = [Hˆ2, HˆT ] = 0, so both Hˆ1 and Hˆ2 are constants of motion. These termsmay be thought of as forming separate thermodynamic reservoirs, each with a unique spintemperature, θ1 and θ2 [A3]. Therefore, the high-temperature density matrix isρ ≈ − 1θ1Hˆ1 − 1θ2Hˆ2.Section 2.5.4 shows how a situation where θ1 6= θ2 could be realized by working in therotating frame, where θ1 and θ2 correspond to the Zeeman and dipolar reservoirs. For nowwe don’t specify what frame we are working in or what Hˆ1 and Hˆ2 are.Our ultimate goal is find ddt( 1θ1,2). Using the fact that time rate of change for the energy in169a reservoir is dEdt= dEdθdθdt, for any reservoir [A4]ddt(1θ)= dE/dt−θ2dE/dθ . (A.1)The derivative in the denominator is relatively straightforward. For a generic reservoir witha Hamiltonian Hˆ and spin temperature θE =Tr{Hˆ exp(−Hˆ/kθ)}Tr{exp(−Hˆ/kθ)}=Tr{Hˆ(1− Hˆ/kθ + Hˆ2/(kθ)2 + · · · )}Tr{1− Hˆ/kθ + Hˆ2/(kθ)2 + · · ·}≈ −1kθTr{Hˆ2}Tr {1}= −1kNθTr{Hˆ2}. (A.2)Where N is the dimensionality of the Hamiltonian. This uses the high–temperature ap-proximation and the assumption that Tr{Hˆ} = 0, which is true for dipolar and ZeemanHamiltonians. And so,dEdθ= 1kNθTr{Hˆ2}. (A.3)Finding dE/dθ is more involved. Assume there are M energy levels in a reservoir (becauseof the possibility of degeneracy, M ≤ N). Then,dEdθ=M∑n=1dpndtEn, (A.4)where pn is the population of the nth level and En is its energy. We use the standard approachto calculate changes in populations: a first-order rate equation [A3,A4]:dpndt=∑m6=n(Wm→npm −Wn→mpn) +∑m6=n,r 6=s(V(m→n),(s→r)pmqs − V(n→m),(r→s)pnqr). (A.5)This is also known as the “master equation”. Here, p and q indicate populations of specificlevels in reservoirs one and two, andWm→n is rate of transitions from levelm to n in reservoirone. V(m→n),(s→r) is the rate of a simultaneous transition in reservoir one from m to n and170in reservoir two from s to r.1We assume that the reservoirs are coupled via energy-conserving interactions, such as theflip-flop transitions from dipolar coupling. Therefore, V(m→n),(s→r) = V(n→m),(r→s) and theenergy difference in both reservoirs is the same: ∆Enm = ∆Ers, where n and s are thehigher-energy states. Then,pnpm= exp(−∆Emnkθ1)≈ 1− ∆Emnkθ1qsqr= exp(−∆Emnkθ2)≈ 1− ∆Emnkθ2.(A.6)If we substitute this high-temperature expansion into the second term of Eq. A.5, we get∑m6=n,r 6=sV(m→n),(s→r)pmqr∆Emnk( 1θ1− 1θ2).Eq. A.4 tells us that we need to know dpndtEn. Calculating this for the second term in Eq. A.5yields:dpndtEn∣∣∣∣∣term2=∑m 6=n,r 6=sEnV(m→n),(s→r)pmqr∆Emnk( 1θ1− 1θ2)=∑m 6=n,r 6=sV(m→n),(s→r)pmqr∆E2mn2k( 1θ1− 1θ2)=∑m 6=n,r 6=sV(m→n),(s→r)∆E2mn2kM1M2( 1θ1− 1θ2). (A.7)Where M1,2 specifies the number of energy levels in reservoirs one and two. On line two, weused the fact that ∑m 6=nEn∆Emn = −∑m6=nEm∆Emn. On line three, we expanded pm aspm =exp(−Em/kθ1)∑M1j=1 exp(−Ej/kθ1)≈ 1∑M1j=1 1= 1M1.These are standard maneuvers when dealing with master equations for NMR relaxation (see1There is an error in Schumacher’s paper on the last line in equation 4. The termW ′nr,mspnqs should readW ′nr,mspnqr. This term relies upon flip-flop transitions from n→ m in reservoir 1 and r → s in reservoir 2.The same term is written correctly later in that paper in equation 6.171Section 5.2 in Slichter [A3]).We can substitute Eqs. A.3, A.5, and A.7 into Eq. A.1, giving [A3]ddt( 1θ1)= −R1(1θ1− 1θ1,L)−R12( 1θ1− 1θ2)ddt( 1θ2)= −R2(1θ2− 1θ2,L)−R21( 1θ2− 1θ1) (A.8)whereR12 =∑m6=n,r 6=s V(m→n),(s→r)∆E2mn2Tr{Hˆ21}M1,R12 =∑m6=n,r 6=s V(m→n),(s→r)∆E2mn2Tr{Hˆ22}M2,R1 =∑n6=mWm→n∆E2mn2Tr{Hˆ21},R2 =∑n 6=mWm→n∆E2mn2Tr{Hˆ22}.