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Selective radio frequency pulses by optimal control

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Title:
Selective radio frequency pulses by optimal control
Creator:
Mao, Jintong, 1944-
Publisher:
[s.n.]
Publication Date:
Language:
English
Physical Description:
viii, 131 leaves : ill. ; 28 cm.

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Subjects / Keywords:
Conjugate gradient method ( jstor )
Experimental results ( jstor )
Fourier coefficients ( jstor )
Fourier series ( jstor )
Magnetic fields ( jstor )
Magnetic resonance imaging ( jstor )
Magnetization ( jstor )
Objective functions ( jstor )
Optimal control ( jstor )
Sine function ( jstor )
Conjugate gradient methods ( lcsh )
Control theory ( lcsh )
Dissertations, Academic -- Physics -- UF
Mathematical optimization ( lcsh )
Physics thesis Ph. D
Pulse amplitude modulation ( lcsh )
Radio frequency ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1987.
Bibliography:
Bibliography: leaves 129-130.
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Jintong Mao.

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University of Florida
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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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18146133 ( OCLC )
AFB3125 ( NOTIS )
AA00004843_00001 ( sobekcm )

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SELECTIVE RADIO FREQUENCY PULSES BY OPTIMAL CONTROL By JINTONG MAO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1987

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ACKNOWLEDGEMENT To Dr. E. R. Andrew, I wish to express my gratitude and deepest appreciation for introducing me to the wonderful research field, Magnetic Resonance Imaging. His intelligent guidance and friendly attitude have been a source of inspiration and encouragement. To Dr. K. N. Scott and Dr. T. H. Mareci, I wish to express my sincere gratitude and special thanlcs for their invaluable aid and advice throughout several years of graduate study. Their suggestions helped to bring this dissertation to its completion. To Dr. W. S. Brey and Dr. N. S. Sullivan, I wish to express my gratitude for giving of their busy and valuable time to the reading of this dissertation. Lastly, I would like to thank ny good friend, Mike Cockman, who gave me immeasurable assistance in correcting my English. His friendship will be remembered forever. ii

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TABLE OF CONTENTS PAGE ACKNOWLEDGEMENT ii LIST OF FIGURES v ABSTRACT vii CHAPTER I. INTRODUCTION 1 II. DESIGNING 180 DEGREE SELECTIVE INVERSION RF PULSE IS AN OPTIMAL CONTROL PROBLEM 9 III. CONJUGATE GRADIENT METHOD AND OPTIMAL 180 DEGREE SELECTIVE INVERSION RF PULSES 15 IV. EXPERIMENTAL STUDY OF THE OPTIMAL 180 DEGREE SELECTIVE RF PULSES 29 V. CONJUGATE GRADIENT METHOD AND THE OPTIMAL 90 DEGREE SELECTIVE RF PULSES 57 VI. EXPERIMENTAL STUDY OF THE OPTIMAL 90 DEGREE SELECTIVE RF PULSES 69 VII. CONCLUSION 86 APPENDIX A. FOURTH ORDER RUNGE-KUTTA METHOD 87 B. FORTRAN PROGRAM FOR ISO DEGREE PULSE DESIGN 88 C. PARAMETER VALUES OF THE PROGRAM IN THE APPENDIX B TO OBTAIN THE OPTIMAL RF PULSES OF FIG. 2, FIG. 11 AND FIG. 14 100 D. FORTRAN PROGRAM FOR REFOCUSING PULSE STUDY 101 E. FOURIER COEFFICIENTS FOR THE FOURIER SERIES REPRESENTATION OF THE OPTIMAL RF iii

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PULSE IN FIG. 2 10A F. FOURIER COEFFICIENTS FOR THE FOURIER SERIES REPRESENTATION OF THE OPTIMAL RF PULSE IN FIG. 14 105 G. FORTRAN PROGRAM FOR FOURIER COEFFICIENTS 106 H. FORTRAN PROGRAM FOR 90 DEGREE PULSE DESIGN 107 I. JOB PROGRAM FOR APPENDICES B AND H 125 J. PARAMETER VALUES OF THE PROGRAM IN THE APPENDIX H TO OBTAIN THE OPTIMAL RF PULSES OF FIG. 20, FIG. 25 AND FIG. 28 126 K. FOURIER COEFFICIENTS OF THE FOURIER SERIES REPRESENTATION OF THE OPTIMAL RF PULSE IN FIG. 20 127 L. FOURIER COEFFICIENTS OF THE FOURIER SERIES REPRESENTATION OF THE OPTIMAL RF PULSE IN FIG. 28 128 REFERENCES 129 BIOGRAPHICAL SKETCH 131 IV

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LIST CF FIGURES FIGURE PAGE 1 A desired slice profile for a 180 degree inversion rf pulse 13 2 180 degree four zero sine function rf pulse and the corresponding optimal selective rf pulse .... 31 3 The computer simulation results of the responses to the rf pulses of Fig. 2 as inversion pulses... 32 4 Computer simulation results of the y-components of the solutions of the 1000 Bloch eqs 34 5 The transverse response to the optimal 180 degree pulse of Fig. 2 when used as transverse magnetization refocusing pulse 35 6 The solid line of Fig. 3 and the line of Fig. 5 are plotted together for comparison 36 7 The transverse response to the four zero sine function pulse of Fig. 2 37 8 The spin echo pulse sequence for a one-dimensional image 39 9 The experimental result of the response to the four zero sine function pulse of Fig. 2 40 10 The experimental result of the response to the optimal 180 degree rf pulse of Fig. 2 41 11 Two 180 degree optimal rf pulses with different desired slice profile 44 12 The responses to the optimal rf pulses of Fig. 11 45 13 The experimental result of the response to the optimal rf pulse of Fig. 11 with WTR = 1/7 46 14 A 180 degree six zero sine function rf pulse and the corresponding optimal selective pulse 48

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15 The computer simulation results of the responses to the rf pulses of Fig. 14 49 16 The experimental result of the response to the six zero sine function rf pulse of Fig. 14 50 17 The experimental result of the response to the optimal 180 degree pulse of Fig. 14 51 18 Relation between the slice thickness and gradient magnitude with 180 degree selective pulse 55 19 A desired 90 degree slice 59 20 A 90 degree three zero sine function pulse and the corresponding optimal pulse 70 21 The simulated responsess to pulses of Fig. 20.... 71 22 The spin echo pulse sequence for a one dimensional image with 90 degree pulse selective 73 23 The experimental result of the response to the sine function pulse of Fig. 20 74 24 The experimental result of the response to the optimal 90 degree pulse of Fig. 20 75 25 Two Q degree optimal rf pulses with different desired slice profile 76 26 The responses to the optimal rf pulses in the Fig. 25 78 27 The experimental result of the response to optimal rf pulse of Fig. 25 with WTR = 1/10 79 28 A 90 degree five zero sine function rf pulse and the corresponding optimal selective pulse 80 29 Simulated responses to the pulses of Fig. 28 81 30 The experimental result of the response to sine function pulse of Fig. 28 82 31 The experimental result of the response to the optimal 90 degree pulse of Fig. 28 83 32 Relation between the slice thickness and gradient magnitude with 90 degree selective pulse 85 VI

