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Characterizing magnetization exchange in healthy human brain and bovine brain Kalantari, Saeed 2013

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Characterizing magnetization exchange in healthy human brain and bovine brain  by Saeed Kalantari  B.Sc. Physics, Amirkabir University of Technology, 2004 M.Sc. Physics, University of Waterloo, 2007  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Physics)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  May 2013  © Saeed Kalantari, 2013  Abstract Multi component T2 relaxation imaging is an established MRI technique for measuring myelin water (MW, water molecules trapped between myelin sheath bilayers). Myelin water fraction (MWF, the fraction of central nervous system water with a short T2) has been quantitatively correlated to histological staining for myelin in central nervous system tissue and hence is considered an in vivo measure of myelin content. Various studies have reported on the measurement of MWF with a diverse range of neurological diseases such as Multiple Sclerosis (MS), Schizophrenia, epilepsy, and Phenylketonuria (PKU). Although T2 relaxation is the main probe for measuring MWF, understanding longitudinal relaxation, T1, is essential in a number ways such as the following: 1) Estimation of the corrections for myelin water fraction that need to be taken into account due to water exchange processes in white matter in vivo is highly dependent on T1 relaxation. 2) Investigating the effect of T1-weighting in MWF measurements at short TR. This is especially important due to recent breakthroughs in developing rapid 3-D whole brain approaches to MWF measurements that are pushing towards shorter and shorter TR times in order to make this technique a valuable clinical imaging tool. First, in vivo MRI data from multi-component T2 relaxation from 57 healthy subjects collected at 3.0 T was analyzed to estimate the corrections which have to be taken into account due to magnetization exchange in white matter. These results showed that these MWF corrections were less than 15% and are uniform across various white matter structures. Next, the variation of MWF as function of repetition time (TR) was investigated using in vivo MRI data collected at 3.0 T from healthy subjects. These results clearly showed that the measured MWF increased as the TR decreased.  ii  Finally, in order to measure T1 as well as the rate of magnetization exchange with higher precision, data from bovine brain white matter was collected using a 4.7 T NMR spectrometer. The results from this study clearly showed that the T1 had two components; therefore magnetization in bovine white matter is not in a fast exchange regime on the T1 timescale.  iii  Preface  The work presented in chapter two is published under the title "Insight into in vivo magnetization exchange in human white matter regions" in Magnetic Resonance in Medicine with Saeed Kalantari as the primary author (Kalantari et al. 66:1142–1151 (2011).) . Coauthors are Dr. Cornelia Laule, Dr. Thorarin A. Bjarnason, Dr. Irene M. Vavasour, and Dr. Alex L. MacKay. I did all of the data analysis, data interpretation, and wrote most of the manuscript for this paper. The work presented in chapter three will be submitted for publication under the title "Variation of Myelin Water Fraction as Function of TR", with Saeed Kalantari as the primary author. Co-authors are Nazanin Komeilizadeh, Ramin Sahebjavaher, Dr. Irene Vavasour, and Dr. Alex L. MacKay. I did the data analysis, data interpretation and experimental planning for this study. The work presented in chapter four will be submitted for publication under the title "Characterizing Longitudinal Relaxation Time T1 in Bovine Brain Ex Vivo " with Saeed Kalantari as the primary author. I did most of the data analysis except the CPMG analysis which was done by Radim Barta. I was also involved in the experimental planning for this study along with Dr. Michal and Dr MacKay. Research Ethics The Ethics Approval Code from the UBC Clinical Research Ethics Board for the data presented in chapter two is H07-70237. The Ethics Approval Code from the UBC Clinical Research Ethics Board for the data presented in chapter three is H06-00282.  iv  Table of Contents  Abstract .................................................................................................................................... ii Preface ..................................................................................................................................... iv Table of Contents .................................................................................................................... v List of Tables .......................................................................................................................... ix List of Figures .......................................................................................................................... x Glossary ................................................................................................................................ xiv Acknowledgements ............................................................................................................... xv Dedication ............................................................................................................................ xvii Chapter 1: Introduction ........................................................................................................ 1 1.1  Basics of NMR ............................................................................................................. 1  1.1.1  T1 and T2 relaxations ............................................................................................. 3  1.1.2  Bloch equations ..................................................................................................... 4  1.2  Magnetic resonance imaging (MRI) ............................................................................ 5  1.3  Myelin and its function in nevous system.................................................................... 7  1.4  Multi-component T2 and myelin water imaging .......................................................... 8  1.4.1  Analysis of multi-component T2 data.................................................................... 9  1.5  Brain white matter regions of interest (ROI) ............................................................. 11  1.6  Two pool model of magnetization exchange ............................................................. 12  1.7  Validation of myelin water fraction ........................................................................... 15  1.8  Overview of this thesis............................................................................................... 17  v  Chapter 2: Measuring magnetization exchange in various micro structures of human brain in vivo............................................................................................................................20 2.1  Introduction ............................................................................................................... 20  2.2  Materials and methods ............................................................................................... 24  2.2.1  Subject information ............................................................................................. 24  2.2.2  MRI experiments ................................................................................................ 24  2.2.2.1 Inversion recovery (IR) experiment ................................................................ 25 2.2.2.2 Inversion recovery simulations......................................................................25 2.2.2.3 Cross relaxation .............................................................................................. 25 2.2.3  Data analysis ....................................................................................................... 26  2.2.3.1 Inversion recovery .......................................................................................... 27 2.2.3.1.1 Experimental data ...................................................................................... 27 2.2.3.1.2 Simulations.................................................................................................27 2.2.3.2 Cross relaxation .............................................................................................. 28 2.3  Four pool model of exchange in white matter ........................................................... 29  2.3.1  Application of the four pool model ..................................................................... 31  2.4 Results..........................................................................................................................35 2.4.1  T1 Measurements ................................................................................................. 35  2.4.1.1 Experimental inversion recovery .................................................................... 35 2.4.1.2 Simulated inversion recovery ......................................................................... 37 2.4.2  Fitting the four pool model to the results ............................................................ 38  2.5 Discussion....................................................................................................................41 2.5.1  T1 scenarios ......................................................................................................... 42  vi  2.5.2 Myelin water exchange .......................................................................................... 43 2.5.3 Correcting MWF for exchange .............................................................................. 44 2.5.4 Exchange with myelin and non-myelin tissue........................................................45 2.5.5 Limitations..............................................................................................................45 2.6  Conclusions ................................................................................................................ 47  Chapter 3: Variation of myelin water fraction as a function of TR................................48 3.1  Introduction ................................................................................................................ 48  3.2  Materials and methods ............................................................................................... 51  3.2.1  Subject information ............................................................................................ 51  3.2.2  MRI experiments ................................................................................................ 51  3.2.3  Data analysis ....................................................................................................... 52  3.3 Results..........................................................................................................................52 3.4  Discussion .................................................................................................................. 54  3.5  Limitations ................................................................................................................. 63  3.6  Conclusion ................................................................................................................. 66  Chapter 4: Characterizing longitudinal relaxation in bovine brain white matter exvivo..........................................................................................................................................67 4.1  Introduction ................................................................................................................ 67  4.2 Materials and methods ................................................................................................67 4.2.1  Samples ............................................................................................................... 67  4.2.2  NMR equipment and experiments......................................................................68  4.2.2.1 Inversion recovery-FID.................................................................................69 4.2.2.2 Inversion recovery-CPMG............................................................................69  vii  4.2.3  Data analysis.......................................................................................................70  4.2.3.1 Inversion recovery-FID ................................................................................... 70 4.2.3.2 Inversion recovery-CPMG .............................................................................. 70 4.2.3.3 Four pool model of white matter .................................................................... 71 4.3  Results ........................................................................................................................ 72  4.3.1  FID ...................................................................................................................... 72  4.3.2  CPMG at long TI ................................................................................................ 74  4.3.3  Inversion recovery- FID...................................................................................... 75  4.3.4  Inversion recovery-CPMG...................................................................................77  4.3.5  T1distributions of water pools..............................................................................78  4.3.6  Application of 4-pool model ............................................................................... 81  4.4  Discussions ................................................................................................................ 84  4.4.1  Simulations for showing minimum required FNR for resolving T1 ................... 84  4.4.2  Limitations .......................................................................................................... 85  4.5  Conclusion ................................................................................................................. 86  Chapter 5: Conclusion ......................................................................................................... 87 References .............................................................................................................................. 89 Appendix ................................................................................................................................ 97  viii  List of Tables Table 2.1  cross-relaxation times between adjacent signal pools for each of the five  examined white matter region, from 57 normal volunteers, corresponding to scenarios I to III. Error estimates are calculated by varying each cross relaxation times until the sum of squares of the fit was increased by 5%. These errors are indicated by a ± symbol..............................40 Table 2.2  measured and cross-relaxation corrected myelin water fractions (%) for five  white matter regions corresponding to the three considered T1 scenario. Standard deviations of the measured MWF, calculated over 57 normal volunteers are shown in brackets. For corrected MWF values, error estimates are followed by “±”. These errors reflect the maximum change in MWF correction estimates when the input parameter for each structure (measured Tcr's) was varied by its standard deviation.............................................................41 Table 3.1 Measured MWF at various TReff times. The standard error associated with each measurement is shown inside parenthesis............................................................................... 53 Table 3.2  lists best estimates of Smw (relative amplitude of myelin water pool), Siew (T1  of fast exchanging myelin pool), T1mw (the T1 associated with myelin water pool), and T1ie (the T1 associated with IEW pool) based on slow exchange model. ...................................... 