(A.9)We have manually inserted the term 1/θ1,L in the equation above, forcing the system to relaxto the lattice temperature. The justification for this is discussed in Slichter [A3] and Slichter& Hebel [A5].The above treatment has been quite general, but now we shall assume explicit forms for theHamiltonians. LetHˆ1 = ∆∑jIˆz,jwhich is a many-spin Zeeman Hamiltonian in a frame rotating at ω, and ∆ = ω0 − ω. Also,Hˆ2 = HˆD,a many-spin dipolar coupling Hamiltonian. With this, we can immediately identify [A3–A5]R1 =1T1and R2 =1T1D.Next, R12 is tackled. This is a constant, so we may find it from comparison to BPP saturation172theory, which is correct for short times [A3]:R12 = W= piω21g(∆),which is defined in Eq. 2.48. We find the last rateR21 = R12Tr{Hˆ21}M1Tr{Hˆ22}M2= WTr{(∆∑j Iˆz,j)2}Tr{Hˆ2D}= W ∆2ω2D,making use of the fact that M1 = M2. Here, ωD is the local field strength, discussed inSection 2.5.Finally, we have the Provotorov Equations (using “Z” for Zeeman and “D” for dipolar):ddt( 1θZ)= − 1T1(1θZ− 1θL,1)−W( 1θZ− 1θD)ddt( 1θD)= − 1T1D( 1θD)−W ∆2ω2D( 1θD− 1θZ).(A.10)The dipolar reservoir lattice temperature θL,2 is extremely hot, so that term is dropped.We can put these into a more useful form. The density matrix for this system isρ = −∆θZ∑jIˆz,j − ωDθD(HˆDωD),so we define the Zeeman and dipolar polarizations as (see Section 2.5.6)pZ = 〈Iˆz〉 = −∆θZpD =〈HˆDωD〉= −ωDθD.Substituting these into Eq. A.10 yields the form of the Provotorov equations introduced inSection 2.5.6.[A1] B. N. Provotorov. A quantum-statistical theory of cross-relaxation. Soviet Physics JETP17315, 611–614 (1962) URL:[A2] M. Goldman. Spin temperature and Nuclear Magnetic Resonance in solids. Interna-tional series of monographs on physics. Clarendon Press, 1970.[A3] C. P. Slichter. Principles of Magnetic Resonance, 3rd Ed. Springer Verlag, 1990.[A4] R. T. Schumacher. Dynamics of interacting spin systems. Physical Review 112, 837–842 (1958)[A5] L. C. Hebel and C. P. Slichter. Nuclear spin relaxation in normal and superconductingaluminum. Physical Review 113, 1504 (1959)174Appendix BCPMG exchange correctionB.1 IntroductionExchange between myelin water (MW) and intra/extra-cellular water (IEW) occurs duringCPMG acquisition. This can cause erroneous observed values of their pool sizes and T2times. The percent error depends on the exchange rate, but in bovine brain it is typically10–20% (see Chapter 5 and ref [B1]). This can be corrected for, however; how to do so isthe focus of this appendix. This correction was described by Bjarnason et al. in ref [B1] andby Bjarnason in ref [B2]. It has been repeated here in a slightly different form for clarity.In the following, we indicate observed values (found from CPMG acquisition with no cor-rection) by a tilde, and set MW as pool 1 and IEW as pool 2. Our goal is to develop twoalgorithms:1. Take the actual T2 times (T2,1 