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Magnetic resonance of human and bovine brain Vavasour, Irene Margaret 1998

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M A G N E T I C R E S O N A N C E OF H U M A N A N D BOVINE  Irene Margaret Vavasour B . Sc. (Physics) University of New Brunswick, 1991 M . Sc. (Physics) University of B r i t i s h Columbia, 1993  A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T O F T H E REQUIREMENTS FOR T H E D E G R E E O F D O C T O R O F PHILOSOPHY  in T H E FACULTY O F G R A D U A T E STUDIES D E P A R T M E N T O F PHYSICS  We accept this thesis as conforming to the required standard  T H E UNIVERSITY *6F BRITISH COLUMBIA  February 1998  @ Irene Margaret Vavasour  BRAIN  in  presenting  degree  this  at the  thesis  in  University of  freely available for reference copying  of  department  this or  partial  fulfilment  British Columbia, and study.  of  I agree  1 further agree  thesis for scholarly purposes by  his  or  her  the  representatives.  may be It  is  requirements that the  for an advanced  Library shall make it  that permission for extensive granted  by the  understood  that  publication of this thesis for financial gain shall not be allowed without permission.  Department of The University of British Columbia Vancouver, Canada  DE-6 (2/88)  head  of my  copying  or  my written  Abstract  Magnetic resonance imaging ( M R I ) has become an invaluable tool for studying brain and its associated pathologies. M u l t i p l e sclerosis ( M S ) is one such pathology and attempts are being made to use M R I to characterise the myelination state of M S lesions. T w o techniques have been proposed which appear to be sensitive to myelination: magnetization transfer ( M T ) and T 2 relaxation. Quantification of these techniques uses magnetization transfer ratios ( M T R ) for M T and myelin water percentages for T relax2  ation. If the two techniques are both related to myelin content then they are expected to be related to each other. It was found by in vivo M R I measurements that white matter from normal volunteers and normal appearing white matter from M S patients had significantly larger M T R s and myelin water percentages than grey matter. However, only a weak correlation was found between M T R s and myelin water percentages i n M S lesions (R=0.5,P=0.005) indicating that each technique provides an independent measure of M S pathology. Since water i n white matter resides i n two m a i n compartments, i n intra/extracellular spaces and between myelin bilayers, it was thought that M T would have a different effect on each water pool. T h i s was examined by combining a T 2 relaxation sequence, which separates the two water pools, w i t h an M T pulse. It was found using in vivo M R I measurements on normal human white matter that the myelin water pool was significantly more affected by an M T pulse than the intra/extracellular water pool (P=0.00001 to p=0.04 for different white matter structures). It was also found that small offset frequencies caused more direct saturation of the myelin water pool than the intra/extracellular  ii  pool resulting i n different contrast. Finally, at long delay times between the M T pulse and the i n i t i a t i o n of the T relaxation sequence (>500 ms), the difference i n M T between 2  the two pools was eliminated indicating exchange w i t h i n that timescale. In vitro experiments on bovine brain were performed on a H - N M R spectrometer. A 1  4-pool model was proposed to explain the different relaxation times measured i n bovine white matter. These pools included intra/extracellular water, myelin water, non-myelin molecules and myelin molecules. Exchange between the myelin water and myelin, and the intra/extracellular water and non-myelin molecules were rapid w i t h the former being slightly faster than the latter. There was no evidence for exchange between the two water pools w i t h i n the timescale of 1 s. For human brain, a diffusion model was proposed to investigate exchange between the water pools. Results showed that variations i n parameters associated w i t h the i n tra/extracellular water pool affected only that pool. Variations i n the myelin water pool, however, influenced the relaxation times and amplitudes of both water pools. Finally, it was found that changes i n the axonal diameter and myelin thickness resulted i n changes in the myelin water percentages and T  2  relaxation times. T h i s could account for some  of the differences i n myelin water percentages and T matter structures i n the human brain.  iii  2  times measured i n different white  Table of Contents  Abstract  ii  List of Tables  ix  List of Figures  x  Acknowledgements 1  Introduction  1  1.1  B r a i n and M y e l i n  1  1.2  M u l t i p l e Sclerosis  1.3  Motivation  7  1.4  Review of N M R and M R I work on B r a i n  9  1.4.1  T  2  Relaxation  9  1.4.2  T  x  Relaxation  1.4.3  Magnetization Transfer  1.5 2  xii  :  .  5  11 12  Overview of Thesis  13  General Theory  14  2.1  Relaxation  14  2.2  Second M o m e n t  17  2.3  Cross Relaxation and Exchange  18  2.4  Magnetization Transfer  19  iv  3  G e n e r a l M a t e r i a l s and M e t h o d s  25  3.1  Samples  25  3.1.1  Bovine brain  25  3.1.2  H u m a n brain  26  3.2  N M R and M R I Equipment  27  3.3  N M R Pulse Sequences and Analyses  27  3.3.1  Free Induction Decay  27  3.3.2  S p i n - S p i n Relaxation  28  3.3.3  Spin-Lattice Relaxation  28  3.3.4  Cross Relaxation  29  3.3.5  T 1 - T 2 Relaxation Dependence  30  3.3.6  Cross-T2 Relaxation Dependence  31  3.4  3.5 4  M R I Pulse Sequences and Analyses  31  3.4.1  Magnetization Transfer  33  3.4.2  T  2  Relaxation  33  3.4.3  T  2  Relaxation w i t h M T  33  Non-negative Least Squares, ( N N L S ) Analysis of Relaxation  34  C o m p a r i s o n of M T R s and M y e l i n W a t e r Percentages  37  4.1  Summary  37  4.2  Introduction . . .  37  4.3  Materials and Methods  38  4.4  Results . .'  39  4.4.1  N o r m a l Volunteers  39  4.4.2  M S Patients  41  4.4.3  N o r m a l Volunteers vs M S Patients  44  v  4.4.4 4.5  Lesions  44  Discussion  46  4.5.1  M T R Values  46  4.5.2  Comparison of Our M T R s and M y e l i n Water Percentages w i t h Other Studies  4.5.3 4.6 5  46  Other Comparisons of M T and T  2  Results  C o n c l u d i n g Remarks  48 48  M T Effects on the Short and L o n g T  2  Relaxation Components of B r a i n  50  5.1  Summary  50  5.2  Introduction  50  5.3  M a t e r i a l and Methods  5.4  . . .  51  5.3.1  Reproducibility  52  5.3.2  B i n o m i a l M T pulse  53  5.3.3  Sine M T pulse  53  5.3.4  M T Frequency Offset  53  5.3.5  Delay Between M T Pulse and 32 Echo Sequence  53  Results  54  5.4.1  Reproducibility  54  5.4.2  B i n o m i a l M T pulse  57  5.4.3  Sine M T pulse  58  5.4.4  M T Frequency Offset  58  5.4.5  Delay Between M T Pulse and 32 Echo Sequence  58  5.5  Discussion  63  5.6  C o n c l u d i n g Remarks  66  vi  6  R e l a x a t i o n Measurements of Bovine B r a i n using M a g n e t i c Resonance  68  6.1  Summary  68  6.2  Introduction  69  6.3  M a t e r i a l and Methods  70  6.3.1  Samples  70  6.3.2  N M R Experiments  6.4  7  .  70  Results  70  6.4.1  Free Induction Decay  6.4.2  CPMG  6.4.3  T i Relaxation  77  6.4.4  Cross Relaxation  77  6.4.5  Ti-T  84  6.4.6  C r o s s - T Dependence  2  70 • • • •  Dependence  7  3  84  2  6.5  Discussion  88  6.6  Conclusion  91  Diffusion m o d e l of T  2  and T i relaxation i n two brain water pools  92  7.1  Summary  92  7.2  Introduction  92  7.3  N u m e r i c a l Methods  93  7.4  N u m e r i c a l Applications  96  7.5  T  simulations  98  7.6  T i simulations  102  7.7  Discussion  102  7.8  C o n c l u d i n g Remarks  105  2  vii  8  Conclusions  106  8.1  T h i s work  106  8.2  O n - g o i n g a n d future work  108  Bibliography Appendix A  109 Source C o d e for T  2  Simulations  viii  121  List of Tables  3.1  C o m p o s i t i o n of bovine brain samples  26  4.1  Comparison of myelin water percentages and M T R s for M S lesions . . . .  45  5.1  A m p l i t u d e of T  54  5.2  M T R s from binomial and sine pulse  57  5.3  M T R s using different off-resonance offsets  59  5.4  M T R s using different M T delays  61  6.1  M y e l i n water percentages for bovine white and grey matter  75  6.2  T  81  6.3  Cross relaxation times for white and grey matter  87  7.1  A x o n diameter and myelin thickness versus myelin water percentage . . .  99  s  2  components for smooth and 4 T  2  model distributions . .  and T ^ for white and grey matter  ix  List of Figures  1.1  Schematic of a nerve cell  2  1.2  E l e c t r o n micrograph of myelin  3  1.3  A c t i o n potential  5  1.4  Different water components from T  2.1  R e l a x a t i o n times and amplitudes as a function of cross relaxation time  2.2  Lineshape for two proton pools i n brain  21  2.3  Power spectrum of different M T pulses  23  3.1  Image w i t h R O I regions shown  32  3.2  T y p i c a l C P M G decay curve and T  4.1  Images from a normal volunteer  40  4.2  M T R s versus myelin water percentages i n white and grey matter  42  4.3  Duncan's multiple range test for myelin water percentages  43  4.4  Images from an M S patient  4.5  M T R vs myelin water percentage i n M S lesions  45  5.1  Difference between smooth and 4 T  55  5.2  T y p i c a l no M T and M T T  5.3  Average M T R as a function of off-resonance offset  60  5.4  Average M T R as a function of delay  62  6.1  Free induction decays for white and grey samples  71  6.2  Second moment as a function of moisture content  72  2  relaxation  2  10 .  distribution  2  35  .  2  model  distributions  x  20  43  56  6.3  T  2  distributions for white and grey samples  73  6.4  T  2  distributions at different r values  74  6.5  T  2  distributions at different moisture contents  76  6.6  T i distributions for white and grey samples  6.7  T i distributions at different moisture contents  6.8  Saturation recovery curves for white and grey samples . .  80  6.9  Solid signal recovery from cross relaxation  82  78 ".  79  6.10 M o d e l of proton pools i n bovine brain  83  6.11 P l o t of T i - T  85  2  dependence  6.12 Cross relaxation dependence of T  components  2  86  7.1  C y l i n d r i c a l model of water pools i n human brain  7.2  Effect of cell radii and myelin thickness on T  7.3  Effect of i n i t i a l T  7.4  Effect of diffusion coefficient on T  2  and T i on T  2  2  and T i times and amplitudes 98  and T i times and amplitudes 2  and T i times and amplitudes  xi  94  100 101  Acknowledgements  I would first like to thank my supervisor A l e x since none of this would be possible without h i m . He was a wonderful supervisor and also a great role model since he's shown me that although research is fun, there are other things i n life that are just as rewarding (not to mention that he's a really nice guy). I would also like to thank m y committee of M y e r , San and D a v i d for all the helpful suggestions and discussions. In conjunction w i t h D a v i d , a thank you must go out to the technologists and the Wendy Morrison at the M S clinic who were instrumental i n getting M S patients as well as scanning b o t h patients and normal volunteers. Next would be K e n who has been invaluable i n everything from programming to analysis to life. (He's also a Babylon 5 fan which puts h i m i n m y good books.) There are a l l the people from room 100 (past and present) who have made the past 4.5 years unbelievably great. In particular, I've enjoyed pottery w i t h E l a n a and Cornelia, talks w i t h E l a n a , Cornelia, Jamie and Traci, Frank is a never ending source of information and always willing to lend a hand (how do you think my thesis got printed?), Jenifer (although she has gone on to bigger and better things) is always ready w i t h great conversation, Denis (who has also moved up i n the world) brought fun and girl guide cookies, R e z a and X i n were usually ready for a game of fooseball and Sophia who is w i l l i n g to listen to me go on about my research. I also need to thank my parents who have instilled i n me a love of physics (as well as the correct physics genes). I guess the apple never falls far from the tree. Last but c e r t a i n l y not least, I need to thank my husband, Jeff, who, from his proposal at the beginning of my P h D to now has been incredibly supportive and loving. (He was also invaluable help i n getting my simulations going.) These past few years have b e e n t r u e l y wonderful. Thanks everyone!  xii  Chapter 1  Introduction  1.1  B r a i n and M y e l i n  The b r a i n is made up of neurons and glial cells which work together to carry out brain function.  T h e neurons form the communication network i n the body while the glial  cells act as a support structure.  A sketch of a typical nerve cell is shown i n Figure  1.1. T h e nerve cell is made up of four different structures, the cell body (which is the metabolic centre of the cell), the axon (which transmits nerve impulses also known as action potentials over long distances), the dendrites (which receive signals from other neurons) and the pre-synaptic terminal (which releases neurotransmitter i n response to action potential i n order to pass on information to the next neuron). A x o n s can propagate for distances up to 1 m and range from 0.2-20 jim i n diameter. T h e axon connects to the cell body at a region called the action hillock. A t this location, the composition of the neuron is unique i n order to allow the initiation of action potentials v i a an integration of a l l signals received by the cell. In order for signals to propagate quickly along the length of the axon, an insulating structure called the myelin sheath is present.  Myelin  is described i n more detail below. A t its terminus, the axon divides into small branches each w i t h a pre-synaptic terminal. These terminals end i n proximity to other dendrites or cell bodies i n order to transmit the signal, through release of neurotransmitters, to the next cell. Therefore, signal propagation i n a neuron begins w i t h neurotransmitters binding to specialised receptors i n the dendrites or cell body where these chemical signals  1  Chapter  1.  Introduction  2  are transformed into electrical signals. These electrical signals then transmit passively to the axon hillock where they are integrated and, if a certain threshold is reached, an action potential is generated. T h i s action potential flows uni-directionally down the length of the axon u n t i l it reaches the pre-synaptic terminals. M o d u l a t e d by the number and frequency of the action potentials, neurotransmitter is released from the pre-synaptic terminal. T h e neurotransmitter then diffuses to the next cell and binds to receptors. T h e whole process begins again. Depending on the type of neuron being fired, an inhibitory or excitatory signal can be delivered to the post-synaptic cell.  > Figure 1.1: A sketch of a typical nerve cell i n the brain. F r o m left to right are the pre-synaptic terminals, the myelinated axon, the cell body and the dendrites. [7]  T h e myelin sheath is a lipid-protein membrane found i n the central nervous system ( C N S ) and peripheral nervous system ( P N S ) of vertebrates [1, 2].  In the C N S , it is  created by specialised glial cells called oligodendrocytes which w i n d themselves tightly around the axon (Figure 1.2) [3, 4]. T h e resulting multilayer is composed of repeating units of membrane-cytoplasmic space-membrane-extracellular space w i t h a thickness of 150-160 A[5]. Dehydrated myelin is composed of 75-80% l i p i d and 20-25% protein [6]. T h i s is quite unusual since other membranes are generally 50% l i p i d and 50% protein.  Chapter  1.  Introduction  3  M y e l i n makes up 50% of the dry weight of white matter and has a relatively low water content of 40% [6]. T h e myelin is present to act as an electrical insulator for neurons and allow conduction of nerve signals to propagate about 100 x faster. There are gaps along the axon where no myelin is present known as the nodes of Ranvier which are important in signal conduction.  Figure 1.2: A n electron micrograph of myelin. T h e bar represents O.lpmx 150000. [1] Since nerve impulses consist of electrical signals, it is important to know how these are created and how they propagate. A nerve cell membrane has a membrane resting potential of -65 m V . (The negative sign indicates that the inside of the cell is negatively  Chapter 1. Introduction  4  charged w i t h respect to the outside of the cell.) T h i s potential is due to the uneven distribution of ions, namely N a , K +  +  , C I " and C a  2 +  , across the membrane.  In signal  transduction, the two most important ions are N a and K . T h e N a concentration is +  +  +  about 10 times lower inside the cell than outside while the K  +  concentration is about  50 times higher inside than outside [7]. These gradients are maintained b y a N a / K +  pump and also by the leakiness of the plasma membrane to K  +  +  but not N a . In the +  membrane, there are voltage gated ion channels which only open when the membrane becomes depolarised. T h e interplay between these channels creates the action potential which propagates along the axon (shown i n Figure 1.3). W h e n the membrane potential is increased by about 10 m V (from -65 to -55), an action potential is fired.  T h e ini-  t i a l depolarisation causes voltage gated N a channels to open and N a rushes into the +  +  cell. T h i s causes a large depolarisation to about +110 m V . T h e N a channels have a +  self-regulating feature which automatically closes them after a short time. A t this time, voltage gated K  +  channels open and K  become repolarised. T h i s efflux of K  +  +  rushes out of the cell causing the membrane to  actually overshoots the i n i t i a l resting potential and  the membrane becomes hyperpolarised. T h e K  +  channels close and the N a / K +  +  pump  restores the ion gradients. T h e overall duration of an action potential is about 1 ms. C I " is often passively distributed across the membrane although there are sometimes pumps which pump the CT~ out of the cell. T h i s ion w i l l also help contribute to the action potential but to a lesser extent. A s the membrane becomes depolarised, it dissipates quickly along the axon. T h i s passive conduction of the depolarisation can cause neighbouring regions of the axon to reach the threshold and a new action potential is triggered. If myelin is present, the rate of dissipation o f the depolarisation is much slower. A t the nodes of Ranvier, N a channels are found i n large concentrations making the triggering +  of action potentials very easy. Therefore, the action potential can hop along the axon from node to node which allows it to travel much faster.  T h i s propagation of action  Chapter 1. Introduction  5  potentials is called saltatory conduction.  0  1  2  3  4  Time (msec)  Figure 1.3: A sketch of an action potential. T h e two solid lines show the flux of N a and K through i o n channels and the dashed line represents the resulting action potential. T h e horizontal reference line is at -65 m V and the top of the action potential reaches about +110 m V . [7] +  +  1.2  M u l t i p l e Sclerosis  In Canada, the number of people afflicted w i t h multiple sclerosis ( M S ) is one of the highest i n the world. It is thought that over 50000 Canadians have the disease. M S is usually diagnosed at an early age, between 20 and 40 years and the ratio of women to men affected is 1.8:1. M S is characterised by the destruction of C N S myelin. T h e disease is thought to be brought on by both environmental and genetic factors. A virus is thought  Chapter  1.  