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SELECTIVE RADIO FREQUENCY PULSES BY OPTIMAL CONTROL By JINTONG MAO August 1987 Chairman: Dr. E. R. Andrew Major Department: Physics Selective radio frequency (rf) pulses are important in both NHR spectroscopy and imaging. Because the Bloch equation that dominates the dynamic character of the magnetization is nonlinear, the design of the rf pulse shapes capable of precisely affecting the magnetization in a well-defined frequency range is not a simple problem. Recently, the concept of optimal control has been introduced for the design of amplitude-modulated rf pulses as a time function to provide the frequency selectivity. Several selective pulse shapes have been developed using these methods; however, very few experimental results that are consistent with the numerical calculations have been presented. In this dissertation we will give a VII

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self-contained and detailed derivation and explanation of how to use the conjugate gradient method to solve the optimal control problem. We will present the simulation results of the optimal 180 degree and 90 degree selective pulses and the experimental results that confirm the numerical calculations. The conclusion is that the optimal 180 degree and 90 degree selective rf pulses found by the conjugate gradient method have much better frequency selectivity than the widely accepted sine function rf pulses. Especially interesting we will prove in experiments and simulations that the responses to a 180 degree selective pulse will be the same no matter how it is used: as a longitudinal magnetization inversion rf pulse or as a transverse magnetization refocusing rf pulse. Thus, the conjugate gradient method has solved the problem of designing rf pulse shapes for frequency selective excitation . viii

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CHAPTER I INTRODUCTION Selective radio frequency (rf) pulses that are able to excite only those spins lying in a well-defined slice are an essential part of most magnetic resonance imaging ( MR I ) rf pulse sequences. Researchers have realized for about ten years that a conventional sine function or a sine function modified by a Gaussian function rf pulse has severe difficulty exciting only those spins lying in a specific slice without affecting the spins outside of the slice, especially for a large tip angle. Many efforts have been made to improve this situation. The publications in the field of selective rf pulse design can roughly be divided into three categories. In the first category an analytical method is used to analyze and to design rf pulses. The representative authors are G. A. Morris and R. Freeman (1), D. I. Hoult (2) and W. S. Warren (3). Morris and Freeman treated the Bloch equation as a linear differential equation and applied the convolution theorem to find the selective pulse sequence called DANTE that consists of a series of non-selective rf pulses. Since the Bloch equation is essentially not linear in rf pulses and DANTE does not combine, a magnetic field gradient that can be used to discriminate the spatial difference, the pulse

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sequence, DANTE, has few applications in MRI. Hoult applied perturbation theory to the Bloch equation and analyzed the rf pulses. Warren applied perturbation theory to the liarniltonian of the magnetic resonance spin system. Since selective 90 degree and ISO degree rf pulses cannot be treated as perturbations, the results of Hoult and Warren must be very rough and restricted in applications. In fact, the selective rf pulses that they found have a worse slice selectivity than that of the optimal selective rf pulses found by the conjugate gradient method that will be discussed in the following chapters. In the second category an analytical method combined with intuitions is used to analyze and to design the rf pulses. The representative authors are D. G. Nishimura (4) and H. Yan and J. C. Gore (5). They used two or more amplitude-modulated rf pulses each of which combined a specific magnetic field gradient and constructed a single selective rf pulse. The selectivity of their pulses is worse than that of the optimal rf pulses and the complication in practice to implement their pulses is higher than the single optimal rf pulses. Because of these reasons Nishimura never uses his own pulses in his later publications and instead he uses the optimal rf pulses. In the third category a synthetic method is used to design the selective rf pulse. "Synthetic" means here that one can start from a desired slice profile and try to find an actual slice profile that will approach the desired

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slice profile as closely as possible by changing the rf pulse shape or phase. The representative authors are M. S. Silver et al. (6), A. 1! . Lent and M. R. Kritzer (7), S. Conolly et al. (8) and M. O'Donnell and W. J. Adams (0). Silver et al. proposed the use of a complex hyperbolic secant excitation function as a 180 degree inversion rf pulse. Their result performs well for magnetization inversion, but they have not discussed the application of their pulses in the transverse magnetization refocusing and the pulse length is about 25 ms which is too long for MRI. Thus, their pulses are not suitable for MRI. Lent and Kritzer, Conolly et al. and O'Donnell and Adams found that the design of selective rf pulses can be reduced to an optimal control problem, or a dynamic optimization problem. Researchers using different methods to solve the optimal control problem found different results and very few experimental results that are consistent with the calculations have been published. The article of Conolly et al. is especially interesting. We try to give a little more detailed comments about their article. Conolly et al. had two goals in their article. First is to give an efficient algorithm to find the optimal selective rf pulses. The method used by Conolly et al. to optimize the objective function is the steepest descent method. The steepest descent method is a less efficient convergent iteration procedure than the conjugate gradient method (10). The results found by

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Conolly et al. are not quite correct because they did not use their optimal rf pulses in their recent article discussing the MR angiography by selective inversion recovery (11) and instead they use a optimal pulse that is quite similar with our optimal pulse (12). A reason that Conolly et al. could not find the correct results may be the slow convergent rate of the steepest descent method. Because of the slow convergent rate of the algorithm the authors had no good chance to conduct many calculations and to find the correct parameters leading to the correct results by using the trial and error method. Another reason that Connolly et al. could not find the correct results may be that they chose an incorrect desired slice profile because they believed that any desired slice profile could be reached by the optimization procedure and they did not realize the relation between the optimal rf pulse shape and the desired slice profile. The second goal of Conolly et al. was to prove the existence of practical selective 90 degree and 180 degree amplitude-modulated rf pulses that can cause the actual magnetization to have any desired magnetization distribution by proving the controllability of the magnetic resonance imaging spin system. Here, "practical" means that the pulse length cannot exceed the maximum rf pulse length used in actual magnetic resonance imaging and the pulse amplitude cannot exceed that which an actual rf pulse amplifier can supply. Thus, the control problem is

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one with a control function, which is the rf pulse in MR I, constrained in both pulse amplitude and pulse length. The authors did prove that the magnetic resonance imaging spin system is controllable, however, the authors did not mention that transferring to any desired distribution may require a no npractical pulse amplitude corresponding to a practical pulse length or a non-practical pulse length corresponding to a practical pulse amplitude, even though that length is finite and definite. In other words, that any desired distribution is reachable was not proven for the control function constrained in both amplitude and length . Conolly et a 1 . realized that the pulse length may be sufficiently long for the proof of the controllability, but the authors did not point out that a sufficiently long pulse length may be not practical. In the discussion of minimum distance, the authors realized that we could trade off some slice definition for weaker pulses, but the authors did not point out that a special slice definition may lead to a stronger pulse and the amplitude of the pulse may be not practical. The practical and reasonable pulse length and pulse amplitude have significant importance for practical rf pulse design. Failing to consider these restrictions in the proof proves nothing for the goal above. The controllability of the spin system without considering these restrictions is only of academic interest. These restrictions are not clearly indicated by

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the paper and the conclusion of the controllability is misleading for practical rf pulse design. Especially, the reader may be misled into thinking that any desired distribution is reachable even with the contrained pulse length and pulse amplitude because the authors had discussed the finite pulse length and the pulse magnitude limit in several places in the paper. Actually, the conclusion of the paper that the optimal control design approach allows for inversion of the Bloch equation for any desired distribution is obviously not correct for a practical rf pulse. For example, one cannot force a slice edge which corresponds to a certain slice thickness to have every finite slope by using a single rf pulse of practical pulse length and practical pulse amplitude. One can only say that in practice some desired magnetization distributions can be reached but others can not. We should also point out that in the article of Conolly et al. the conclusion that all optimal 90 degree pulses have the same shape is not correct. This mistake is because the authors did not pay any attention to the relation between the optimal rf pulse shape and the profile requirement for a desired slice, for example, the requirement for a definite slice edge slope of a desired slice profile with a certain slice thickness. A different requirement for the slice edge slope leads to a different optimal rf pulse shape. This conclusion can also be applied to the optimal 180 degree rf pulses. Ke will give