58 Table 3.3 lists best estimates of T1MW1 (the T1 of slow exchanging myelin pool), T1MW2 (T1 of fast exchanging myelin pool), FMW1(the fraction of slow exchanging myelin pool), and MWF TR = ∞ (myelin water fraction at long TR) assuming the hybrid myelin model..............61 Table 4.1 The estimated cross relaxation times (Tcr) corresponding to hard and soft inversion pulses. Error estimates are calculated by varying each cross relaxation times until the sum of squares of the fit was increased by 5%. These errors are indicated by a ± symbol......................................................................................................................................82  ix  List of Figures Figure 1.1 Electron micrograph of a cross section of a myelinated axon...............................8 Figure 1.2 Five white matter locations: the genu (GU) and splenium (SP), the posterior internal capsules (IC), and the major forceps (MJ) and minor forceps (MN) that are widely used in this thesis.....................................................................................................................12 Figure 1.3 Simulation of exchange between MWF and IEWF using the two pool model. MWF and IEWF and T2A and T2B are plotted as a function of cross relaxation time Tcr.......15 Figure 1.4 Correlation between MWF and the optical density of Luxol fast blue (LFB, an established stain for myelin) has been measured in grey matter (GM), lesion, diffusely appearing white matter (DWM), and white matter (WM).......................................................16 Figure 2.1 Schematic representation of the four pool model of white matter......................23 Figure 2.2 Mono-exponential fit to the IR data (with SNR(T1) of 85) from major forceps of one healthy normal volunteer acquired with a single-slice fast gradient echo sequence obtained at 15 TI times. The residual sum of squares of the fit was 141.2. The extracted single T1 was 760 [35] ms and its amplitude was 261 [23]. The values in the square brackets are the associated 95% confidence intervals of the fits...........................................................36 Figure 2.3 Bi-exponential and mono-exponential fits to the synthetically generated IR data with SNR(T1) of 85 and 15 TIs. The residual sum of squares of the fit was 83.7 for the biexponential fit and 256.3 for the mono-exponential fit. Extracted T1MW and T1IE from biexponential fitting were 652 ± 483 ms and 948 [2511] ms, and their associated amplitudes were 165 [437] and 105 [598] respectively. The single T1 extracted from mono-exponential fitting was 1007 [106] ms. The values in the square brackets are the associated 95% confidence intervals of the fits.................................................................................................37 x  Figure 2.4 (a-e) Myelin water and IE water signals from (a) genu, (b) splenium, (c) minor and (d) major forceps, and (e) internal capsules of 57 normal volunteers for all three scenarios investigated. The analytical solution corresponding to each T1 scenario is shown by a line. Filled circles and triangles represent the experimentally measured myelin water and IE water data. Standard deviations are shown in error bars.........................................................39 Figure 3.1 Plots of MWF at various TReff times from 165 ms to 665 ms measured across five different brain white matter structure in vivo. Standard errors are shown as error bars..54 Figure 3.2 Synthetically generated fit showing MWF vs. TReff assuming both MW and IEW pools have the same T1= 1000 ms..................................................................................55 Figure 3.3 Synthetically generated MWF fit (assuming that the myelin water has a shorter T1 compared to the I/E water T1) plotted along with the experimental MWF data vs. TReff..57 Figure 3.4 Hybrid myelin model showing MWF vs. TReff assuming the hybrid myelin model........................................................................................................................................61 Figure 3.5 Plot of MWF from putamen (gray matter) measured at various TReff times from 165 ms to 665 ms in vivo. Standard errors are shown as error bars........................................62 Figure 3.6  Plot of MW and IEW from major forceps measured at various TReff times from  165 ms to 665 ms in vivo.........................................................................................................64 Figure 3.7 Iterative simulation using the four pool model. Each of the four signal pool calculated at TReff =165 ms for 10 iterations...........................................................................65 Figure 3.8 Iterative simulation using the four pool model. Each of the four signal pool calculated at TReff =665 ms for 10 iterations...........................................................................66  xi  Figure 4.1 Schematic representation of the four pool model of white matter. Sizes of the four compartments are scaled to correspond roughly to the relative numbers of protons in each pool in white matter.........................................................................................................72 Figure 4.2 Free induction decay of bovine brain sample at long TI after the hard inversion pulse. A straight line (red line) was fitted to the data at times 0.2 ms < t < 0.8 ms................73 Figure 4.3 The free induction decay of bovine brain sample following the hard inversion pulse.........................................................................................................................................73 Figure 4.4 The free induction decay of bovine brain sample following the soft inversion pulse. A straight line (red line) was fitted to the data at times 0.2 ms < t < 0.8ms ................74 Figure 4.5 The T2 distribution of bovine brain sample following a hard inversion pulse at the longest inversion time of TI = 9.3 s...................................................................................74 Figure 4.6 The aqueous and non-aqueous signal extracted from FID curves following the hard inversion pulse.................................................................................................................76 Figure 4.7 T2 distribution of bovine brain sample following a hard inversion pulse at the shortest inversion time of TI = 450 μs.....................................................................................77 Figure 4.8 A-B The amplitudes of each of the water pools are shown as a function of inversion time TI in figure A. Figure B gives a zoomed view on the zero crossing of each component................................................................................................................................78 Figure 4.9 T1 distribution of myelin water pool from bovine brain sample following a hard inversion pulse.........................................................................................................................79 Figure 4.10 T1 distribution of intra- / extra- cellular water pool from bovine brain sample following a hard inversion pulse..............................................................................................80  xii  Figure 4.11 T1 distribution of long T2 water pool from bovine brain sample following a hard inversion pulse.........................................................................................................................80 Figure 4.12 T1 distribution of all water pools from bovine brain sample following a hard inversion pulse.........................................................................................................................81 Figure 4.13 Fit to the experimental hard inversion pulse FID data from the aqueous and nonaqueous pool. The residuals sum of squares for this fit is 1.2 x 103........................................83 Figure 4.14 Fit to the experimental soft inversion pulse FID data from the aqueous and nonaqueous pool. The residuals sum of squares for this fit is 9.7 x 103........................................83  xiii  Glossary B0 B1 BW CNS CPMG CSF EPI FOV GM GE IE IEW IR LT2F M MF MR MRI MS MT MW MWC MWF NM NMR NNLS ODE PD RF ROI SAR SE SNR T T1 T2 TCR TE TI TR VTR WM  Main magnetic field strength Radio frequency field strength Bandwidth Central nervous system Carr-Purcell-Meiboom-Gill (spin-echo pulse sequence) Cerebrospinal fluid Echo planar imaging Field of view Grey matter General electric Intra and extra-cellular Intra and extra-cellular water fraction Inversion recovery Long-T2 fraction Myelin Mobile fraction Magnetic resonance Magnetic resonance imaging Multiple sclerosis Magnetization transfer Multiple water Myelin water content Myelin water fraction Non myelin Nuclear magnetic resonance Non negative least squares Ordinary differential equations Proton density Radio frequency Region of interest Specific absorption rate Standard error Signal-to-noise ratio Tesla Spin-lattice relaxation time Spin-spin relaxation time Cross relaxation time Echo time Inversion time Repetition time Variable repetition time White matter  xiv  Acknowledgements First and foremost, I would like to thank Dr. Alex MacKay for his supervision of my PhD researches during my studies at UBC. I have learnt a great deal from him during these years. I especially enjoyed his amazingly relaxed yet scientifically curious character. I feel enormously privileged to be one of his graduate students. I also thank each member of my supervisory committee: Dr. Carl Michal, Dr. Piotr Kozlowski, Dr. Stefan Reinsberg, Dr. David Li, and Dr. Qing-San Xiang, for kindly accepting to be in my supervisory committee and for their valuable inputs in our discussions of my projects. I also thank the UBC MRI Research Center technologists for their help in scanning subjects. Additionally I thank the people who volunteered to be scanned during my research projects. I need to give thanks to the people both at the UBC MRI Research Center as well as the UBC Department of Physics and Astronomy who helped me in one way or another throughout my research projects. I also thank UBC IT staff for their help, especially Hadi Susanto. I thank Dr. Carl Michal for his collaboration for the work presented in chapter four. I thank the Multiple Sclerosis Society of Canada for their financial support of my research. Special thanks to our lovely 'British Queen' Linda Chandler for her efforts in keeping UBC MRI Research Center up and running. Thanks to the secretaries of the Department of Physics and Astronomy especially Oliva Cruz. I would like to express my sincere gratitude to my parents, my sisters, and my brother for their support throughout my entire academic journey and helping me achieve my goals in life.  xv  Last but definitely not least, my biggest thanks go to my beloved partner, Nazanin. Her constant support and love have helped me stand strong and stay focused on my goals during all these years of being together and no words can possibly be able to express my gratitude. I specially need to thank her for the beautiful peaceful heaven she has built for me and her everlasting love and support. Without her, the writing of this thesis and finishing up the work would have been much harder if not impossible. Thank you my dearest for all these years of support and sheer love and I love you more than anyone can possibly imagine!  xvi  Dedication  To Charles Darwin, Richard Feynman, Richard Dawkins, Bertrand Russell, and Nazanin Komeilizadeh  xvii  Chapter 1:  Introduction  Magnetic resonance imaging (MRI) is one of the most widely used imaging modalities for studying the human body. The fact that MRI is a non-invasive imaging modality is partly the reason that it is one of the most widely used imaging techniques for studying the human brain. An amazing characteristic of MRI is that one can choose from a wide repertoire of different MRI pulse sequences that are designed to optimize the desired contrast. For instance, quantitative T2 imaging is an established MRI method to study brain white matter with exquisite detail and precision. In this chapter we briefly go over the basics of magnetic resonance imaging with an emphasis on quantitative T2 imaging techniques.  1.1  Basics of NMR  NMR is founded on the interaction between an external magnetic field (Bₒ) and the nuclei with a non-zero intrinsic magnetic moment. If placed in an external magnetic field, the magnetic moments of these nuclei can either align parallel or anti-parallel to the external magnetic field thus generating two energy states corresponding to spin up and spin down states. The energies associated with each of these states known as Zeeman energies are given by the following equations:  (1.1) As the population of the spins that are parallel to the magnetic field is slightly larger than the population of the spins in the anti-parallel direction, in equilibrium, there is a net magnetization, Mₒ , the magnitude of which is given by the following approximate equation:  1  |Mₒ| = Bₒ ρ  2  where ρ is the proton density of the sample,  ɤ2 /4 T kB ,  (1.2)  is Plank's constant divided by 2π, T is absolute  temperature, ɤ is a constant called the gyro-magnetic ratio, and kB is Boltzmann's constant. At the macroscopic level, it has been observed that, this net magnetization precesses about the external static magnetic field with an angular frequency of ωₒ which is commonly known as the Larmor frequency. This precession is very similar to the precession of a spinning top tilted slightly off-axis due to the force of gravitation. This precessional frequency, ωₒ, is shown to be proportional to the external magnetic field, given by the Larmor equation: ωₒ (r) = ɤ Bₒ (r)  (1.3)  where r is the position vector in the 3 dimensional space. Conventionally, this magnetic field Bₒ is considered along the z-axis and the equations governing the magnetization vector Mₒ , are written in a frame of reference that is rotating at the Larmor frequency and usually denoted by x', y', z'. To flip this magnetization vector Mₒ to any desired angle, a radio frequency (RF) oscillating magnetic field B1 is applied in an orthogonal direction with respect to Bₒ. Assuming the magnitude of the RF pulse in the rotating frame (B1) is constant, the magnitude of this flip angle (θ) is given by the following equation written in the rotating frame: θ=ɤ  . B1 . t  (1.4)  where t is the time duration that the B1 pulse is applied. After this RF pulse is applied, the system is said to be in an excited state or simply excited. The precession of the magnetization vector will generate electromagnetic radiation which can be picked up and analyzed in order to produce the image. These RF pulses are normally applied on the sample by a set of coils  2  known as transmitting coils and the response of the sample to this excitation is picked up by another set of coils known as receiving coils. In some cases though, both transmitting and receiving are done by the same set of coils.  1.1.1  T1 and T2 relaxations  After the B1 pulses (RF pulses) are applied to flip the net magnetization vector, Mₒ , to the perpendicular plane (also known as transverse xy-plane), the excited protons will start to relax back to their original equilibrium states. This phenomenon is generally referred to as relaxation. In more detail, this relaxation phenomenon is composed of two distinctive behaviors, one being the dephasing of the magnetic moments associated with each excited nucleus right after the applied RF pulse, and the other is their realignment along the z axis as they emit back the energy they absorbed from the RF pulse. The dephasing of the spins in the transverse plane is mainly due to the spin-spin interactions among the magnetic moments of nuclei as they randomly move within the sample. This interaction is characterized by a time constant T2 and is commonly referred to as spin-spin relaxation. In mathematical language the time constant T2 is the time at which the transverse magnetization irreversibly decays to 1/e of its maximum magnitude. This relaxation time constant is a very good characteristic of the magnetic resonance (MR) properties of the sample under study as it is very sensitive to the micro-structure of the sample. T1 relaxation, on the other hand, is due to the transfer of absorbed energy from the excited nuclei to the surrounding environment known as the lattice, and is thus given the term spin-lattice relaxation. It is noteworthy that in spin-spin interaction there is no loss of energy that was  3  initially absorbed from the RF pulse, while during a spin-lattice interaction there is the exchange of energy from the excited spin system to the surrounding lattice.  1.1.2  Bloch equations  The Bloch equations are a set of first order differential equations that describe the behavior of the magnetization vector M with respect to time. Right after the application of an RF pulse to flip the magnetization to the orthogonal transverse plane, the z-component of magnetization vector, Mz will grow to reach its equilibrium magnitude according the following equation:  d  M  dt  z    [M  o   M z]  (1.5)  T1  The solution to this first order differential equation would be of the following form:  M z = M o + [ Mz(0) - Mo] exp(-t / T1)  (1.6)  In the transverse plane on the other hand, the x-component of magnetization is coupled with y-component of magnetization and the y-component of magnetization is coupled with the xcomponent of magnetization as shown by the following Bloch equations:  d dt  d dt  M  M  x  y        M  x  T2  M   M  y  Bo  (1.7) (1.8)  y  T2   M x B o  assuming that the effective magnetic field would be of the form Beff = Bo k where k is the unit vector along z-axis.  4  In order to simplify the solution to the transverse magnetization, we consider a perfect 90 ̊ flip about the x-axis for which one can assume that at t=0 (the centre of the pulse) the magnetization vector would be of the form M = 0 i + Mo j + 0 k , thus the solutions (in absence of B1 field) will be:  Mx = Mo sin(ωt) exp(-t/T2)  (1.9)  My = Mo cos(ωt) exp(-t/T2)  (1.10)  Mz = Mo [1- exp(-t/T1)]  (1.11)  where the transverse component can be written even more concisely using complex terms:  Mxy = Mo exp(iωt) exp(-t/T2)  (1.12)  This expression explicitly shows that at t=T2 , the transverse component of magnetization has decayed to e-1 of its original magnitude just after the RF pulse has been switched off. It should be mentioned that equations 1.9,1.10, and 1.12 assume a perfectly homogeneous Bₒ.  1.2  Magnetic resonance imaging (MRI)  In essence, MRI utilizes the NMR phenomenon namely linear dependence of frequency on the net local magnetic field (equation 1.3). The spatial encodings are implemented by small variations in local magnetic fields known as magnetic field gradients. First 2D slices are produced by the combination of an excitation RF pulse and simultaneous slice-select gradient. The bandwidth (∆ω) of this RF pulse determines the thickness of the excited slice (∆z) and its frequency determines the location of the excited slice as shown by the following equation:  ∆ω = ɤ . Gz . ∆z  (1.13) 5  where Gz is the strength of the magnetic field gradient along the z-axis and ∆ω is the bandwidth of the pulse. As the RF pulse is applied when the gradient is on, only the spins that are in resonance with the frequency range of the RF pulse will be excited . For example if this gradient is applied along the z-axis, the excited slice is a 2-D plane orthogonal to the zaxis. Then the in-plane MR signal is encoded in terms of the spatial frequencies of the object using frequency-encoding gradients and phase-encoding. Frequency encoding is similar to slice selection and is conventionally assumed to be along the x-axis. When signal is acquired during a frequency encoding gradient, the location of signal is encoded by the NMR frequency. The phase encoding gradients are another set of gradients that once they are applied, the phase of the spins will encode their location along the direction they were implemented. During the phase encode gradient, the precession frequency of the spins will increase or decrease an amount depending on their location along the direction in which the gradient was applied. When the phase encode gradient is switched off, the excited nuclei will precess with their original frequency, but will have different phase angles. The amplitude of the phase encode gradient is incremented between data acquisitions giving rise to incremental phase changes which depend on the positions (along the direction in which the gradient was applied) for each spin and can be utilized to extract their locations along that direction. In spin warp MRI, lines of raw data (k-space) are acquired with the same frequency encoding, but varying amounts of phase encoding. So by applying these series of magnetic field gradients we could then provide a strategy to implement spatial encoding as desired to collect the MRI data.  6  Then the collected spatial frequency data (also known as k-space) is 2D Fourier transformed in order to generate the image.  