Introduction  •6  to play a role but none has been definitively linked [8, 9]. There are two forms of M S : a chronic form (90% of cases) which manifests as a continuous relapsing/remitting cycle of demyelination and an acute form (10% of cases) where there is rapid and progressive demyelination. T h e chronic form usually consists of relapses and then subsequent recovery of most disabilities. After several years, complete recovery no longer occurs and the disease becomes known as secondary progressive. Demyelination arises from an autoimmune response targeting myelin antigens which are thought to derive from myelin basic protein or some other myelin specific protein [1]. These areas of demyelination are referred to as lesions. E a r l y lesions have been correlated w i t h the breakdown of the blood b r a i n barrier near small vessels. T h e lesions are then believed to become inflamed followed by demyelination, gliosis and finally axonal loss. Lesions can be visualised by computerised tomography ( C T ) and magnetic resonance imaging ( M R I ) [10, 11, 12] although the i m ages are generally non-specific to the state of the lesion. M S lesions become infiltrated by immune cells such as macrophages, neutrophils and lymphocytes. T h e macrophages appear to destroy myelin sheaths by digesting the outer layers u n t i l a l l myelin is gone. Strangely, this cellular attack w i l l stop at a boundary and the white matter beyond looks relatively normal. There is no known cause for this behaviour. N o r m a l l y the demyelinated tissue would be repaired but the disease also affects the body's ability to replenish the myelin. Oligodendrocytes do attempt to remyelinate some neurons but they are both slow and inefficient. A l t h o u g h lesions are the main locations for disease, there is evidence that normal appearing white matter may also have small abnormalities which are difficult to see w i t h M R I (hence the term normal appearing). These abnormalities may add to the overall lesion load and contribute to disabilities acquired by the patient.  Chapter 1. Introduction  1.3  7  Motivation  M y e l i n is found wrapped around axons i n brain white matter and helps speed up nerve signal conduction.  Unfortunately i n certain diseases such as multiple sclerosis ( M S ) ,  myelin breaks down. Magnetic resonance imaging ( M R I ) is a fairly new technique which has proven useful i n detecting M S lesions. However, M R I is not capable of detecting the pathological state of the lesion and correlation between lesion load and disability has been small [13, 14, 15]. Therefore efforts are being made to produce an M R I sequence which w i l l be able to probe the state of the lesion and be able to quantify the amount of myelin. D r . A l e x M a c K a y has developed ah in vivo T relaxation pulse sequence which allows 2  the separation of different water pools i n the brain. One of these pools is thought to be associated w i t h myelin. Initial experiments on brain [16, 17] showed that white matter had larger quantities of this "myelin water" than grey matter. Also, M S lesions were found to have greatly reduced amounts of myelin water compared to the surrounding tissue.  These experiments were very exciting since they showed that one component  from T relaxation seemed to be related to water associated w i t h myelin which may be 2  related to myelin content. Elsewhere, the group of Dousset et al. began applying the technique of magnetization transfer ( M T ) to lesions [18]. T h e i r initial findings showed that mainly edematous lesions produced a different M T effect than demyelinated lesions. These results indicated that M T might also be able to differentiate between different types of lesions. Neither technique has fully validated its ability to measure myelin.  A study was  carried out where an M S - l i k e disease (experimental allergic encephalomyelitis or E A E ) was induced i n primates and M T was subsequently performed [19]. F r o m histological data, lesions w i t h more demyelination were indeed found to have a larger change i n  Chapter  1.  Introduction  M T t h a n non-demyelinated lesions. T  8  2  relaxation has been performed on guinea pigs  w i t h E A E [20] and histological samples showed that the lesions were demyelinated which correlated well w i t h their reduced myelin water peak. Therefore, there is evidence that b o t h these techniques are influenced by myelin content. Since b o t h techniques are thought to be related to myelination, we decided to compare them. If they are indeed proportional to the amount of myelin, then a linear correlation between them would be expected. T h i s study was done on normal volunteers and M S patients. It was hoped that we could determine which technique was better at quantifying myelin. F r o m the results of T relaxation, it was evident that there were different water pools 2  i n the brain. These water pools have different interactions w i t h the macromolecules and therefore each pool should have a different M T effect. T h e T  2  relaxation pulse sequence  was combined w i t h the M T sequence i n order to monitor the effects of M T on each water pool separately. Since most groups assume that there is only one water pool i n the brain, this work would improve modelling i n this area and also help i n understanding the mechanisms involved w i t h M T i n the brain. In vitro H - N M R experiments were performed to measure interactions between the 1  different water pools as well as the surrounding macromolecules i n order to provide a better understanding of the mechanisms of relaxation present i n brain tissue.  This  i n t u r n could lead to improvements i n M R pulse sequences which could provide novel information i n the diagnosis of different diseases. Finally, a model for relaxation was proposed that involved diffusion between the two water pools.  N u m e r i c a l simulations were carried out to t r y and determine the effect  of different parameters (such as diffusion coefficients and cell size) on the relaxation amplitude and time of each pool.  9  Chapter 1. Introduction  1.4  R e v i e w of N M R and M R I work on B r a i n  1.4.1  T  2  Relaxation  T h e T relaxation decay curve from brain arises from a l l the water i n the brain [20, 21]. 2  Previous studies [16, 17, 22, 23] have shown that three water compartments may be distinguished on the basis of T  2  time: a long T  2  component assigned to cerebrospinal  fluid, an intermediate T component assigned to extracellular water and cytoplasm, and 2  a short T component assigned to myelin water (Figure 1.4). T h e myelin water is thought 2  to be trapped between myelin bilayers where the short T time derives from interactions 2  between the water and the molecules i n the myelin bilayers. T h e amount of water between myelin bilayers is expected to be proportional to the amount of myelin. In general, we ignore C S F which does not contribute much to the signal from white and grey matter so that there are two water pools i n normal C N S tissue and they are resolvable on the basis of their T relaxation. 2  Measurements of T  2  relaxation times i n brain are not always done rigorously which  leads to conflicting results i n the literature [24]. T  2  relaxation pulse sequences must  include more than 4 echoes and have echo spacings of 10 ms or less i n order to measure b o t h the infra/extracellular and myelin water. In white matter, the intra/extracellular water T  2  varies for the different structures (e.g. 86 ms for the internal capsules and 71  ms for the minor forceps) which is likely due to differences i n myelin content [17]. In certain cases, the T  2  distribution for white matter shows two peaks i n the 80-100 ms  region which may be separation of the intracellular and extracellular water pools. T h e T  2  of infra/extracellular water i n grey matter is also found to be of the order of 70-90  ms. Some groups induced cerebral edema i n animals i n order to determine the effect on T i and T  2  relaxation [25, 26, 27]. W h e n edema was induced, the longer T  2  component  Chapter 1.  Introduction  10  Figure 1.4: A T distribution of white matter showing the different water components i n the brain. T h e area under each peak is proportional to the number of protons i n that environment. 2  11  Chapter 1. Introduction  split into two peaks representing intracellular and extracellular water. T h e extracellular water peak was larger and at longer T which was attributed to the extra water present 2  from edema. T h e intracellular peak remained largely unaffected by the edema. A n o t h e r experiment on cats studied, the effect of gliosis [28]. T  2  times were found to be largely  unchanged even i n the presence of edema as well as white matter packed w i t h glial fibrils a n d other structures.  T h i s lack of change i n T  2  was attributed to the efficient  cross relaxation between the water and the extra cytoplasmic structures resulting i n no net T  2  change. Finally, i n animals induced w i t h E A E , edema i n lesions caused the  intra/extracellular water T to increase [29, 30, 31]. 2  In multiple sclerosis, the T of normal appearing white matter was found to be slightly 2  elevated compared to normal controls [32]. In lesions, large variations i n T were found 2  which were thought to be due to different underlying tissue composition [33, 34, 35, 36]. In some cases, the large T  2  component had split into two peaks and was thought to  represent axonal loss leading to a larger extracellular space [37, 38]. However, other studies have not been able to confirm this result [39, 40].  1.4.2  T i Relaxation  Unlike T relaxation which is able to differentiate different water pools, T i relaxation of 2  human b r a i n appears to yield only one relaxation time. In a myelinated crayfish nerve, the T i was found to be about 1.2 s [22] which is slightly less than that expected for pure water. In the squid giant axon, a similar T i of 1.5 s was found [41]. In human brain however, the T i of white matter was about 600 ms whereas the T i of grey matter was 1 s [42]. A g a i n , only mono-exponential relaxation was found. T h e difference i n T i between white and grey matter is thought to be due to myelin and i n particular interactions between myelin water and myelin molecules [21, 43, 44]. Supporting this theory are experiments measuring T i from newborns and adolescents which showed T i times of 1.6  12  Chapter 1. Introduction  s a n d 500 ms respectively for white matter and 1.6s and 800 ms for grey matter [45]. T i relaxation times were linearly correlated w i t h water content i n brain tumour samples [46]. W i t h edema, the T i of white matter was found to increase more rapidly than grey matter. T h e suggested reason i s that excess water i n grey matter i s taken up by cells where the T i is reduced by the proteins present i n the cytoplasm. In white matter, the excess water remains i n the extracellular spaces which is free of such molecules. In animal models of edema, T i was also increased [25, 26, 27, 29, 30, 31]. In gliosis, T i was increased as opposed to T which d i d not change resulting i n a decrease of T relative to 2  2  T i [28]. In M S patients, T i increased i n lesions but was still mono-exponential and therefore probably not as useful as multi-exponential T  2  measurements [34, 35, 36]. There were  however, large variations i n T i between lesions attributed to different underlying tissue structure. N A W M was also found to have an increased T i [32].  1.4.3  M a g n e t i z a t i o n Transfer  Magnetization transfer ( M T ) is a relatively new M R technique which provides a novel form of contrast. It was developed by Wolff and B a l a b a n [47] and based on a technique discovered b y Forsen and Hoffman [48]. M T has become widely used to study many diseases, i n particular multiple sclerosis [16, 18, 47, 49, 50, 51, 52, 53, 54, 55, 56]. M T [47] utilises the fact that there is continuous magnetization exchange between two proton pools i n the brain: the motionally restricted pool which arises from non-aqueous tissue and the mobile pool from water [21]. M R I can only directly detect signal from the mobile pool. If the magnetization from the motionally restricted pool is disturbed by an M T pulse, then the effect of exchange can be seen on the mobile pool as a decrease i n signal [48]. T h i s effect is called magnetization transfer and is quantified through a magnetization transfer ratio ( M T R ) (further defined i n Chapter 2). It is expected that  13  Chapter 1. Introduction  brain volumes which have a larger number of motionally restricted protons w i l l show a greater signal decrease upon application of an M T pulse and therefore a greater M T R . Protons associated w i t h myelin would be part of the motionally restricted pool. In this way, the state of myelination of a lesion can be probed using M T .  1.5  Overview of Thesis  First, the groundwork w i l l be set by reviewing some general theory i n Chapter 2. T h i s w i l l be followed by the general materials and methods used i n a l l experiments (Chapter 3). Further details on the materials and methods is given i n the individual results chapters. Three different studies are presented i n the next three chapters.  Chapter 4 compares  M T R s and myelin water percentages i n normal volunteers and M S patients. Chapter 5 extends the work by incorporating both M T and T  2  relaxation into one sequence and  studying the effect on normal volunteers. In Chapter 6, in vitro ^ - N M R studies are presented from bovine brain. M a n y different • experiments were carried out i n order to determine how different proton pools interacted and over what timescale. Simulations of T  2  and T i relaxation are presented i n Chapter 7 for human brain by assuming that  diffusion occurs between the different water pools.  F i n a l l y i n Chapter 8, conclusions  are made about how a l l the studies relate to each other. A l s o , future experiments are suggested as well as some that are already underway.  Chapter 2  General T h e o r y  2.1 1  Relaxation  H nuclei are spin 1/2 particles and possess magnetic moment y and angular momentum  J . These two quantities are related by  A i  = J  (2.1)  7  where 7 is known as the gyromagnetic ratio. The sum of all the magnetic moments in a sample gives the total magnetization denoted by M . If a magnetic field, B , is applied to the system, a torque results such that j  = 7(MxB).  (2.2)  This equation produces precession of the magnetization around the external magnetic field at a rate known as the Larmor frequency given by UJ = 0  7  5  (2.3)  o ;  If a radio frequency (rf) pulse, B\, is applied at this frequency, the energy will be absorbed and the net magnetization will tip by an angle 9 = 7-Bi£ where t is the length of time p  p  that the rf field is applied. When the magnetization is perturbed from its equilibrium state, it tries to return via a process called relaxation.. One form of relaxation, denoted T i , brings the magnetization back to equilibrium along the direction of the external magnetic field (normally assigned 14  Chapter 2.  lb  General Theory-  to the z-axis). The other, denoted T , destroys any net magnetization in a plane perpen2  dicular to the external field (x-y plane). These terms are added to equation 2.2 in order to obtain the Bloch equation = 7  7  (  M  <  B  )  ^  t  -  -  -  |  (2.4)  where M is the net magnetization at equilibrium. 0  For spin 1/2 particles, the most important relaxation mechanism is the dipole-dipole interaction. The dipolar Hamiltonian is given by  #12  =  2  ^ [ I i  • h - 3(Ix • n)(I • h)]  (2.5)  2  where I is the angular momentum, r is the separation between the two spins and n is the unit vector in the direction joining them. If we transform n into polar coordinates with angle 6 and <p with respect to the external magnetic field, the Hamiltonian can be written as H  12  = ^ ^ ( A + B + C + D + E + F)  (2.6)  where A = B = -\(I^2 C = ~\{Ithz D ——|  I I {l-3cos 8) 2  lz  +^  2z  + 2  ) ( l - 3cos 0) 2  + /iz/2+)  sin 9cos  sin 9 cos 9e  i4>  ( / f / 2 , + /i2/2-)  £ = F = ~(Jf  9e~*  sin 9e~ * 2  2i  J -) sin 9e 2  2  .  (2.7)  2i<t>  The terms A and B only allow for transitions between equal Zeeman energy states, the C and D terms allow one of the spins to flip (with energy transition of hu ) and the E 0  Chapter 2. General  16  Theory  and F terms allow both the spins to flip (with energy transition of 2hu ). 0  Consequently,  T i relaxation (which involves a net change in energy of the system) only results from the last 4 terms. T relaxation arises from all terms. 2  If motion (in the limit of M r 2  -C 1, where M is the second moment of the dipo-  2 c  2  lar broadened lineshape and r is the correlation time for the motion) is present, then' c  the dipolar interaction becomes averaged. In this case, relaxation can be described by Redfield's Theory [57].. For spins at a constant separation r and varying in orientation isotropically, relaxation can be defined as ±  ¥  = ^ [ J M  i67^  = 2  [ 6 J ( 0 )  +  1  0  + 4J(2»o)]  J  {  U  o  )  +  (2.8)  4 J ( 2 c J o ) ]  -  ( 2  9 )  where J(u>) is the spectral density and equal to the Fourier transform of the correlation function.  Usually, the correlation function is assumed to be exponential with a time  constant r so equation 2.8 and 2.9 become c  JL--3_2^! T  x  ~ 10 r  1 1+U*T*  6  Tc[  J_ - A l ! ^ ! T  2  ~ 20^  For short correlation times (U T 0  C  T  4  c  T  2  _12  2  0  5 c  L  3  +  2j  '  (  c  }  2  1+U*T*  +  <C 1), one has ^ = ^ = M r . J-l -12 2  In bulk water, r ~ 10  l + 4w r  +  c  1 + 4^r 2 c  J  (  2  '  l  l  j  (2.12)  s and T i and T are between 1 and 3 s. In general for tissue, 2  < T i due to interaction between water and macromolecules. For brain tissue, we expect  more than one stochastic process to influence the relaxation and therefore equations 2.8 and 2.9 would involve more than one r . This would lead to much more complicated c  expressions.  Chapter 2.  2.2  General  17  Theory  Second M o m e n t  The moments of a lineshape function f(u) can be defined as M. -  ( 2  Jo f{u)duj  ,3)  The so-called second moment comes from n =2. For a rigid lattice, the second moment can be calculated from •M,->'J(J where  l  +  )  i  ;  (  1  -  3  f  (  2  .  1  4  )  is a vector describing the relative position of two protons and 8jk is the angle  between the applied magnetic field and this vector.  If one averages over all angles,  equation 2.14 reduces to M  = | ^ /(/ + l)E-i-2  2  7  (2- ) 15  In the presence of motion, the lineshape becomes narrowed and the measured second moment will be smaller. (The true second moment never changes.)  This is known as  motional narrowing. Unfortunately, equation 2.15 is very hard to apply experimentally. Instead, the LoweNorberg theorem relates f(u)  to a free induction decay function F(t) F(t)=  Jo  f(u)cos(ojt)du  (2.16)  where / °° f(u))dw =1 is assumed. If cos(u>t) is replaced by its Taylor's series, then we get 0  oo m  ±2n  - £ ( - D »  m  M  2  n  .  (2.17)  Therefore, a free induction decay can be used in order to determine the second moment.  Chapter 2.  2.3  18  General Theory  Cross Relaxation and Exchange  Cross relaxation can be represented by a modification of the Bloch equations to incorporate exchange of magnetization between two pools of protons (A and B) [58]. The longitudinal magnetization in pool A is designated M  and in pool B M .  A  B  The fully  relaxed magnetizations are designated Myi and MB . The rate constant for exchange 0  from pool A to pool B is K  0  and from pool B to pool A is K ^ . The reciprocal of the rate  A  constants give the lifetimes that protons spend in each pool, i.e. When exchange occurs, pool A loses ^ B loses  = ^ and K  B  —  magnetization to pool B per unit time and pool  magnetization to pool A per unit time. The modified Bloch equations are  given by  dM dt  M  =  A  - M TIA  dM  -  M  =  B  BO  dt where T\  A  _MA  A  Ao  T  T  A  _MB_  M  B  TIB  (2 18)  MB  +  B  +  ^  M  A  T  T  B  A  and TIB are the longitudinal relaxation times as defined above. The general  solution to these coupled equations is [48] = M  M (t) A  M (t)  = M  B  B  O  +  R C  l  A  +  l  ^  A  -  + C -  Xlt  A  X  O  l  ie  (2.20)  + C e-  X2t  2  e - ^ + C  ^ + ^'V^  KB  K  (2.21).  B  where Ci =  R  I  A  + Ai  C  2  =  R  I  A  (M (0)  A  X2  A  - M ) AO  — A2  + ~ (M (0) KA  XI  a  Al  and  K  — A2  - -K*-(M {0) B  - M ) BO  (2.22)  Ai — A2  - M ) AO  - - ^ - ( M A l — A2  B  ( 0 ) - M ) BO  (2.23)  Chapter 2.  General  = {(RIA  XI,  2  [(R  +K  1A  R±  A  +R  A  and R\  B  19  Theory  + K)  2  lB  B  + K  + R  A  - A(R R 1A  1  B  +  + Ri K  1B  A  K )± B  + R K )) ' }/2. 1 2  A  lB  B  (2.24)  are the reciprocals of the longitudinal relaxation time for pool A and B  respectively. A cross relaxation time, T  c r  , can be defined between the pools as follows [59] PK A  A  = PK B  B  = -J-  (2.25)  cr where P  A  and P  are the probabilities of finding a proton in pool A and B respectively.  