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a detailed discussion of this problem in the Chapter IV of this dissertation. We should mention that the reason why an actual slice profile obtained by an optimal rf pulse always has some errors when comparing with the desired slice profile is because any desired distribution is not reachable by a practical rf pulse, not because of the formulation of minimum distance as claimed by the Conolly et a 1 . in their article. In practical rf pulse design the squared error objective function cannot achieve zero by simply increasing the pulse length of a practical rf pulse. Thus, the main results of the article of Conolly et a 1 . are basically incorrect even though Connolly et a 1 . are one of the first to use the optimal control concept to design the selective rf pulses. In this dissertation, we try to give a detailed and self-contained derivation and explanation of the application of the conjugate gradient method (10,12,13) to design the selective rf pulses. We will discuss the experimental procedure and give the experimental results that are consistent with our theoretical calculations. Comparing the experimental results with the calculated results, one can conclude that the optimal control method has solved the selective rf pulse design problem that has existed since the appearance of the magnetic resonance imaging. The results presented in this dissertation can be

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8 considered as an expansion of our previous results (12,14,15). Chapter II will be devoted to show how the design of a 180 degree selective rf pulse can be reduced to an optimal control problem. Chapter III will give a detailed derivation of how to use the conjugate gradient method to solve the optimal problen. Chapter IV will give the calculated results for several special cases of 180 degree selective rf pulses and compare the calculations with the experiments. Chapter V will give a detailed derivation and procedure of how to use the conjugate gradient method to design the 90 degree selective rf pulses. Chapter VI will discuss several special cases of the 90 degree selective rf pulse and compare the calculations with the experiments. In the discussion in the folio w in g chapters, some variables may be ignored for brevity from their relevant functions if they are evident from the context. For example, the magnetization component Mz(t,l,U) has three variables, t, 1, and U, and sometimes we just write i i z ( t , 1 , U ) as M z ( t ) or M z ( t , 1 ) , where t is time, 1 is an integer for frequency step numbers and U is the rf pulse used as an input to the magnetic resonance spin system.

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CHAPTER II DESIGNING 180 DEGREE SELECTIVE INVERSION RF PULSES IS AN OPTIMAL CONTROL PROBLEM The dynamic equation of the spin magnetization is the Dloch equation dM _ _ Ml*i + M2*j (M3 M0)*k = r *y t x II dt T2 Tl [1] where Ml, M2 and M2 are x, y and z components of the spin magnetization vector M ; MO is the equilibrium value of the magnetization; K is the applied magnetic field; Tl and T2 are the longitudinal and transverse relaxation times, respectively; i, j and k are the unit vectors of the x, y and z axes; the asterisk, * , means multiplication; the bar, -, means vector; r is the gyro magnetic ratio. Because the Bloch equation, Eq. [ 1 ] , is very complicated for designing a selective rf pulse, Me only consider the situations in which the relaxation and diffusion effects can be ignored when the selective rf pulse is acting in a pulse sequence and Eq. [1] can be reduced to the following equation: dM = r*M x II. dt [2] The applied magnetic field can be expressed as H = HO + Gz*z*k + Hl(t)*I [3] where HO is the main magnetic field which is assumed in

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10 N X I I' = the z direction; Gz is the magnetic field gradient applied along the z direction; Ill(t) is the selective rf pulse applied along the x direction on resonance. If one consider the problem in the rotating frame, the applied magnetic field in the rotating frame is H = Gz*z*k + Hl(t)*I. [4] Because of the folio wing relation: i j k Ml M2 M3 HI Gz*z = M2*Gz*z*i Ml*Gz*z*"J + H1*M3*J + Hl*M2*k, [5] the Bloch equation without relaxation in the rotating frame is [6] where H(z) = z * G z is the magnetic field that is caused by the magnetic field gradient Gz; U(t) = -r*Hl(t) , Ill(t) is the selective rf pulse and it can be considered as an input to the Bloch equation. For convenience of discussion, we can call either Hl(t) or U(t) as the selective rf pulse. For convenience of the following discussion, one can write Ml, M2 and M3 more detailed as Ml(t,l,U), M2(t,l,U) and M3(t,l,U), respectively. An ideal selective rf pulse will excite only those spins lying in a well-defined slice without any effect to ' 111 r * H ( z ) HI d -H2 = -r*H(z) -U(t) M2 dt M3 U(t) . M3

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11 the spins outside of the slice. This requirement can be expressed by a desired magnetization profile. For the selective 180 degree inversion rf pulses, the desired magnetization can be expressed as ;;o w2 < i*dw D(l*dw) = < -MO -wl < l*dw < wl [7] MO l*dw < -v2 where D ( 1 * d w ) varies linearly from -w2 to -wl and from wl to v 2 ; dw is the frequency stepsize; 1 * d w = r * H ( z ) , where 1 = 1 , 2 ,...., L and L, is the total frequency steps. In the actual calculation, the choice of L depends on the compromise between the precision requirement and the computer time. If L is too small, for example L = 32 as O'Donnell and Adams (9) chose, there will not be enough precision in final calculational results to reflect the physical nature of the problem to be solved. This might be the reason that O'Donnell and Adams could not find the correct results even though they were the first to use the conjugate gradient method. If L is too large, the computer time will be too long. Because one should try several different values for each parameter to find the correct one for a special calculation, if L is too large, one will lose the patience to try several times and lose the chance to find the correct results. In the following calculation L is chosen as 159 in the optimization procedure of designing the 90 degree and 180 degree selective rf pulses; L is chosen as 1000 in the discussion of the 180 deeree

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12 selective refocusing rf pulses, because in the situation of the refocusing rf pulses one needs nuch nore details to understand the physical nature of the problem. The difference, w 2 wl, is the width of the edge of the desired slice profile. The thickness of the slice can be measured from the middle point of a slice edge to the middle point of the other edge of the slice and it can be expressed as 2*wl + 2*(w2-wl)/2 = w2 + wl . An important parameter for the slice quality can be defined as the ratio of the width of the slice edge to the thickness of the slice (WTR) and it can be expressed as V.'TR = (w2-wl ) / ( w2+wl ) . The parameter WTR will be discussed in detail in Chapters IV and VI. Also, the desired magnetization can be expressed graphically as shown in Fig. 1. The purpose of designing a selective rf pulse is to find an optimal selective rf pulse, K 1 ( t ) , that causes the magnetization at the end of the pulse, M3(tf,l,U), to approach the desired magnetization, D(l*dw), as closely as possible, where tf is the end of the ISO degree pulse. In optimal control theory (13), a mathematical function, which is called the objective function and is a measure of the difference between the desired response and the actual response, must be formed to quantify the quality of the control. One can use the following objective function to measure the difference between D ( 1 * d w ) and M 3 ( t f , 1 , U ) :

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13 i i i i i 0) x: 4-> CD O dJ 4-1 l-l CO 0J 0) 00 -o a o> c 00 o> • C\J 1—1 U C7> i •H — to .-1 CN X « u u co c 3 o a) Ci 1 km u. •H O ^ CM U-l K^y * i o x: 1-1 4-> O a, -a -h •H 4-> o S C3 u M •H a r-l JC 01 K> E-i u •H a • r-H 01 0) CO (4 (0 •H r-t 0) 10 3 X! 01 Q. 4-1 T3 U-l M-l < l-i o • C CO f—t O CO •H 0) • W C cc v* .* •H o> o Cn > -H C -C

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14 tf J(U) » ( dt S' [K3(t,l,U)-D(l*dw)]**2 J tO 1 + S' Rl*[M3(tf ,1,U)-D(l*dw)3**2 1 [8] where tO is the beginning of the 130 degree rf pulse; El are some numerical values to be chosen; the primed capital letter, £', means summation; [K3(t,l,U)-I}(l*dw)]**2 means squared error distance. Thus, the purpose of designing a 130 degree inversion rf pulse is to find a rf pulse, U (t), which will minimize the J(U(t)) under the restriction of Eq. [6] with the initial condition Ml (tO) = 0, M2 (tO) = 0, H3 (tO) = MO. This goal can be reached efficiently by the conjugate gradient method, which will be discussed in the folloving Chapters III and V.