1.3  Myelin and its function in nervous system  Neurons are the functional building blocks of the nervous system both at the central (CNS) and peripheral (PNS) levels. Each neuron is composed of a cell body (where the nucleus resides) and an axon which is an elongated tube evolved to transmit the action potential from one neuron to the next. Action potentials in the form of electrical signals are received at the cell body via its dendrites and are then passed on along the axons to the next neuron. In order to prevent the attenuation of this action potential along the axon as well as accelerate the speed of neuronal signal transmission, each axon is wrapped around with multiple layers of insulating sheaths made of lipid-protein membranes called myelin. Myelin sheaths in the central nervous system are generated by a special class of glial cells known as oligodendrocytes. Myelination and demyelination of neurons in the central nervous system (CNS) have crucial implications on the functions of healthy neurons. Nerve signal propagation speeds are accelerated by more than an order of magnitude in the presence of myelin. Therefore accurate assessment of myelin is of great importance. In other words, quantitative and reproducible measurement of myelin content is vital for characterizing brain damage and also evaluating the effectiveness of therapy of demyelinating diseases such as multiple sclerosis (MS). In CNS, water molecules trapped in between myelin sheath bilayers [known as myelin water, shown in figure 1.1 (1)], intra /extra cellular water and cerebrospinal fluid (CSF) each  7  have their own distinctive T2 values. This phenomenon is the basis on which Myelin Water imaging has been founded.  Figure 1.1: Electron micrograph of a transaction of a myelinated axon (Norton, and Cammer 1984).  1.4  Multi-component T2 and myelin water imaging  The most common MRI technique to measure the T2 relaxation time of a given sample is a multi spin echo pulse sequence developed by Carr, Purcell, Meiboom, and Gill (2,3), thus commonly known as the CPMG pulse sequence. This pulse sequence is composed of a 90º excitation RF pulse about the x-axis, followed by a train of equally spaced 180º refocusing pulses along the y axis. Each of the 180º pulses will generate an additional echo. Using this strategy one can generate a smoothly decaying curve known as the T2 decay curve. It has experimentally been observed that pure, homogenous liquid samples usually  8  generate a characteristic T2 decay curve to which a single exponential function can be fitted. Non-homogenous samples on the other hand contain nuclei that are located in different compartments and different compartments can have unique T2 properties. These distinct T2 characteristics can be utilized to introduce contrast between various biological tissue types. Mackay et al. (4–6) have introduced a new quantity called myelin water fraction (MWF, the fraction of central nervous system water with a short T2) as a marker of myelin. Myelin water fraction has been shown to quantitatively correlate with myelin content measured by histological staining methods using Luxol fast blue marker for myelin (7,8).  1.4.1  Analysis of multi component T2 data  One of the most promising methods to analyze the multi component T2 decay curves is the so called Non Negative Least Square (NNLS) approach (4,5). The main goal of the NNLS approach is to find a set of non-negative amplitudes, Sj, that minimize the error associated with fitting the T2 decay curve with exponential functions. In order to do this, signal intensities of the T2 decay curves, Yi, at each echo are written as a linear combination of exponentials plus an error term with the following form:  Yi   m   S j exp(  j 1  t  i )  T2j  i ,  i = 1, 2, 3, ..., N  (1.14)  where m is the number of elements in the T2 partition, N is the number of collected echoes, ti are the echo times, Sj are the signal amplitudes for each partitioned relaxation time T2j , and εi is the error associated with echo time ti. The choice of these parameters depends on the nature of the sample under study and its T2 relaxation behavior. For myelin water imaging it  9  is common to make use of m~ 150 equally spaced relaxation times on a logarithmic scale from 10 ms to 1.5 s. The term to be minimized is the following:   2     m    j  1  2 S j  μ > 0  ,  (1.15)  where μ defines the extent of regularization so that large μ gives rise to smoother T2 distributions and small μ leads to spikier smoother T2 distributions. Regularization is done by varying the parameter μ in the above expression. χ2 is the misfit and is mathematically defined in equation (1.16a). The value of χ2 is an indicator of the quality of the fit in the sense that χ2 >> N means that the dynamics of the curve are not well represented in the fits and thus some of the curve details are not accounted for and χ2 << N on the other hand the fitting function might be fitting the noise and consequently might introduce artifacts in the T2 distribution.    2  n  [Y i  1 i       t m i )] 2 exp( S  j j  1 T2 j 2  (1.16a)  In order to avoid these problems and produce robust T2 distribution plots, the following energy constraint is included to bound the values of χ2 as the following (9):  1 . 02   2  min    2   1 . 025   2  min  (1.16b)  10  2  where (χmin) is the misfit at μ = 0. The amplitude of each of the T2 peaks is calculated as the ratio of the summation of Sj within the desired range of T2 over the total Sj as shown by the following expression: l S  j j  k m S  j j  1  (1.17)  where k and l are the lower and upper T2 bounds of the desired T2 range for which the amplitude is being calculated. For reference, typical T2 ranges for human brain in vivo measurements at 1.5 T and 3.0 T are as follows: myelin water 10 ms-40 ms, intra/extra cellular water 50-150 ms, and CSF 1000 ms-2000 ms. Fitting exponentials is a complex and sophisticated area of mathematics and most algorithms are highly dependent on prior knowledge and initial guesses. One of the most important advantages of using the NNLS approach to analyze multi component T2 decay curves is that it does not require any a priori assumptions about the number of major T2 peaks one should expect in a given T2 decay curve thus making NNLS a less biased and more reproducible algorithm for analyzing these types of datasets.  1.5  Brain white matter regions of interest (ROI)  The brain regions of interest (ROI) that were analyzed throughout this thesis are the following five white matter locations: the genu (GU) and splenium (SP), the posterior internal capsules (IC), and the major forceps (MJ) and minor forceps (MN). These regions (shown in figure 1.2) were chosen because they have widely varying MWFs and can be conveniently measured on a single MR slice. Additionally SP and GU are parts of the corpus  11  callosum that connects the two brain cerebral hemispheres, and IC is part of the motor control cortical spinal tract system.  Figure 1.2 Five white matter locations: the genu (GU) and splenium (SP), the posterior internal capsules (IC), and the major forceps (MJ) and minor forceps (MN) that are widely used in this thesis. 1.6 Two pool model of magnetization exchange 1.6 Two pool model of magnetization exchange A major goal of this thesis is to investigate magnetization exchange between different tissue compartments in brain. In order to measure the rate of magnetization exchange, one first needs to develop a model that describes the interactions between the each proton pool with its neighboring pools. In chapter two of this thesis, a four pool model will be developed to model magnetization exchange in white matter. To introduce the mathematics required for characterizing magnetization exchange, the much simpler model of only two interacting proton pools will be introduced here. This two pool model of magnetization exchange has previously been formulated (10). According to this model, cross relaxation between the two spin systems can occur whenever 12  two protons, one in each pool (system 1 and system 2) are in close proximity. Each pool can be considered a separate compartment in which the magnetization is uniform at any given time. The Bloch equations governing this exchange of magnetization can be written in the following way using the reduced magnetization of Edzes and Samulski (10) : d dt  mi   R1i mi  ki mi  ki m j  [1.18]  Where mi    [ M i  M i ( )]  and R  1i  [1.19]  2 M i ( )  1    [1.20]  T 1i  where Mi denote the time-dependent magnetization along the z-axis in each signal pool and Mi (∞) are the equilibrium values of the corresponding pool. The k1 and k2 are the directional exchange rate constants of proton pool 1 and 2 respectively. T11 and T12 are the spin-lattice relaxation times for spin system one and spin system two respectively. The fundamental assumption for writing these Bloch equations is that the magnetization of each pool is considered to be homogenous within each pool at any time. The solution to the coupled differential equations 1.18 and 1.19 is of the form:         mi  Ci exp( R1 t )  Ci exp( R1 t )  [1.21]  where  2 1/ 2 2 R1  R1i  R1 j  k i  k j  [( R1i  R1 j  k i  k j )  4k i k j ]  [1.22]  and  R  R k i i 1 1 C i   mi ( 0 )  [ m i ( 0 )  m J ( 0 )] R  R R  R 1 1 1 1   [1.23]  13  pi ki  p jk  j  [1.24]  assuming pi is the fraction of the protons in spin system i. Cross relaxation time Tcr can also be defined at the following equation: Tcr   1 ki    1  [1.25]  kj  Tcr is a useful quantity because it provides a pool size independent measure of exchange rate. As expected in the special case of no exchange (k1 , k2 = 0) this solution would simplify to equation 1.5 where the magnetization in each pool is not dependent on the T1 relaxation of its neighboring proton pool.  M i  M i (  )  ( M i ( 0)  M i (  )) exp(  t ) T 1i  [1.26]  The reason that the two pool model is discussed here is that it demonstrates many of the same features of more sophisticated models such as four pool model of brain white matter that was employed in the coming chapters to measure the rate of magnetization exchange in brain white matter. The simple two pool model may be applied to the problem of myelin water exchange. The initial conditions chosen, crudely based on experimental data, were: population of myelin water, MWF = 0.20, T2A= 15ms, population of intra/extracellular water, IEWF= 0.80, T2B = 120ms. The cross relaxation time, Tcr, was varied from very slow exchange, 84 s to very fast exchange, 2 ms. The results are displayed in figure 1.3. In the slow exchange regime (long Tcr), the system behaves as two independent water reservoirs. In the fast exchange regime (short Tcr), a single component is measured with zero MWF and an average T2 time, where (1/T2 )= (MWF/T2A) + (IEWF/T2B)  [1.27]  14  The interesting result is that the transition between the two exchange regimes occurs when Tcr = ~ 100 ms. This result is not strongly dependent on the initial value of MWF. It is worth mentioning that this crude model did not taken into account the exchange of magnetization between the non-aqueous pools and aqueous pools.  Figure 1.3 Simulation of exchange between MWF and IEWF using the two pool model. MWF and IEWF and T2A and T2B are plotted as a function of cross relaxation time Tcr.  1.7  Validation of myelin water fraction  Histological staining has been used to validate the use of myelin water fraction (MWF) as a marker for myelin content. For this reason, Luxol fast blue (which is an established stain for phospholipids which are abundant in myelin) has been employed to assess the correlation  15  between myelin water fraction (measured on a 7 T scanner) and the optical density of the staining in a fixed brain sample. As figure 1.4 below (Courtesy of Dr. Laule) shows there is a strong correlation between these two quantities (MWF and the optical density of the myelin staining) especially which the white matter structures of the brain proving that MWF is a reliable marker for myelin in white matter. This suggests that exchange does not play a large role in determining the measured myelin water fraction.  MWF  0.3  0.2  GM Lesion DWM WM Fit  0.1  0 0  0.05  0.1 0.15 LFB OD  0.2  0.25  Figure 1.4 Correlation between MWF and the optical density of Luxol fast blue (LFB, an established stain for myelin) has been measured in grey matter (GM), lesion, diffusely abnormal white matter (DWM), and white matter (WM). (Courtesy of Dr. Laule)  16  1.8  Overview of this thesis  This thesis focuses on measuring magnetization exchange as well as understanding the behavior of longitudinal relaxation in various regions of brain white matter. The overarching goal is to better understand the physical interpretation of the myelin water fraction and to ascertain how reliably the MWF can serve as a marker for myelin content in vivo.  A primary role of measuring magnetization exchange is to assess the rate of movement of water between myelin water and intra/extra cellular water pools. In fact if the rate of magnetization exchange approaches the fast exchange regime, then the measured myelin water fraction would be an underestimation of the myelin content in the white matter. In fact one of the major goals of this thesis is to address the question: Do the large variations in MWF in white matter indicate differences in myelin content or differences in exchange rate? A preliminary work in this area is the studies by Seymour Koenig (11) that related the rate of exchange with the thickness of myelin sheaths in nervous system. The Koenig research related to the T1 time scale which is substantially longer than the T2 time scale. In a previous ex-vivo study on bovine brain white matter the MWF value was found to be practically constant between 24C and 37C. Since water diffusion is known to increase substantially across this temperature range, those results suggested that the myelin water fraction is not substantially affected by exchange with intra/extra cellular water.  17  In chapter two, the four pool model of brain white matter is presented in detail. This model is applied to measure the rate of magnetization exchange in human brain in vivo using MRI data collected at 1.5 T. Additionally the current status of literature and the role of longitudinal relaxation in measuring magnetization exchange is discussed. Various quantities such as cross relaxation time (Tcr), directional exchange rate constants (k), non-directional exchange rate constants (R=1/Tcr) and residence times (1/k) have been used to describe the magnetization exchange. In the analysis of the data presented in chapters two and four, we mainly used the cross relaxation times (Tcr) as they do not depend on the population sizes and thus represent the phenomenon of exchange more intuitively.  In chapter three, MWF was measured at various repetition times (TR) in different brain white matter structures using multi component T2 relaxation imaging in healthy human white matter in vivo MRI at 3.0 T. The main goal of this study was to find out if the MWF was a function of TR. If white matter has two T1 components, one would expect the T1 weighting at short TR to influence MWF. This is important, not only to better understand T1 in white matter but also because new MR pulse sequences designed for measuring MWF across large regions of brain in vivo are required to operate at shorter TR times. A number of hypotheses that could potentially justify the observed results are presented, discussed and analyzed.  In chapter four we set to investigate the fundamental question that whether T1 relaxation is a mono exponential or a multi exponential phenomenon in white matter. We analyzed ex vivo bovine brain data collected at 37 ̊ C and 4.7 T using a home built NMR spectrometer in Dr Michal’s laboratory in UBC Department of Physics. By using inversion recovery combined  18  with free induction decay or T2 decay curve measurements we were able to apply the four pool model in order to characterize magnetization exchange in bovine brain white matter.  The concluding chapter discusses the overall findings of the individual chapters and the significance of the thesis research.  19  Chapter 2: Measuring magnetization exchange in various micro-structures of human brain in vivo Most of the content of this chapter is published in Magnetic Resonance in Medicine, 66:1142–1151 (2011). 2.1  Introduction  The exquisite contrast provided by magnetic resonance imaging in central nervous system tissue plays a key role in clinical MRI; however, the mechanisms producing this contrast are still not well understood. On a microscopic level, central nervous system tissue is not homogeneous; for example, myelinated and unmyelinated regions can be distinguished in white matter using electron microscopy. Since myelin contains multiple bilayers, water molecules in myelinated regions undergo different dynamics from those in unmyelinated regions. Consequently, water in the myelin sheath has different MR properties than water in intra/extracellular (IE) spaces. Many ex vivo studies (12–15) reported a uniquely short T2 for water trapped inside the myelin bilayers compared with water in other physical spaces, a property that has also been observed in white matter in vivo (5,6,16,17). The assignment of the myelin water signal was validated in several studies (7,8,18–20) which compared the myelin water fraction (MWF, the fraction of central nervous system water with a short T2) with histological measures of myelin content. It has been noted that regions with dense parallel fiber bundles, such as the spinal cord and the cortical spinal tract have a higher MWF, while regions with more randomly oriented fiber tracts, like frontal white matter have a lower MWF (5).  