and T2,2) and actual sizes (M0,1 and M0,2) as inputs.Return the observed values (T˜2,1, T˜2,2, M˜0,1, M˜0,2) as outputs. Note that we use thisnotation for the pool sizes instead of M(∞) as in the rest of this thesis since M → 0as t→∞ in the CPMG.2. Takes the observed values (T˜2,1, T˜2,2, M˜0,1, M˜0,2) as inputs. Return the actual values(T2,1, T2,2, M0,1, M0,2) as outputs.B.2 Equations from a two pool modelTo develop these algorithms, the strategy will be to first derive equations linking the actualand observed values, then determine how to convert between the two. Using a vector M(t)175to represent the actual magnetization,M(t) = M1(t)M2(t) , (B.1)the time evolution for these two pools is represented byddtM(t) = −k12 −R1 k21k12 −k21 −R2M(t) (B.2)= AM(t)where Ri = 1/T2,i and R1 > R2 (since pool 1 is MW). As Bjarnason pointed out, we do nothave to include the non-aqueous pools; their short T2 times means they act like transversemagnetization sinks. They return no magnetization to the aqueous pool, so their effects canbe incorporated into R1 and R2.We can simplify matters by noting that total magnetization remains constant before andafter the correction,Mtot = M˜0,1 + M˜0,2= M0,2 +M0,2. (B.3)Usingk12M0,1 = k21M0,2andTcr = k−112 + k−121 ,we writek12 =MtotM0,1Tcrk21 =MtotM0,2Tcr= Mtot(Mtot −M0,1)Tcr .To solve Eq. B.2, in which the matrix A contains the dynamics, we use the well-knownequations for the eigenvalues λ± and eigenvectors v± of a 2x2 matrix:λ± = T2 ±√T 24 −D (B.4)176v± = λ± − A2,2A2,1= λ± + k21 +R2k12 (B.5)whereT = Tr(A)= A1,1 + A2,2= −k12 − k21 −R1 −R2andD = Det(A)= A1,1A2,2 − A1,2A2,1= (k12 +R1)(k21 +R2)− k12 − k21= k12R2 + k21R1 +R1R2.After some simplification,λ± = −12(k12 + k21 +R1 +R2)± 12√(R1 −R2 + k12 − k21)2 + 4k12k21. (B.6)Assuming T˜2,1 < T˜2,2, the eigenvalues are related to the observed T2 times viaT˜2,1 = − 1λ−T˜2,2 = − 1λ+,(B.7)since |λ−| > |λ+|The formal solution to Eq. B.2 isM(t) = c+v+eλ+t + c−v−eλ−t M1(t)M2(t) = c+v+,1eλ+t + c−v−,1eλ−tc+v+,2eλ+t + c−v−,2eλ−t . (B.8)Here, v+,1 is the first component of the v+ eigenvector (and similarly for the other terms),and c± are constants defined below. Connecting this equation to the observed pool sizes,177the total CPMG signal isM1(t) +M2(t) = {c+v+,1 + c+v+,2} eλ+t + {c−v−,1 + c−v−,2} eλ−t,where the prefactors in {} are the observed pool amplitudes:M˜0,1 = c+v+,1 + c+v+,2 (B.9)M˜0,2 = c−v−,1 + c−v−,2= Mtot − M˜0,1.(B.10)If the actual values are known, the c± constants are determined by the t = 0 initial condition:M(0) = M1(∞)M2(∞) = M0,1Mtot −M0,1 = c+v+ + c−v−= V c+c− ,(B.11)where V is a matrix whose columns are v+ and v−. This leads toc+ =1Det(V ) (V2,2M0,1 + V1,2(Mtot −M0,1))c− =1Det(V ) (−V2,1M0,1 + V1,1(Mtot −M0,1)). (B.12)With that, we have derived the necessary equations. Now we will see how they are applied.B.3 Algorithm 1: Actual to observed valuesGiven the actual T2 times (T2,1 and T2,2) and pool sizes (M0,1 and M0,2), along with a Tcrtime, Eqs. B.6 and B.7 give the observed T2 times T˜2,1 and T˜2,2. The observed pool sizes,M˜0,1 and M˜0,2, are found using Eqs. B.9 and B.10.B.4 Algorithm 2: Observed to actual valuesGoing the other way (finding the actual T2 values and pool sizes from the observed