B  If the signal from pool B is brought to zero and the T i times for both pools are ignored, then the system will return to equilibrium through a signal increase in pool B from pool A of the form S{t) = (1 - e~ ). t/TB  Experimentally, T is measured and related to T B  C T  (2.26)  .  As an example, white matter is thought to consist of two water pools, myelin water and intra/extracellular water. The amplitudes and T i times have been estimated as 16% and 200 ms for myelin water and 84% and 800 ms for intra/extracellular water. A plot of the change in relative amplitude and T i measured in the presence of cross relaxation as a function of the cross relaxation time is shown in Figure 2.1.  2.4  M a g n e t i z a t i o n Transfer  Magnetization transfer (MT) is equivalent to cross relaxation and used in MRI [47]. In brain tissue, two pools of protons are assumed, one mobile pool associated with water and one motionally restricted pool associated with macromolecules. There is very little motion involved in the motionally restricted pool so the frequency distribution is broad and the signal decay time is short. In the mobile pool, motion averages the dipole-dipole  Chapter 2.  General  1.0  20  Theory  p o o  O o o  O  o  o  8  S  o|  c/>0.8  CD T3  E  CO  Eo.6 co  •• •  •  T  O  P,  •  CO CD  1 (  T.1 m  A  E  io.4H CO X JTO  o DC  0.2  • 0.0  •  11111  0.01  A -l—i  i  A  111HI  A  •  A  A  A  -i—nr  0.1 1 Cross relaxation time  A  A  TTT  10  Figure 2.1: A white matter model of the change in T i relaxation time and amplitude of myelin water (labeled with m) and intra/extracellular water (labeled with i) as a function of the cross relaxation time. The true amplitudes and T i times for the different pools are 16% and 200 ms, and 84% and 800 ms.  Chapter 2.  General  21  Theory  interaction and the lineshape narrows with an associated increase in T relaxation time. 2  The lineshapes of the two pools are shown in Figure 2.2. These two pools continuously undergo exchange of magnetization.  mobile  pool  Figure 2.2: Representation of two proton pools in brain: motionally restricted macromolecules (broad line) and mobile water (narrow line). Both curves are centred on the water resonance, v . Note that the intensity of the mobile pool has been reduced to fit on the plot. 0  Unlike in H - N M R , with MRI it is not possible to measure the motionally restricted 1  component of the signal. It is possible however, to perturb the motionally restricted pool and then monitor the effect of magnetization transfer (or cross relaxation) on the mobile signal. If the motionally restricted signal is kept at saturation (signal equals zero), then the M B term in equation 2.18 disappears and the magnetization in pool A will relax with  Chapter 2.  General  a time constant, T i  22  Theory  s a t  , given by 1  1  1  (2.27)  This leads to the solution M  A  where t  sat  = (M  - M t)e-  +  tsat/Tlsat  Ao  Asa  M  Asat  (2.28)  is the time for which pool B is kept saturated and M t Asa  =  (2.29)  M . Ao  T  1A  M t Asa  is the final magnetization attained by pool A if pool B is kept saturated.  There are two methods of saturating pool B. One is the application of a long offresonance pulse which will only affect the broad macromolecular pool. By continuous application of this pulse, the broad pool continuously undergoes dephasing and therefore, no net magnetization remains. The power spectrum of the sine pulse is shown in Figure 2.3. The second method is to apply a short on-resonance binomial pulse. The net angular displacement of this pulse is zero so protons with a long T will not undergo relaxation and 2  their magnetization will be brought back to its initial position. Protons with short decay times will dephase during the course of the pulse and will not experience a net zero degree pulse but a randomisation of their magnetization. Therefore, the macromolecules, which have short T 2 S , become dephased while the water is brought back to its initial position. The power spectrum of a binomial pulse superimposed on the spectrum of the two brain proton pools in shown in Figure 2.3. In practice, it is very hard to attain full saturation of pool B while keeping pool A unaffected. Therefore, compromises are made which cause the above equations to become invalid. Instead a more qualitative approach is taken where the difference in the image intensity before and after the M T pulse is observed and related to the underlying tissue  Chapter 2.  General Theory  23  Figure 2.3: The power spectrum of both a sine and binomial M T pulse is sketched superimposed on the spectrum of the two proton pools in brain.  Chapter 2.  24  General Theory  structure. This change is quantified by means of a magnetization transfer ratio (MTR) defined as MTR  = (l -  x 100%  (2.30)  where M refers to the signal intensity from an image acquired with an M T pulse and M s  0  refers to the signal intensity from an image acquired without an M T pulse. Decreases to the M T R would occur due to a reduction in the macromolecular pool or an increase in the water pool.  Chapter 3  General Materials and M e t h o d s  3.1 Two  Samples sets of experiments were carried out, one on bovine brain and the other on human  brain. The bovine brain was used for were performed  3.1.1  in vivo  in vitro  ^-NMR  experiments. MRI  experiments  on human brain from normal volunteers and MS patients.  B o v i n e brain  Bovine brain was obtained within three hours of slaughter (GrandMaison beef farm, 5175-184 St, Surrey) and immediately placed in phosphate buffered saline (Oxoid, lot# R025361-002) cooled with ice. As soon as possible, the brain and buffer were placed in a refrigerator (4°C). White and grey matter tissue samples were cut from the brain, dried on a paper towel and placed in a 10 mm o.d. NMR in an N M R  tube. The samples were then placed  spectrometer and a variety of experiments were carried out. When a sample  was not in use, it was stored in the refrigerator. Samples were allowed to warm up to room temperature for half an hour before the experiments were started. Samples were cut from the brain just prior to their placement in the spectrometer. Therefore, samples were removed from the brain over the course of 3-4 days. Three bovine brains were used with 6 samples taken from the first brain (labelled Samples 1-1, 1-2, 1-3, 1-4, 1-5, 1-6), 4 samples from the second brain (labelled 2-1, 2-2, 2-3, 2-4) and 6 samples from the third brain (labelled 3-1, 3-2, 3-3, 3-4, 3-5, 3-6). The composition of the different samples is  25  Chapter 3.  General Materials  and Methods  26  Brain 1 Sample Composition  1-1 MG  1-2 W  1-3 W  1-4 G  Brain 2 Sample Composition  2-1 W  2-2 G  2-3 W  2-4 G  3-1 W 71.0 0.3  3-2 W 73.2 68.2 59.9 37.7 13.3  3-3 G 83.8 .0.4  3-4 G 84.0 77.0 52.8 2.5  Brain 3 Sample Composition Moisture Contents  1-5 G  1-6 W  3-5 . 3-6 W G 85.4 72.0 7.5 36.0 6.9  Table 3.1: Composition of samples obtained from three separate bovine brains. M G = mostly grey, W = white, G = grey. shown in Table 3.1. Dehydration experiments were carried out on samples from the third brain by placing the samples in an oven with a drying agent (anhydrous CaSO^) at 30°C. For complete dehydration, the samples were left under vacuum for 48 hours. Samples were weighed before and after each dehydration step. Moisture contents were calculated as the weight of liquid in the sample over the total weight of the sample. The moisture contents used are shown in Table 3.1.  3.1.2  H u m a n brain  For one experiment (chapter 4), ten normal volunteers (age 20-47) and nine clinically definite MS patients (age 30-56) were examined. Five MS patients were relapsing-remitting (R/R) and four were secondary progressive (SP) in their clinical course. For the second experiment (chapter 5), only normal volunteers were used. More details appear in  Chapter 3.  General Materials  27  and Methods  chapter 5.  3.2  N M Ra n d M R I E q u i p m e n t  The N M R experiments were performed on a modified Bruker S X P 4-100 N M R spectrometer operating at 90 MHz with a lips receiver deadtime. The data acquisition and analysis system included a locally built pulse programmer [60], a Rapid Systems digitizer and an I B M compatible computer. The 90° pulse length was 1.8 ps and the 180° pulse length was 3.9 ps. Temperature was set using a Bruker B-ST 100/700 temperature controller accurate to ± 0 . 5 ° C . MRI experiments were done using a 1.5 T General Electric Signa clinical scanner operating at the 5.4 software level.  3.3  N M RPulse Sequences a n d Analyses  In this section, all pulse sequences used on the N M R spectrometer are described. The method of analysis associated with each sequence is also mentioned. More details of the individual experiments are described in later chapters.  3.3.1  Free Induction  Decay  The free induction decay (FID) was used to separate the signal from solid and mobile protons. The solid signal arose from dipolar coupled protons in macromolecules. The mobile signal came from protons in water or other isotropically moving protons. The pulse sequence was: 9 0  x  - T / 2 - ( 1 8 0  y  - f )  8  - r f l  ••  Chapter 3. General Materials  with r=200ps.  28  and Methods  The FID signal for dipolar coupled protons can be fit to a moment  expansion equation given as: S(t) « 5(0)(1 - M t /2\ 2  2  + M t /4\ 4  A  - M t /6\)  (3.1)  6  6  where M , M , M are the second, fourth and sixth moments of the lineshape and S(0) 2  4  6  is the total signal at t=0. The brain FID decay curve from 17 to 42/xs from the centre of the 90° pulse was fit to this expression using a non-linear function optimisation program minimising x [61]- The eight 180° pulses were included in order to minimise the effect 2  of magnetic field inhomogeneity. The intensities from the tops of the 8 echoes were fit to a sum of two exponentials and extrapolated to t=0 in order to obtain the initial mobile signal, M(0). In order to determine the second moment of only the solid protons, M' , the 2  mobile signal (assumed to have an M of zero) was removed using M .= M2g( f-M(o)' 2  2  0  The difference in total signal intensity of the FID. between two different hydration levels and the corresponding mass difference was used to estimate the proton density of the solid (non-aqueous) brain tissue.  3.3.2  S p i n - S p i n Relaxation  The mobile signal was further characterised using a Carr-Purcell-Meiboom-Gill (CPMG) [62, 63] sequence with 4320 180° pulses and echo spacings of 100 /is, 200 ps or 400 ps. 736 echoes were collected; the first 224 echoes were collected at every echo and the last 512 echoes were collected every 8  th  echo. The first 4 decay points were discarded since  there was a large amount of scatter in the intensity of the initial points [64].  3.3.3  S p i n - L a t t i c e Relaxation  The spin-lattice relaxation time (Ti) was determined using a modified inversion recovery pulse sequence (IR) and a partial saturation recovery pulse sequence (SAT). The IR pulse  Chapter 3.  General Materials  and  29  Methods  sequence was: 90 x  I80  X  - r -  TR  90.  T  -  TR  Eighteen r values were used for IR in brain 1 and 2: 1, 3, 5, 10, 15, 25, 40, 60, 90, 140, 200, 300, 500, 750, 1000, 1500, 2000, 3000 ms. In brain 3, 30 r values were used: 1, 2, 3, 5, 7.5, 10, 15, 20, 25, 30, 40, 50, 75, 100, 125, 150, 200, 250, 300, 400, 500, 750, 1000, 1500, 2000, 2500, 3000, 4000, 5000 ms. The second FID signal was subtracted from the first in order to obtain a resultant positive signal, S(t), that decayed to zero at long r. For a system with only mono-exponential relaxation, this signaldecays as S(t) =  2M e~ ^ T  Tl  0  where M is the total magnetization along the z-axis. The solid and mobile (assumed to 0  be liquid) parts of the signal were separated for each r. The mobile signal was taken as the average signal from 80 to 90 ^s after the 90° pulse. The solid signal was found by subtracting the liquid signal from the total signal (averaged between 15 and 25 ps from the 90° pulse). For the S A T pulse sequence, an FID was used with r=200yus and T R s of: 200, 300, 500, 600, 750, 1000, 1500, 2000, 3000, 4000, 5000, 7000 ms. SAT was only measured for brains 1 and 2. The signal intensity was then fit to the expression S(t) = M ( l — 0  e~ ^ ). TR  Tl  This equation assumes a mono-exponential T i .  3.3.4  Cross  Relaxation  The cross relaxation time,  between the solid and mobile protons was measured using  the pulse sequence [65]: 90 - ti - 90x  x  90 -t x  x  - T - 9 0  x  90. - r - 90* T  - T R  TR  Chapter 3.  General Materials  30  and Methods  The ti value was 400/^s which allowed for complete dephasing of the solid signal. The r time allowed exchange between the solid and mobile protons causing the reappearance of the solid signal after the third 90° pulse. The r times used were 1, 2, 5, 10, 15, 25, 50, 75, 100, 150, 200 ms for brains 1 and 2 and 1, 2, 3, 5, 7.5, 10, 15, 20, 30, 50, 75, 100, 150, 200 ms for brain 3. The first trace was subtracted from the second in order to eliminate T i effects from the solid signal but not the water. The average signal from points 80 to 90 fxs was subtracted from the average signal from points 16 to 26 fxs in order to characterise the solid signal intensity.  3.3.5  Ti-T  2  Relaxation Dependence  The two-dimensional T i - T  2  dependence was determined with the modified C P M G se-  quence: 90*  ~^Y~  (180„ - TE)  TE 180 - r - 90 - — x  x  n  -TR  - (180„ - TE)  n  -  TR  with TE=200^xs and the first 15 r values (Brain 1 and 2) or all 18 r values (Brain 3) of the' Brain 1,2 IR sequence above. The second curve was subtracted from the first curve in order to obtain a positive T i relaxation signal which decays to zero. The C P M G part of the sequence was collected exactly the same as before (Section 3.3.2). The array of T 1 - T 2 decay data was simultaneously fit using a modified NNLS algorithm [66] with a logarithmically spaced set of 20 T times from 0.01 to 5s and 20 T times from 0.001 to x  2s. This sequence was only performed on white matter samples.  2  Chapter 3.  3.3.6  General Materials  Cross-T  2  31  and Methods  Relaxation Dependence  The two-dimensional cross-T dependence was determined with the modified C P M G 2  sequence: 90 - t - 90_ - r - 90  TW  (180„ - TE) 2 TE 90 - h - 90* - T - 90 - — - (180,, - TE) x  x  x  x  x  x  -  n  -  n  TR TR  with ti=400/xs, TE=200//s and the same r values as the cross relaxation experiment for Brain 3. The second curve was subtracted from the first curve in order to eliminate T i effects from the solid. The C P M G part of the pulse sequence was collected exactly the same as before (Section 3.3.2). Each C P M G curve was analysed in the same manner as the spin-spin relaxation data using NNLS (decribed in 3.5). The T distributions were 2  found to have 2 or three components assigned to myelin water ( T between 1 ms and 2  30 ms), intra/extracellular water ( T between 30 ms and 200 ms) and buffer ( T greater 2  than 200 ms).  2  For each r value, the area under the myelin water peak and the area  under the intra/extracellular water peak were calculated. This resulted in decay curves of r times and peak areas. These curves were fit to mono-exponential relaxation models. This experiment was applied only to the Brain 3 samples. T i - T  2  relaxation curves were  also decomposed into 2 or 3 T components in the same manner for comparison to the 2  cross-T decay curves. 2  3.4  M R I Pulse Sequences and Analyses  MRI experiments included magnetization transfer (MT), 32 echo T  2  relaxation and 32  echo T relaxation with an M T pulse. Each sequence as well as the analysis is described 2  below. In all cases, a transverse slice through the base of the genu and splenium of the corpus callosum was examined. Volumes of interest were outlined in eleven structures,  Chapter 3.  General Materials  and Methods  32  including five white matter or normal appearing white matter (NAWM) structures: the genu and splenium of the corpus callosum, the posterior internal capsules and the major and minor forceps, and six grey matter structures: the putamen, the head of the caudate nucleus, the thalamus, the cingulate gyrus, the insular cortex and cortical grey matter. Outlines of these regions are shown on an image in Figure 3.1.  Figure 3.1: Image with each brain structure outlined to show ROIs chosen for MRI analysis. On this image, white matter regions are darker on the image than grey matter. A water tube is shown on the right-hand side of the image.  Chapter 3. General Materials and Methods  3.4.1  33  M a g n e t i z a t i o n Transfer  For the magnetization transfer measurements, a spin echo sequence was used with sequence parameters: repetition time T R 1 s, echo time T E 16 ms, 2 averages, slice thickness 5 mm, F O V 220 mm and matrix size 256x128. Ten slices were acquired but only the slice matching that of the T relaxation measurement was used in this analysis. This 2  sequence was run twice, once with no M T pulses and once with two 19 ms sine pulses at 2 kHz off-resonance with total flip angle amplitude equivalent to a 1060° pulse applied preceding each spin echo. This sequence was optimised to obtain the greatest difference in M T R between normal white and grey matter. M T R images were produced by plotting the M T R at each voxel. A binomial pulse was used in Chapter 5. This pulse consisted of six 1 ms long 121 on-resonance pulses with spacings of 400 us.  3.4.2  T  Relaxation  2  For the T  2  relaxation measurements, a single slice 32 echo sequence was used.  This  sequence consisted of a 90° slice selective pulse followed by 32 rectangular composite 180° pulses flanked by slice-select crusher gradient pulses of alternating sign with descending amplitude for elimination of stimulated echoes and signal from outside the selected slice [67]. Sequence parameters were: repetition time T R 3s, echo spacing 10ms, slice thickness 5mm, F O V 220mm, bandwidth ± 3 2 k H z , matrix size 256x128 and 4 averages.  3.4.3  T  2  R e l a x a t i o n with M T  The pulse sequence consisted of the preparatory M T pulse followed by the 32 echo sequence mentioned above (Section 3.4.2). The only difference was that only 2 averages were done.  Two types of preparatory M T pulses were employed: 1) two 19 ms sine  pulses at 2 kHz off-resonance (unless otherwise specified) or 2) 6 121 binomial pulses,  Chapter 3.  General Materials  34  and Methods  each binomial pulse with a duration of 1 ms. Five different studies were carried out and described in chapter 5.  3.5  Non-negative Least Squares ( N N L S )  Analysis of Relaxation  The C P M G (NMR) and 32 echo (MRI) decay curves were decomposed into an arbitrary number of exponentials using a modified non-negative least-squares fitting routine (NNLS) [66]. The range of T times included 100 points from 1 ms to 5 s (15 ms to 2 s 2  for 32 echo). Non-negative least-squares analysis gives a discrete T up of delta functions as the lowest x fit to 2  2  distribution made  decay curve. The true T distribution  a  2  in brain is more likely composed of a continuous distribution of relaxation times. To accomodate this, the x value was allowed to increase from 1-2% (0.1-0.3% for 32 echo) 2  by minimising solution roughness as well as x [17, 66, 68]. This produced smooth T 2  2  distributions which were more robust in the presence of noise. A typical T distribution 2  is shown in Figure 3.2. The corresponding decay curve is shown in the inset. Both the discrete and smooth T distributions are shown. Myelin water percentages were calcu2  lated by dividing the intensity of the T distribution from 1-50 ms (15-40 ms for 32 echo) 2  by the intensity between 1-700 ms (excluding CSF). Myelin water maps were obtained by calculating the myelin water percentage at each pixel in the image and plotting the intensity. Geometric mean T times [17] (mean on a logarithmic scale) for the entire T 2  2  distribution and for the range from 50 to 900 ms (excluding the short T component and 2  cerebrospinal fluid) were calculated for MS lesions. Analysis of the T relaxation with M T data was slightly different. Within each white 2  and grey matter structure, regions of interest were drawn and the decay curves from all pixels averaged. The average curve was then decomposed into a maximum of four exponentials at 20, 80, 120 and 2000 ms.. The four exponentials were chosen to match  Chapter 3.  