20  While quantitative T2 studies of white matter reveal different T2s for water in myelin and water in the IE spaces, most (6,13,21) but not all (22,23) T1 studies in normal white matter reported single component T1 relaxation in vivo. This discrepancy between T1 and T2 could be accounted for by magnetization exchange. Magnetization exchange between proton sites can occur through spin diffusion (mediated by dipolar interactions between adjacent hydrogen nuclei), chemical exchange (movement of hydrogen nuclei between different molecules), or by self-diffusion (diffusion of hydrogen nuclei or water molecules to different sites). The observation of multi component T2 but only a single component T1 in white matter can be accounted for by self-diffusion of water molecules between the myelin sheath and the IE spaces at a rate which is fast compared with the T1 timescale of approximately 1 s. If magnetization exchange is fast on the T1 timescale, then it is possible that such exchange could also influence measured T2 and amplitudes; increased exchange would result in decreased MWF (24–27). The observation that the more heavily myelinated fiber tracts have larger MWFs, which is nominally attributed to increased myelin, could hypothetically be accounted for by postulating decreased water exchange in regions containing thicker myelin sheaths. If this exchange scenario were true, then the observed variations in MWF between regions might reflect differences in exchange rates rather than differences in myelin content. In fact, a very recent study (28) on rat spinal cord found the MWF underestimated histologically measured myelin content in regions of spine where the myelin sheath was thin. Therefore, it is important to better understand magnetization exchange in cerebral white matter and that is the primary goal of this study. Four different studies (24–26,29) have used a four pool model to simulate the MR behavior of white matter including the effects of magnetization exchange. The four pools, illustrated in 21  Figure 2.1, are nonaqueous myelin (M), myelin water (MW), nonmyelin water (IE), and nonaqueous nonmyelin tissue (NM). This model is the least complex model of brain white matter that incorporates myelin and myelin water. The choice of these special pools is rooted from insights from our current understanding of brain white matter architecture. For instance, the fact that this model does not allow direct interaction between non-aqueous myelin and non-aqueous non-myelin tissue can be justified from inspection of high resolution electron micrographs of myelin sheaths (figure 1.1). This model was also previously applied in three separate ex vivo studies on bovine white matter (24) and bovine optic nerve (25) to estimate exchange between myelin water and IE water. Harrison et al. (30) were the first to use the four pool approach to understand T2 in white matter. Their results suggested that exchange occurred between the two white matter water pools in a time greater than the T2 time scale but shorter than their measurement timescale of 7 s. The studies by Bjarnason et al. (24) and Stanisz et al. (25), which used different experimental approaches, reported exchange times between myelin water and IE water of 560 ms and 2064 ms respectively. Based upon these estimated exchange times, measured MWFs should be increased by up to ∼15% to reflect the “true” myelin water content of white matter. One might expect exchange rates in human CNS in vivo to be different from ex vivo animal models; however, two studies (17,31) that investigated in vivo cerebral white matter exchange rates, but did not use the four pool model, also estimated myelin water exchange times in the hundred ms range. Deoni et al. (17) estimated a myelin residence time of about 100 ms.  22  Figure 2.1: Schematic representation of the four pool model of white matter. Sizes of the four compartments are scaled to correspond roughly to the relative numbers of protons in each pool in white matter.  Recently, Labadie et al. (22,23) in a careful T1 relaxation study, measured two T1 components (200 ms and 1.250 s at 4 T) in human white matter in vivo. These two T1 components had similar amplitudes to the T2 components associated with myelin water and IE water. The existence of two component T1 relaxation would seem to suggest that magnetization exchange in white matter is relatively slow on the T1 timescale. However, in another inversion recovery (IR) experiment on rat brain (32), a two component decay curve (20 ms and 1.5 s at 4.7 T) was also observed in both white matter and grey matter. In the rat brain study, the bi-exponential behavior was convincingly described as the result of a magnetization transfer between nonaqueous protons and water (33). In summary, the observation of two components in IR measurement in white matter does not necessarily mean that exchange between myelin water and intra and extracellular water is slow on the T1 time scale. This study applied the four pool model to in vivo human MR data from five white matter regions. The MR experiment consisted of an off resonance pulse followed by a variable delay and a multi-echo T2 measurement. The analysis differed from previous four pool studies in 23  two ways:1) previous work assumed single component T1, where this study examined three different T1 scenarios and 2) this work employed an analytic solution to the Bloch equations describing magnetization evolution of the four pool model. This study differed from the Harrison et al. (30) study in that a relatively brief MT pulse was used rather than a 7 s long continuous wave pulse. Specific aims of this study were to estimate magnetization exchange times for five different white matter regions, to determine whether variations in MWF could be explained by differences in exchange rates and to investigate how multi-component T1 relaxation affects the interpretation of MWF in white matter.  2.2.  Materials and methods  2.2.1  Subject information  For the IR experiment, five healthy normal volunteers (2 female, 3 male, average age = 35.2 years, range = 22–51 years) were scanned. Fifty-seven normal volunteers (37 male, 20 female, average age = 28.6 years, range = 18–54) underwent MR examinations for the cross relaxation study. Informed written consent as approved by the Clinical Research Ethics Board of the University of British Columbia was obtained from all subjects.  2.2.2 MRI experiments All experiments were performed on a 1.5T GE Signa clinical MR scanner (version 5.7 of hardware and software).  24  2.2.2.1 Inversion recovery (IR) experiment T1 was measured from a single transverse slice through the base of the genu and splenium of the corpus callosum using a IR prepared SPGR (Spoiled Gradient Echo) sequence (repetition time TR = 10 seconds , TE = 8 ms, matrix size = 256 by 128, slice thickness = 5 mm, 1 average, flip angle = 12 degrees, 128 phase encodes per IR prep) with 15 inversion time TIs (50 ms, 100 ms, 150 ms, 200 ms, 250 ms, 300 ms, 400 ms, 500 ms, 750 ms, 1000 ms, 1250 ms, 1500 ms, 1750 ms, 2000 ms, 2500 ms). The time required to collect all 15 TIs, plus three datasets with no inversion pulse, was approximately 3 min. The T1 data was acquired from five healthy volunteers.  2.2.2.2 Inversion recovery simulations We synthetically generated two component IR data (15% with T1 = 230 ms and 85% with T1 = 1200 ms) with 15 TIs. Gaussian noise was added to give an SNR of 85 (similar to the experimentally measured IR data). Additional two component IR data was generated at successively higher SNRs to explore what SNR might be required in order to for the analysis to be able to reliably separate the two T1 components.  2.2.2.3  Cross relaxation  Cross relaxation was assessed in a single transverse slice through the base of the genu and splenium of the corpus callosum for each volunteer. In addition to localizers, the MR  25  examination included multiple experiments for each volunteer. The initial experiments employed a combined magnetization transfer (MT)—T2 relaxation sequence consisting of a preparatory 19 ms single cycle sinc MT pulse (+ 2000 Hz off-resonance, maximum B1 of 0.11 mT, flip angle = ~ 6.3 π, and band width of ~ 100 Hz) followed by delay times of either (a) 18, 51, 84, 118 ms, (b) 84, 118, 218, 318 ms, or (c) 318, 468, 618, 768 ms. Approximately 20 subjects were scanned for each combination of the above delay times (a, b, or c). At the end of the delay, there was a multi-echo imaging sequence (34) consisting of a 90° slice selective pulse followed by 48 rectangular composite 180° (90°x-180°y-90°x) pulses. The off resonance pulse was typical of the type used in clinical magnetization transfer pulse sequences. For the last experiment, the multi-echo T2 relaxation sequence was run without the MT preparation pulse. To minimize out-of-slice signal and stimulated echo artifacts, z-axis gradient crushers were applied on either side of the refocusing pulses (35); these crushers decreased linearly in strength and alternated in direction along the echo train. The multi-echo imaging parameters were: TR = 3800 ms, echo spacing = 10 ms, FOV = 22 cm, matrix size = 64 × 64, slice thickness = 5 mm, 2 signal averages. The scan time required for the cross-relaxation measurements was approximately 45 min.  2.2.3 Data analysis For both experiments, regions of interest were drawn for five brain white matter regions: the genu and splenium of the corpus callosum, the posterior internal capsules, and the major and minor forceps.  26  2.2.3.1 2.2.3.1.1  Inversion recovery Experimental data  Conventional mono-exponential [S = S0 (1 − f exp(−TI/T1))] and bi-exponential IR models were applied to fit the IR data from each white matter region in order to extract single component and two component results from each region. These T1 s, which were extracted by fitting to magnitude data after negating signal acquired at TI times before the zero crossing, may inherently contain errors arising from inaccuracies in estimating the precise location of zero crossing. Fitting the absolute value of the equation to the magnitude signal was also tried and showed very similar results. One way to avoid this would have been to collect the phase data and thus preserve the polarity of the IR data, but that would also require sophisticated phase wrapping corrections. The T1s from each region from each volunteer were extracted separately and then the mean and standard deviations were calculated across all five volunteers. The signal to noise ratio associated with the IR experiment, SNR(T1), was defined as the ratio of S0, to the standard deviation of the residuals of the fit to the mono-exponential model. While it is true that this definition of SNR(T1) is dependent on the model that is used to fit the data, it provides us with a lower bound on the signal to noise of the data. 2.2.3.1.2 Simulations To investigate the minimum SNR(T1) required to robustly differentiate two component T1 relaxation from single component T1 relaxation, mono-exponential and bi-exponential models were applied to fit the synthetically generated IR data and the corresponding T1 and their associated 95% confidence intervals were extracted. This simulation was repeated with  27  successively higher SNR(T1) values, until it resulted in a decay curve to which bi-exponential components could be robustly fitted. The fit was defined to be robust when the extracted T1 values with standard deviations were not overlapping and had no dependence on initial fit estimates.  2.2.3.2 Cross relaxation A four exponential model with the following T2s (31) of 20, 80, 120, and 2000 ms, were fitted to the T2 decay curves at each delay time and for the no MT pulse experiment for each region of interest, extracted from the 57 healthy subjects, using non-negative least squares (4–6,36). Because of the variability found in the intensity and position of the short T2 component due to the low signal to noise ratio resulting from taking only two signal averages, only four T2s were used instead of using a large (∼100) number of T2s, which is typically done for multi-exponential non-negative least squares analysis. These components were chosen to match the T2s measured in previous in vivo studies of normal human brain, where 20 ms corresponds to myelin water, 80 and 120 ms correspond to IE water (31) and 2000 ms corresponds to cerebrospinal fluid. The two IE T2 times are required to handle the range of T2 times one measures from different white matter structures. It was found in a previous study (25) that using 80 ms and 120 ms was equivalent to 80 ms and 200 ms. However, in this study, when a single IE time (82.5 ms) was used in an initial calculation, the variation of MWFs across the different white matter structures varied substantially from previous results, suggesting that variations in IE T2 times were influencing the estimated MWFs. IE water magnetization and myelin water magnetization were determined for each region of interest from each subject. The scaled magnetization was determined for each scan  28  as a ratio of the variable of interest, myelin water (20 ms), or IE water (80 ms and 120 ms), to the total water signal from the no MT pulse experiment. Scaling to the no MT pulse experiment was done to remove any factors the scanner could introduce when collecting data from different volunteers on different days. Scaled magnetizations were averaged from each volunteer, from each region, at each delay time (total of nine data points for each white matter region) and the graphs of scaled magnetization versus delay time were generated and a four pool model was applied to determine the cross relaxation times. The no MT pulse experiment was defined as the t = 0 data point.  2.3 Four pool model of exchange in white matter The four-pool model (30), depicted in figure 2.1, was used to describe the magnetization interactions between aqueous (myelin water (MW) and IE water) and nonaqueous (myelin (M) and nonmyelin (NM)) compartments. In this model, the interaction between spins from each of the four pools and the lattice, through T1 relaxation, was also considered. Based on their T2 relaxations, we were able to discriminate between the MR signals from myelin water and IE water pools; however, we had no experimental data from either of the nonaqueous pools. Time evolution of the zcomponent of magnetization in each of the proton signal pools can be described using the Bloch equations 2.1– 2.4 listed below, assuming that the time scale for magnetization within each proton pool to come to a common state is much shorter than the inter-pool exchange time scales (27).  29  d dt d dt  d dt  M M  k12 M M   [ M M  M M ()]  M MW  k21M MW   M IE  k32 M IE   d dt  M  NM  M  T1  [ M MW  M MW ()] MW  T1  [ M IE  M IE ()]    k 43 M   k21M MW  IE T1  NM    [M  NM  [2.1]   k23M MW  k12 M M  k32 M IE  [2.2]   k34 M IE  k23M MW  k43M NM  M T1  NM  (  )]  NM   k 34 M  [2.3]  [2.4]  IE  Mi denote the time-dependent magnetization along the z-axis in each signal pool and Mi (∞) are the equilibrium values of the corresponding pool. The kij, depicted in figure 2.1, are the directional exchange rate constants between each two adjacent signal pools. T1 s are the spin lattice relaxation times for each of the four pools. The above four equations must be solved simultaneously in order to determine the kij. The analytically solved solution of the nonhomogeneous system of ordinary differential equations (ODEs) of 2.1–2.4 is extremely lengthy (over 400 pages) but contains only four different exponential decay times. More details about this analytic solution can be found in appendix 1.  30  2.3.1 Application of the four pool model As shown in figure 2.1 each of the proton pools considered in this model exchanges magnetization with its two neighboring pools, therefore three different exchange processes can be defined between adjacent proton pools: nonaqueous myelin and myelin water, myelin water and IE water, and IE water and nonaqueous non-myelin. These three exchange rate constants can be more concisely described by defining three cross-relaxation times (TCR) which are a function of magnetization exchange rate constants transferring the magnetization between the two adjacent proton pools (equations 2.5-2.7)  Tcr  M     Tcr  D  Tcr  IE  1 k12 1 k 23 1 k34      1  [2.5]  k21 1  [2.6]  k32 1 k43  [2.7]  where TCRM is the cross-relaxation time between the nonaqueous myelin and myelin water pools, TCRD is the cross-relaxation time between myelin water and IE water pools (mainly due to diffusion), and TCRIE is the cross-relaxation time between the IE water pool and nonaqueous NM pool. The value of these cross relaxation times is that they depend solely upon the interactions between the two pools involved and not upon the relative numbers of  31  spins in each pool. Directional exchange rate constants (kijs) responsible for each of the three existing exchange processes are mutually related (37) according to the following equations:  1 M  T cr 1  D  T cr 1  IE  T cr  M      P k 12 P  P P   P  M  MW  MW  MW  k 23  P  IE      P P  P    P  MW   P  IE    MW  P P  IE  k 21   P  M  P  IE  P k 34  MW  NM  IE  [2.8]  MW  k 32  P  IE  k 43   P  [2.9]  [2.10]  NM  where PM, PMW, PIE, and PNM are the probabilities of finding a proton in myelin tissue, myelin water, IE water, and nonaqueous NM signal pools, respectively. If one defines the mobile proton fraction (MF) as the ratio of the total mobile protons (MW and IE water) to total protons (solid and mobile), then the probability of finding a proton in a specific environment can be estimated as  Alternatively, MF   P MW = MF x MWF  [2.11]  P IE = MF x (1- MWF)  [2.12]  P M = P NM = 0.5 (1- MF)  [2.13]  1  where F is the ratio of the total semisolid protons to total 1 F water protons defined in many MT studies. PM and PNM represent the probabilities of finding 32  a proton in the nonaqueous myelin pool and the nonaqueous non-myelin pool respectively. These values were assumed to be equal for this study based upon wet lab work by Norton et al (1) and upon our private correspondences with Dr. Leif Hertz and Dr. Bruce Trapp, who are experts in the field of non-neuronal brain cells. Based on equations 2.11- 2.13, the mobile proton fraction and MWF determine all the Pis. These values were then substituted into equations 2.8 – 2.10 to set directional exchange rate constants kij and kji in terms of the three independent cross relaxation times TCR. The constrained parameters supplied to the Bloch equations 2.1– 2.4 were the T1 s and the initial and final (equilibrium) magnetization associated with each signal pool in our four pool model of white matter. For the aqueous proton pools, equilibrium and initial values of the magnetization were obtained from our experimental data. As our experiments were not sensitive to the signal from nonaqueous (i.e., myelin and nonmyelin) pools, it was difficult to estimate the magnetizations of these pools, so we used the values measured in an NMR study, at 37°C, done by Bjarnason et al.