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CHAPTER III CONJUGATE GRADIENT METHOD AND OPTIMAL 150 DEGREE SELECTIVE INVERSION RE PULSES To make the objective function, Eq. [8], easier to handle, one can define a new variable, M4(t,l,U), by the equation dM4(t,l,U) = [K3(t,l,U) D(l*dw)]**2 T9] dt with the initial condition H4(tO,l,U) = 0. Substituting Eq. [9] into the objective function, Eq. [8], it becomes ( tf d J(U) = S' \ — M4(t,l,U)dt + S*Rl*[M3(tf , 1 ,U)-D(l*dw) ]**2 1 Ug dt 1 = S' {M4(tf ,1,U) + Rl*[M3(tf t l f U)-B(l*dw)]**2}. [10] 1 Combining Eq. [6] with Eq. [9], the dynamic equations become dMl(t,l,U) = l*dw*M2(t,l,U) dt dM2(t,l,U) = -l*dw*Kl(t,l,U)-U(t)*M3(t,l,U) dt dM3(t,l,U) = U(t)*M2(t,l f U) dt dM4(t,l,U) = [M3(t,l,U)-D(l*dw)]**2, [11] dt which can be called the extended Dloch equation. The 15

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16 [12] initial condition for E q . [11] is Ml(tO) ] [ K2(t0) M3(t0) MO M4(t0) J L To make the discussion clearer, the dynamic equations can be generally written as — Mi(t,l,U)=fi(Ml(t,l,U),M2(t,l,U),M3(t,l,U),H4(t,l,U),U) dt [13] where i = 1, 2, 3 and 4 , and the objective function can be generally written as J(U) = J( Ml ( tf ,1,U ),..., M4(tf,l.U) H 1 ( tf, L,U ),..., H4(tf,L,U)) [14] To minimize the objective function, J ( U ) , one can consider a family of selective rf pulses, U(t)+nS(t), which are labelled by the variable q . The tine function S(t) is a modification to the selective rf pulse U ( t ) . If the minimizing selective rf pulse is U(t), the rf pulse which makes the numerical value of the objective function, J(U(t) + qS(t)), as snail as possible will accompany the variable q = 0. One can apply the standard method of the elementary calculus to determine the minimum. A necessary, but not sufficient, condition for a ninimum of a functional is the vanishing of the first order derivative of the functional. Thus, to find the minimum of the objective function, one lias

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17 dJ( U(t)+qS(t) ) = i=0 [15] Equation [15] suggests that one should find an expression dJ(U(t)+qS(t)) for first. dq Using the general forn of the objective function, Eq . [14], and the chain rule of the calculus, one can hav< dJ(U+qS) L 4 dJ(U+qS) d,"i ( tf , 1 , U+qS ) = s'S' dq q = 1 i d:Ii( tf , 1 ,U+qS ) dq q = 0. [16] To find each expression of the rip, ht side of E q . [16], one can integrate Eq. [13] and obtain Mi(t.l.U) Mi(tO,l,U) + fi(Ml MA,U)dv. [17 J to Replacing U by U+qS, Eq. [17] becomes Mi(t,l,U+qS) = Mi(tO,l,U+qS) +\ f i(Ml f . . . ,M4, U+qS)dv. > to [18] Differentiating E q . [IS] with respect to q, one obtains — Mi(t,l,U+qS) dq d , t 4 dfi dMj dfi d(U+qS) = — Mi(tO,l,U+qS) + 1 [ S' + ]dv dq J tO j dMj dq dU dq t 4 dfi dlij dfi [ s' + 5(v) ]dv tO j dMj dq dU [19] where — Mi ( tO , 1 , U+qS ) = because Mi ( tO , 1 , U+qS ) are the dq

PAGE 26

18 equilibrium magnetizations which are constants. Differentiating both sides of Eq. [19] with respect to t, one can finally derive a set of differential equations d dMi(t,l,U+qS) 4 dfi dMj(t ,1 ,U+qS) dfi = s' + S(t), [20] dt dq j dMj dq dU where i = 1, 2 , 3 and 4 . By the matrix form, the Eq. [20] can be rewritten as dfi dfi i r d:;i — — — . # # # — — — BMI dill dh'4 dq d dt dMl(t,l,U+qS) dq dM4(t,l,U+qS) dq df4 dMl df4 dM4 dH4 dq " df 1 " dU + • • df4 . dt! . (t) [21] or in a compact form, Eq. [21] can be written as d dM(t,l,U+qS) dM = fm(t,l,U+qS) — + fu(t,l,U+qS)S(t) dt dq dq [22] where M has four components: Ml, M2, M3 and M4; dMl dM dq fm dq • dM4 dq dfi dMl • df4 dMl [23] dfi dM4 df4 dM4 [24]

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19 fu [25] df 1 dU df4 dU The solution of Eq. [22] is dM(t,l,U+qS) dM(tO,l t U+qS) , t = K(t,tO) + \ K(t,v)fu(v)S(v)dv dq dq 3 tO [26] where K ( t , 1 ) is the n x n matrix solution of the matrix differential equation dK(t.v) = f m (t)K(t,v); dt K(v,v) = I where I is an unit matrix. This can be proved easily by differentiating botli sides of Eq. [26] with respect to t and combining Eq. [27] as the following: d d_M dK(t,tO) dM(tO) dt dq dt dq t dK(t,v) fu(v)S(v)dv + K(t,t)fu(t)S(t) tO dt [27] fra(t)K(t f tO)dM(tO) dq fm(t)K(t,v)fu(v)S(v)dv + fu(t)S(t) to dM(tO) ( t = fm(t)[K(t,tO) + 1 K(t,v)fu(v)S(v)dv] dq ) tO

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20 + fu(t)S(t) dM(t) = fm(t) + fu(t)S(t). dq The left side of above equation and the right side of the last line of above equation consistute an equation which is identical to the Eq. [22], Thus, the proposal that Eq . [26] is the solution of Eq . [22] is proved. Since i ! ( 1 , 1 , U + q S ) is the equilibrium magnetization, dii(tO) dq [26] becomes dM(tf ,l,U+qS) dq tf thus = ; letting t = tf and combining Eq. [25], K ( t f , v ) f u ( v , 1 , IJ + qS ) S ( v ) d v to tf dv S(v) tO tf dv S(v) to K 1 1 ... K 1 4 KA1.. .K44 4 df j S' Klj j dU dfl diJ df4 L dU [28] 4 df j S 1 K4j j " dU where Kij is the ith row, jth column element of the matrix K. In the component form, Eq. [2 8] can be written as