General Materials  and Methods  35  Figure 3.2: Typical T distribution for bovine white matter. The corresponding decay curve is shown in the inset. Both discrete (spikes) and smooth (curves) distributions are shown. 2  Chapter 3.  General Materials  36  and Methods  the T relaxation values found in our previous brain studies [17, 23]. In these previous 2  studies, decay curves were decomposed into an arbitrary number of exponentials using a modified non-negative least-squares (NNLS) technique. Unfortunately, the lower signal to noise ratio obtained with only two averages for our 32 echo images caused fluctuations in the intensity and position of the short T  2  component.  Therefore, the number of  degrees of freedom was decreased in order to make the short T component signal more 2  stable. Results from our modified NNLS solution with % misfits of 5% greater than the 2  minimum and a myelin window of 15-40 ms gave similar results as the four exponential solutions. Also, replacing the 120 ms T with a 200. ms T did not change the results. 2  2  The four proton water pools were separated into the short T component (signal at 20 2  ms) and the long T component (combined signal from 80 and 120 ms). The total signal 2  was the sum of the short and long T components and signal from C S F (signal with T 2  2  at 2 s) was ignored. Magnetization transfer ratios were calculated for the short, long and the total T component signals 2  M T R = ( l - ^ ± ) x 100% V ' Area „ J where Area  s  and Area  0  are the intensities of the given T  2  component or components  acquired with and without an M T pulse, respectively. NNLS was also used to determine T i distributions from T i decay curves. The range of Ti times included 100 points from 1 ms to 10 s. The x  2  from 1-2% to obtain smooth distributions.  value was allowed to increase  Chapter 4  C o m p a r i s o n of M T R s and M y e l i n Water Percentages  4.1  Summary  Magnetization transfer and T relaxation experiments were performed on five white and 2  six grey matter brain structures from ten normal volunteers and nine multiple sclerosis patients. Thirty MS lesions were also analysed. Magnetization transfer ratios and myelin water percentages were compared. Both techniques showed a significant difference between the average of white and grey matter of the normal volunteers as well as the average of normal appearing white matter and grey matter of the multiple sclerosis patients. The average magnetization transfer ratio and myelin water percentage for lesions were significantly lower than those of normal appearing white matter in MS patients. Myelin water percentages and magnetization transfer ratios were uncorrelated in white and grey matter but showed a small (R=0.5,P=0.005) but significant correlation in multiple sclerosis lesions. In summary, the myelin water percentage and the magnetization transfer ratio provide quantifiable but largely independent measures of multiple sclerosis lesion pathology.  4.2  Introduction  In multiple sclerosis (MS), lesions show up as clearly defined bright areas on proton density or T -weighted M R images [51, 54, 56]. 2  Unfortunately, it is not possible to  determine lesion pathology (e.g. edema, demyelination or gliosis) from conventional M R  37  Chapter 4.  Comparison  of MTRs  and Myelin  Water  38  Percentages  images [53]. Two recently developed in vivo M R techniques have the potential to provide more specific information about MS pathology: magnetization transfer (MT) [18] and measurement of the T  2  decay curve [16].  Both techniques provide results which vary  over a wide range for different lesions and each has been hypothesised to be related to myelination [16, 18, 49, 50, 52, 55]. A n early study by Dousset et al. [18] found, in guinea pigs with experimental allergic encephalomyelitis (EAE), that E A E lesions had slightly reduced M T effects and showed no signs of demyelination. They also found, in MS patients, that MS lesions had a very pronounced reduction in magnetization transfer compared to normal white matter. A study showing a more convincing link between M T and demyelination was again done by Dousset et al. [19] who showed histological correlations between M T R s and heavily demyelinated and necrotic lesions in primates injected with lysolecithin.  The myelin  water percentage has also been found to vary for different multiple sclerosis lesions [16]. Stewart et al. [20] showed that, in guinea pigs with E A E lesions, spinal cord samples with histological evidence of demyelination also showed reduction in the amplitude of the short T component. 2  Both M T and T decay curve analyses promise to provide specific information about 2  multiple sclerosis pathology. If the magnetization transfer and the myelin water percentage are both related to myelination, then they should also be related to each other. The goal of this study was to apply both techniques to MS patients and normal volunteers to determine whether the information from these two techniques was correlated.  4.3  Materials and Methods  All experiments were done on a 1.5 T General Electric Signa clinical M R scanner. M T and T  2  relaxation (described in Section 3.4) were performed on ten normal volunteers  Chapter 4.  Comparison  of MTRs  and Myelin  Water Percentages  39  and nine MS patients. ROIs (as outlined in Section 3.4) were drawn on all images. For each ROI, the M T R and myelin water percentage was determined. In the MS patients, 30 lesions were also examined (16 from R / R patients and 14 from SP patients). All statistical comparisons were done using the Student's t-test. A two-tailed test was used to determine significant differences and a one-tailed test was carried out to determine significance when comparing normal to MS results since the hypothesis was that results from MS should be lower than results from normal brain. For comparisons between the means from the five different white matter structures, a single factor anova test was used.  Significance in all cases was indicated by P values less than 0.05.  If  a difference in the mean was found from the anova test, Duncan's multiple range test was applied to the white matter structures to determine which pairs were significantly different.  4.4  Results  4.4.1  Normal  Volunteers  A typical M T R image and myelin map from a normal volunteer are shown in Figure 4.1 (left and centre respectively). The corresponding proton density weighted image is shown in Figure 4.1 (right). In the M T R image and myelin map, white matter regions appear brighter than grey matter regions consistent with the hypotheses that myelin has a larger pool of motionally restricted protons and contains compartmentalised water. Figure 4.2 shows M T R s and myelin water percentages for each structure averaged over the ten normal volunteers (closed circles). The average M T R of white matter was significantly different than grey matter (P<0.00001). The anova test between the five white matter structures showed that the mean M T R s were not significantly different. The average myelin water percentage of white matter was significantly different than  Chapter  4.  Comparison  of MTRs  and Myelin  Water  Percentages  40  Chapter 4.  Comparison of MTRs and Myelin Water Percentages  41  grey matter (P<0.00001). The five white matter structures had significantly different mean myelin water percentages (P<0.00001). Results from Duncan's multiple range test on myelin water percentages are shown in Figure 4.3 with structures written in ascending order and ovals joining structures without significant differences. The ranking of white and grey matter structures according to their M T R or their myelin water percentage was different.  There was no correlation between M T R s and  myelin water percentages in white matter (R=0.126,P=0.39) or grey matter (R=-0.014,P=0.9).  4.4.2  M S Patients  A n M T R image and a myelin map from an MS patient (R/R) are shown in Figure 4.4 (left and centre respectively). The corresponding proton density weighted image. Figure 4.4 (right), is also given where the lesions are clearly seen as bright areas. Figure 4.2 shows the average M T R and myelin water percentage for each structure excluding lesions (open circles). The ranking of the white matter structures between MS and normal was almost the same for T  2  relaxation (two structures swapped) but  not the same for M T (three structures were ranked differently). None of these changes in rank were significant according to the anova and Duncan's multiple range tests. In MS patients, the average M T R of N A W M and the average M T R of grey matter were significantly different (P<0.00001). The mean MTRs between N A W M structures were not significantly different with the anova test. The average myelin water percentage of N A W M was significantly different than grey matter (P<0.00001). The means of the five N A W M myelin water percentages were significantly different (P<0.00001). Results from Duncan's multiple range test on myelin water percentages are shown in Figure 4.3 with structures written in ascending order and ovals joining structures without significant differences. As with the normal volunteers, the rankings of white and grey matter structures  Chapter 4.  Comparison  2018 -  of MTRs  I  and Myelin  Water Percentages  GREY  WHITE  ^ • 0 6 Hi  42  1_  f  c  S  f  6H  <X>  4-  f  2 -  * 8  2019 -  18-  0 K  1  8  0)  H  ^15 14 13  0  12 H  ^  *>« 4? c<*  Figure 4.2: A plot of the average myelin water percentage (top) and the average M T R (bottom) for five white matter and six grey matter structures. Results from both normal volunteers (closed circles) and MS patients (open circles) are shown. Error bars correspond to the standard error.  Chapter 4.  Comparison  of MTRs  and Myelin  Water Percentages  43  Normal  ascending myelin water percentage  Figure 4.3: Duncan's multiple range test for myelin water percentages from normals and MS patients. Ovals join structures with no significant difference.  Figure 4.4: A n M T R image (left), a myelin water map (centre) and a proton-density weighted image (right) from an MS patient.  Chapter 4.  Comparison  of MTRs  and Myelin  Water  44  Percentages  between M T and T relaxation were not consistent. There was no significant correlation 2  between M T R s and myelin water percentages for N A W M (R=0.154,P=0.3) and grey matter (R=0.253,P=0.065).  4.4.3  N o r m a l Volunteers vs M S Patients  The results from normal volunteers and MS patients were compared. For all white matter structures, N A W M had a lower M T R and lower myelin water percentage than normal white matter. The probability of either of these occurring by chance is only 3%. A onetailed t-test was used to determine whether M T R s or myelin water percentages from MS structures were significantly lower than normal structures. Structures with significant differences are indicated by a star on Figure 4.2 and include the minor forceps for both M T R and myelin water percentage and the splenium and genu of the corpus callosum for M T R .  4.4.4  Lesions  For this study, the most important comparison is that between M T R and myelin water percentages in lesions.  Values are listed in Table 4.1 and plotted in Figure 4.5. The  average values of normal white matter, N A W M and grey matter are shown for comparison with lesion values.  There was a significant correlation between myelin water  percentage and M T R but it was surprisingly moderate (R=0.5,P=0.005). The M T R in lesions was slightly more correlated with the geometric mean T  2  tribution (R=-0.59,P=0.0005) and with the geometric mean of T (R=-0.54,P=0.001) than with the myelin water percentages.  of the entire T 2  2  dis-  between 50-900 ms  Chapter 4. Comparison  of MTRs  Structure R/RMS SP MS Average lesions Average White Average Grey  and Myelin  45  Water Percentages  Myelin Water (%) normal MS 3.8(0.9) 5.6(0.6) 4.6(0.5) 11.8(0.7) 10.6(0.7) 3.0(0.3) 3.1(0.3)  . M T R (%) normal MS 7.7(1.2) 13.3(0.8) 10.3(0.9) 18.9(0.2) 17.6(0.3) 13.1(0.2) 13.0(6.3)  Table 4.1: Comparison of results for myelin water percentages and magnetization transfer ratios from MS lesions. (Average white and grey are included for reference.) Numbers in parentheses are standard errors.  T  MTR (%)  15  20  Figure 4.5: A plot of the myelin water percentage versus the M T R for MS lesions. Lesions from patients with relapsing-remitting MS (closed circles) and from patients with chronicprogressive MS (open circles) are differentiated.  Chapter 4.  4.5  Comparison  of MTRs  and Myelin  Water  Percentages  46  Discussion  The main goal of this study was to compare M T R values and myelin water percentages in . brain of normal volunteers and MS patients. Both methods claim to be related to myelin content and therefore one might expect the results to be correlated with one another.  4.5.1  M T R Values  Brain M T measurements in the literature [18, 49, 50, 52, 55, 69, 70, 71, 72, 73] have been acquired in a variety of ways so that absolute M T R values vary considerably from study to study. Because gradient echo sequences use much less rf power to produce an image, the incremental effect of an M T pulse is more potent for gradient echo M T sequences than spin echo M T sequences. Consequently, gradient echo M T sequences produce much larger M T R values for normal white matter. However, a recent abstract by Berry et al. [74] showed that M T R s for normal white matter were linearly related to the effective flip angle (FAsat) divided by the repetition time between saturation pulses (Tsat). Our M T R values for normal white matter, which were obtained with a ratio of FAsat/Tsat equal to 10.6, fell onto Berry's linear relation within experimental error. Therefore, although our spin echo M T sequence yielded smaller white matter M T R values than many other studies, especially those obtained with gradient echo M T sequences, our results should be representative of all M T studies.  4.5.2  C o m p a r i s o n of O u r M T R s and M y e l i n Water Percentages w i t h O t h e r Studies  Several groups have reported M T R values from different normal white matter regions [50, 52, 55, 69, 71, 72]. The ranking of their white matter structures was the same as ours except in one case. Although, Mehta et al. [72], like us, found that the genu of the  Chapter 4.  Comparison  of MTRs  and Myelin  Water Percentages  47  corpus callosum had the largest M T R , the other structures were ranked differently. This group was the only one to use an on-resonance binomial M T pulse which may account for the difference in ranking. Three groups compared normal white matter to grey matter and found a significant difference in the average M T R [18, 50, 72] consistent with our results. In MS patients, three groups found that the average M T R of N A W M was lower than normal white matter [18, 49, 52], as did our group, but one group did not find it significant [50]. Several groups examined MS lesions and all found that the average M T R of lesions was significantly lower than N A W M [18, 49, 50, 70], as was found in our experiments. There was a wide range of M T R values for the lesions. These lesions were further separated into lesions from patients with relapsing-remitting and secondary progressive'MS. Three of the groups did not find a significant difference in the M T R between the two MS classes [49, 50, 70] but one group found that lesions from R / R patients had a higher M T R than lesions from SP patients [18]. This result is the opposite of ours; we found that lesions from SP patients had a higher M T R than lesions from patients with R / R MS. This finding could be due to the large variation in types and ages of lesions found within a single MS patient. In SP MS, lesions are expected to be older than lesions from R / R patients. Tomiak et dl. [73] found that lesions older than one year had a larger M T R than younger lesions. They attributed this result to older lesions being mainly gliotic and therefore having less water to exchange with the motionally restricted protons. The findings in that study would support a larger M T R in lesions from SP patients as opposed to R / R patients. Only one other systematic study of myelin water percentages has been carried out with normal volunteers. In our earlier study [17], the average myelin water percentage was 11.3% in white matter and 3.1% in grey matter for normal volunteers. These values were not significantly different than our present results even though the previous study was carried out on a different scanner and employed a different slice thickness (10 mm).  Chapter 4.  Comparison  of MTRs  and Myelin  Water Percentages  48  The ranking of the white matter structures was the same as this study except for the major forceps.  In the previous study, the optic radiation was included in the major  forceps but, in this study, since the slice chosen for most MS patients did not include the optic radiation, it was not included. Grey matter structures were ranked slightly differently probably due to the small myelin water percentage in grey matter being more affected by the noise.  4.5.3  O t h e r Comparisons of M T and T  2  Results  In a recent study of maturing brains [75], a qualitative inverse correlation was found between T and M T R for brains at different myelination stages. The mono-exponential 2  T  2  measured in that experiment was estimated from a dual echo pulse sequence and  therefore cannot be directly related to the T times or myelin water percentages reported 2  in this study. A comparison between T times measured from a 16 echo sequence and M T R s in le2  sions from patients with optic neuritis showed a significant correlation (r=-0.413, P<0.05) [40] which is similar to our results.  4.6  Concluding Remarks  In a clinical setting, the M T sequence has advantages over T  2  measurements since it  is easy to implement in a multi-slice fashion and has shorter acquisition times. The T  2  relaxation was measured on a single slice because a multi-slice study would have corrupted the T decay curves with M T effects due to off-resonance irradiation from other excited 2  slices.  Chapter 4.  Comparison  of MTRs  and Myelin  49  Water Percentages  T relaxation results show two distinct water compartments within the mobile compo2  nent of normal white matter. This finding suggests that the two-pool model normally employed for characterising M T is inadequate; a three pool model (macromolecules, myelin water and extracellular/intracellular water) would be more representative. Magnetization transfer is expected to be stronger for myelin water than the intracellular/extracellular water. Furthermore, direct saturation effects, which become more prominent as the M T offset frequency is moved closer to resonance, should be different for the two mobile pools since, they have different linewidths. Further studies are required to determine how M T and T relaxation results relate to 2  brain pathology. If both techniques are indeed primarily measuring myelin content, then we would expect the M T R s and myelin water percentages to be strongly correlated. This study demonstrates that the relationship is much more complex since, in MS lesions, only about one fifth of the variance in the myelin water percentage was accounted for by the M T R . We found a slightly higher correlation between M T R and geometric mean T than 2  with the myelin water percentage. If the myelin water percentage is indeed a measure of myelin content, then the M T R must also be dependent on other aspects of the tissue. In summary, the M T R and myelin water percentage provide quantifiable but largely independent measures' of MS lesion pathology. More investigation is required before we can reliably relate these two parameters to the actual pathological changes.  Chapter 5  M T Effects on the Short and L o n g T  5.1  2  Relaxation Components of B r a i n  Summary  A T relaxation sequence was modified by including a preparatory magnetization transfer 2  pulse in order to determine the M T effect on each of the two T components separately. 