(24). It was assumed that both nonaqueous pools had the same initial magnetization. The magnetization of each of the aqueous pools at the longest delay time (∼768 ms) was assumed to be close to the steady state magnetization in that pool and was substituted for the Mi(∞) in the equations 2.1– 2.4. This assumption is supported by the observation of asymptotic behavior of the magnetization at longer delay times for most of the curves. As the directional exchange rate constants (kijs) responsible for each of the three existing exchange processes are mutually related (37) (equations 2.8 - 2.10), the analytical solution of the Bloch equations 2.1– 2.4 contained only three independent parameters (kijs) that were determined by fitting to the experimental data. Once these three kijs were determined, the  33  three corresponding cross-relaxation rates of each of the defined exchange processes were calculated from equations 2.5 - 2.7. Due to uncertainties in the literature regarding single versus two component T1 in white matter, we examined three different T1 scenarios.   Scenario I: It was assumed that all four proton pools had the same T1, although each white matter region had its own distinct T1 as determined experimentally from the IR experiment on five healthy controls.    Scenario II: Both nonaqueous proton pools (myelin and nonmyelin) were assumed to have a short T1 of 150 ms (consistent with a two state (nonaqueous and aqueous protons) fast exchange model and a white matter T1 of 0.7 s) and both aqueous pools (myelin water and IE water) were given a T1 of 2 s which is close to the spin-lattice relaxation time of bulk water protons at 1.5 T. This model is equivalent to that used by several authors to account for the dependence of T1 on water content (38–41). In this scenario, all white matter regions were treated equally in terms of T1.    Scenario III: Myelin related and nonmyelin related proton pools were separated in terms of their T1. We assumed the T1 associated with nonmyelin pools (nonaqueous NM and IE water) to be similar to the gray matter T1 of ∼1200 ms at 1.5 T (6). The T1 of myelin related pools (nonaqueous myelin tissue and myelin water) were derived using equation 2.14:                                                            1  WM T1    MWF T1  MW    IEWF T1  IE  [2.14]  34  where T1WM, T1MW, T1IE are the T1 values of white matter, myelin water, and IE water respectively. Using the average MWF and IEWF (ratio of IE water signal to total signal), 14% and 84%, respectively, which were directly extracted from the experimental data in the current study, equation 2.14 yields a T1MW of ∼230 ms at 1.5T. Having calculated the cross-relaxation times for each of the T1 scenarios, the “true” MWF values that would be measured in the absence of exchange were then estimated. This was done by “turning off” the exchange between the myelin water and IE water pools (i.e., setting TCRD = ∞) in the Bloch equations 2.1– 2.4 and solving for the magnetization of the two water pools analytically. These new analytical solutions were functions of the new equilibrium magnetizations of the aqueous pools (M′i(∞), magnetizations in the case of no exchange) because we solved the Bloch equations 2.1– 2.4 assuming the M′i(∞) to be unknown parameters. To solve for these new equilibrium magnetizations, we assumed TCRD approaches infinity whereas TcrM and TcrIE were unchanged. We also required two data points from the magnetization recovery curves. For this reason, we chose the two earliest data points, at 18 ms and 51 ms time delays respectively, because we speculated that signal at these early times was less affected by exchange processes. 2.4  Results  2.4.1  T1 measurements  2.4.1.1 Experimental inversion recovery The mono-exponential model fitted the data from each region robustly, with no dependence of extracted T1 on the initial estimates of the fitting parameters. The T1s, reported as mean ±  35  standard deviation, were: 723 ± 21 ms for genu, 726 ± 19 ms for splenium, 761 ± 26 ms for minor forceps, 760 ± 27 ms for major forceps, and 828 ± 32 ms for internal capsules. On the other hand, the bi-exponential fit of the IR data generated a wide range of overlapping T1 estimates and the results were highly dependent on initial estimates of the fitting parameters. Figure 2.2 shows the mono-exponential model fit to the IR data from major forceps acquired from one subject. The T1 extracted by applying a mono-exponential model fit to this data was 760 [35] ms, and its amplitude was 261 [23] (the values in the square brackets are the associated 95% confidence intervals of the fit). The SNR(T1) for this IR data was 85 ± 34. When applying a bi-exponential model to fit the same data, the extracted T1MW and T1IE were 652 [483] ms and 948 [2511] ms, and their associated amplitudes were 165 [437] and 105 [598], respectively. The very high errors in these fitted T1s, indicate the unreliability of the extracted values.  Figure 2.2: Mono-exponential fit to the IR data (with SNR(T1) of 85) from major forceps of one healthy normal volunteer acquired with a single-slice fast gradient echo sequence obtained at 15 TI times. The residual sum of squares of the fit was 141.2. The extracted single T1 was 760 [35] ms and its amplitude was 261 [23]. The values in the square brackets are the associated 95% confidence intervals of the fits.  36  2.4.1.2  Simulated inversion recovery  We applied the bi-exponential model to fit the synthetically generated two component IR data (see figure. 2.3) and the extracted T1MW and T1IE were 360 [1035] ms and 1385 [2054] ms and their associated amplitudes were 51 [256] and 238 [150] respectively.  Figure 2.3: Bi-exponential and mono-exponential fits to the synthetically generated IR data with SNR(T1) of 85 and 15 TIs. The residual sum of squares of the fit was 83.7 for the bi-exponential fit and 256.3 for the mono-exponential fit. Extracted T1MW and T1IE from biexponential fitting were 652 [483] ms and 948 [2511] ms, and their associated amplitudes were 165 [437] and 105 [598] respectively. The single T1 extracted from mono-exponential fitting was 1007 [106] ms. The values in the square brackets are the associated 95% confidence intervals of the fits.  37  When a mono-exponential model was fit to the same synthetically produced IR data the extracted T1 was 1003 [109] ms and its amplitude was 260 [14] which is the weighted mean of the two T1s from the bi-exponential model fit (the values in the square brackets are the 95% confidence intervals of the fits). The overlapping extracted T1MW and T1IE estimates from the bi-exponential fits made it impossible to determine whether the IR data was biexponential or mono-exponential in nature. Based on fits for a range of SNR(T1) values, we concluded that with 15 TIs we would need either an SNR(T1) approaching 850 or more TIs (>128) to robustly differentiate between mono-exponential and bi-exponential T1 relaxation. This indicates that all three T1 scenarios considered in this study are equally plausible for a T1 data acquisition with SNR(T1) of 85 and 15 TIs.  2.4.2 Fitting the four pool model to the results Figure 2.4 a–e shows the myelin water and IE water signals as functions of delay time for all five examined cerebral white matter regions along with the fitted curves from the analytical solutions to the four pool model for each of the three T1 scenarios. The three cross-relaxation times (TCRM, TCRD, and TCRIE) for each scenario are listed in table 2.1. The fits to the experimental data arising from the three T1 scenarios considered here were not significantly different from each other (the least square error averaged over all five examined regions were 0.0025 ± 0.0011 for scenario I, 0.0023 ± 0.0015 for scenario II, and 0.0018 ± 0.0011 for scenario III meaning that it is not possible to choose the best T1 scenario based on this investigation.  38  Figure 2.4 (a–e): Myelin water and IE water signals from (a) genu, (b) minor forceps, (c) major forceps, (d) splenium, and (e) internal capsules of 57 normal volunteers for all three scenarios investigated. The analytical solution corresponding to each T1 scenario is shown by a line. Filled circles and triangles represent the experimentally measured myelin water and IE water data. Standard deviations are shown in error bars. Variations of the standard deviations for each structure is partially due to the different number of subjects that underwent each section of the experiments. 39  Cross-relaxation times Genu  Internal  Minor Forceps  Major Forceps  Splenium  Capsules Scenario I TCRM (ms)  74 + 23  173 + 33  75 + 23  86 + 31  113 + 20  TCRD (ms)  1351 + 322  1321 + 348  1383 + 271  1479 + 358  1281 + 342  TCRIE (ms)  303 + 34  279 + 41  256 + 38  231 + 39  155 + 32  TCRM (ms)  75 + 33  159 + 32  73 + 35  79 + 29  92 + 32  TCRD (ms)  2731 + 795  2795 + 741  2951 + 814  3412 + 836  3126 + 982  TCRIE (ms)  342 + 48  264 + 42  269 + 51  224 + 40  149 + 39  TCRM (ms)  72 + 22  141 + 37  71 + 27  75 + 37  98 + 25  TCRD (ms)  4090 + 703  5850 + 732  4751 + 732  3993 + 729  4532 + 844  TCRIE (ms)  371 + 45  273 + 41  295 + 33  220 + 43  153 + 39  Scenario II  Scenario III  Table 2.1: Cross-relaxation times between adjacent signal pools for each of the five examined white matter region, from 57 normal volunteers, corresponding to scenarios I to III. Error estimates are calculated by varying each cross relaxation times until the sum of squares of the fit was increased by 5%. These errors are indicated by a ± symbol.  40  Genu  Internal  Minor  Major  Capsules  Forceps  Forceps  Splenium  Corrected-Scenario I  21 +9.8  16 +5.2  18 +12  11 +6.9  14 +7.5  Corrected-Scenario II  20 +10  15 +5.1  17 +12  9.9 +7.1  14 +7.8  Corrected-Scenario III  19 +9.7  14 +4.9  16 +13  9.7 +6.5  13 +8.2  Measured  18 (9.2)  14 (4.7)  16 (11)  9.3 (6.2)  13 (7.1)  Table 2.2: Measured and cross-relaxation corrected myelin water fractions (%) for five white matter regions corresponding to the three considered T1 scenario. Standard deviations of the measured MWF, calculated over 57 normal volunteers are shown in brackets. For corrected MWF values, error estimates are followed by “±”. These errors reflect the maximum change in MWF correction estimates when the input parameter for each structure (measured Tcr's) was varied by its standard deviation.  2.5  Discussion  The main focus of this study was to obtain in vivo insight into magnetization exchange in various human cerebral white matter regions. A four pool model of white matter was capable of fitting the decay curves arising from the combined MT/T2 experiment for all five white matter regions, for all three T1 scenarios.  41  2.5.1  T1 scenarios  Unfortunately it was not possible, with this data, to distinguish which T1 scenario best describes spin lattice relaxation in white matter. Quantitative measurement of white matter T1 requires high signal to noise and many inversion times; the IR data displayed in figure 2.2 with 15 TI times and SNR(T1) of 85 does not support the assignment of two T1 components. As an example, Labadie et al. (23) who found two T1 components used 64 TI times in his measurements. Figure 2.3 shows simulated IR T1 data with two T1 components 230 ms and 1200 ms, an amplitude ratio of 1 to 6 and signal to noise equivalent to that in figure 2. 2. Analysis of this simulated dataset did not support a model with two T1 components. Therefore, it is not surprising that previous literature did not find multi-component T1 relaxation in white matter. We chose to analyze three scenarios for T1: scenario I assumed mono-exponential behavior of T1 relaxation in all four proton pools; scenario II separated aqueous protons from nonaqueous tissue protons; and scenario III separated myelin related protons from nonmyelin related protons. While scenarios II and III gave slightly different TCRs, they both fit the experimental data reported by Labadie et al. (22,23). Scenario II is the most fundamental and realistic model and it has already received wide usage as a justification for using T1 as a measure of water content (38–41). More research is required before T1 relaxation in white matter is fully understood. We note that results from a recent study (42) of progressive magnetization transfer in white matter were best interpreted with a T1 model similar to scenario II where a nonaqueous T1 of 171 ± 22 ms was estimated.  42  2.5.2  Myelin water exchange  A fundamental goal of this study was to determine how much magnetization exchange between myelin water and IE water affects the MWF. Table 2.1 indicates that for all three T1 scenarios, exchange played a minor role in the MWF. Estimated TCRD times varied little between white matter regions which have substantially different MWF values. Therefore, it seems unlikely that regional variations in MWF are caused by regional variations in exchange; hence one can conclude that MWF in brain is largely determined by myelin content, rather than by exchange which may be influenced by myelin thickness. Values for TCRD obtained from scenario I agree approximately with exchange rates previously derived from application of the four pool model to ex vivo results using the same single component T1 assumption (24,25) suggesting, surprisingly, that exchange rates do not differ between in vivo and ex vivo. Figure 2.4 a–e show a consistent pattern of initial sharp decrease followed by an increase in MWF. The shortest exchange time between myelin water and nonaqueous myelin, TCRM, causes the rapid drop in the myelin water signal at times less than 100 ms. The myelin water signal increase at later times is accompanied by a decrease in IE water signal, a behavior pattern which is in complete agreement with the equations 2.1- 2.4 describing the evolution of the magnetization in each signal pool, as there are two processes that determine the evolution of the magnetization. One is the exchange of magnetization with neighboring proton pools and the other is T1 recovery. When T1 recovery proceeds rapidly, meaning short T1 for myelin water, magnetization exchange is no longer required to explain the increase in MWF with delay time. The MWF increase can be interpreted as evidence of myelin water  43  and IE water exchange (scenario I), or as T1 recovery of myelin water, either directly (scenario III) or indirectly through the myelin nonaqueous tissue (scenario II). The results in table 2.1 can also be reported in terms of myelin residence time which were 270.2 ms, 546.2 ms, 818 ms for scenarios I, II, and III in genu respectively. 2.5.3 Correcting MWF for exchange Most current MWF measurement techniques assume the magnetization exchange time scales between the myelin water signal pool and the IE water pool, which is mainly due to diffusion, are long compared with the duration of the T2 measurement. Having measured the cross-relaxation times for each of the considered T1 scenarios, we were able to estimate the cross-relaxation correction to increase the accuracy of current MWF measurements. Corrections to the MWF values (due to magnetization exchange) were a 12–15% increase for scenario I, a 6–8% increase for scenario II and a 3–4% increase for scenario III. In all cases, we note from table 2.1 that the estimated TCRD times were longer than the last delay time, hence these corrections have wide uncertainties. In summary, all corrections are less than 15% and for scenarios II and III, the estimated corrections are negligible when compared with the measurement errors. We should note also that at the longest delay time of 768 ms the magnetization may not yet have completely reached equilibrium.  The results for myelin water exchange reported here do not agree with those reported by Does et al. (28). This discrepancy may be due to the fact that myelin morphology in rat spinal cord is different from the myelin morphology in human brain. Additionally, the estimation of magnetization exchange rate is dependent on the white matter model that is used to fit the  44  data, and as the Does group used a very different model of white matter compared to our four model, it is expected that the findings do not necessarily match.  2.5.4 Exchange with myelin and nonmyelin tissue In all cases TCRM was less than TCRIE, which is not surprising since exchange is faster in myelin where the water and tissue are much more tightly coupled than in NM, where the concentration of nonaqueous tissue is much smaller. Furthermore, this study employed the Zimmerman-Brittin model (27) which assumed that each compartment had essentially uniform magnetization at all times. A more rigorous approach might adopt a Brownstein-Tarr model (43) to deal with the effects of diffusion on the distribution of magnetization within the myelin sheath and also within the larger intra- and extra-cellular spaces. Such an approach would be nontrivial to implement since: 1) it would require detailed knowledge of water diffusion coefficients in myelin and in the other spaces and 2) it might involve separately modeling the myelin and nonmyelin regions of white matter. A similar model was used previously to interpret T2 relaxation in wood (44). In that model, the cellular water T2 was found to be determined by the wood cell radius.  