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21 dMi(tf ,1,'J+qS) f tf 4 df j(v,l,U+qS) = \ dv S(v) S' Kij . dq 3 tO j dU Combining Eq. [16] with Eq. [29], one gets dJ(U+qS) dJ dMi(l) = S' S' dq 1 i dMi(l) dq dJ ( tf df j(l) = s* S' \ dv S(v) S' Kij 1 i drii(l) J tO j dU [29] tf dfj(l) dv S(v) S' S 1 S' dJ ij to tf to l j du i d;;i(i) dv S(v) G(v,U+qS) L A dfj(l,U) k dJ(v,U+qS) where G(v,U+qS) = S' S* S' Kij. 1 j dU i dMi(l) [30] [31] The function G(v,U+ qS) is called the gradient of the objective function. Equation [30] is the wanted expression for dJ(U+qS)/dq. It will be used to prove that the conjugate gradient algorithm is able to minimize the objective function, J(U). With the conjugate gradient algorithm, as with any convergence algorithm, the first element of the convergent sequence, U(0,t), is simply guessed. The remaining members of the sequence are derived from the following Eqs. [32] [35], Equations [32] [35] are called the conjujate gradient algorithm (13). S(0,t) = -G(t,L'(0,t)), [32] U(i+l,t) = U(i,t) + qS(i,t), [33] where q > is chosen to minimize the objective function

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22 J ( U ( i , t ) + qS(i,t)) and the minimizing q can be written as q(i), i.e. U(i+l,t) = U(i,t) + q(i)S(i,t), S(i+l,t) = -G(t,U(i+l,t)) + b(i)S(i,t), [34] b(i) = tf i tO tf dt G(t,U(i+l,t))**2 > 0. [35] tO dt G(t,U(i,t))**2 One can use the method of reduction to absurdity and prove that the algorithm Eqs. [32] [35] will make tiie objective function smaller and smaller. Eventually the objective function will reach a minimum. One can start from any guessed initial rf pulse U ( i , t ) . One can assume that no q > exists for which J(U(i,t) + qS(i,t)) < J(U(i,t)). [36] Therefore one must have J(U(i,t) + qS(i,t)) > J(U(i,t)). [37] The right side of E q . [32] can be moved to the left side and divided by q > 0, one has J(U(i,t) + qS(i t t)) J(U(i,t)) q riJ(U(i,t) + qS(i,t)) dq According to Eq. [30] and combining Eq . [34], S(i,t) = -G(t,U(i,t)) + b(i-l )S(i-l , t) , one can have dJ(U(i) + qS(i)) dq q = > 0. q = [3S]

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23 tf dv S(i,v) G(v,U(i) + qS(i)) tO q»0 tf dv S(i,v) G(v,U(i)) to tf to tf dv (-G(v,U(i)) + b(i-l)S(i-l,v))G(v,U(i)) tf dv G(v,U(i))**2 + b(i-l) \ dv S(i-1 , v)G(v,U(i) ) . tO 1 to [39] One can combine Eq. [30] with Eq. [33] and find the value of the second term of the right side of Eq. [39] as the f ol lowing : dJ(U(i-l)+qS(i-l)) q-q(i-l) tf i tO tf to dq dv S(i-l,v)G(v,U(i-l) + qS(i-l)) dv S(i-l,v)G(v,U(i)). q=q(i-l) [40] Since q is chosen to minimize J(U(i-l) + qS(i-l)) and the minimizing q can be written as q(i-l), one has dJ(U(i-l) + qS(i-l)) dq q=q(i-l) Therefore the value of the second term of Eq. [39] i tf = 0. [41] dv S(i-l,v)G(v,U(i)) = 0, [42] tO Substituting Eq . [42] into Eq . [39], one can have dJ(U(i) + qS(i>) dq q = tf dv G(v f U(i))**2 < 0. [43] to

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24 One may notice that E q . [43] contrr. diets E q . [38] unless 0(v,U(i)) = 0. Thus if G(v,U(i)) is not equal to zero, then conjugate gradient algorithm will make the objective function smaller and smaller. To apply the conjugate gradient algorithm, one needs to find an expression for G(t,U(i)) which can be valued from the objective function and the Ploch equation by the numerical procedure. One can start from E q . [31] and define a new variable dJ Qj(t) = S' — Kij i dMi [44] or by matrix form 01 02 03 . Q4 . 1:11 ... K4i L K14 ... K44 dJ dMl dJ d!14 J In a compact form, Eq. [44] can be written as Q(t) = K + (tf,t)Jm where + means transpose; 01, Q2, 03 and Q4 are the components of the vector 0(t); the letter m of Jm means differentiation with respect to Ml, M2, M3 and M4 . To obtain a soluble equation for Q(t), letting S(t) = and setting y(t) = df!(t)/dq, Eq.[22] becomes dy(t) = fm(t)y(t). dt According to Eq. [26], the solution of Eq. [46] is [45] [46]

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25 y(t) = K(t,tO)y(tO), [47] letting t = tf, tO = t, E q . [47] becomes y(tf) = K(tf,t)y(t). [43] Combining E q . [47] and E q . [48], one can get y(tf) = K(tf ,t)X(t,tO)y(tO). [49] Letting t = tf, Eq. [47] becomes y(tf) = K(tf,tO)y(tO). [50] Comparing Eq. [50] with Eq. [49], one can conclude that K(tf,tO) = K(tf ,t)K(t,tO). [51] Assuming that tO and tf are constants, differentiating Eq. [51] with respect to t and cor. bin in g Eq. [27], one obtains dK(tf,t) dK(t.tO) = K(t,tO) + K(tf,t) dt dt dK(tf.t) = K(t.tO) + K(tf,t)fr.(t)K(t,tO) dt d?:(tf,t) = ( + K(tf,t)fm(t))K(t,tO). [52] dt Since K ( t , 1 ) is not equal to zero, one gets from Eq. [52] dX(tf ,t) = _K(tf ,t)fm(t). [53] dt According to Eq. [27], the terminal condition of Eq. [53] is K(tf,tf) = I. [54] Taking the transpose of Eq. [53] and Eq. [54] produces dK + (tf,t) = -fm+K + (tf,t); dt

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26 K+(tf,tf) = I. [55] Postmultiplying both sides of En. [55] by Jm gives dK+(tf,t) j n , = -fm+K+Ctf ,t)Jm; dt K + (tf,tf)Jm = Jn. [56] By the definition of Q(t), Eq. [45], Eq. [56] becomes dQ(t) = -fra + Q( t ) ; dt 0(tf) = Jm. [57] Equation [57] is the soluble equation for C(t) and is called an adjoint equation of the Eq. [27]. After solving Eq. [57], one can finally get the gradient by Eq. [31] dfj(l.U) G(t,U) = S' s' Qj(t). 1 j dU [53] -fn + = Considering the extended Eloch equation, Eq. [11], the definition equations, Eq. [24] and Eq. [25], become l*dw -l*dw -U(t) U(t) -2(M3(t)-D(l*dw)) -H3(t) M2(t) Considering the objective function, Eq. [10], the terminal condition of Eq. [57] becomes [59] fu(t) = [60]