2  Two types of M T pulses were examined: an on-resonance binomial pulse and an offresonance sine pulse. The effect of varying the offset frequency of the M T sine pulse was also determined. The time between the sine M T prepulse and the initiation of the T  2  relaxation sequence was increased and the effect on each T component was seen. The 2  M T effect was significantly larger on the short T component than the long T component 2  2  (P<0.00001 to P<0.04). This differential M T effect was removed at time delays of greater than 500 ms. The binomial M T pulse and small frequency offsets caused substantial direct saturation which preferentially affected the short T component. In summary, M T affects 2  the short T component in brain more than the long T component except at long delays 2  2  between the M T pulse and the initiation of the T relaxation sequence. 2  5.2  Introduction  Most models of M T in the brain have included only two pools of protons [47, 72, 76, 77, 78, 79], motionally restricted and mobile. The "bulk" water (mobile) interacts with the macromolecular matrix (motionally restricted) through some water surface layer but all the water has equal access to the matrix. From T relaxation, it is clear that the 2  50  Chapter 5. MT Effects on the Short and Long T Relaxation 2  Components  of Brain  51  mobile pool can be further separated into two distinct water pools. Therefore, a model including at least 3 proton pools should be used to better understand the M T signal [80]. Exchange would take place between the two water pools and between the myelin water and the motionally restricted pool (consisting mainly of myelin). In normal M T sequences, the signal from the two tissue water pools is not separable and therefore some sort of average M T is measured. However, a 32 echo sequence preceded by an M T pulse is able to distinguish the M T effect on each of the T components. 2  The goal of this study was to look for differences in magnetization transfer effects between the two brain tissue water pools. Since the microscopic local environment for water between myelin bilayers is very different from that of intra and extracellular water, one might expect the two pools to exhibit different M T effects. The experimental protocol consisted of a preparatory M T pulse applied before a 32 echo T relaxation measurement 2  sequence. Two types of M T pulses were investigated: namely an on-resonance binomial pulse and a 19 ms off-resonance sine pulse. With the sine pulse, the effect of changing the offset frequency of the M T pulse was investigated. Also, the effect of varying the delay between the end of the sine M T pulse and initiation of the multiecho T measurement 2  sequence was examined. M n C l doped water tubes with T s of 20 and 80 ms were also 2  2  placed within the image. The choice of water T was in order to mimic the previously 2  measured T of the different water pools in the brain [16, 17] namely myelin water and 2  intra/extracellular water.  5.3  Material and Methods  MRI measurements were done on a 1.5 T General Electric Signa clinical M R scanner operating at the 5.4 software level. Four M n C l doped water tubes (two with T 80 ms 2  and two with T 20 ms) were placed next to the head of each volunteer. 2  2  Chapter 5. MT Effects on the Short and Long T Relaxation 2  Components of Brain  52  Relaxation decay curves for all experiments were fit to a 4 T model using an NNLS 2  algorithm as detailed in Section 3.4. The relaxation times were then separated into the short T component ( T =20 ms) and the long T component ( T = 80 + 120 ms). In 2  2  2  2  the case of the reproducibility data, decay curves were also fit using the modified NNLS algorithm with a 0.1-0.3% increase in x • 2  All statistical comparisons were done using the two-tailed Student's t-test. Probabilities less than 0:05 were considered significant. If a structure had no short T component 2  when no M T pulse was present, then the person was not included in any average for that structure or the t-test calculation. Also, if an M T R resulted in a negative value (likely due to noise), the M T R was set to zero and included in the average. This occurred in one white matter structure for two different volunteers using the sine M T pulse. In the experiment with a varied frequency offset, this occurred in the minor forceps for one volunteer (at 500, 750 and 1000 Hz) and in the genu (at 750 Hz). In the experiment where a delay was put between the M T pulse and the 32 echo sequence, negative M T R s were found in the splenium and minor forceps for delays > 500 ms. These negative M T R values ranged from -1.2 to -55 with an average of -12.6. Five different experiments using the T relaxation with M T sequence were done and 2  described below.  5.3.1  Reproducibility  Seven experiments with a sine M T pulse were repeated at the same sitting on one volunteer as well as one case with no M T . Each white matter structure was analysed using the normal 0.1-0.3% smoothing NNLS algorithm and the 4 T method described in chapter 3. 2  This was used to test whether the 4 T method was appropriate for relaxation analysis. 2  Chapter 5. MT Effects on the Short and Long T Relaxation 2  5.3.2  Components  of Brain  53  B i n o m i a l M T pulse  Fifteen normal volunteers (age 25-48) were studied using a binomial M T pulse followed by a 32 echo train. Experiments were done in pairs, without and with M T . M T R s were then determined for both the short and long T  2  components. Only one volunteer was  scanned with the four water tubes.  5.3.3  Sine M T pulse  Eleven normal volunteers (age 23-48) were studied using a sine M T pulse followed by a 32 echo train. The delay between the end of the sine M T pulse and the initiation of the 32 echo train was 18.5 ms and the frequency offset was 2000 Hz. Experiments were done in pairs, without and with M T . M T R s were then determined for both the short and long T  2  components.  5.3.4  M T Frequency Offset  A total of three scans were done on two normal volunteers (i.e. one volunteer was done twice) using a range of off-resonance offsets for the sine M T pulse. Offsets used were 2000, 1000, 750, 500 and 300 Hz. Only one of the scans included the 300 Hz experiment while only-the other two scans included the 750 Hz experiment. Again, an experiment with no M T was done in order to allow the calculation of M T R s at each offset.  5.3.5  Delay Between M T Pulse and 32 Echo Sequence  One normal volunteer was studied twice using a protocol that varied the time between the end of the sine M T pulse and the initiation of the 32 echo sequence, called the M T delay. Experiments consisted of a sequence with no M T and five other sequences with delays of 18.5, 218.5, 518.5, 1018.5 and 2018.5 ms. M T R s were determined for each delay.  Chapter 5. MT Effects on the Short and Long T Relaxation 2  Short T . Smooth 4 T model 2  Structure  2  195(19) 84(31) 196(25) 94.2(7.5) 181(15)  Genu Splenium Minor Forceps Major Forceps Internal Capsules  237.1(3.6) 100(25) 206.2(6.8) 106.5(9.5) 167(10)  Components of Brain  54  Long T Smooth 4 T model 939.7(1.8) 959(20) 995.7(4.0) 1028(20) 989.3(2.6) 969(25) 1015.4(4.3) 999.0(2.8) 980(21) 985.6(3.9) 2  2  Table 5.1: Amplitudes of the short and long T components for the smooth and 4 T model distributions in normal white matter. Standard errors are given in parentheses. 2  5.4  2  Results  5.4.1  Reproducibility  Seven experiments were repeated on one volunteer in order to determine the reproducibility. The amplitudes of the short and long T  components are shown in Figure 5.1 and  2  Table 5.1 comparing the normal 0.1-0.3% smoothing NNLS solution and the 4 T method. 2  In all cases except for the splenium, the 4 T  2  analysis resulted in a smaller amplitude  range. In the case of the splenium, movement between scans is thought to have affected the results thereby leading to unreliable comparisons. The amplitude resulting from the 4 T  2  model was always near the middle of the range of amplitudes for the smooth analysis.  All other experiments were analysed using the 4 T model. 2  A typical example of both T distributions determined with and without M T is shown 2  in Figure 5.2. Two cases are shown: one using a smooth NNLS solution (smooth curves) and the other using the 4 T fit to the same data (spikes). The solid lines (labelled no 2  M T ) refer to the 32 echo experiment without an M T pulse present and the dotted lines (labelled M T ) refer to the 32 echo with M T experiment. The M T axis has been shifted in order to make them visible beneath the no M T lines. A decrease in intensity can be seen between the no M T and M T distributions.  Chapter 5. MT Effects on the Short and Long T Relaxation 2  Components of Brain  55  1200 H  Splenium  Genu  Major F  Minor F  Int Caps  Figure 5.1: The difference between the smooth and 4 T model is shown for one no M T (solid symbol) and seven M T (open symbol) cases from white matter of one volunteer. Data in the lower plot comes from the short T component and data from the upper plot comes from the long T component. Note the change in scale along the vertical axis between the two plots. 2  2  2  Chapter 5. MT Effects on the Short and Long T Relaxation 2  600-  Components  of Brain  56  no MT MT  500 -  o  1  1 2  Ik,  i 3  ft  i i i i i i I 4 5 6 7 8 9  - i — 2  0.1  1 3  1—i—I 4 5 6  i i i I 7 8 9  1  r 2  T (s) 2  Figure 5.2: Example" of T distributions without and with an M T pulse. The curved solutions were obtained by fitting the decay curves using the modified smooth NNLS algorithm. The spiked solutions were, obtained by restricting the allowed T values to 4. . The no M T curves refer to 32 echo experiments without an M T pulse and the M T curves are experiments with an M T pulse. Note the axis of the M T 4 T solution was shifted slightly to the right to make it visible under the no M T solution. 2  2  2  Chapter 5. MT Effects on the Short and Long T Relaxation 2  Binomial  Sine  Structure Genu Splenium Minor Forceps Major Forceps Internal Capsules Average W h i t e 80 ms tube 20 ms tube Genu Splenium Minor Forceps Major Forceps Internal Capsules Average W h i t e 80 ms tube 20 ms tube  Components of Brain  57  Average M T R (%) P value Total Long T Short T 16.1(1.3) . 0.0001 14.2(1.4) 25.0(1.9) 19.8(1.1) 0,00001 16.7(1.1) 35.5(2.7) 12.5(1.4) 0.04 16.8(2.2) 12.2(1.3) 13.0(1.2) 15.6(1.2) 0.00001 34.7(1.6) 0.00001 14.1(1.2) 20.0(1.3) 45.4(3.4) 0.00001 16.9(1.4) 14.0(1.3) 31.5(3.5) 37.4 40.3 0.005 12.0(0.6) 8.7(1.0) 22.2(2.9) 0.024 7.7(1.0) 10.2(0.9) 24.4(5.4) 0.007 10.4(0.6) 7.4(0.8) 22.4(3.6) 0.032 9.9(0.6) 7.9(0.7) 23.6(5.4) 10.0(0.6) 0.005 6.6(0.9) 23.6(4.0) 0.001 7.8(0.8) 10.6(0.6) 23.1(2.9) 2.4(0.8) • 48(0.7) 2  2  Table 5.2: Average M T R s of white matter structures for 15 volunteers using binomial (top) and sine (bottom) M T pulses (offset—2000 Hz and delay=18.5 ms). Decay curves were analysed using the 4 T model. P values refer to a comparison between short and long T components. Numbers in parentheses are standard.errors. 2  2  5.4.2  B i n o m i a l M T pulse  Fifteen normal volunteers were examined using the binomial M T pulse sequence. The average M T R for the short and long T components and the average total M T R is shown 2  in the top half of Table 5.2 for each structure.  P values comparing the M T R s from  short and long T components are also given. The M T R of the short T 2  2  component  was significantly higher than the M T R of the long T component for all white matter 2  structures.  The signal from the 20 ms and 80 ms water tubes was lower due to the  binomial pulse with the 20 ms water tube showing a larger M T R .  Chapter 5. MT Effects on the Short and Long T Relaxation 2  5.4.3  Components of Brain  58  Sine M T pulse  In this experiment, eleven normal volunteers were scanned with the sine M T pulse sequence. The average M T R for the short and long T components and the average total 2  M T R is shown in the bottom half of Table 5.2 for each structure. P values comparing the M T R s from short and long T components are also given. The M T R of the short 2  T  2  component was significantly higher than the M T R of the long T component for all 2  white matter structures. The average M T R of the 20 ms water tube was significantly higher than that of the 80 ms water tube although the effect was small in both.  5.4.4  M T Frequency  Offset  The effect on each of the T pulse was determined.  2  components of varying the offset frequency of the M T  Results for both short and long T  2  components are shown in  Table 5.3 for each white matter structure and in Figure 5.3 for the average over the five structures. At all offset frequencies, the M T R for the short T  2  component (excluding  the minor forceps at 750 Hz) was larger than that for the long T component. As the 2  offset frequency was decreased to 300 Hz, the absolute increase in M T R for the short T  2  component became much larger than that for the longer component but the fractional increase was about the same. The reason for the dip in M T R at 750 Hz is not understood. The increase in calculated M T R with decreasing offset frequency was much larger for the 20 ms T water tube than for the 80 ms water tube. 2  5.4.5  D e l a y B e t w e e n M T P u l s e a n d 32 E c h o  Sequence  The length of the delay between the sine M T pulse and the initiation of the 32 echo sequence was varied from 18 ms to 2 s in five steps. Results for both the short and long T  2  components for each white matter structure are shown in Table 5.4.  For both T  2  Chapter 5. MT Effects on the Short and Long T Relaxation 2  Component Short  Long  Total  Structure Genu Splenium Minor Forceps Major Forceps Internal Capsules Avg White water tubes Genu Splenium Minor Forceps Major Forceps Internal Capsules Avg White water tubes Genu Splenium Minor Forceps Major Forceps Internal Capsules Avg White  300 33.2 53.0 36.5 55.4 43.3 44.3 20.4 13.9 11.0 11.3 8.9 5.7 10.2 8.3 18.5 17.5 16.9 15.1 15.2 16.6  Components of Brain  offset (Hz) 500 750 1000 31.4 18.7 7.6 43.4 44.9 58.6 23.4 10.8 19.3 43.4 ' 35.6 35.6 32.2 40.6 27.9 39.5 25.9 30.1 11.1 23.7 13.7 11.9 16.5 16.1 12.4 10.4 14.5 13.4 18.3 19.0 11.1 14.2 11.2 12.1 11.9 9.8 15.1 14.1 11.3 5.4 8.5 5.5 19.8 21.5 17.9 18.2 18.6 19.2  14.0 17.4 11.7 14.5 15.4 14.6  13.8 15.9 13.9 14.4 14.9 14.6  59  2000 19.7 41.1 23.8 27.2 17.4 25.8 3.2 8.2 6.4 7,8 7.7 7.2 7.5 2.6 10.9 11.9 10.8 10.3 9.7 10.7  Table 5.3: Average M T R s of white and grey matter structures using different offresonance offsets. Short and long component water tube M T R s are for the 20 ms and 80 ms tubes respectively.  Chapter 5. MT Effects on the Short and Long T Relaxation 2  Components  of Brain  60  50 - T  500  1000  1500  2000  F r e q u e n c y offset (Hz)  Figure 5.3: A plot of the average M T R of the short and long T components in white matter as a function of off-resonance offset. The data were collected from three scans of two volunteers. 2  Chapter 5. MT Effects on the Short and Long T Relaxation 2  Component Short  Structure Genu Splenium Minor Forceps Major Forceps Internal Capsules Avg  Long  White  water tubes Genu Splenium Minor Forceps Major Forceps Internal Capsules Avg  White  delay (ms) 518.5 1018.5 16.2 12.0 6.3 0.0 3.5 6.5 2.8 1.4 6.6 6.8  61  2018.5 7.7 10.8 1.3 4.6 14.2  18.5 18.1 21.5 14.4 17.7 18.3  218.5 21.3 26.4 14.4 16.6 25.2  18.0  20.8  7.7  4.7  7.7  4.2 9.4 8.2 9.0 9.1 8.4  2.9 12.5 11.6 12.5 12.1 10.5  1.9 9.2 10.8 9.6 10.1 9.4  1.3 5.0 6.9 6.0 7.1 4.9  2.9 2.2 4.4 3.3 ,5.2 1.9  8.8  11.8  9.8  6.0  3.4  1.8 11.1 10.4 10.1 10.2 10.4  2.2 14.3 14.1 12.9 12.7 13.5  4.4 10.6 9.6 9.0 8.8 8.8  4.9 6.4 5.4 4.8 5.8 4.7  5.1 3.1 5.4 2.6 3.8 4.4.  10.5  13.5  9.4  5.4  3.9  water tubes Genu Splenium Minor Forceps Major Forceps Internal Capsules Avg  Total  White  Components of Brain  Table 5.4: Average M T R s for the short and long T component in brain using different M T delays. Short and long components water tube M T R s are for the 20 ms and 80 ms tubes respectively. 2  components, the M T R at short delays was larger than that at longer delays. For delays of 200 ms or less, the M T R of the short T component was much larger than the M T R 2  of the long T component. For delays of 500 ms or greater, the M T R values of both T 2  2  components were similar. Figure 5.4 shows the average M T R for' both T components in 2  white matter as a function of the delay. Neither water tube showed appreciable change in M T R although the 80 ms water tube, for unknown reasons, had an increased M T R at longer delay times.  Chapter 5. MT Effects on the Short and Long T Relaxation 2  T — ; 0  ~i 500  —\ 1000 MT delay (ms)  Components of Brain  62  r~  1  1500  2000  Figure 5.4: A plot of the average M T R of the short and long T component in white matter as a" function of M T delay. The data were collected from two scans of the same normal volunteer. 2  Chapter 5. MT Effects on the Short and Long T Relaxation  Components of Brain  2  5.5  63  Discussion  The conventional method of analysing T relaxation data was not employed because the 2  amplitudes for the two T components were not stable probably due to a low signal to 2  noise ratio. Instead, a model which incorporated only 4 T  2  times at 20, 80, 120 and  2000 ms was fit to the decay curves. These values were chosen to correspond to T  2  values  measured in normal brain. The amplitudes associated with each T component had much 2  less variance than the smooth model and the values were centred within the amplitude range of the smooth solutions. It is important to note that although this model is helpful in normal human brain, disease can cause changes in relaxation time and this model would no longer be valid. Since it is impossible to separate the contributions to the M T R from each of the two tissue water pools in a conventional M T sequence, the measured M T R is a weighted average over the short and long T proton signal components. By combining the prepa2  ration M T pulse with a 32 echo sequence, this study separately measured the M T R for each water pool. The M T R of the short T  2  component was found to be significantly  larger (P<0.00001 to 0.04) than that of the long T  2  component for all white matter  structures and for both types of M T preparation pulses. A larger M T R is an indication of closer association between water and motionally restricted non-aqueous protons such as lipids and proteins. Therefore, the differential M T R effects observed here support the assignments of the short T  2  component to water trapped between myelin bilayers and  the longer T component to water from intra and extracellular water. 2  Another possible difference between the two water pools is the amount of direct saturation due to an M T pulse. Direct saturation is a decrease in magnetization of the water pool due to direct absorption from the M T pulse and not as a result of transfer of magnetization between proton pools. In an ideal experiment, there should be no direct  Chapter 5. MT Effects on the Short and Long T Relaxation 2  Components of Brain  64  saturation but this is never the case. The two doped water tubes were included in the image slice in order to look for direct saturation effects in the M T experiments. One would expect direct saturation effects in tissue to be similar to those in water phantoms possessing similar T and T times. The 20 ms T tube (Ti =800 ms) represented the x  2  2  short T component observed in brain and the 80 ms T (Ti =200 ms) tube represented 2  2  the long T component. We note that when the offset frequency of the M T pulse was 2  decreased, the 20 ms component was affected much more than the 80 ms component. Similar direct saturation effects are expected for the short and long T components in 2  vivo. Comparison between the M T R results from binomial and sine M T pulses show an interesting difference. With the binomial M T pulse, the M T R s for the short T compo2  nent varied among the different white matter structures. On the other hand, the M T R values obtained with the sine M T pulse did not show this variance. The M T R for the long T component showed no significant variation for either type of M T pulse. The 2  ranking of binomial pulse M T R s for the short T component coincide with the ranking of 2  the myelin water percentages in Chapter 4 for different white matter structures. These different behaviours with the two M T pulses is tentatively attributed to direct saturation. The binomial M T pulse produces a very large amount of direct saturation as measured from the water standards (Table 5.2) and this direct saturation preferentially affects the myelin water. This direct saturation effect is accentuated at higher myelin water contents. Since the measured M T R is a combination of the M T effect and direct saturation, different white matter structures will have different MTRs. The sine M T pulse did not produce much direct saturation and therefore the measured M T R values were simply due to M T which appeared to be uniform among different white matter structures. This similarity between white matter M T R s was also found in Chapter 4 where a sine pulse was used.  