2.5.5 Limitations One of the limitations of this work is that the applied 4 pool model assumes a single exchange process between myelin water and the intra/extracellular water pools. A more accurate model would have separated the intra and extra cellular pools, however our measurements were not sensitive to this separation. Additionally these experiments were 45  insensitive to exchange processes which happened on time scales much faster than the timescale of the measurement. i.e. if myelin water was exchanging at a rate with Tcr faster than say 10 ms, the experiments could provide no information on such a fast exchange regime and the initial values used in the 4 pool model would include the affects of that fast exchange process. Some of the MWFs measured in this study were slightly different from those reported in other studies. There are three potential explanations; volume averaging due to the large voxel sizes used in the multi-echo sequence, the use of only two averages, and the use of only four T2 input in the non-negative least squares analysis. Since the dipolar broadened signal from the nonaqueous proton pools (myelin and nonmyelin) decay in less than 100 μs, they are not easily detectable by MRI. Furthermore, even if it had been possible to measure the nonaqueous signal, separation of the myelin and nonmyelin signals would have been problematic. Consequently, to solve equations 2.1- 2.4 we had to estimate the equilibrium values for magnetization in nonaqueous pools and assume that the signal from nonaqueous myelin tissue was equal to that of the nonaqueous NM. Our assumed values for the nonaqueous signals were commensurate with those used in a previous ex vivo study (24). We note that the composite pulses used for refocusing the magnetization in the crossrelaxation experiment could have also caused a magnetization transfer effect due to absorption by the nonaqueous protons. We expect that this would not have affected the results substantially because the MT effect should have been similar for all equilibration times. 46  The composite decay curves used in this measurement contained MWF results from 57 volunteers aged from 18–57 years. Since a small but significant age dependence was reported in frontal white matter (45), this might have influenced the results from the minor forceps. 2.6  Conclusions  The evolution of the MR signals from the myelin water pool and the IE water pool following an off resonance pulse was well characterized by four pool model Bloch equations. Because the SNR of our IR experiment was insufficient to differentiate between mono-exponential and bi-exponential T1 relaxation, three different T1 scenarios were considered. All three T1 scenarios yielded similar exchange times between myelin water and IE water for five different white matter regions, indicating the observed variations in MWF between white matter regions is not due to water exchange rates. For single component T1 relaxation, magnetization exchange was estimated to affect the measured MWF values by 12 % to 15%. For the other two scenarios with bi-exponential T1 relaxation, magnetization exchange effects on MWF were less than measurement error. These results support the use of MWF as an accurate in vivo measure of myelination.  47  Chapter 3: Variation of myelin water fraction as a function of TR  Much of the material in this chapter will be submitted for publication in the journal NeuroImage.  3.1  Introduction  MRI is an imaging tool that is sensitive to central nervous system (CNS) pathology and thus widely used for studying tissue both in vivo and ex vivo. A key basis for this sensitivity to pathologic changes of the CNS lies on the fact that magnetic resonance properties of brain, for example T2, are influenced by the microscopic structure of tissue. Accurate measurement and analysis of T2 decay curves from brain tissue has shown that as many as three different water pools can be distinguished based on their unique T2 relaxation times (5,14,15,34,46). Myelin water (MW, water molecules trapped in the myelin bilayers) have the shortest T2 times of 10 ms - 40 ms, Intra/Extra cellular water (IE water, water molecules in the intra and extra cellular spaces) have T2's from 60 ms - 100 ms, and more freely moving water molecules such as cerebro-spinal fluid (CSF) have T2 ~2 seconds. The myelin water fraction (MWF, the fraction of central nervous system water with a short T2) was found to be quantitatively correlated to histological staining for myelin in central nervous system tissue (7,8,19) and hence is considered an in vivo measure of myelin content. Various studies have reported on the white matter MWF in a diverse range of neurological diseases such as Multiple Sclerosis (MS) (47–50), Schizophrenia (45), and Phenylketonuria (PKU) (51,52).  48  Recent breakthroughs in developing rapid 3-D whole brain approaches to MWF measurements (53–58) have made this technique a potential clinical tool for diagnosis and management of subjects with neurological disorders as well as providing a means for assessment of the efficacy of potential therapeutic interventions. It is important to point out that making MWF available in a clinical time frame may require the use of shorter TR times and consequently exposure of the acquired T2 decay curve to T1-weighting. Therefore a comprehensive understanding of T1 relaxation in brain, specifically white matter, has become crucial. The literature on T1 in white matter is not yet mature. In particular, two phenomena which influence T1 in brain are not yet well understood:1) the time scale of magnetization exchange between the myelin water and intra/extracellular water pools in white matter and 2) reequilibration of magnetization between non-aqueous tissue and water following an MRI inversion pulse. Bjarnason et al. (in bovine brain ex vivo) (24), and Kalantari (in human brain in vivo) (59) reported that MW to IEW magnetization exchange was relatively slow with a characteristic time of about 1 s. However, Dula et al. (in rodent spine ex vivo (28)) and Harkins et al. (in rodent spine in vivo (60)) demonstrated that this exchange process occurred at a significantly faster rate in nerves with thinner myelin sheaths. If this exchange process is slow on the brain T1 timescale of about 1 s, both T1 and T2 could be multi-component and the amplitudes of the T2 components should accurately reflect the relative populations of myelin and intra/extracellular water. However, if this exchange process acts on a timescale less than the brain T1 timescale, T1 relaxation will approach a single component and the amplitudes of the T2 components will be altered towards less MW signal and more IEW signal.  49  The signal from non-aqueous tissue in brain (lipids, proteins, etc) has a dipolar broadened line shape which causes it to decay to undetectably-small levels in less than 100 s (15,24). Typical human MRI sequences (excluding ultrashort TE sequences) use inversion pulses which last several ms; while these pulses produce rotations of close to 180o to water molecules, they have a nearly negligible effect on the magnetization of non-aqueous molecules. However, subsequent re-equilibration of magnetization of the two spin species by energy conserving processes results in changes in the water signal that could be mistaken for T1 relaxation. This phenomenon was used extensively by Gochberg et al. (33,61) as a magnetization transfer measure and was also demonstrated by Prantner et al. (28) in a rat brain model. Most of the current literature on T1 in white matter reports single component T1 relaxation, however there are several reports from careful studies reporting that T1 relaxation in brain has two components (22,23,62). The main goal of this preliminary study was to investigate the behavior of the measured MWF as a function of TR in human brain in vivo. It is important to note that for multi spin echo sequences the effective TR, TReff, is not defined by its usual definition of the time from the start to finish of the sequence but rather the length of time from the last 180 ̊ refocusing pulse to the beginning of the next sequence. This is because each of the 180 ̊ refocusing pulses invert the z-component of magnetization thereby preventing the growth of signal due to T1 relaxation. Thus the time that the system really has to recover back to its equilibrium is basically the time after the last 180 ̊ pulse to the beginning of the next sequence. This study involved measuring MWF in white matter structures in 5 healthy normal volunteers as a function of effective TR times from 165 ms to 665 ms. The results were  50  expected to address two key questions: 1) Does the measured MWF in white matter depend upon TReff? 2) Would this experiment enable us to distinguish between fast MW/IEW exchange, which should presumably yield a largely TReff independent MWF, and slow MW/IEW exchange, which should lead to an increase in MWF as TReff is shortened?  3.2  Materials and methods  3.2.1 Subject information Five normal volunteers (average age = 37.2 years, range =25-64) with no known neurological disorders or MR-visible brain abnormalities underwent MR examinations twice. Informed written consent as approved by the Clinical Research Ethics Board at the university of British Columbia was obtained from all subjects.  3.2.2 MRI experiments All MR examinations were conducted using an eight-element phased-array head coil on a 3.0-T MR scanner (Achieva 3.0T, Philips Medical Systems, Best, The Netherlands). A multi echo T2 sequence with 32 spin echoes was repeated at five TR times: 1100, 1200, 1300, 1500 and 1600 ms which correspond to effective TReff times of 165 ms, 265 ms, 365 ms, 565 ms, and 665 ms. A 32 echo sequence with a constant 10ms echo spacing could not be used here due to specific absorption rate (SAR) limitations. Hence we designed the sequence to include 16 pulses at echo spacing 10 ms to ensure the capability of measuring the myelin water signal followed by 16 pulses at echo spacing 50 ms running to the last echo at 960 ms.  51  The last 180 ̊ refocusing RF pulse was 935 ms later than the initial 90 pulse, hence TReff= conventional TR – 935 ms. Other multi-echo imaging parameters were: number of slices = 7, slice thickness = 5 mm, FOV = 24 cm, matrix size = 256 x128. The scan time was ~ 13 min for the shortest TReff of 165 and ~ 42 min for the longest TReff of 665 ms. For each subject, data acquisition was carried out in two sessions 2-5 days apart.  3.2.3 Data analysis Each subject's images were registered to the TReff = 665 ms data using FLIRT (FMRIB's Linear Image Registration Tool) with 6 degrees of freedom and mutual information approaches (63). Regions of interest (ROI) were drawn for five white matter structures (genu (GU) and splenium (SP) of the corpus callosum and posterior internal capsules (IC), and the major forceps (MJ) and minor forceps (MN)) and the T2 decay curve at each ROI was extracted. These T2 decay curves were analyzed using a NNLS algorithm to generate the T2 distributions (for detailed explanation of the NNLS routine please refer to chapter one).  3.3 Results Myelin water fraction (MWF) values from splenium (SP), genu (GU), posterior internal capsules (IC), major forceps (MJ), and minor forceps (MN) measured at each TReff are shown below in table 3.1.  52  WM structure  TReff  165 ms  265 ms  365 ms  565 ms  665 ms  MJ  SP  IC  MN  GU  0.12685  0.21034  0.20832  0.076668  0.15465  (0.013)  (0.013)  (0.011)  (0.011)  (0.013)  0.10735  0.18221  0.16729  0.061835  0.13332  (0.012)  (0.012)  (0.012)  (0.011)  (0.013)  0.083164  0.16466  0.15621  0.046829  0.12134  (0.012)  (0.012)  (0.012)  (0.011)  (0.013)  0.075835  0.15534  0.13629  0.043667  0.11231  (0.013)  (0.012)  (0.013)  (0.011)  (0.013)  0.072659  0.15124  0.12971  0.041332  0.10569  (0.012)  (0.012)  (0.012)  (0.011)  (0.013)  Table 3.1 MWF at various TReff times. The standard error associated with each measurement is shown inside parenthesis.  53  3.2 Figure 3.1. Plots of MWF at various TReff times from 165 ms to 665 ms measured across five 3.3 different brain white matter structure in vivo. Standard errors are shown as error bars.  Figure 3.1 and table 3.1 show the MWF is a very sensitive function of TReff with increases between the longest and the shortest TReff of 154 % to 172% across the investigated brain white matter structures.  3. 4 Discussion The most conclusive finding of this study is the demonstration that the measured myelin water fraction increased appreciably with decreasing TReff when TReff was shorter than about 600 ms. As the drive to reach faster MWF imaging with whole brain coverage may result in sequences with shorter TR times, it should be noted that going below the 600 ms TReff threshold could be problematic. 54  We would like to understand why MWF increases with TReff. We list below several potential scenarios: 1) Fast magnetization exchange between myelin water and intra/extracellular water 2) Slow exchange or no exchange between myelin water and intra/extracellular water 3) More complex models involving equilibration of magnetization between non-aqueous and water protons and/or a hybrid myelin model where some myelin water undergoes fast exchange within intra-extracellular water while the rest of the myelin water does not exchange on the T1 time scale. If MW and IEW were in fast exchange on the T1 timescale, they should have the same T1. In that case, the plot of MWF vs. TReff for any structure simplifies to a straight line since both water pools undergo the same T1 weighting at all TReff. Figure 3.2 demonstrates that the experimental results are clearly inconsistent with this scenario.  Figure 3.2 Fast exchange model showing MWF vs. TReff assuming both MW and IEW pools have the same T1= 1000 ms. 55  Another scenario is the existence of slow exchange or no exchange on the T1 time scale which implies that the water signal has two T1 components whose relative proportions will change in favor of the shorter T1 component at shorter TReff. To visualize this scenario in more detail, we modeled the behavior of MWF vs. TReff as two distinct T1 times. Based on this model the myelin water (MW) and intra-extra cellular water (IEW) have distinct T1s. The equation that is used to fit the data based on this model is equation [3.1] as shown below:  [3.1] where [3.2]  [3.3]  The T1mw and T1ie are the longitudinal relaxation associated with MW and IEW respectively. Smw and Siew are the relative amplitude of the total MW and IEW respectively. T1mw , T1ie , Smw and Siew were the four variables that were allowed to vary based on this model. T1 used for myelin was approximately 200ms which is in agreement with literature estimates on myelin water T1 's (42,64). By restricting the myelin T1 to values greater than 200ms, this model did not include cross-relaxation with the non-aqueous protons which, as a ‘T2’ like process not involving exchange of energy with the lattice would be expected to have a characteristic time similar to the brain tissue water T2 time scale of about 50ms. Indeed, initial investigations of this cross relaxation process in model systems  56  demonstrated cross relaxation times in the 10’s of ms range (33). Because the number of parameters was close to the number of points, many fitting programs are unable to deal with the consequent correlations between parameters. Therefore, the data was fitted to equations 3.1 to 3.3 manually. As is shown in figure 3.3, this model depicted a much better fit to the experimental data from all five white matter structures compared to the single component T1 model of figure 3.2. While the MWFs measured at TReff 655 ms are in approximate agreement with numbers obtained with much longer TReff values (e.g. Whittall et al 1997), at longer TReff, the predicted MWFs are unrealistically small. Therefore we are forced to abandon the simple two component T1 scenario as an explanation for the experimental data.  Figure 3.3 Slow exchange model showing MWF vs. TReff assuming that the myelin water has a shorter T1 (but longer than 200ms) compared to the I/E water T1.  57  Smw  Siew  T1mw  T1ie  MJ  1  30  190  1100  SP  1  12  290  1100  IC  1  14  230  1000  MN  1  30  200  1000  GU  1  18  270  1050  Table 3.2 lists best estimates of Smw (relative amplitude of myelin water pool), Siew (T1 of fast exchanging myelin pool), T1mw (the T1 associated with myelin water pool), and T1ie (the T1 associated with IEW pool) based on slow exchange model.  We conclude that models in which the water in white matter is either in fast or slow exchange (with T1 for myelin water > 200ms) on the T1 time scale cannot account for the observed variation of MWF vs. TReff. The plots in figure 3.1 have two distinct regimes: a) the short TReff regime where an abrupt increase in MWF with decreasing TReff and b) the long TReff regime where there is a relatively level behavior of MWF with increasing TReff. The short TReff regime can be fitted by a component with a short T1 relaxation time; this component could arise from cross- relaxation with the non-aqueous component (33,61) or it could result from myelin which has a short T1. The long TReff regime could be fitted by contributions from spins which are undergoing fast exchange as shown in figure 3.2.  