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27 Q(tf) = : El*(M2(tf)-P(l*dw)) ] [61] According to E q . [60] and Eq. [61], the gradient of the objective function, Eq. [Si 1 ], becones L G(t,U(i)) = S' (-M3(t,l)Q2(t,l) + M2(t,l)Q3(t,l)). [62] In summary, the procedure for designing the ISO degree selective inversion rf pulses is in the following : 1. Guessing an initial 180 degree rf pulse, for ex an pie, a 150 degree truncated sine function rf pulse, integrate trie Eq. [11] under the initial condition Eq. [12] and find the first value of the objective function, J(U,0). 2. According to the solution of Eq. [11] and using the Eq. [59] and Eq. [61], construct the adjoint equation Eq. [57] and integrate Eq. [57] backwards in time from tf to tO to get the vector function 0(t). 3. Construct the gradient of the objective function by Eq. [621. 4. Using Eq. [32] and Eq. [23], construct a new rf pulse, U(t). The numerical coefficient, q, is chosen to minimize the objective function J for the given U(i,t) and S(i,t) and the minimizing q can be found by using the Golden Section Search (16). The new rf pulse is use:! to find a new value of the objective function, J(n). Comparing the new value with the old value of the objective function,

PAGE 36

28 J(n-l), stop here if J(n) > J(n-l), or construct a new 2 r a c! i n e t and continue by u s i n g E q . [34] if J ( n ) < J ( n 1 ) , To solve the extended i:loch equation, Eq. [11], and the adjoint equation, Eq. [52], numerically, the fourth or ( sr Runge-Kutta method can be used which is in the Appendix A . The four steps above can be implemented by a Fortran program that is in the Appendix B. The parameter values used in the program of Appendix B to obtain the optimal rf pulses in Fig. 2, Fig. 11 and Fig. 14 of Chapter IV are in a), b) and c) of Appendix C.

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CHAPTER IV EXPERIMENTAL STUDY OF THE OPTIMAL 180 DEGREE SELECTIVE RF PULSES A detailed derivation and procedure of how to use the conjugate gradient method to design the 180 degree selective inversion rf pulses is already given in the Chapter III. In this chapter several special cases of 180 degree selective rf pulse will be discussed and the simulated results will be compared with the experimental results. The experimental results have confirmed the simulation prediction that the optimal 180 degree selective rf pulses have much better selectivity than the widely accepted sine function rf pulses in the both situations of longitudinal magnetization inversion and transverse magnetization refocusing. An important conclusion of this chapter is that the responses to a selective 180 degree rf pulse will be the same no matter how it is used: as a selective inversion pulse or as a selective refocusing pulse. Thus one can apply the design procedure for the 180 degree selective inversion rf pulse to obtain the selective rf pulse for the transverse magnetization refocusing. Before starting a computation, one should choose a desired slice profile and an initial trial function. A desired slice profile for a 180 degree selective inversion 29

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30 rf pulse is already shown in Fig. 1, The slice thickness is measured in frequency unit. As it was already pointed out in the Chapter II, the important parameter of the slice quality is the ratio of the width of the slice edge to the thickness of the slice (WTR) and it can be represented by WTR = ( w2-wl ) / ( wl+w2 ) , where w2-wl is the width of the slice edge and wl+w2, which is equal to 2*v;l + 2*(w2-wl)/2, is the half height thickness of the slice measured from the middle point of an edge of the slice to the middle point of the other edge. It is obvious that the smaller the WTR value, the better the slice quality, if all other conditions are the same. A four zero truncated sine function, as shown in Fig. 2 (dotted line), is chosen as an initial trial function for the conjugate gradient algorithm. The inversion response to the 180 degree four zero sine function rf pulse is shown in Fig. 3 (dotted line). If the slice thickness of a desired slice profile is identical with that of the response to the four zero sine function pulse and if one chooses WTR = 2/9, as shown in Fig. 1, the corresponding optimal inversion rf pulse is shown in Fig. 2 (solid line) and the inversion response to the optimal rf pulse is shown in Fig. 3 (solid line). Since a selective inversion rf pulse will be used as a selective refocusing rf pulse in the following discussion, the solution of the Bloch equation without relaxation in the refocusing situation will be considered here. For explanation, one can consider a simple situation in which

PAGE 39

31 epn)!|duiv ©A|jB|9y 4-1 CD O co co c to CC en jz o i— i OJ 1 — I 0) 4J •H 3 CO c 4-1 P. i— 1 CD J* JZ u 3 > U 4J c Uh C •H •H 1-1 3 Ih 4-1 C s <4-4 VM u 4-1 I— 1 1-4 a> I-l u co H 0) CO c E c CD u a H H o CO •H -H ca 4J •H iH 4-> a. 4J i-l CO C o o u ec CD Ih c E "3 -a 0) 01 s •H 01 -H N .n <4-l 4-1 l-i 4-1 O•H CO l-i 09 u o CO -H 3 c E c ID O •H •H OCX) CD "4-1 CO 0} c CO 4-J o •H 0) i— 1 OJ x E o — X! 3 ID c u c H s. u H d) o ca o N a • y-i QJ 4-) 09 '-» t-i "3 U 0) CD p. 3 Ih C i-H c H O 1H CO CC s 4-4 O iH E 1 — 1 u •H I-l 01 T3 4J 0) 0) OJ u •H CujC 4-1 U jG i-I o 4-1 01 CC 4-1 O E u co ai o co -3 T3 wx: 4-J Ih C 4J CO o CB a OJ a, 00 co c CO 1 — 1 ^— s i—l -H c 0) OJ 3 CO o J3
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32 o o o o *CO 0) CO o 4-1 W -3 0J C B c O O •H O. B.|H co a> to oj X OJ T3 Ih a-) CD 4-1 0> co c 4J x: CG u O j-> TT OJ •o 1+-I Cv c CO o co •H X! 3 i— 1 4-1 CO ^ •u OJ -c •a N 1-1 I* •H c z a d .— I to w co c a ^ n CN >. >-. o • o X! co • CM 3 c c jj X! CO o H •H • 3 •H c fc CC 9 4J CO • •H • 03 X! 09 U-l fl-, Ui— I 3 0) o 3 CO U-l E CN rH CD O •H 3 CO CO • a.tH oj OC 3 CO li •H 4-1 PUiH 00 Exl-i 3 4J vm a. 3 IW c u c O c <4-l E •H OJ u O Efl (0 e (J OJ Ih •H CD co 0) rH C ai i— i >• •H x: 3 c •a «h H a. H •H • M-4 a> O 0J o o o o co • in 0) > •H CO 4-1 JJ 0J o cox; u x: -o V •H 4J CD 4J 1 Cju 1—1 CJ o CD o x: 4J CO 4J 4J

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33 there is no selective rf pulse and there is only a z-direction magnetic field gradient. In solving the Bloch equation without relaxation in the rotating frame by computer, there are 1000 points in the gradient dimension and each point corresponds to a solution of the Bloch equation without relaxation in which the specific z-direction magnetic field is determined by the gradient. If the same initial transverse magnetization along the y-direction is used for each solution, the y-components of the solutions for 1000 points can be plotted together and are shown in Fig. 4. The dephasing action of the gradient is not included in this simulation. If the optimal selective rf pulse of Fig. 2 is used as a selective refocusing pulse for the transverse magnetization, the response to the optimal pulse is shown in Fig. 5. For comparison, the solid line of Fig. 3 and the line of Fig. 5 are plotted together in Fig. 6. Two lines of Fig. 6 fit together very well. If the sine function rf pulse of Fig. 2 is used as the selective refocusing pulse for the transverse magnetization, the response to the sine function pulse is shown in Fig. 7 (dotted line). The solid line of Fig. 7 is the dotted line of Fig. 3. Two lines of Fig. 7 fit together very well also. Thus, the responses to a selective 180 degree rf pulse will be the same in either situation of use: as a refocusing pulse or as an inversion pulse. In the following, the selective ISO degree rf pulses will be used experimentally as transverse