values)is less straightforward. The observed values (T˜2,1, T˜2,2, M˜0,1, and M˜0,2) are typically taken178from a CPMG acquisition when the system starts in equilibrium. The first step is to findthe R1 and R2 values (the inverse of the actual T2 times). Using Eq. B.6,λ+ + λ− = −(k12 + k21 +R1 +R2)=⇒ R1 = −(k12 + k21 +R2 + λ+ + λ−) (B.13)and(λ+ − λ−)2 = (k12 − k21 +R1 −R2)2 + 4k12k21=⇒ R2 = −12√(λ+ − λ−)2 − 4k12k21 − k21 − 12(λ+ + λ−). (B.14)Note the choice of the negative root:√(k12 − k21 +R1 −R2)2 = −(k12 − k21 + R1 − R2).This ensures that when Tcr → ∞ (causing k12 → 0, k21 → 0, λ+ → −R2, and λ− → −R1),then the RHS of Eq. B.14 correctly becomes −12((−R2)− (−R1))− 12(−R2)− 12(−R1) = R2.Now, Tcr and M0,1 are the only remaining unknown parameters (M0,2 is found via Eq. B.3).In the case of fitting the four pool model (Chapter 5) Tcr is a fit parameter. At eachstep in the solver iteration, Tcr will have some trial value, so M0,1 is found by solving thetranscendental equation given in Eq. B.9. We found that a bracketed root finder like SciPy’simplementation of the quadratic Brent algorithm [B3] worked well.[B1] Bjarnason, T., Vavasour, I., Chia, C. & MacKay, A. Characterization of the NMRbehavior of white matter in bovine brain. Magnetic Resonance in Medicine 54, 1072–1081(2005). URL[B2] Bjarnason, T. The Effect of Cross Relaxation on the NMR Behaviour of Bovine WhiteMatter. UBC Master’s thesis (2005). URL[B3] Brent, R. P. Algorithms for Minimization Without Derivatives. Englewood Cliffs, NJ:Prentice-Hall, (1973). Ch. 3-4.179Appendix CCircuit analogies in NMR relaxationC.1 IntroductionCertain relaxation problems have one-to-one correspondence with electric circuits. Theseamusing circuit analogies offer no new physics, but they do provide an interesting way oflooking at certain problems which may be more intuitive. Circuit analogies have been usedbefore by Bloch to describe the Nuclear Overhauser Effect [C1]. Here, we apply them to thefour pool model and Provotorov equations. R1 R2 R3 R4R12 R23 R34C1 C2 C3 C4VFigure C.1: The equivalent circuit of the four pool model.180C.2 Four pool modelThe four pool model equations aredMMdt=− MM −MM(∞)T1,M− kM,MWMM + kMW,MMMWdMMWdt=− MMW −MMW(∞)T1,MW+ kM,MWMM − kMW,MMMW−kMW,IEWMMW + kIEW,MWMIEWdMIEWdt=− MIEW −MIEW(∞)T1,IEW+ kMW,IEWMMW − kIEW,MWMIEW−kIEW,NMMIEW + kNM,IEWMNMdMNMdt=− MNM −MNM(∞)T1,NM+ kIEW,NMMIEW − kIEWMNM.The circuit in Fig. C.1 is described by equivalent equations. Consider the node where C1,R1, and R12 meet. The current flowing into the node from the capacitor is dQ1/dt, whereQ1 is the charge on the capacitor. Kirchoff’s node rule givesdQ1dt= V −Q1/C1R1+ Q2/C2 −Q1/C1R12= −(Q1 − C1V )R1C1− Q1C1R12+ Q2C2C12= −(C1R12)−1Q1 − (Q1 − C1V )R1C1+ (C2R12)−1Q2.Comparing this to the expression for dMM/dt above, we make the connectionsQ1 = MM(C1R12)−1 = kM,MW(C2R12)−1 = kMW,MMM(∞) = C1VT1,M = R1C1.Similar expressions will apply for the other pools.In this analogy, magnetization is charge, and our goal is to quantify all of the componentvalues. We do this by putting different initial charges on the capacitors, and then observinghow it returns to equilibrium.181  CD CDRD RD CZCZRDZRDZ RDZRZRZ(A) (B)rf applied at +∆ or -∆ rf applied at ±∆ simultaneouslyV VFigure C.2: The equivalent circuit of the Provotorov equations with Ω = 1.C.3 Provotorov equationsThe Provotorov equations also lend themselves to an analogous circuit, shown in Fig. C.2Aand B. In this case, only with Ω = 1 does the circuit provide an easy analogy. The ProvotorovEquations (Eq. 2.47) for the Zeeman and dipolar magnetizations MZ and MD are thendMZdt= −(MZ −MZ(∞))T1−WMZ +WMDdMDdt= −MDT1D−WMD +WMZ .