Chapter 5. MT Effects on the Short and Long T Relaxation 2  Components of Brain  65  Recently, many studies have employed off resonance M T pulses with the off-resonance M T pulse moved to smaller frequency offsets since this leads to larger M T R s [81, 82]. This could also cause more direct saturation of the mobile pool. This was found experimentally and in particular, water with a short T was preferentially saturated at small offset values 2  (water tubes in Table 5.3). Therefore, in brain white matter, the myelin water will be more affected by small offsets than intra/extracellular water. For clinical applications this effect may be desirable, however it should be appreciated that the measured M T R at small offsets will be a combination of M T and the different direct saturations of the water. In the previous chapter, the offset used for measurement of M T R s was 2 kHz and therefore a negligible amount of direct saturation was expected. Only one study has previously combined a C P M G sequence with an M T pulse and it was carried out on bovine white matter [80]. Although they found two T  2  components  in white matter, they found no difference in their M T effect. This result seems to conflict with those of the present study, however the experimental approaches of the two studies were different. The M T prepulse used in the bovine brain study was a 7 s long rectangular pulse designed to produce complete saturation of the motionally restricted proton pool. During this 7 s, the two water T components would have had ample time 2  to exchange magnetization and thereby average the overall M T effect. Hence, a common average M T R was measured for the two components. In our experiment, the two 19 ms sine pulses (or the six 1 ms binomial M T pulses) did not accomplish complete saturation of the motionally restricted protons and the 32 echo T measurement sequence was 2  applied before the two tissue water pools in white matter had sufficient time to come to equilibrium. This hypothesis is further supported by the M T delay experiment. At shorter delays, M T R s were different for the short and long T  2  components.  At longer  delays, the M T R of each T component was essentially the same indicating that mixing 2  of the two pools had averaged out the magnetization.  Chapter 5. MT Effects on the Short and Long T Relaxation 2  Components  of Brain  66  The M T experiment in white matter involves several exchange processes: a separate exchange of magnetization between motionally restricted protons and water in each of the two tissue water pools, and exchange of magnetization between the two tissue water pools. This work enables us to establish crude limits for these exchange times. Exchange between motionally restricted protons and water is believed to occur rapidly since a magnetization transfer effect can be measured after a delay of about 20 ms from the end of the sine M T pulse. Exchange between the tissue water pools is much slower. The timescale for exchange between the two water pools must be long on a T timescale (i.e. 2  greater than 100 ms) since we observe the separate T relaxation components for each 2  pool. However, exchange must be fast on a T i timescale (less than 700 ms) since only one relaxation component is found. Also, the differential M T effect between the short and long T components disappears after about 500 ms. 2  5.6  Concluding  Remarks  In all cases (except for long M T delays of 1-2 s), the M T R for the short T component 2  was significantly larger than the M T R for the long T component. This was expected 2  due to the closer interaction between myelin water and myelin than between non-myelin and intra/ extracellular water.  The introduction of a delay between the M T and 32  echo pulse sequences allowed the magnetization of the two water pools to equilibrate and only one M T R was found. This occurred by 500 ms of delay showing that the two water pools exchange in a time faster than 500 ms but slower than 100 ms (the T of 2  infra/extracellular water). These results show the importance of using a model with more than 2 proton pools when trying to explain M T in brain. These results also demonstrate the difference between using a binomial and a sine M T pulse to produce saturation. With binomial pulses, white matter structures are found to have different M T R s which  Chapter 5. MT Effects on the Short and Long Ti Relaxation  Components of Brain  67  correspond to the same ranking as myelin water percentages. Therefore, binomial pulses, which produce both M T and direct saturation, may be more sensitive to differences in myelin water content.  Small frequency offsets for the sine M T pulse also produce a  differential M T effect on the two water pools which is likely due to a combination of M T and direct saturation. Exploitation of this effect may improve contrast between myelin water and intra/extracellular water which in turn may help in visualising pathology such as multiple sclerosis lesions.  Chapter 6  R e l a x a t i o n Measurements of Bovine B r a i n using M a g n e t i c Resonance  6.1  Summary  In vitro relaxation times were measured from white and grey matter samples excised from three different bovine brains. T  2  relaxation distributions showed 4 peaks which  were attributed to buffer solution ( T >500 ms), intra/extracellular water ( T between 2  2  50 ms and 500 ms) and 2 to myelin water ( T <50 ms). The myelin water percentage 2  was larger for white matter ( a v e r a g e = 1 4 . 3 ± 0 . 9 ) than grey matter ( a v e r a g e = 4 . 7 ± 0 . 6 ) . T i relaxation of the solid signal was mono-exponential for grey matter but bi-exponential for white matter. A T i - T dependent measurement showed that there was no cross relaxation 2  between the different water pools as distinguished by T i and T . Cross relaxation times 2  in fully hydrated white matter were measured to be 193 ms between intra/extracellular water and non-myelin molecules and 66 ms between myelin water and myelin. These experiments showed that a 4-pool model with cross relaxation between the water and macromolecules but no exchange between the two water pools (within 1 s) is appropriate when describing bovine white matter.  In grey matter, the cross relaxation time was  measured to be about 200 ms between the intra/extracellular water pool and the nonmyelin molecules.  68  Chapter 6. Relaxation  6.2  Measurements of Bovine Brain using Magnetic Resonance  69  Introduction  Many biological systems have been examined using H nuclear magnetic resonance (NMR) 1  and in particular spin-spin (T ) and spin-lattice (Ti) relaxation. These techniques al2  low the differentiation of different proton pools. Previous in vivo T relaxation studies 2  [16, 17] have shown that water in human brain has three different relaxation times which are associated with three different water compartments: cerebrospinal fluid ( T  2  >ls),  intracellular and extracellular water ( T between 80 and 120 ms) and water tightly asso2  ciated with myelin bilayers ( T < 50 ms). In vivo measurements on human white matter 2  have shown mono-exponential T i relaxation [17]. In vitro experiments are advantageous since, signal from both the motionally restricted and mobile protons can be separated.  Previous studies on guinea pig brain [20] and  crayfish abdominal nerve cord [22] showed multicomponent T relaxation again associated 2  with different water pools. This study examined bovine white and grey matter using a wide variety of N M R pulse sequences. Mobile and solid fractions were determined as well as the second moment of the solid signal. T i and T relaxation times were measured and,, for white matter, their 2  interdependence.  Cross relaxation times between different water and macromolecular  pools were found. Finally, N M R properties were studied as a function of tissue hydration. The purpose of this study was to characterise the behaviour of water within the brain, to separate the H - N M R signal into different solid and mobile (water) components and 1  to determine interactions between the solid and mobile protons, as well as the mixing of different water pools.  Chapter 6. Relaxation  6.3  Measurements  of Bovine Brain using Magnetic Resonance  70  Material and Methods  6.3.1  Samples  Samples from three bovine brains were used as indicated in Section 3.1.1. Their composition is shown in Table 3.1  6.3.2  N M R  Experiments  A variety of N M R experiments were conducted on the brain samples. These included FID, IR, S A T , cross relaxation, T 1 - T 2 relaxation and cross-T relaxation experiments. 2  The repetition time, T R , for all experiments was 7 s unless otherwise stated. Sample 1-2 and 1-5 were measured at 37°C using free induction decay, C P M G and inversion recovery experiments (described in Section 3.3). All.other experiments were carried out at 24°C. The integrity of the samples was monitored by repeating the free induction decay and C P M G sequences at least once per day. No changes were seen during the course of the experiments (less than 1 week). Standard errors are presented in parentheses.  6.4  Results  6.4.1  Free Induction  Decay  The FID was collected for all samples. shown in Figure 6.1.  Typical FIDs for white and grey matter are  The top two curves represent fully hydrated samples and the  lower two curves represent dehydrated samples. The total mobile and solid signals and the second moment were determined from the moment expansion. The average second moment for white matter was 1.86(0.15) x 10 s~ and for grey matter 1.99(0.13) x 10 9  s  - 2  2  9  at an ambient temperature of 24°C. This was slightly larger than the second moments  measured from the white and grey samples at 37°C,, 1.31 x 10 s 9  - 2  and 1.38 x 10  9  s  - 2  Chapter 6. Relaxation  Measurements of Bovine Brain using Magnetic Resonance  71  respectively. The second moment at different moisture contents for both white and grey matter samples is shown in Figure 6.2. From the initial rapid decay of the FID, the solid signal fraction was found to be significantly larger in white matter than grey matter. The average water content of fully hydrated white matter was 72.1(1.1)% and grey matter 84.4(0.9)% as measured from the weight at full hydration and at complete dehydration. From the changes in mass and signal intensity during the hydration study, the average proton density for the solid was found to be 0.0811(0.007) gH/gSolid for white matter and 0.0578(0.018) gH/gSolid for grey matter.  200  400  600  Time (us)  800  1000  Figure 6.1: Free induction decays for white (solid) and grey (dashed) matter samples. The upper two curves are for fully hydrated samples (71.0% for white and 83.8% for grey) and the lower two curves are for dehydrated samples (0.3% for white and 0.4% for grey). Note that only the first four echoes are included in order to better resolve the initial rapid decay.  Chapter 6. Relaxation Measurements of Bovine Brain using Magnetic Resonance  4.5  -  O  white  •  grey  72  O 4.0 •4—*  o  o 3.5 - | o ? 3.0  c o  H  o  o  2.0  n—: 20  o o 1 40  Moisture Content (%)  T 60  ,O CD 80  Figure 6.2: A plot of second moment (M' ) versus moisture content for white matter (open circles) and grey matter (closed circles). 2  Chapter 6. Relaxation  6.4.2  Measurements  of Bovine Brain using Magnetic Resonance  73  CPMG  Typical T distributions calculated from the r=200m's decay curves for white and grey 2  matter samples are shown in Figure 6.3. The results of fully hydrated samples consistently showed four or five peaks with T ranges of 2-8, 10-70, 80-120 and >120 ms. T 2  2  distributions at r =200 and 400 us showed similar behaviour whereas at r =100 ps, the distribution was much broader (Figure 6.4).  The myelin water percentages calculated  for each sample are shown in Table 6.1. The myelin water percentages for fully hydrated samples were significantly higher in white matter than grey matter (P<0.00002). As the tissue was dehydrated, the T of the peaks shifted to lower values and the intensity of all 2  the peaks decreased. Dehydration of white and grey matter samples is shown in Figure 6.5.  •  10000-1  '-]  Figure 6.3: Typical T distributions for white (solid) and grey (dashed) samples at full hydration. 2  Chapter 6. Relaxation  Measurements  of Bovine Brain using Magnetic  Resonance  74  Figure 6.4: T distributions for a hydrated white matter sample at 3 different r values. The r =100 ys curve was only smoothed by 0.1-0.3% whereas the r =200 and 400 ps were smoothed by 1-2%. 2  Chapter 6. Relaxation  Measurements  of Bovine Brain using Magnetic  White Sample (MC%) myelin water (%) 17.4 1-2 11.1 1-3 15.7 1-6 12.1 2-1 17.5 2-3 15.0 3-1 (71.0) 11.4 3-2 (73.2) 3-2 (68.2) 11.5 10.6 3-2 (59.9) 10.1 3-2 (37.7) 14.0 3-6 (72.0) 14.9 3-6 (36.0) 14.3(0.9) average  Resonance  75  Grey Sample (MC%) myelin water (%) 6.4 1-1 1-4 5.8 4.9 . 1-5 2-2 4.6 2-4 6.6 2.2 3-3 (83.8) 3.7 3-4 (77.0) 3-4 (52.8) 4.6. 5.4 3-5 (85.4)  average  4.7(0.6)  Table 6.1: Myelin water percentages for white and grey matter samples. Moisture contents are included for brain 3 samples. The average values are for fully hydrated samples only and the standard error is included in parentheses.  Chapter 6. Relaxation  Measurements  of Bovine Brain using Magnetic Resonance  84.0%  73.2%  T  0.001  76  TTTnll  0.01  0.1  I' I I I III  IH+l | ^  I  ' II  0.001  ITTl[  " I  I I 111 llf  0.01  YhTTTTT  0.1  52.8%  37.7% —' "Q. E <  iinh• i1 1i \(\w\ 0.001  A  0.01  0.001  0.001  0.01  I'I  0.1  2.5%  13.3%  H+4J  r i  I I llll|  0.1  1 4 I lllll|  1 I I I llll|  0.01  0.1  T (s) 2  1 I I l l l l l l l l l l — * ? > l I I KTTl|  0.001  1 I I lllll|  0.01  1 I I II  0.1  T (s) 2  Figure 6.5: T distributions at different moisture contents. Plots on the left-hand side are from a white matter sample and plots on the right-hand side are from a grey matter sample. (Note that the amplitude of the fully hydrated sample has been reduced by a factor of 2.) 2  Chapter 6. Relaxation  6.4.3  Measurements of Bovine Brain using Magnetic Resonance  77  T i Relaxation  A n example of the T i distribution for white and grey matter is shown in Figure 6.6A and B. Both the liquid and solid intensities are included. In grey matter, the solid peak overlapped the liquid peak whereas in white matter, the solid peak was shifted to the left of the liquid peak. The T i distributions as a function of moisture content are shown in Figure 6.7. Saturation recovery curves for white and grey matter are shown in Figure 6.8. Monoexponential fits to the data are also shown.  6.4.4  Cross  Relaxation  The reappearance of the solid signal as a function of T\ for a 90 — r — 90 — T\ — 90 experiment [65] is shown in Figure 6.9 for a white and grey matter sample. This signal was then fit to the form S(t)  —  a + 6(1  — e~  l )  T  Ts  where  a is  the correction for the T  decay of the mobile signal at short times and b is proportional to the amplitude of the solid signal. The measured T is related to the cross relaxation time, To-, by: s  P  1  P  s w Per Ts T L  1  1  w  where P and P s  are the probability of a proton being in the solid and water fraction  w  respectively and T  w  is the water relaxation time due to cross relaxation. A schematic of  the different proton pools in bovine brain is shown in Figure 6.10. The signal intensity from each pool was defined as S  mw  from 1-50 ms), S  m  for myelin water (the area under the T distribution 2  for solid myelin (half of the solid FID signal), S{ for intra/extracellular w  water (the area under the T  2  distribution from 50-200 ms) and S  nm  for non-myelin  macromolecules (half of the solid FID signal). Since the signal intensity, in the absence of relaxation, is directly proportional to the number of protons in a pool, these intensities can be used to determine P . In bovine brain, two different types of cross relaxation were s  2  Chapter 6. Relaxation  0.01  Measurements of Bovine Brain using Magnetic Resonance  2  -i—i—i i i 111 3 4 5 6 7 8  0.1  T, (S)  78  10  100-  80 H  •g 60Q.  E <  40-  20  H  04 0.01  Figure 6.6: Typical T i distributions showing the separation of liquid and solid protons. Plot A is from a white matter sample and plot B is from a grey matter sample both at full hydration.  Chapter 6. Relaxation  Measurements  of Bovine Brain using Magnetic Resonance  'i  79  84.0%  0.01  0.01  0.1  0.1  1  0.1  1  52.8%  37.7%  /  t  /  0.01  0.1  13.3%  i  *  k  0.01  -ml—i—i  \  0.1  11  inil  0.01  1  2.5%  +ftj  1  "iTi M(s)t mill  +  0.01 M i n i  1 11 1  llTlf^*l ''7'l-l C  0.1  Ti (s)  1 1 1 1 III  1  Figure 6.7: T i distributions of solid and liquid protons at different moisture contents. Plots on the left-hand side are from a white matter sample and plots on the right-hand side are from a grey matter sample. (Note that the amplitude of the fully hydrated sample has been reduced by a factor of 2.)  Chapter 6. Relaxation  Measurements of Bovine Brain using Magnetic Resonance  80  Figure 6.8: Plot of the saturation recovery curve for a white and grey matter sample. The curves shown come from the equation S(t) = Mo{l - e ). ( _ t / T R )  Chapter 6. Relaxation  Measurements of Bovine Brain using Magnetic Resonance  Grey Sample (MC%) T (ms) 1-1 18.5 3-4 (77.0) 22.9 3-4 (52.8) 13.7 3-5 (85.4) 33.1  White Sample (MC%) 1-2 1-6 3-1 (71.0) 3-2 (73.2) 3-2 (68.2) 3-2 (59.9) 3-2 (37.7) 3-6 (72.0) 3-6 (36.0)  T  s  (ms) 18.9 21.3 20.2 24.6 19.8 16.8 9.8 25.5 10.1  T™ (ms) 37.2 41.9 52.6 48.7 35.6 27.2 12.8 58.9 18.4  T i (ms) 144 165 174 212 141 105 36.7 230 58.0  s  81  T t (ms) 206 219 83.1 194  Table 6.2: T measured in Goldman-Shen cross relaxation experiment for white and grey matter samples. Moisture contents are included for brain 3 samples. Calculated cross relaxation times are also included. s  assumed: one between the myelin water and solid myelin pool (T™) and the other between the intra/extracellular water and non-myelin molecule pool (T^.). For the first case, P  m s  was calculated by:  pm  §jn ^m ^mw  =  , ,  i  A small normalisation correction was applied to S  mw  since the C P M G and FID ex-  periments did not yield exactly the same total mobile intensities. This normalisation consisted of multiplying S  mw  by the ratio of the total FID mobile signal and the total  C P M G signal at t= 0. In the second case, P was calculated by: s  pr = T s  q  q  •  The same normalisation correction was applied to Si . w  (6-2) T  c r  was then calculated as the  ratio of T to P for each case. T and both cross relaxation times for all samples are s  s  s  listed in Table 6.2. As the brain was dehydrated, both cross relaxation times decreased.  Chapter 6. Relaxation  Measurements of Bovine Brain using Magnetic Resonance  82  Figure 6.9: A plot of the solid signal recovery in a cross relaxation experiment. A white and grey sample are presented. The cross relaxation fits to these data are also shown.  Chapter  6. Relaxation  Measurements  of Bovine Brain using Magnetic  White  Resonance  Intra/Extra Cellular Water (Siw.T7.TV) Diffusion (very slow)  Myelin Water l mw 1 •' 2 I s  Cross relaxation (To/)  1  Cross relaxation  (V)  Myelin Molecules  Non-myelin Molecules  <s .Tr.T )  /Q  m  m  Grey  -j- nm -p nm*  2  Non-myelin Molecules /Q  -r- nm -i- niriv  ^nm''i  Intra/Extra Cellular Water  (s^r,^)  ''2  >  Cross relaxation (TJ)  Figure 6.10: Model of the different proton pools in bovine brain  83  Chapter 6. Relaxation  6.4.5  T!-T  2  Measurements  of Bovine Brain using Magnetic Resonance  84  Dependence  A plot of the T 1 - T 2 dependence of a white matter sample is shown in Figure 6.11. From this experiment, water with a short Ti was also found to have a short T . There was 2  only one off-diagonal element found in one sample and its intensity was very low. The 3-D plots were divided into two regions, one consisting of peaks at short Ti (< 750 ms) and T (< 40 ms) and the other of peaks at long Ti (> 800 ms) and T (> 40 ms), and 2  2  the ratio of their intensities averaged to 20%.  