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34 -n O o o o CO 4-1 c 0! C O 4-> CX 3 E O JZ U 4J 1 -H o co •H CO x: 4J 3 CO 0) 1~ c o •H 4-> a c o H 4J CO 3 D" 0) Xi u c c •3 CO M C •H -H 4-1 CJ e c o CC-H cc 4J E co 3 CD 3" co u CO o C t-i co m 4J x: u i-H CO cc a> •H 4-J t-l H O C U-4 3 U N 4-1 111 cd x: C 4J co cc >. E r-l B C O o a o E -H CO 4J co u X! o cc a) c O -H • •.4-1 4J 4-J O C C CJ o X U -H •H *3 3 CO I U N CO cu E cc CO CD I* x: -h 4-1 "3 l •3 >> U C <4-l CO CD x: oo "3 4J C CD -H •H 00 4-1 i-H C CO eu o 4-> B.r-1 O CO cc i-

PAGE 43

35

PAGE 44

36 All? •M** .•.-.-.., L o o o o e line of Fig. 5 ordinate represents ations . *"* J3 N 4J 0) -H N X J3 4-> %* 3 an on . magn o > CO c . -H 0) 00 J-c CD 3 •H CO 1h o fa D. u. . The solid line of ted together for co gitudinal and trans o vO JJ C o o o o • t-H rH o T 00 O. fa 0) 4J I* O CO ^3

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37 o "-"/.".". ::::;-.-;:::rA~. ."."".":";*. " ~.^ll"~ : Ji wr"1"!"".' "I""*-w».'.'.'".'."".":::r.v;:»««'.'•V.V"/.",'. :vvv-.-.v.-.v.-.-.'. ":::"-••:. ~:::;-.-.t:>.~ ".V.V.V.V.* '.V.v"::=i-.w '.'.""".:•:::;-.-.-_-.-.-.-.•.-..•.. '".'."ill" -"•V.V.""r**^* " :::: * :: =»__ o o o o 2 i a • a j= CO -C fre 4J o u •H c CO a 4-1 •H a « a. CO w r^ jN •a 3 cc •H o CD 12C 4-1 u CO H 0) CD 3 <4-l T3 c N I-. Ui OC C o CO U OJ oc E 3 si c 0) O -? H -3 0) VM K H CO /N 3 1(1) 0) u • OJ x: c m > 4-1 •H U-l CO 1— 1 CJ • c o u OC co 4J — H Ih CD c fc 4-1 0) 4J o go 4-1 •i-f 14-1 "O c o 4J O c o -c CO CO c. s-^t-: a. 1 (0 •w c i-H ai CNI 4-> -H CO m 01 r-i c • c •H 01 DC 00 -o "O CO •H CO CD 5 iU. E 4J 4-) ai 4-1 •H > <-J CD c 00 CO c M -c c c U. o CO OJ 0! a' rH u CO > .3 4-> — i CO 4J J= 3 c 4-1 0) a. CG CO o x: u i.3 E-h K-l 4-1 u 01 CO • CD c 4J r^ c > H c o H iH ai • -H 4J CO OC 4-> U T3 cu •H u OJ H u b C i— 1 .— 1 Q. 3 (U o CU 14-1 K CO t-

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38 magnetization refocusing pulses and the experimental results will be compared with the simulation results when the 180 degree pulses are used as longitudinal magnetization inversion pulses. The Fortran program for selective refocusing rf pulse study is in Appendix D. The instrument used in the experiments is the CE CSI-II 2T spectrometer /imager system. The sample is a tube of water and it is along the main magnetic field ( z-direction) . The tube length is 100 mm and the tube diameter is 10 mm. A spin echo pulse sequence, as shown in Fig. 8 , is used in the experiments. In the pulse sequence, the 90 degree rf pulse is not selective and 180 degree refocusing rf pulse is frequency selective. The pulse sequence in Fig. 8 results in a one-dimensional projection image in the z-direction. In the experiments, the non-selective 90 degree pulse length is 165 us, the four zero sine function selective 180 degree pulse length is 4.5 ms and the six zero sine function selective 180 degree pulse length is 4.5 ms as well. The selective 180 degree rf pulse lengths can be reduced by increasing the output power level of the rf pulse amplifier. The echo time, TE, is 30 ms and the acquisition data block size is 1024 points. The experimental results for the 180 degree four zero sine function rf pulse and the corresponding optimal rf pulse are shown in Fig. 9 and Fig. 10, respectively. Comparing Fig. 9 and Fig. 10 with Fig. 3, one can see that

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39 0) > •H j-i 4J o C CD 01 H •H (U a w CO Sn • cc 4.) c 1 TO c 0) 1-t 11 o •H CO ^s M H a •H O J-> CO 4-1 14-1 U u C OJ DC 0) u u s o •H CC o^»""^ c -a c cu 01 1 •H CO ( 3 K CO CT to • V 0) cu x; 4-1 ) CO V a. c i < y M ai 3 cu co l-i / O.X! M I Eh -a O 0) DC si • i-i C u o u H "V s 01 CO 0) H CO <4-t 3 c E 0) U •H •H M o D. 4-1 10 i— 1 0) i) CC .o U (1) c ^ J= o c TD f H H CO C f w u CO I • c V cc 3) 1—1 4-1 • N — E CO > a m r 1 ^^^^ • 11 C0T3 D H •H 1 0) o Cn 0) 4-> CO C c h N o •H cc

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40 o — o o " o ^~ cu -C 4-1 • O JJ • oo cu •H CO fc c o VM a. O CO cu cu >H CO 1— 1 cu 3 -C c J-> ***. Hh U-l Ih N o 0) w J-J cu rH u >• 3 CO o u CO cu c 0) T3 e (-1 3 a e O CO o o o o 0) o (c H X cc I— ( •u c c cu • N 00 •H 1tL, 3 O 4-1

PAGE 49

41

PAGE 50

42 the experimental results are very close to the computer simulations . The reason why there is not any signal in the experimental results of Fig. 9 and Fig. 10 that corresponds to the oscillating magnetization of Fig. 6 and Fig. 7 is the gradient dephasing action. Applying a pulse sequence in which there is no 180 degree selective refocusing pulse and the rest is the same as in Fig. 8, one cannot get any signal experimentally, because the applied gradient of the pulse sequence will dephase the magnetization. By solving the Rloch equation without relaxation, the y-component of the magnetization for the pulse sequence without selective rf pulse is already shown in Fig. 4 . Thus, Fig. 4 corresponds to zero signal obtained experimentally by the pulse. sequence without selective pulse. By applying the pulse sequence of Fig. 8, the selective refocusing 180 degree pulse will select only a portion of the oscillating magnetization of Fig. 4. The refocusing gradient has the refocusing action only to this portion of the magnetization. The magnetization that is not selected by the 180 degree pulse will be dephased. Thus, one can only obtain the signal from the selected and refocussed magnetization. If the thickness of a desired slice is identical with that of the response to the four zero sine function rf pulse and if one reduces WTR, that is, one tries to produce a sharper edge, for example, one chooses WTR = 1/7,