(C.1)Panel A shows the case for single-sided irradiation. Here, we have for the charge across eachcapacitor (following the analysis from the previous section)dQZdt= −(QZ − V CZ)CZRZ+ (CZRDZ)−1QD − (CZRZ)−1QZdQDdt= − QDCDRD− (CDRDZ)−1QD + (CZRZ)−1QZ.(C.2)The D and Z subscripts refer to dipolar and Zeeman reservoirs respectively. ComparingEqs. C.1 and C.2, we can make the connections:V CZ = MZ(∞)QZ,D = MZ,DCZRZ = T1CDRD = T1D(CDRDZ)−1 = (CZRDZ)−1 = W.182In ihMT, this would be the case for a single off-resonance prepulse (an S+ or S− prepulse).In panel B, the equivalent circuit behaves like the Zeeman and dipolar reservoir for an Sdualprepulse: they are uncoupled.This analogy makes it clear why WT1D is a key parameter for ihMT. This isWT1D = CDRD(CDRDZ)−1= RD/RDZIf RDZ →∞ (W = 0), there is obviously no ihMT. If RD ≈ 0 (T1D ≈ 0), then the dischargeof CZ behaves the same in both cases, and there is also no ihMT. In the intermediate case,if RD ∼ RDZ , then the discharge of CZ will behave differently in both cases, and there isihMT.[C1] Bloch, F. Dynamical theory of nuclear induction. II. Physical Review, 102 104–135(1956). URL DModel of ihMT using pulse-trainprepulsesThe pulse-train variety of ihMT experiments use prepulses consisting of trains of shapedpulses. Typically, in the S+ or S− experiments, these shaped pulses are all at the offset+∆ or −∆, respectively. In the Sdual experiment, these shaped pulses alternate between+∆ and −∆. In the following, we show a simple model of the behavior of a many-spinsystem under these prepulses. As with the model of CW prepulses presented above, weignore magnetization transfer to aqueous protons.Fig. D.1 shows our simple model of pulse-train prepulses. We make the assumption ofrectangular pulses so that while the RF is on the Provotorov equations in Eqs. 6.16 and 6.19apply. As discussed earlier in Chapter 7, we can add an extra dimension to eliminate theinhomogeneous term from T1 relaxation, leading todρ±dt= W−1− 1WT1Ω 〈Iz〉0T1Ω −Ω2 − 1WT1D00 0 0ρ±= Aρ±.(D.1)where A is the coefficient matrix. The solution to this equation for an arbitrary initialcondition vector ρ(0) = (ρ1, ρ2, 1) and an irradiation length τ1 is [154]ρ+(t) = P+ρ+(0)P+ = F(τ1)F−1(0),(D.2)184 Osets+ ++ ---S-S+Sdual,1( )n1Experiment Matrix representation(          )n-PR PR -(          )n+PR PR -(          )n+PR PR+- +Sdual,2(          )n+PR PR-2 1 2Figure D.1: A model of ihMT prepulses of the pulse-train variety. With the assumptionof square pulses, the sequence can be modeled as a product of matrices. An actual ihMTexperiment consists of many cycles (n ∼ 100–1000).where the matrix F(τ1) has columns composed ofF(τ1) =[v1eλ1τ1 v2eλ2τ1 v3eλ3τ1]. (D.3)Here, v1,2,3 are the eigenvectors of A and λ1,2,3 are the eigenvalues, given by Eq. 6.21. Thematrix for irradiation