6.4.6  Cross-T  2  Dependence  Plots of cross-T relaxation are shown in Figure 6.12A and B. Plots of T i - T relaxation 2  2  are also shown as a comparison. The cross-T decay curves arising from the myelin water 2  in hydrated white matter were fit lo an exponential with an average relaxation value, T™, of 36 ms. In contrast, the relaxation value for the T - T x  2  decay curves from the myelin  water gave a value of 371 ms. The cross-T decay curves for the infra/extracellular water 2  from hydrated white matter were found to have an average T ^ = 178 ms. For the T i - T decay curves, the infra/extracellular water was found to have an average T ; r e  a x  2  = 797  ms. In grey matter, the cross-T decay curve for the myelin water could not be fit to 2  an exponential since there was only a small short T component present. The relaxation 2  time for the intra/extracellular water was found to be 175 ms which was the same as white matter. Measured relaxation rates in the cross-T experiment and the calculated 2  cross relaxation times for each water pool are shown in Table 6.3 for some samples. The T  s  values presented in the table were calculated from the P , P  this experiment.  m  nm  and T^s measured in  Chapter 6. Relaxation  Measurements of Bovine Brain using Magnetic Resonance  85  Figure 6.11: A plot of the T i - T dependence of a white matter sample. Plot A shows a 3D view of the data while plot B shows a bird's eye view of T i versus T . 2  2  Chapter 6. Relaxation  0.00  Measurements of Bovine Brain using Magnetic Resonance  0.05  0.10  Tau (s)  0.15  86  0.20  Figure 6.12: Plots showing the cross relaxation dependence of myelin and intra/extracellular water in white matter (closed circles). Plot A represents the myelin water component and plot B the intra/extracellular water. Exponential fits to each data set are also shown. The data from the T dependence of myelin and intra/extracellular water are shown for reference (open circles). Note, only the first 200 ms of the T i and T data are shown. x  2  Chapter 6. Relaxation  white  grey  Measurements  Sample (MC%) hydrated (72.1) 3-2 (68.2) 3-2 (59.9) 3-4 (52.8) 3-5 (85.4)  T™ (ms) 37.0(6.6) 29.1 61.0  -  of Bovine Brain using Magnetic  T L (ms) 175(9.4) 83.8 162 231 175  T ? (ms) 29.0(3.2) 36.6 98.3  -  T ? (ms) 22.5(1.5) 13.8 30.9 27.1 18.2 m  Resonance  T S (ms) 66.0(8.7) 65.7 159 -  -  87  T i (ms) 193(10.9) 97.9 193 258 194  Table 6.3: Measured T times for both water pools using the cross-T2 experiment. The calculated cross relaxation are also included for some white and grey samples. The hydrated sample is an average over the three fully hydrated white matter samples from Brain 3. Standard errors are shown in parentheses. w  Chapter 6. Relaxation  6.5  Measurements of Bovine Brain using Magnetic Resonance  88  Discussion  The main difference between white and grey matter is myelin which makes up about 50% of the dry mass of white matter tissue [6]. Therefore, the relative amount of solid signal is expected to be twice as big in white matter as in grey matter. This was found to be the case for both the FID experiment and the T i experiment where solid and liquid signals were separated (Figures 6.1 and ??). Water content for human white and grey matter have been found to be 70% and 80% respectively [6]. The average water content of fully hydrated grey matter samples was larger than that of white matter although the absolute values are not necessarily accurate since this was an in vitro experiment. The water content of myelin is 40% which gives us an estimate of the myelin water percentage in white matter being 16%. bovine white matter.  This is fairly close to the average of 14.3% measured in  The estimated proton densities found for both white and grey  matter fall within the proton densities of its constituents: lipids (0.13 gH/gLipid) and proteins (0.04-0.06 gH/gProtein). The mobility or orientational order of the solid protons is measured using the second moment. Both white and grey matter had similar second moments indicating that the amount of order in both tissues is similar. The second moment of phospholipid bilayers has been previously measured as 5xl0 s~ for the liquid crystal phase and 3.5xl0 8  2  for the gel phase [83] and typical proteins as 5xl0  9  s~  2  [84].  9  s  - 2  The second moment of  fully hydrated bovine brain tissue falls between the second moment of liquid crystalline phospholipids and proteins. The T distribution generally showed four peaks for fully hydrated.samples. In white 2  matter, two of the peaks were centred about 100 ms and may represent separation of the intracellular and extracellular fluid. All peaks below 50 ms were attributed to myelin water due to the short T relaxation time. Although the shape of the T distribution was 2  2  Chapter 6. Relaxation Measurements of Bovine Brain using Magnetic Resonance  89  broader for the r = 100 ps experiment than the r =200 or 400 ps, the T times between 2  experiments were indistinguishable and the myelin water percentages were similar (24.1, 22.6 and 18.1 for the 100, 200 and 400 ps experiments respectively). When the samples were dehydrated, the myelin water percentage stayed fairly constant indicating that all the water in the sample was decreasing at the same rate. This was surprising since it was expected that water in the intra/extracellular spaces would evaporate before more tightly bound water trapped between myelin bilayers. The T i of both the solid and liquid components of grey matter coincided at approximately 1.2 s. In white matter, the T i of solid was measured at about 500 ms while the liquid peak was at 800 ms. The similarity in grey T i s is thought to arise from averaging between the intra/extracellular water pool and the non-myelin macromolecular pool (Figure 6.10). In white matter, only myelin water is in close contact with the solid and therefore much less averaging must take place. This allows differences in the T i values for the solid and mobile pools. A difference in T i for the two water pools was seen by another group [80] but no explanation as to the mechanism was given. The single T i value found for the two liquid pools in this work is attributed to the inability of NNLS to separate peaks which are within a factor of two in relaxation time. No cross peaks were found between the T i and T components which indicated that there was no exchange 2  on the timescale of 1 s between the two water pools. This was different than results found from in vivo human brain where a single T i component indicated mixing between the two pools. It should be noted however that these experiments were carried out at room temperature and that the rate of diffusion would be increased at physiological temperatures. Since there was no mixing of the two water pools, each cross relaxation time, T™ and T L , was calculated for the separate proton pools (Figure 6.10). In the case of the solid signal, half the intensity was attributed to myelin and half to other macromolecules  Chapter 6. Relaxation  Measurements of Bovine Brain using Magnetic Resonance  90  since myelin comprises about 50% of the dry weight of white matter [6]. Two different relaxation times, T™ and T ^ , were measured from the cross-T2 dependence. These values as well as P  m  and P  were used to calculate the expected T™ and T " (equal to 29 ms m  n m  and 22.5 ms respectively). Since both values were quite close, only one T  S  time should  have been measurable in the Goldman-Shen cross relaxation experiment. The individual cross relaxation times, T™ and T^,, were calculated from the single T and the probability s  of protons being in either pool, P  m  and P  n m  - In grey matter, it was assumed that all  the solid signal belonged to the non-myelin molecule pool and all the mobile signal to intra/extracellular water. The expected T "  m  was calculated and found to be 18 ms. This  was slightly different than the measured value of 33 ms. The reason for this discrepancy is unknown. Cross relaxation times have been measured in wood and found to be 1 ms in red cedar and 4 ms in hemlock [85]. These are an order of magnitude smaller than the cross relaxation time in myelin probably due to the water in the cell wall having much greater interactions with the solid protons than water in myelin. The decay of the solid is governed by dipolar interactions between protons which causes the spins to dephase quickly. Cross relaxation between the solid and myelin water protons contributes to the T relaxation time of the mobile signal. Therefore, T 2  W  is an  upper limit for the T time of mobile protons, i.e. 2  —  > —  -  1  If we assume that cross relaxation is the only mechanism responsible for T relaxation 2  in grey matter and that all the water is in contact with all the solid, then the expected T  2  time is about 175 ms. Since the measured value was about 100 ms, cross relaxation is  not the only mechanism for T relaxation. In white matter, the assumption was that the 2  main source of cross relaxation was between the myelin water and the myelin. If we then calculate the expected T for myelin water, the result is around 37 ms. This is slightly 2  Chapter 6. Relaxation  Measurements  of Bovine Brain using Magnetic Resonance  91  larger than the measured value of about 20 ms. Therefore, mechanisms other than cross relaxation also affect the T relaxation times in white and grey matter. 2  6.6  Conclusion  A model for T i and T  2  relaxation of water in bovine brain has been proposed which  incorporates 4 proton pools: intra/extracellular water, myelin water, non-myelin macromolecules and myelin macromolecules. During the course of an N M R experiment (<1 s), the two water pools do not exchange. There is, however, rapid exchange between the myelin water and myelin pools (T™ = 66 ms) and slightly slower exchange between the intra/extracellular and non-myelin pools ( T ^ = 193 ms). The myelin water protons are found to have a short T value partly due to cross relaxation with the myelin. Although a 2  simple T i experiment is not able to distinguish the two water pools, a T i - T experiment 2  is able to separate the myelin water based on its short T i and T . 2  Chapter 7  Diffusion model of T  7.1  2  and T i relaxation i n two brain water pools  Summary  A diffusion model of myelin water and intra/extracellular water was used to simulate T i and T  2  relaxation data from white matter in human brain. Relaxation times and  amplitudes were calculated from models where different parameters including diffusion coefficients, and cell sizes were varied. Generally, changes in the intra/extracellular pool parameters had little effect on the relaxation while changes to the myelin pool parameters caused all relaxation times to vary. The diffusion coefficient of the myelin water pool was estimated to be about 0.0002 //m /ms. Increases in the myelin thickness and axonal 2  diameter (in the same proportion) caused increases in the myelin water percentage and T  2  relaxation time of the myelin water component which could account for the experimental differences in myelin water percentage between different white matter structures.  7.2  Introduction  Relaxation in biological tissue is multi-exponential with time constants shorter than those found in bulk water. Mechanisms describing the relaxation have been proposed incorporating diffusion between different proton pools. A preliminary model by Brownstein and Tarr [86] involved a single water pool which relaxed due to interaction with strongly relaxing surfaces. This model worked well for certain biological cells such as wood [87] but did not take into account the role of cytoplasmic components of the cell or the presence  92  Chapter 7. Diffusion model of T and 7\ relaxation in two brain water pools 2  93  of different water pools. Belton and Hills [88] expanded the diffusion model to include two water pools separated by a permeable or semi-permeable membrane. Their approach used the Laplace transform method to obtain an analytical solution for diffusion. In human brain, two water pools (myelin water and intra/extracellular water) have been suggested on the basis of T relaxation experiments [16, 17]. The intra/extracellular 2  water pools are combined into one since they can not be distinguished on the basis of in vivo relaxation measurements.  Exchange between the myelin water and in-  tra/extracellular water is thought to occur within 500 ms since T i relaxation is monoexponential [17] and M T delay measurements show averaging at long times (see Chapter 5). The likely mechanism for this exchange is diffusion. In this chapter, a diffusion model is applied to the two water pools, one in a cylindrical shell and the other inside the cylinder, in order to simulate in vivo T and T i relaxation 2  data (Figure 7.1).  The diffusion-Bloch equation is solved numerically for cylindrical  coordinates since axons are cylindrical in nature. Diffusion coefficient's, cell radii and T  2  and T\ relaxation times are input into the model and the measured output consists of T  2  7.3  and T i relaxation times and amplitudes.  Numerical Methods  The effect of diffusion on the magnetization, M(r, i), of the two water pools can be described by the diffusion-Bloch equation  where D is the diffusion coefficient of the water. This equation was discretised (in time steps A i and radius steps A r ) using the Crank-Nicholson [89] method to obtain Af  r | t +  i - M  Ttt  At  =  M i, r +  t  - 2M , + M - i , r  (Ar)  t  2  r  t  1 M rAr  r + 1 | t  -  2Ar  Af r  M  ^  (7.2)  Chapter 7. Diffusion model of T and T i relaxation in two brain water pools 2  94  Figure 7.1: A schematic showing the two different cylindrical pools of water in the brain and their associated parameters.  Chapter  7. Diffusion model of T and T i relaxation 2  in two brain water pools  95  where r and t are the radius and time interval indices. This equation can be reduced to the form M , r  t + 1  = [F — G ] M _ r  r  M  + [1 - 2F - ^ ] M , + [F + G ]M r  t  r  (7.3)  r+u  J-2  where DAt  and  (A^  '  F=  (7 4)  Other definitions in equation 7.3 are A; ?2 =  {  0<r<a  An! T  2 m  ;  _  a < r < o ~ " , a <r <b  (7-7)  where a is the position of intra/extracellular water and myelin water boundary and b is the outer boundary of the region (Figure 7.1. The initial magnetization (at t=0) is defined as  «*-<™ °-;-<°» KM ; 0  ^  a <r < b  where K is the partition coefficient between the concentrations of the two water pools. The solution is symmetric and continuous about r = 0 which implies that ^f\o  = 0.  Using the forward Euler method of discretisation, we can obtain expressions for M +\. r  At the other boundary, r = b, we again impose symmetry and Qjjr\b = 0. backward Euler method of discretisation, we can obtain expressions for M _ i . r  Using a At the  boundary, r — a, we must invoke Fick's Law or continuity of flux which gives  -K^D^U-  (7.9)  Chapter 7. Diffusion model of T and T \ relaxation in two brain water pools  96  2  Since the density of water is different on either side of the boundary and density is related to K, a i f  2 / / 3  term was added to the equation in order to correct for flux through different  unit areas. This can be discretised using a forward Euler method to get  _ y Ma<-M _ K  iD  a  u  _ W  =  D  a + u  Ar Ar  1  -W , a  Ar Ar  m  where the magnetizations are defined as 0<r<a '  W(r,t);  a<r<b  X  K  1  Again at the boundary r = a, the partition balance equation gives > W, a  t+1  = KM , .  (7.13)  a t+1  where the K is included to account for the different densities of water on either side of the boundary. When equations 7.10, 7.11 and 7.13 are combined, the resulting expression is a forward "marching" in time equation for the magnetization at r = a. The stability condition for this equation is At <  7.4  •  N u m e r i c a l Applications  Most of the parameters which are involved in the numerical simulation are unknown for brain. Therefore, the best initial guesses were used and then each input parameter was varied.  Myelin water makes up 16% of the total water in white matter whereas  intra/extracellular water is 84%. The partition coefficient can be calculated from these water contents, the model cell radii, a and b, and the expected magnetization such that 016_o^4_ 0.84 (b - o ) M „ 2  2  {  '  Chapter 7. Diffusion model of T and T\ relaxation 2  where M  0  in two brain water pools  97  is the initial magnetization as defined in equation 7.8. Data on nerve axon  diameters were very hard to obtain but one paper gave values for the corpus callosum [90]. This was used to estimate the initial radii, a and b, to be 1.0 pm and 1.2 pm respectively which corresponds to a i f of 0.4. This K is equivalent to the moisture content of the myelin water pool which has been measured at 40% [6]. The diffusion coefficient of bulk water at room temperature is about 2.0 /im /ms. The 2  diffusion coefficient of the intra/extracellular water was guessed to be about 100 times less than free water, giving Di = 0.02 //m /ms. Using the diffusion equation, D = r /2t 2  2  and the above cell radii, the diffusion coefficient of myelin water was estimated to be D  = 0.00002 pm /ms.  This value of D  2  m  results in minimal mixing of the two water  m  pools in 100 ms and partial mixing in 500 ms which models MRI T and T i experiments. 2  Also estimated were T  = 20 ms and T  2m  2i  ments. The T i relaxation times were set to T  = 200 ms from MRI T relaxation experi2  i 7 n  = 200 ms and T I J = 800 ms since myelin  water was believed to have a lower value than intra/extracellular water. The step sizes for the T  2  simulations were A r =0.005 pm and At =0.00005 ms.  Enough time steps were taken to simulate a 32 echo decay curve. For T i simulations, A r =0.005 /mi and At =0.0005 ms. Eight data points were collected at times of 10, 20, 50, 100, 200, 500, 1000 and 2000 ms to produce a decay curve. Both the T and Ti decay 2  curves were fit to bi-exponential models to obtain relaxation times and amplitudes for each pool. The fitting algorithm chose all pairs of T i and T relaxation times from an 2  input of 100 possible relaxation times and, using an NNLS algorithm, output the pair of relaxation times and amplitudes which gave the lowest x 2  Chapter 7. Diffusion model of T and T\ relaxation in two brain water pools 2  7.5  T  98  simulations  2  Variations in radii were done with a constant K = 0.4 but changing a and b such that a/b remained constant.  The effect on each pool is seen in Figure 7.2. T  reduced by decreases in the myelin thickness but T myelin thickness.  The amplitude of each T  2  m  was greatly  2 i  was only reduced at a very small  component was fairly constant except at  2  small myelin thickness which caused the amplitude of the myelin water pool to decrease and the intra/extracellular water pool to increase.  1.0 H CO CD "D  A  |-0.8-  A  CO  c A  CO  A  A  A A  A  A  A  co CD •o  A  A  A  A  200-1  Q. E CO T3 C CO  150  A T A Amp,  H  2  "co"  E  CO CD  E  I co  0.4-  X  CO CD  T , A  m  •  T  O  Amp  2  m  m  CO m  E 100-| c o  m  P m  to X  «  •  • • • •  •  o  O  O  O 0.0-1  Amp;  O  0.2O  T„  A  •  •  cc Q_  •  i  O  O  CC 5 0 - | 3  Q.  o  o  1  1  1  1  0.2  0.4  0.6  0.8  04  -r  o  o  o  o  o  0.6  0.8  -  0.2  0.4  b-a (^.m)  b-a (nm)  Figure 7.2: A plot showing the effect of changing the cell radii and the myelin thickness on the T and T i relaxation times and amplitudes. The cell radius and myelin thickness were increased proportionally in order to keep K constant at 0.4. The amplitude of the relaxation components are shown in % for T and fractions for T i . 2  2  The effect of changing the initial T  2 i  or T  2  m  is shown in the lower half of Figure 7.3.  Decreasing the initial T * only decreased the T j relaxation time. Decreasing the initial 2  2  Chapter  7. Diffusion model of T and T i relaxation  in two brain water pools  2  Structure  axon radius (a) pm  Simulations myelin thickness (b) pm  Minor Forceps Genu Major Forceps Splenium Int Capsules  1.1 2.0 2.0 3.0 5.0  1.32 2.4 2.4 3.6 6.0  myelin water (%) 6.9 10.7 10.7 12.1 13.4  99  myelin water (%) experimental 7.3 10.7 10.5 12.6 18.1  Table 7.1: Myelin water percentages from simulations with different axon diameters and myelin thickness are shown and compared to experimentally measured myelin water percentages for different white matter structures. The parameters used were Di = 0.02 and D = 0.0002 pm /ms, and T = 200 and T = 30 ms. The input myelin water percentage was 15% for all cases. 