58  We speculate that white matter may present a hybrid behavior which accounts for both the short and long TReff regimes of the curves. The short TReff behavior may arise from crossrelaxation with non-aqueous protons (because the two spin systems are not in equilibrium after the 90 ̊ RF pulse) or it might arise from myelin water protons with a short T1 time. The long TReff regime can be accounted for by regions of myelin in which myelin water and intra/extracellular water undergo fast exchange on the T1 time scale. This hybrid myelin model is in harmony with recent histological studies in which myelin sheaths were shown to be very permeable where they were populated with Schmidt-Lanterman Clefts (SLC) while at other parts they seemed to be rather isolated from the intra/extra cellular water pools. According to this model (65) myelin bilayers show two distinct behaviors. We call the former myelin pool fast exchanging myelin (FEM) and the latter slow-exchanging myelin (SEM). Thus one can assume these two myelin pools have two distinct MR properties such as T1 and T2.  The equation that is used to fit the data based on this model is equation [3.4 ] as shown below: [3.4]  where  [3.5]  59  [3.6]  and  [3.7]  MW1 and MW2 are the fast exchanging and slow exchanging myelin pools respectively, and FMW1 is the fraction of slow exchanging myelin pool. Smw and Siew are the relative amplitude of the total MW and IEW respectively. T1mw1, T1mw2, and T1ie are the longitudinal relaxation of fast exchanging myelin pool, slow exchanging myelin pool, and intra-extra cellular water pool. T1mw2 and T1ie were assumed to be equal and thus FMW1, Smw, and Siew were the three variables that were allowed to vary based on this model. Figure 3.4 shows a simulation optimized to fit the data from all five investigated white mater structures. Due to the small number of data points and the large number of parameters for this model, the system is under-determined and thus robust fitting was not an option so we had to fix the two longitudinal relaxations T1MW1 (the T1 associated with the slow exchanging myelin pool) and T1MW2 (T1 associated with the fast exchanging myelin pool and/or with cross-relaxation) at 50 ms and 1300 ms respectively and find the optimsed value of MWF . The T1MW1 and T1MW2, and MWFTR=∞ (myelin water fraction at long TR) values for most optimum parameters are shown in table 3.2. We note that the choice of 50 ms for TMW1 is also consistent with a cross-relaxation mechanism for the MWF vs TReff behavior.  60  Figure 3.4. Hybrid myelin model showing MWF vs. TReff assuming the hybrid myelin model.  T1MW1 (ms)  T1MW2 (ms)  FMW1  MWFTR=∞  MJ  50  1300  0.044  0.057  SP  50  1300  0.096  0.135  IC  50  1300  0.0769  0.106  MN  50  1300  0.024  0.033  GU  50  1300  0.0476  0.091  Table 3.3 lists best estimates of T1MW1 (the T1 of slow exchanging myelin pool), T1MW2 (T1 of fast exchanging myelin pool), FMW1(the fraction of slow exchanging myelin pool), and MWF TR = ∞ (myelin water fraction at long TR) assuming the hybrid myelin model. 61  As the results rejected both the fast exchange regime and the slow exchange regime, we needed to look further to understand relaxation in white matter. We were left with two potential models of white matter: 1) myelin in white matter has a hybrid character whereby part of the myelin is in a fast exchange regime and the rest of the myelin is in a slow exchange regime (65) or 2) the MWF vs. TR increases were due to cross-relaxation between non-aqueous tissue and water giving the appearance of two T1 components. To address this question, we have done one more analysis: we looked at the MWF vs. TR plot for a grey matter structure which has close to zero MWF. If model 2) were correct, we would expect to have MWF increase at short TR to about ~1/2 the extent that the white matter ROI's increased with short TR, since grey matter has about half the amount of non aqueous protons. If model 1) were correct, then for a grey matter structure with no myelin water signal we would expect MWF to be constant as effective TR is shortened.  Figure 3.5 Plot of MWF from putamen (gray matter) measured at various TReff times from 165 ms to 665 ms in vivo. Standard errors are shown as error bars. 62  Figure 3.5 depicts the measured MWF for a grey matter structure, the putamen, at various TR times are constant suggesting that the data points toward a hybrid model for myelin, although more work will be required to elucidate this further. Thus the concluding message is that myelin is more complicated than we thought and consequently more sophisticated models are needed to justify its behaviour.  3.5  Limitations  The T2 decay curves measured in this study were not corrected for stimulated echo artifacts due to B1 field inhomogeneities for the following reasons: 1) small errors in MWF due to sub optimal 180 ̊ pulses is unlikely to change significantly the variations of MWF with TR, 2) the investigated brain structures were chosen in locations which normally have close-to-perfect 180 ̊ refocusing RF pulses, and 3) correction of stimulated artifacts in a T2 decay curve which does not have equal echo spacing would have been very challenging. As mentioned above, the best fits for the experimental data contained more parameters than data points, therefore, the model presented in Figure 3.4 and Table 3.2 should be considered as speculative. Utilizing the myelin water (MW) and intra-extra cellular water (IEW) data for fitting the parameters is potentially advantageous compared to utilizing the MWF (as some characteristics of the data may have been lost due to working with the ratios), this approach for technical reasons not yet understood, gave rise to increasing MW and IEW signals with increased with decreasing TR as shown in figure 3.6. This result was unexpected; however it could presumably be a result of cross relaxation between the water and the non-aqueous  63  signal. To explore this further would require application of the four pool model used in Chapters 2 and 4.  Figure 3.6 Plot of MW and IEW from major forceps measured at various TReff times from 165 ms to 665 ms in vivo. In an effort to see if we could apply the four pool model approach to the data presented in this chapter we implemented a series of step-wise iterative simulations. In these simulations we set the magnetization initial condition of the myelin water (MW) and intra-extra cellular water (IEW) to be zero at each iteration while the magnetization initial condition of the nonaqueous myelin pool (M) and the non-aqueous non-myelin pool (NM) were started with a non-zero value for the first iteration and were obtained as the output of the four pool model solutions at the desired TReff times for the consequent iterations. The T1 values used for this simulation were 150 ms and 2 s for the aqueous and non-aqueous pools respectively (i.e. Scentario II in Chapter 2) The initial values used for the first iteration (estimated from Chapter 4 data) were 0.075, 0, 0, and 0.075 corresponding to non-aqueous myelin pool,  64  MW pool, IEW pool, and non-aqueous non-myelin pool respectively and the equilibrium values were 0.075, 0.08, 0.77, and 0.075 respectively. Figure 3.7 and 3.8 show the results of these step-wise iterative simulation at TReff 165 ms and 665 ms respectively. As these figures show all four pools depicted an oscillating behaviour, and so this approach was not successful.  Figure 3.7 Iterative simulation using the four pool model. Each of the four signal pool calculated at TReff =165 ms for 10 iterations.  Figure 3.8 Iterative simulation using the four pool model. Each of the four signal pool calculated at TReff =665 ms for 10 iterations. 65  3.6  Conclusion  This work demonstrates that measured myelin water fraction increases substantially as TReff is shortened. This has significant consequences for the design of pulse sequences for measuring MWF. The rate of this increase was larger at shorter TReff as well. Although this study was inconclusive as which of the proposed hypotheses are better able to justify the observed phenomenon, it provides a lower limit for TReff above which MWF could be considered independent of TReff.  66  Chapter 4: Characterizing longitudinal relaxation in bovine brain white matter ex vivo  4.1  Introduction  Accurate measurement of longitudinal relaxation in white matter has crucial implications for myelin water imaging using multi component T2 relaxation. For instance in chapter two we discussed in detail that in order to estimate the magnetization exchange in white matter we would need to understand the longitudinal relaxation behavior in white matter. A main goal of this study was to investigate whether white matter T1 relaxation in brain is a mono-exponential or a multi-exponential phenomenon. In order to measure longitudinal relaxation time T1 in white matter with high precision and accuracy, we studied white matter samples from Bovine brain using a 4.7 T NMR spectrometer. As this spectrometer was capable of measuring the signal from all protons in brain tissue, a secondary goal was to characterize the signals from both aqueous and non-aqueous pools in order to better understand magnetization exchange between these two proton pools.  4.2 4.2.1  Materials and methods Samples  Bovine brain was obtained within 3 hr of slaughter from Grandmaison Beef Farm, Langley, British Columbia, and cooled with ice during transport or placed in a 4°C refrigerator. White matter tissue samples were cut from the brain and placed in an NMR tube. The 0.1 mL samples were taken from a location on the bovine brain similar to where the splenium is  67  located in human brain then placed in the NMR spectrometer. The experiments were carried at 37° C. In order to test the integrity of the samples, the first experiment was repeated at the end of each set of experiments. The results were shown to be very reproducible suggesting the samples did not deteriorate during the course of our experiments. 4.2.2  NMR equipment and experiments  The NMR experiments were performed on a home-built 4.7 T NMR spectrometer. The data acquisition system included a SpinCore Technologies (Gainsville, FL) Pulseblaster™ pulse programmer, a digitizer, and a PC computer. Temperature was set using a Bruker (VT1000) temperature controller that was used in conjunction with a controlled air flow device.  The NMR experiments involved an initial inversion pulse followed by either a free induction decay or a CPMG data collection. The FID enabled measurement of the non-aqueous signal while the CPMG enabled measurement of the T2 components of the signal from water. Two types of inversion pulse were employed: a ‘hard’ pulse for which both non-aqueous and water protons received a 180o rotation and a ‘soft’ pulse which had little effect on the nonaqueous magnetization but provided a 180o rotation for the water protons. The ‘soft’ pulse was representative of a typical MR imaging inversion pulse.  68  4.2.2.1  Inversion recovery-FID  For the hard inversion pulse, the 180o RF pulse length was 2.4 s. Soft inversion pulses were composed of a three-lobe sinc function with a duration of 3 ms. Spectral width was 1 MHz for the inversion recovery experiments and 8192 data points were collected. We collected 2 averages for the inversion recovery experiments and collected FIDs for 98 inversion times, TI (the first 20 TIs were collected with 0.4 μs spacing, and the rest were collected by a geometric factor of 1.1 (meaning that the ratio of the next TI to the preceding one was 1.1). The first TI was at 178 μs and the last TI was at 9.3 s. The receiver recovery time was about 8 μs.  4.2.2.2  Inversion recovery-CPMG  For the CPMG experiments we collected 49 TI s starting from 0.005 s and ending at 7.69 s, increasing by a geometric factor of 2.2 (meaning that the ratio of the next TI to the preceding one was 2.2) and 4 averages. The echo delay (time between the middle of the 90 ̊ pulse and middle of the first 180 ̊ pulse) was 1 ms and the time interval between centers of subsequent 180 ̊ pulses was 2 ms. We collected 160 echoes out to a TE time of 320 ms. For each echo, we calculated the average of 50 pts with a dwell time of 5 μs centered on the echo peak.  Data were analyzed using in-house written software developed in Matlab (The MathWorks, Natick, MA, USA) and Mathematica (Wolfram Research, Inc. Champaign, IL, USA).  69  4.2.3 4.2.3.1  Data analysis Inversion recovery-FID  In order to extract the aqueous signal from the solid signal in the FID curves, a straight line was fitted to the data for times from 250 μs to 0.8 ms as the decaying exponential function can be approximated with a straight line at sufficiently-short times (based on Taylor's expansion theorem). Then the intercept of this line with the vertical axis was calculated and assumed to be the intercept of the aqueous signal. This aqueous signal was then subtracted from the intercept of the total signal at t=0 in order to find the contribution of the nonaqueous signal to the total FID signal.  This approach was based on the fact that the non-aqueous signal had decayed to undetectably-low levels at times longer than 250 s after the 90 pulse. The signal from aqueous and solid pools were then fitted to inversion recovery signal equations for mono- or double- exponential T1 behavior.  4.2.3.2  Inversion recovery-CPMG  The Inversion Recovery-CPMG data was analyzed using a Non Negative Least Squares (NNLS) approach which was explained in detail in chapter one.  First the IR-CPMG data from the shortest (5 ms) and the longest (7.69s) TIs were analyzed for the T2 distribution and shown to be composed of two major peaks corresponding to myelin water and intra/extracellular water plus an additional T2 peak at long T2 times. Subsequently, all 49 IR-CPMG curves were fitted to two fixed T2 components  70  (corresponding to those measured from the shortest and longest TI times) plus an additional floating long T2 component, hypothesized to be the residual extracellular water arising from cutting the brain sample.  4.2.3.3  Four pool model of white matter  In order to estimate the cross relaxation times, TCR due to magnetization exchange in bovine brain, we used the four pool model (displayed again in figure 4.1) of white matter that was introduced in detail in chapter two. The advantage of using this model over fitting IR curves with exponentials is that this approach accounts for any exchange processes involved in the transfer of magnetization between water pools and non-aqueous pools thus providing a more accurate estimate of T1 relaxation. In fact for the data acquired with a 3 ms inversion pulse, the use of the more conventional inversion recovery equation [M = M0(1-f e-TI/T1 + e-TR/T1)] would have been inappropriate due to the large re-equilibration of magnetization immediately following the inversion pulse. To estimate initial conditions for solving the differential equations governing the evolution of magnetization in the four pool model we used the experimental data from the FID and the CPMG measurements. The ratio of the myelin to non-myelin non-aqueous pools was taken from fitting the non-aqueous data extracted from the FID curves. The initial values for the aqueous pools (MW and IEW) on the other hand were estimated using the myelin water fraction extracted from the CPMG data.  71  Figure 4.1 Schematic representation of the four pool model of white matter. Sizes of the four compartments are scaled to correspond roughly to the relative numbers of protons in each pool in white matter.  4.3 4.3.1  Results FID  A FID curve collected at very long TI is shown in figure 4.2. As the figure shows the decay curves can be considered a straight line at times 0.2 ms < t < 0.8 ms reflecting only the contribution from the aqueous pools. Immediately following a hard inversion pulse, magnetization from both aqueous and non-aqueous pools was inverted as shown in figure 4.3. In case of the 3 ms soft inversion pulse on the other hand, only the magnetization from the aqueous pools was inverted while the magnetization from the non-aqueous pools remained on the positive z-axis as shown in figure 4.4.  72  Figure 4.2 Free induction decay of bovine brain sample at long TI after the hard inversion pulse. A straight line (red line) was fitted to the data at times 0.25 ms < t < 0.8 ms.  Figure 4.3 The free induction decay of bovine brain sample following the hard inversion pulse.  73  Figure 4.4 The free induction decay of bovine brain sample following the soft inversion pulse. A straight line (red line) was fitted to the data at times 0.25 ms < t < 0.8 ms.  4.3.2 CPMG at long TI  Figure 4.5 The T2 distribution of bovine brain sample following a hard inversion pulse at the longest inversion time of TI = 9.3 s  74  The T2 distribution collected at very long TI= 9.3 s is shown in figure 4.5 above. A quantity named fit to noise ratio (FNR, the proton density arising from the NNLS fits divided by the standard deviation of the residuals of the fit) is calculated to quantitatively measure the quality of the fits. The FNR can be an approximate estimation of the signal to noise ratio of the IR-CPMG data. Note that the NNLS algorithm fits with sums of exponential functions which are smooth functions and therefore are incapable of fitting the fluctuations due to random noise. FNR was about 19,000 for this data. As shown in this plot, the CPMG data collected at very long inversion time clearly depicted the MW and IEW water pools and also showed a third component with T2 longer than 1 s. 4.3.3 Inversion recovery-FID Figure 4.6 below depicts the aqueous and non-aqueous signal extracted from FID curves following the hard inversion pulse. Fitting these data to the exponential functions revealed that the aqueous signal possessed a mono-exponential behavior with amplitude Aaq = 6025 (100 %) and longitudinal relaxation T1aq = 1341 ms. Figure 4.6 below depicts the aqueous and non-aqueous signal extracted from FID curves following the hard inversion pulse.  75  Fitting these data to the exponential functions revealed that the aqueous signal possessed a mono-exponential behavior with amplitude Aaq = 6025 (100 %) and longitudinal relaxation T1aq = 1341 ms. Signal from the non-aqueous pools on the other hand was best fitted by two T1 components with amplitudes: A1NA = 211 (23 %), A2NA = 703 (77 %), associated with T1NA1 = 110 ms and T1NA2 = 703 ms respectively. We hypothesize that the non-aqueous pool with A1NA = 211 (23 %) and T1NA1 = 110 ms to represent non-aqueous myelin because the second, larger component had a longer T1 time suggesting exchange with the intra/extra cellular water pools.  