at −∆ is P− = P+(∆→ −∆). During the delay τ2 between pulses, amatrix R describes the dipolar relaxation:R =− 1T10 〈Iz〉0T10 − 1T1D00 0 0 . (D.4)As shown in Fig. D.1, we can now represent a pulse-train prepulse sequence of n cycles byproducts of matrices. The Zeeman magnetization at the end of the pulse train is〈Iz〉+(n) = ((P+ RP+ Rρ0)n)1〈Iz〉−(n) = ((P−RP−Rρ0)n)1〈Iz〉dual,1(n) = ((P+ RP−Rρ0)n)1〈Iz〉dual,2(n) = ((P−RP+ Rρ0)n)1,(D.5)where the subscript 1 indicates the value of the first (Zeeman) component of the vector.Finally,Non-aqueous ihMTR = 〈Iz〉+ + 〈Iz〉− − 〈Iz〉dual,1 − 〈Iz〉dual,22〈Iz〉0 . (D.6)Although we do not consider this here, some recent studies with pulse-train prepulses also185used dual irradiation during τ1 to measure and increase sensitivity to T1D [127, 133]. Thiscould be included through the equationdρdualdt= W−1− 1WT10 〈Iz〉0T10 −Ω2 − 1WT1D00 0 0ρdual. (D.7)This approach can also be used to model shaped pulses by discretizing them as rectangularpulses. Also, coupling to the aqueous protons could be included as well. Here we onlyconsider the effect of rectangular pulses on isolated non-aqueous protons..Fig. D.2 shows the results of simulations using this pulse-train model. Unless otherwisestated, the following parameters were used: ∆ = ωD/2pi = 10 kHz, T1 = 1 s, τ1 = τ2 = 3ms, n = 41, and g(2pi∆) as a Gaussian with standard deviation of ωD. Fig. D.2A replicatesFig. 6.2A. The plot is discretized because the prepulse time increases in steps of 2(τ1 + τ2),which is the time for one cycle. Because there is a delay of time τ2 between pulses in thepulse train, the effective transition rate Weff isWeff = W τ1τ1+τ2 . (D.8)The relative rates term for the pulse-train model is WeffT1D, and for the equivalent value ofWT1D in the CW model, the ihMT behavior is nearly identical.Fig. D.2B replicates Fig. 6.2B using the pulse-train model. Again, very similar behavioris seen. This plot also shows the importance of inter-pulse period τ2. The non-aqueousihMTR will have a multiplicative term of exp(−τ2/T1D)n, so the ihMTR is suppressed unlessτ2 & T1D.Finally, Fig. D.2C plots the offset frequency dependence, as in Fig. 6.2C for the CW model.A similar trend is seen, albeit with a smaller effect, at shorter overall pulse lengths. This islikely because the effective irradiation time is actually 25/2=12.5 ms, which is too short togenerate a large ihMT effect.186Figure D.2: Simulation of non-aqueous ihMT in non-aqueous spin system using pulse-trainprepulses. Coupling to aqueous protons has not be included. (A) shows the dependenceon prepulse duration, which is discretized to fit in an integer number of cycles. This showssimilar behavior as in the CW model (see Fig. 6.2A). (B) shows the dependence on theeffective relative rates term WeffT1D. Again, the behavior is very similar to the CW modelin Fig. 6.2B. Finally, (C) shows the offset-dependence, similar to Fig. 6.2C. Because theduty cycle is only 50%, when τ = 25 ms, the ihMT repsonse is significantly smaller thanin the CW case. Unless otherwise indicated, in all plots ∆ = ωD/2pi = 10 kHz, T1 = 1 s,τ1 = τ2 = 3 ms, n = 41, and g(2pi∆) as a Gaussian with standard deviation of ωD.187


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