2  m  T  2 i  2  caused a decrease in both T j and T  2 m  2  2 m  m  . The amplitudes were fairly constant for  both cases. The diffusion coefficients of each pool were varied separately and the effect is shown in the lower half of Figure 7.4. Changing Di caused slight fluctuations in the relaxation time of each pool. When D  m  was increased, T j decreased dramatically while T 2  2  m  decreased  slightly. However, at high values of D , the amplitude of the myelin water component m  became 0. Since the amplitude of the myelin water component was very dependent on the axon diameter and the myelin thickness, these two parameters were varied in order to obtain myelin water percentages equal to the different white matter structures in chapter 4. Results are shown in Table 7.1. pm /ms 2  and T  2 i  = 200 and T  experimental data.  For these simulations, A 2  m  = 0.02 and D  m  = 0.0002  = 30 ms which resulted in better matches to the  Chapter 7. Diffusion model of T and Ty relaxation in two brain water pools  100  2  o o.o D; (|im  D  /ms)  S2003  •  "5. E  •  A  A  A  m  (urn  /ms)  A  A A Ampi  CO  "g 150co  • O  A  E  Amp  co CD  £ 100-  8  •O  •O  •  O OH.  o  A  A  A  A  50 •  3 Q. 3  A  •O  CC  A  •O  co x CO CD  A  A  A  A  c o  m  i  \ 10""  10'  5  9 1 10'  3  D| (nm  •  i  r 10"  2  /ms)  10"  1  10"  6  10"  10'"  5  D  m  (\im  •  •  O  r  10'  3  o  O  10  /ms)  Figure 7.3: A plot showing the effect of changing the initial T i (upper) and T (lower) on the simulated T and T i relaxation times and amplitudes. On the left, T j was varied and on the right, T changed. The amplitude of the relaxation components are shown in % for T and fractions for T i . 2  2  m  2  Chapter 7. Diffusion model of T and T\ relaxation in two brain water pools 2  101  to 1.0-  CD TJ  A  3  A  "5.  A  A •  A  •  i °- • 8  A  C  • ' 0.6  CD  H  E  A  .2 0.4 • co  Ampi  A.T„ • T  X  l m  «  O  Amp  m  CD  DC . -0.2-  • •  3 Q. 3  o o.o —I  o  •••• •  o  o  0.6  0.8  o  o  o  i 1.2 0.0  1.0  Tn (S)  i 100  1  \  200  T  2i  300 (ms)  O  o  r  0.1  r~  I  1  0.2  0.3  0.4  0.5  T .30  T  T  40  50  T (s) im  1 400  1—T 500  0  r 10  20_  T2m , (ms) m  Figure 7.4: A plot showing the effect of changing the diffusion coefficient on the T i (upper) and T (lower) relaxation times and amplitudes. On the left, D, was varied and on the right, D was changed. The amplitude of the relaxation components are shown in % for T and fractions for T i . 2  m  2  Chapter 7. Diffusion model of T and T\ relaxation in two brain water pools 2  7.6  102  T i simulations  As with the T simulations, a and b radii were varied with a/b constant. The effect on each 2  pool is seen in Figure 7.2. Both Tu and T  i  m  were reduced as the myelin thickness was  reduced except at very small thickness when T  X m  increased dramatically. The amplitude  of the myelin water pool paralleled the behaviour of T i . m  The effect of changing the initial T I J or T  i  m  is shown in the top half of Figure 7.3.  Decreasing the initial T I J only decreased the Tu relaxation time. Decreasing the initial Ti  T O  caused a decrease in both T i , and T i . The amplitudes were fairly constant for the m  first case but the amplitude of the myelin water pool decreased and then increased for the second case. The diffusion coefficients of each pool were varied separately and the effect is shown in the top half of Figures 7.4. Increasing Dj caused slight decreases in the relaxation time of each pool. The amplitude of the myelin water pool was also slightly decreased. When D  m  was increased, T  2 i  decreased while T  2  m  increased resulting in a single relaxation time.  The amplitudes of each component also varied.  7.7  Discussion  The goal of these simulations was to examine the trends in T i and T relaxation and am2  plitudes by varying parameters which are not available experimentally. Although myelin is composed of multiple bilayers, the model treated the whole structure as uniform with the parameters D, T i and T taking into account the bilayers. Also, since intracellular 2  and extracellular water pools are not separable from in vivo T relaxation measurements, 2  both pools were combined into the intracellular pool. Certain parameters between the intracellular and extracellular water pools are expected to be slightly different, such as  Chapter 7. Diffusion model of T and T\ relaxation in two brain water pools  103  2  diffusion coefficients, but since only one compartment can be measured in vivo, these differences were averaged into one pool. Also, the extracellular water pool is approximately five times smaller than the intracellular water pool and therefore would have a smaller effect on the results. Consequently, this model is only a first approximation of diffusion in white matter. One of the main assumptions when measuring myelin water percentages in vivo was that the amplitude of the short T  2  component was equal to or at least related to the  amount of water in the myelin water pool. With the simulations, we were able to test this hypothesis. When either Tu or T , was changed, there was no change in the amplitude 2  of either pool. Therefore, any pathological problem which may cause a change in the T i or T of the intra/extracellular water should not affect the myelin water pool. Similarly, 2  when T  2  m  was increased there was no change in the amplitudes of the T  2  components.  It should be noted that the expected amplitude for the myelin water T component was 2  15% and not the 7% output by the simulation. From the simulations, it was obvious that changing the properties of the myelin water pool influenced the relaxation times and amplitudes to a much greater extent than did varying the intra/extracellular water pool parameters. This was expected since the myelin water pool acted like a relaxation sink for water and therefore has a greater influence on the intra/extracellular water pool. This was further supported by increases in the myelin water diffusion coefficient causing decreases in the intra/extracellular water relaxation time indicating more mixing of the two pools. Evidence for partial mixing of the water pools was found at diffusion coefficients of Di = 0.02 and D  = 0.00002 pm /ms since changes in T 2  m  2  m  and T  l r r a  caused T  2 i  and  Tij to also change. This is likely due to exchange between the water pools averaging the relaxation times. However, at this value of D , m  two T i relaxation times were still  discernible which was not consistent with experiment. A D  of 0.0002 pm /ms caused 2  m  Chapter  7. Diffusion model of T and T\ relaxation 2  in two brain water pools  the two water pools to become completely averaged. This larger value of D  m  104  also caused  the amplitude of the short T component to decrease slightly indicating that there was 2  some exchange within the T timescale. Therefore, short T 2  2  amplitudes measured ex-  perimentally may be slightly lower than the actual values. The "best guess" value of Di used in the simulations was lower than that found in the literature of 0.5 /jm /ms [91] 2  although this was not for human nerve. Fortunately, changes in Di had little to no effect on output relaxation times and amplitudes and therefore these results should still be valid for larger values of Dj. Future simulations will incorporate a more realistic diffusion coefficient. Exchange was also seen with small values for the myelin thickness which caused large decreases in T , suggesting that a larger portion of the myelin, water was able to 2  interact with the intra/extracellular water. In T i simulations, the small myelin thickness caused the T i s of the two water pools to become similar, again indicating averaging. Since myelin is fairly impermeable to water, a possible mechanism for myelin water exchange is that it diffuses circularly around the rings of the bilayer until it reaches either the intra or extracellular space.  This time to travel around the bilayers scales  as the square of the number of layers. If we assume that myelin water has a diffusion coefficient equal to free water (2.0 //m /ms) in between the bilayers and the radius of the 2  cell is l'/xm, then the time to travel from the inside to the outside of the myelin (about 10 bilayers thick) is about 1 s. This may account for the averaging found within the T i timescale but not the T timescale in human brain. 2  A very exciting result seen from the simulations is that the T of the intra/extracellular 2  water changes depending on the thickness of the myelin and the axonal diameter. Smaller axons with thinner myelin sheaths were expected to have shorter T ; . Also, from Table 2  7.1, it can be seen that smaller axons with thinner myelin also have smaller apparent myelin water percentages even though the input myelin water percentages was equal to 15% for all cases. From experiment, it was found that different white matter structures  Chapter 7. Diffusion model of T and T i relaxation in two brain water pools 2  105  have different measured myelin water percentages and can be ranked accordingly. This ranking was preserved when looking at the geometric mean T [17]. Therefore, different 2  myelin water percentages and different T values measured in vivo may be a consequence 2  of different sizes of axons and myelin thickness. It should be pointed out that these simulations involved a model where the ratio of the axon diameter to myelin thickness remained fixed which is likely not the case in brain. From the simulations, we have found that the myelin water percentage is related not only to the amount of myelin water but also to the morphology of the axons as well as parameters such as the diffusion coefficient.  7.8  Concluding Remarks  Relaxation in human white matter can be modelled as diffusion between two water pools. Changes to the diffusion coefficient and relaxation time of the myelin water pool can cause dramatic differences in the relaxation times and amplitudes of both water pools. The diffusion coefficient of the myelin water pool was estimated to be 0.0002 pm /ms since 2  this value caused complete averaging of the two water pools during the T i timescale but not the T timescale. The diffusion coefficient of the intra/extracellular water pool could 2  not be estimated since changes in its value had little effect on the output relaxation times and amplitudes. A proportional increase in the diameter of the axon and the myelin thickness also increases the myelin water percentage which may account for the differences in myelin water percentage found between different white matter structures. The T relaxation time of the myelin water component also increased as the diameter of 2  the axon and the myelin thickness increased.  Chapter 8  Conclusions  8.1  This  work  In this thesis, human brain was examined using MRI and bovine brain was examined using ifl-NMR. Measurements that were supposedly related to myelin content were explored in both normal and multiple sclerosis brain.  Since there are two water pools in the  brain, the M T effect on each was investigated.  Finally, experiments in bovine brain  were done to explore how different proton pools interacted and over what timescale these interactions occurred. Simulations were also carried out to examine how different water pool parameters could affect the amplitude and value of relaxation rates. Magnetization transfer ratios and myelin water percentages were uncorrelated in white and grey matter from both normal volunteers and MS patients. In lesions, there was a small (R=0.5,P=0.005) but significant correlation. The lack of strong correlation was surprising since it indicates that the methods can not both be proportional to myelin contents. The effect of M T is different for each brain water pool with the short T  2  component  being affected significantly more than the long T component (P<0.00001 to P<0.04). 2  This finding helps strengthen the argument that the short T component is myelin wa2  ter which would be closely associated with myelin molecules.  With different types of  M T pulses, different amounts of direct saturation were seen and the amount of direct saturation appeared to be proportional to the amount of myelin water.  106  This allowed  Chapter 8.  107  Conclusions  the binomial M T pulse (which produced large amounts of direct saturation) to differentiate between different white matter structures while the sine M T pulse (small direct saturation effect) did not. In bovine brain white matter, a 4-pool model with cross relaxation between the myelin and myelin water (named myelin pools) and the intra/extracellular water and non-myelin molecules (named non-myelin pools) was found to represent the relaxation data. Surprisingly, no exchange between the two water pools was found within the timescale of our measurements (within Is). In grey matter, a 2-pool model was invoked which only included the intra/extracellular water and the non-myelin molecules.  The cross relax-  ation time between the myelin water and myelin was found to be 66 ms and between intra/extracellular water and non-myelin to be 193 ms for fully hydrated white matter. For grey matter, the cross relaxation time was found to be about 200 ms. One of the main differences between the bovine and human brains was the rate of exchange between the two water pools. As stated earlier, in bovine brain, there was no exchange between these two pools over the timescale of our measurement.  In human  brain, exchange was found within about 500 ms. Since the bovine brain was removed from the cow before imaging, it is possible that these differences are due to in vivo versus in vitro studies rather than differences between the brains. Simulations which modelled the exchange between the water pools in human brain MRI experiments through diffusion showed that relaxation times were highly dependent on the diffusion coefficient of the myelin water pool but not that of the intra/extracellular water pool. Differences in the size of the myelin cylinder compared to the axon cylinder caused changes in the relaxation time of the intra/extracellular water pool but not much change to the myelin water pool. The amplitude of the myelin water component was dependent on the axon diameter and myelin thickness. Therefore, different white matter structures may have different myelin water percentages due to different axon cell sizes. ^  108  Chapter 8. Conclusions  8.2  On-going a n d future  work  Currently, a serial study has been started which will examine lesions over the course of one year using M T , T  2  relaxation and T i relaxation. The patients have at least one  enhancing lesion which is viewed by all three techniques. T i times will be used to correct for relaxation during the scan and therefore accurate water contents can be obtained. Comparisons are planned between all techniques as well as monitoring the progression of lesions over time. Further experiments using T relaxation with an M T prepulse will be done in order 2  to get better statistical numbers. In particular, more normal volunteers will be examined using different frequency offsets and delay times between the M T pulse and the initiation of the T  2  relaxation sequence.  A time constant associated with the rate of exchange  between the two water pools may be obtained from the delay experiment. Since the assumption that myelin water is water trapped between myelin bilayers has not been completely substantiated, biochemical analyses of bovine brain white matter samples will be done in conjunction with ^ - N M R measurements. It is hoped that a correlation between biochemically measured myelin content and the myelin water percentage will be found. Finally, fixed brains have been obtained from a number of deceased MS patients. These brains will be scanned using the T relaxation and M T sequences and then cut 2  through the appropriate slice in order to perform myelin staining.  The pathology of  lesions found in the slice will be compared to the MRI results to see if correlations can be found. This may ultimately determine which MRI method best represents the state of the lesion as well as giving correlations between the pathology and magnetization transfer ratios and myelin water percentages.  Bibliography  [1] P. Morell, editor. Myelin.  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Med.,  Appendix  S o u r c e C o d e for T  2  A  Simulations  c This program w i l l c a l c u l a t e the U matrix from the set of c d i f f u s i o n equations i n a c y l i n d e r c c c a l l s TRIDIAG c PROGRAM EULER_EQN c r e a l * 8 a,b r e a l * 8 Df,Dm,T2f,T2m,Mo,k real*8 deltar.deltat integer nr,nt,counter,tester r e a l * 8 m(10000),n(10000),p(10000),q(10000) r e a l * 8 Ff.Gf(10000),Fm,Gm(10000) r e a l * 8 sum,sump,sumq,sumtotal real*8 decayp(32),decayq(32),decayt(32) character*50 filename c c Input the parameters c read(5,*)filename write(6,*) 'Enter Filename ',filename read(5,*) a,b write(6,*) 'Input radius of axon and axon+myelin:',a,b read(5,*) Df.Dm write(6,*) 'Input d i f f u s i o n constant of area 1 and area 2',Df,Dm read(5,*) T2f,T2m write(6,*) 'Input T2 of area 1 and area 2',T2f,T2m read(5,*) Mo write(6,*) ' I n i t i a l magnetisation',Mo read(5,*) k write(6,*) ' P a r t i t i o n c o e f f i c i e n t ' , k read(5,*) d e l t a r  121  122  Appendix A. Source Code for T Simulations 2  write(6,*)  'radius  read(5,*)  deltat  write(6,*)  'time  read(5,*) write(6,*)  step  step  size',deltar size',deltat  nt 'Number o f t i m e  points',nt.  nr=IIDINT(b/deltar) c c C a l c u l a t e F and G f u n c t i o n s c Ff=Df*deltat/(deltar**2) Fm=Dm*deltat/(deltar**2) c  w r i t e ( 6 , * ) ' F f and F m ' , F f , F m Gf(1)=0  c  open(unit=Gstuff,File='Gstuff.dat',status='new') Do 10 if  i=2,nr+l ( i . l e . IIDINT(a/deltar+l))  Gf(i)=Df*deltat/(2*(i-l)*  *  deltar**2) if  ( i .ge. IIDINT(a/deltar+D)  Gm(i)=Dm*deltat/(2*(i-l)*  *  deltar**2)  c 10 c  w r i t e ( G s t u f f , * ) i , ' Gf and G m ' , G f ( i ) , G m ( i ) Continue Close(unit=Gstuff)  c c c Define  initial  t  and m m a t r i c e s  c Do 15 i = l , n r + l  15  if  ( i . l e . IIDINT(a/deltar+l))  m(i)=Mo  if  ( i .ge. IIDINT(a/deltar+l))  n(i)=K*Mo  Continue  c c Use T r a p e z o i d a l R u l e t o add up m a g n e t i z a t i o n c da=IIDINT(a/deltar+l) sump=0 sump=deltar**2*(0*m(l)+nr*m(nr+l))/2 sumq=0 sumq=deltar**2*(0*n(l)+nr*n(nr+l))/2 Do 4 5  i=2,nr  sump=sump+(i-l)*deltar**2*m(i) sumq=sumq+(i-l)*deltar**2*n(i)  Appendix A. Source Code for T Simulations 2  45  Continue sumtotal=sump+sumq write(6,*)j,sump,sumq,sumtotal  c c  c c c c c  counter=0 open(unit=IData,File=f ilename, status^new') npoints=32 stdev=1.0 write(IData,*)npoints,stdev open(unit=Test,File='test.dat',status='new') Do 50 j=l,nt write(6,*) j  c * c * c  c 20 *  Do 20 i=l,nr+l i f ( i .eq. 1) then p(i)=(l-2*Ff-deltat/T2f)*m(i)+2*Ff*m(i+l) write(6,*)i,p(i) else i f ( i . I t . IIDINT(a/deltar+l)) then p(i)=(Ff-Gf(i))*m(i-l)+(l-2*Ff-deltat/T2f)*m(i) +(Ff+Gf(i))*m(i+l) write(6,*)i,p(i) else i f ( i .gt. IIDINT(a/deltar+l) .and. i . I t . nr+1) then q(i)=(Fm-Gm(i))*n(i-l)+(l-2*Fm-deltat/T2m)*n(i) +(Fm+Gm(i))*n(i+l) write(6,*)i,q(i) else i f ( i .eq. nr+1) then q(i)=(l-2*Fm-deltat/T2m)*n(i)+2*Fm*n(i-l) write(6,*)i,q(i) endif Continue p(IIDINT(a/deltar+l))=(Dm*q(IIDINT(a/deltar+2))+ K**(2/3)*Df*p(IIDINT(a/deltar)))/(K**(2/3)*Df+K*Dm) q(IIDINT(a/deltar+1))=K*p(IIDINT(a/deltar+1)) write(6,*)IIDINT(a/deltar+l),p(IIDINT(a/deltar+l)) write(6,*)IIDINT(a/deltar+l),q(IIDINT(a/deltar+l))  c c c c Use Trapezoidal Rule to add up magnetization c da=IIDINT(a/deltar+l)  Appendix A. Source Code for T Simulations 2  40  sump=0 sump=deltar**2*(0*p(1)+nr *p(nr+1)) sumq=0 sumq=deltar**2*(0*q(l)+nr*q(nr+l)) Do 40 i=2,nr sump=sump+(i-l)*deltar**2*p(i) sumq=sumq+(i-l)*deltar**2*q(i) Continue sumtotal=sump+sumq write(6,101)j,sump,sumq,sumtotal write(Test,102)j,sump,sumq,sumtotal  c c c c Save r e l e v a n t decay points c tester=nt/JFIX(nt*deltat)*10 i f ( j / t e s t e r * t e s t e r .eq. j ) then counter=counter+l decayp(counter)=sump decayq(counter)=sumq decayt(counter)=sumtotal write(6,*)j,counter,decayp(counter) write(6,*)j,counter,decayq(counter) write(6,*)j,counter,decayt(counter) write(IData,101)counter*0.01,decayt(counter) endif c  30 c 50 c c c 101 102  Do 30 i=l,nr+l m(i)=p(i) n(i)=q(i) continue continue close(unit=IData) close(unit=Test) Format(2E15.6) Format(2X,I2,2X,F13.10,2X,F13.10,2X,F13.10) stop end  

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