Figure 4.6. The aqueous and non-aqueous signal extracted from FID curves following the hard inversion pulse.  76  4.3.4. Inversion recovery-CPMG Non negative least square (NNLS) approach was used to fit the data for short and long inversion time TI s. It was observed that adding a third pool at long T2 time improved the quality of the fits, so while the T2 from the myelin water and I/E water pools were fixed, we added a floating long T2 component to the fits of the decay curves corresponding to the 49 TI times. As shown in figure 4.5 and figure 4.7, the MWF values following the hard inversion pulse measured at the shortest (TI= 450 μs) and longest inversion time (TI= 9.3 s) were 0.049 and 0.051 respectively. The amplitudes from each of these three observed T2 signal pools were plotted as a function of inversion time TI as shown in figure 4.8 (A-B). As can be seen from this figure, the amplitudes from each of these pools cross the zero line at different times, thereby indicating that in this bovine brain white matter sample the MW and IEW had different T1 times.  Figure 4.7 T2 distribution of bovine brain sample following a hard inversion pulse at the shortest inversion time of TI = 450 μs.  77  Figure 4.8 A-B The amplitudes of each of the water pools are shown as a function of inversion time TI in figure A. Figure B gives a zoomed view on the zero crossing of each component.  4.3.5 T1 distributions of water pools In order to investigate the T1 behavior of each of the water pools observed in the T2 distribution, we treated the signal from each of these water pools separately using NNLS to obtain their T1 distribution. In order to do this each data point was subtracted from the infinity value (the data point at the longest TI) so that the curve would look like a T2 decay  78  Figure 4.9 T1 distribution of myelin water pool from bovine brain sample following a hard inversion pulse.  curve. The myelin water pool T1 distribution showed a predominant peak at 496 ms and two very small peaks at 114 ms and 1.91 ms (figure 4.9). The intra/extra cellular water pool on the other hand showed a single component T1 distribution of 1.47 s (figure 4.10) and the water pool that was observed at the very long T2 values in the T2 spectrum showed a single component T1 of 2.86 s (figure 4.11). Interestingly when these three water pools were added, it was not possible to distinguish different T1 peaks using NNLS, instead a single component T1 was observed at 1.47 s (figure 4.12)..  79  Figure 4.10 T1 distribution of intra- / extra- cellular water pool from bovine brain sample following a hard inversion pulse.  Figure 4.11 T1 distribution of long T2 water pool from bovine brain sample following a hard inversion pulse.  80  1.34  Figure 4.12 T1 distribution of all three water pools from bovine brain sample following a hard inversion pulse.  4.3.6  Application of 4-pool model  We used the four pool model of the brain white matter to estimate the cross relaxation times (Tcr) by fitting the model to the FID data for aqueous and non-aqueous pools. The calculated T1 values for each of the four pools (T1M = 110 ms , T1MW = 496 ms , T1IEW = 1470 ms, and T1NM = 703 ms) were used as an input in the corresponding Bloch equations of the four pool model (these were the constrained parameters). The residual sum of squares of the optimized fit to the FID experimental data was 1.2 x 103 for the hard inversion pulse. The same approach was also adopted to treat the data from the soft inversion pulse and similar Tcr 's were found. The residual sum of squares of the optimized fit to the FID experimental data was 9.7 x 103 for the soft inversion pulse. Figures 4.13 and 4.14 show the fit to the hard inversion pulse and soft inversion pulse FID data. The corresponding extracted Tcr 's for the hard inversion pulse and soft inversion pulse FID are reported in table 4.1. Cross relaxation  81  time error estimates are calculated by varying each cross relaxation times (Tcr) until the sum of squares of the fit was increased by 5%. In order to test the independence of each of the cross relaxation time on the other two Tcrs, for the purpose of assessing parameter covariance, we adjusted one Tcr from its best fit value until the sum of squares of the residuals increased by 5%. Then this Tcr was fixed and the other two Tcrs were floated to achieve the best fit. We noticed that applying this approach, the second fit produced same Tcrs that were generated previously by the original fits. As the new sum of squares was the same as before (5% off the best fit) we concluded that the correlations between fitted parameters are small.  Cross relaxation times  TCRM (ms)  TCRD (ms)  TCRIE (ms)  Hard inversion pulse  212.3 + 93  5965 + 673  197.6 + 71  Soft inversion pulse  189 + 56  5864 + 597  157 + 85  Table 4.1 The estimated cross relaxation times (Tcr) corresponding to hard and soft inversion pulses. Error estimates are calculated by varying each cross relaxation times until the sum of squares of the fit was increased by 5%. These errors are indicated by a ± symbol.  82  Figure 4.13 Fit to the experimental hard inversion pulse FID data from the aqueous and non-aqueous pool. The residuals sum of squares for this fit is 1.2 x 103.  Figure 4.14 Fit to the experimental soft inversion pulse FID data from the aqueous and nonaqueous pool. The residuals sum of squares for this fit is 9.7 x 103.  83  4.4  Discussion  The main focus of this research was to investigate the number of T1 components in bovine brain white matter and to find out how exchange influenced the T1 curves. The free induction decay curves allowed for the separation of the aqueous pools from the non-aqueous pools. The IR-CPMG data on the other hand provided us with the T2 distribution as a function of TI. Using these data, we could more reliably apply the four pool model of white matter to estimate cross relaxation times (Tcr 's) as well as longitudinal relaxation time associated with each water pool found in the T2 distribution. To our knowledge this is the first observation of bi-exponential T1 relaxation of the nonaqueous signal in white matter. To the extent that the four pool model is correct, these two components arise from the myelin and non-myelin non-aqueous tissue. Therefore, one might consider this to be a direct measure of non-aqueous myelin. Additional research is required before this assignment can be confirmed.  4.4.1  Simulations for showing minimum required FNR for resolving T1  The FNR values associated with the T1 distributions for this study were in the 102 to 103 range and yet as mentioned earlier, when the signal from all water pools were analyzed using NNLS, only a single T1 peak could be observed. However, when the data from each T2 water pool was separated using the CPMG decay curve, the MW and IEW peaks had distinctly different T1s suggesting that we would need higher FNR levels to robustly resolve separate T1 peaks. In order to estimate the required FNR we performed a simulation in which random noise was added to synthetically generated bi-exponential data with the known MW and IEW T1 times. We then used NNLS to analyze this synthetic data. The criteria we used for a  84  correct distribution in this simulation were: a) The estimated T1 value associated with each T1 peak must be within 0.1s of the simulated T1 and b) amplitude ratio of small peak to the big one must be within 20% of the original ratio. Using this criteria we observed that we would need the enormous threshold FNR of 3.6 x 105 in order to be able to resolve each T1 peak and it's amplitude with the accuracy we obtained using the CPMG T2 data. We suspect the discrepancy for the threshold FNR level for T1 compared to T2 lies on the fact here we have set a very strict criteria for deciding an acceptable T1 resolving in our simulations.  4.4.2 Limitations Ideally we would like to elucidate the behavior of longitudinal relaxation in human brain white matter in vivo, but technical and ethical difficulties have forced us to use bovine brain ex vivo instead. Another limitation we faced during this research is that the applied four pool model of white matter treats the intra and extra cellular water in the same way, although these two pools differ in their physical location. Clearly, the assignment of the two nonaqueous T1 components to myelin and non-myelin protons was speculative and requires further investigation. One of the limitations of this work is that the applied 4 pool model assumes a single exchange process between myelin water and the intra/extracellular water pools. Additionally these experiments were insensitive to exchange processes which happened on time scales much faster than the timescale of the measurement. i.e. if myelin water was exchanging at a rate with Tcr faster than say 10 ms, the experiments could provide no information on such a fast exchange regime and the initial values used in the 4 pool model would include the affects of that fast exchange process. However, because MW and IEW were separable on a T1 time 85  scale of several 100 ms it seems unlikely that the system is undergoing exchange on the much shorter T2 time scale. It should be mentioned that the fitting of a straight line to water signal from the soft inversion pulse data at short TI times was sub optimal compared to the straight line fitting situations for the long TI times for the soft inversion pulse data and all the TI times for the hard inversion pulse data. It should also be mentioned that the rather low MWF values reported in this study are partially due to the fact that our brain samples were taken from young cows (under thirteen months old) as we had no access to mature cows' brain samples. 4.5  Conclusion  The findings of this study clearly demonstrate two component T1 relaxation in bovine brain white matter measured in an ex vivo setting. The relatively long estimated cross relaxation times between myelin water and intra/extra cellular water pool also suggests that exchange is a rather slow process in bovine brain. The fact that the extracted Tcr 's associated with the hard inversion pulse FID is very similar to those extracted from the soft inversion pulse FID, shows that the four pool model approach is able to reliably estimate the rate of magnetization exchange regardless of the initial conditions. It should also be mentioned that since virtually all MRI pulses are soft pulses (similar to what was used in this study) thus in an imaging setting, non-aqueous and aqueous spins systems are almost never at equilibrium immediately after a radio frequency pulse. Consequently the usual expressions for estimating T1 may not always be valid.  86  Chapter 5: Conclusion  The main focus of this research was to characterize longitudinal relaxation and exchange in white matter. More specifically, we were trying to investigate if T1 relaxation in white matter is a mono- or multi-exponential phenomenon. Additionally, the role of exchange in measuring MWF was studied. The current literature on exchange in white matter is contradictory: Distinctly different values of MWF have been measured for different white matter structures, experimental work by Mark Does (28) suggests that the measured MWF includes exchange which is a function of myelin sheath thickness, work by Labadie (22,23) suggests white matter has two T1 components and therefore cannot be in fast exchange, work of Gochberg/ Prantner (32,33,61) suggests that the two measured T1 components are due to cross relaxation between non-aqueous tissue and water. Using a four pool model of brain white matter, the work presented in chapter two clearly showed that the measured cross relaxation times for magnetization exchange between myelin water and IE water did not differ from structure to structure in white matter but it had two important limitations: a) it could not distinguish between exchange and T1 differences between myelin water and IE water and b) it was not sensitive to water exchange processes in a very fast exchange regime.  The work presented in chapter three investigated the effects of TReff on MWF. The concluding message is that myelin is more complicated than we thought and consequently more sophisticated models are needed to justify its behaviour.  87  In order to be able to unambiguously characterize longitudinal relaxation using high SNR data we collected ex vivo data from a 4.7 Tesla NMR spectrometer using bovine brain white matter. Our results, presented in chapter four clearly show that T1 relaxation is a biexponential phenomenon in bovine brain white matter ex vivo. More specifically the data presented in this chapter showed that for bovine brain white matter, the T1s of myelin water and IE water were clearly different. It also pointed out that these two T1s would be very hard to separate in a human in vivo measurement and that the non-aqueous signal also had two T1 components. Additionally it showed that the magnetization exchange measured in bovine brain ex vivo is a relatively slow process as the estimated Tcr 's were relatively long compared to the T2 times of the water components of white matter.  Overall this study lends support to the application of four pool model in describing the magnetization exchange in brain white matter in both in vivo and ex vivo studies on bovine as well as human brain. Despite the fact that comprehensive understanding of longitudinal relaxation in white matter in vivo is an elusive and challenging feat, the results of this study clearly support a bi-exponential T1 behavior in brain white matter.  The results of this thesis support the use of myelin water fraction as a reliable technique for measuring myelin content, however, further research is required to answer the fundamental question: Why do different white matter structures have different myelin water fractions?  88  References  1.  Norton WT, and Cammer W. Myelin. 2nd ed. New York: Plenum Press; 1984.  2.  Carr H, Purcell E. Effects of Diffusion on Free Precession in Nuclear Magnetic  Resonance Experiments. Physical Review. 1954 May;94(3):630–8. 3.  Meiboom S, Gill D. Modified Spin-Echo Method for Measuring Nuclear Relaxation  Times. Review of Scientific Instruments. 1958;29(8):688. 4.  Whittall KP, MacKay AL. Quantitative interpretation of NMR relaxation data.  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Labadie C, Lee J-H, Rooney WD, Jarchow S, Aubert-Frécon M, Springer CS Jr, et al.  Myelin water mapping by spatially regularized longitudinal relaxographic imaging at high magnetic fields. Magn Reson Med. 2013 Mar 6; 65.  Velumian AA, Samoilova M, Fehlings MG. Visualization of cytoplasmic diffusion  within living myelin sheaths of CNS white matter axons using microinjection of the fluorescent dye Lucifer Yellow. Neuroimage. 2011 May 1;56(1):27–34.  96  Appendix Matrix representation of the exchange differential equations  The differential system of equations [A.1] below describe magnetization transfer in the four pool model of brain white matter. The time-dependent magnetization in each of the four proton pools is coupled with one or two other time-dependent magnetizations of the neighboring signal pools and it is assumed that this system will reach its equilibrium state in which the equilibrium magnetization values are Mi(∞). These equations must be solved simultaneously to extract the kijs which were then used to calculate the three previously defined cross-relaxation times. The analytical solutions of the system of differential equations [2.1]-[2.4] fall under the category of a non-homogeneous linear system of four first order differential equations with constant coefficients. The matrix representation of this system of ordinary differential equations (ODE) has the form  [A.1]  Where the coefficient matrix C is represented as    97  [A.2]  Using equations [2.8]-[2.10], k21, k32, and k43 can be replaced by (k12 Pm / Pmw ), (k23 Pmw / Pie), and (k34 Pie / Pnm ) respectively in order to limit the number of unknown parameters to three directional transfer rates k12, k23 , and k34.  The general solution of this system of non-homogeneous ODEs consists of the general solution of the homogeneous part plus a particular solution of the non-homogeneous system. The solution of the homogeneous system of ODEs has the form C1 x (1) (t) + C2 x (2) (t) + C3 x (3) (t) + C4 x (4) (t) where vector functions x  (1)  (t),…, x  (4)  [A.3]  (t) are linearly independent solutions of the  corresponding homogenous system. These vector functions have the form x = ε -eλt , where ε and λ are eigen-vectors and eigen-values of the equation: (C - λ I) ε = 0  (I is the identity matrix)  [A.4]  The four eigen-values that construct the general solution of the homogeneous system are the roots of the following quartic equation (fourth-order polynomial equation)  98  [A.5]  The T1s of each of the proton pools were used in the quartic equation [A.5] as known variables. Equation [A.5] was then solved analytically using Wolfram Mathematica software version 6.0.3.0 (copy right Wolfram Research Inc.). In order to find a particular solution of the non-homogeneous system of ODEs of [A.1], the equilibrium magnetization values (Mi(∞)) for each of the four proton pools were plugged into the system of equation [A.1] and the particular solution of the system was obtained analytically. The analytical solution of the non-homogeneous system of ODEs of [A.1] contained three independent parameters (k12, k23 and k34) that needed to be extracted by fitting this solution to the experimental data. Once these kijs were determined from the fit, cross-relaxation rates corresponding to each of the defined exchange processes were determined. To verify the accuracy of these solutions, they were tested in limiting situations of no exchange (Tcr 's approaching ∞) to get 4 isolated T1 times as well as for the fast exchange limit where a weighted average T1, [1/T1 = sum( Pi/T1i)] was obtained.  99  

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