PAGE 51

43 the optimal rf pulse obtained is shown in Fig. 11 (dotted line). The corresponding response to the dotted line pulse of Fig. 11 is shown in Fig. 12 (dotted line). One can notice that even though there are sharper edges on the slice, there are more artifacts on the top of the slice and outside of the dice. Thus one may conclude that for a particular pulse length there is a minimum value of WTR, represented by WTRmin, at a certain slice thickness. If WTR > WTRmin, one can always find an rf pulse to which the response will be very close to the desired slice. If WTR < WTRmin, one can obtain a sharper slice edge, but the price is that there are more artifacts on the top of the slice and outside of the slice, as shown in Fig. 12. The above observation has been proved by experimental results as shown in Fig. 10 and Fig. 13. If one changes the pulse length while retaining the shape, the slice thickness will be changed. For example, if the pulse length of a IPO degree four zero sine function rf pulse is doubled and the pulse shape is still four zero sine function, the slice thickness will be half, if all other conditions are the same. However, the value of WTR, which determines the sharpness of a slice, does not depend on the pulse length and will not be changed if only the pulse length is changed. From the discussion above one may conclude that one cannot make a slice as sharp as one wants by a single selective rf pulse at. a certain slice thickness. The property from the discussion above can be attributed to

PAGE 52

44 in * 4-1 If4 (U c tu CD •H u i— 1 r-J 0J cd cc ai CO X! 4-1 E c 4-> VM •H •H •o H 4J t-H oj c T) Q. u 4J c "O •H x: a> 00 CD 4-1 B 4J CU CD •H ra 4J -c C s c o CO c 0J c. CO 4J jr CO 0) C • 4J 0) en v cs kl -I DO c 3 1 • a; SJ D. u OC X! XI c •H s 4J y-i cj Pn ki u a. <4-l <4-l do o .— i CD c 1— 1 to E TO c 3 4-1 E •H 4-1 a. co •H i— I a x: ^^ 4-1 X! i«-i 4J • E Q.-U 4-) ki o •H x: i— 1 00 ^H 4-> • H OJ O IS TO •h r(1) C/j E S \ k. 0) -H 1— 1 ec • E 4J 1— 1 01 OJ cs CL ca II a i— 1 H n c u •H « O "4-1 OJ ki 4J H 00 o jr 0) C 3 I— 1 kl 4-1 XI 0) Q. 4.) -a 4-i o CO o •H 3 > SJ •H c XI H u CC CO •H XI •H « • iH u n 0J 1—1 CO •H 4J co n i— 1 -C C CO -H -G 3 o CD i-H • CD CO c cc kl u OJ .* -3 •H •H CO k< O -H Pn 09 T— 1 arid a; 3 CU X! O •a a. u 4J CC epnj!|duuv eAijeisy

PAGE 53

45 — o o o o o o o o 0) U-t c o •H r-l oo V -a oo •H !-H iH a O D. oo tM ai >H x; 4-1 f-H CO o E 4-> ^^ •H N 4J 00 X CUTS w O c o >. u o. o c 9 ja 00 j_) CD 3 Ih CT u 9 4-1 o *(J U. 00 0) 0) 00 c c •H o 1—1 D, 00 •o 0) •H u i— 1 o CD 00 Xi H V • H CN i-H • • i— i >— i • I— 1 .—1 oo •H • • &-, QO O0 2 l £, &-.

PAGE 54

46 o o o o o o o o o d ai I-C ai cc <4-i co -i x: -h o • 4-1 cu e o 4-> 4J U O i-i O H 1C CD 4-> 3 x: x: co • CO 4-> 4J 4J 00 CU 01 CJ -H co u co x: co tt.. e cc CC4-I O CU 3 -H <4-l D.JC C O 4J O WHO XI u CU -H 4J CO 4-1 U • 4-1 CC r-h c cu x: ai ^-•c 0) (-1 4-1 JS —> c > o 4-1 O tJ E C II CJ co <4-l • cu x: N X O KH CJ U 4J E-h CO Du CO *^ u2 iJ 03 0) i-H C X ID U >, 3 r (u CO 1-H CJ K 4-1 E CU i—l c CU i-H -H cu xz co U 3 U CO 4J 3 CU iH CU a i— 1 •-< c 3 « x: • CO -H X c d' 4-i 4-1 CD u C • O «*-i 3 CO "WOO QJH U x: 3 cu 4J w -a H O.T3 am c c CU 00 CO • U-> 3 CJ -a cn u x cu c •— i -a cu o t-H 0) 1-4 4-1 • CO c o> CU Ot E-H a. cu •rl -H CC • l-c O t4-1 4J CC CO -H Q.X) •H X: r-l O O En CO CO

PAGE 55

47 the nature of the spin system. By the terminology of control theory, one properly could conclude that not every desired slice profile is reachable by a practical rf pulse. If one tries to improve the slice quality, that is, to reduce the value of WTR, it can be noticed that one must choose a desired slice that has a broader slice thickness if the pulse length is fixed. Because the conjugate gradient method, like most other iterative numerical procedure, can only find a local minimum, one can start the computation from a trial pulse shape to which the response will have a nearly equal slice thickness to that of the desired slice. One can choose a sine function shaped pulse with more zeros as the initial trial function because the pulse of sine function with more zeros will have a broader frequency response for a fixed pulse length. For example, one can choose a desired slice that has the slice thickness which is identical with that of the response to a six zero sine function rf pulse and WTR = 1/7, the corresponding optimal rf pulse is shown in Fig. 14 (solid line). The simulated response to the optimal pulse is shown in Fig. 15. The experimental results are shown in Fig. 16 and Fig. 17, respectively. The slice edge in Fig. 17 is sharper than that in Fig. 10; however, the slice in Fig. 17 has a broader frequency range. If one wants the same spatial slice thicknesses for two slices at a specific value of WTR, the broader frequency range requires a higher gradient. One can notice

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48 epnijidwy eAijeiey O co ro CO co jc, c 0) r— 1 d a_ o 03 c •H tH i— 1 V _* x: 4-1 I-, s > U 4-> U CX-H •H -H C iH 4-> C s 3 CO <4-4 O 4J CO c 03 O E •H 0) rH cu O T3 J3 3 CU o — h C E-i O. 14-> 0) O CO N a. • in 0) X) 00 <-» U O 0) X 0) 0) (0 •H I-. C rH O 3 CO kl •H CO 00 rH E rH DS a u •H H a> -O 4J 0) 3; M o •H O.JS oox: rH O 4-1 1m cu 4J O CU •a 03 01 O 4J "S w.C u (U o c 4-1 E CO CO 03 CU CO r— I co c 03 U /~s rH -H C CO • < a* 3 CO o a.r~ c CX 4-> CU *»«. • •H -3 CO 0) i— I CU -C CU CU tH 4J l-i "O 4-1 03 CO pt, o 00 0) rH rH X) CU CO »H 3 3 ^^ "O 3 o o, o.

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49 — o o o o o o o o 2 I > O w h cu co X! c j-i a >. o o •H c H c 4J cu o CO jC • 3 i-H * a a 3 CD E 00 3 •H C.1H Ctl. S o o O •H VH o CO CO o CU l-> u n C H 3 C -H a-riH m <4-l CD T3 t— i •H i-H • 1) tJ O CCJ3 U CO •H •u> 0) pt. i-t CU O D£ 4J CO *J

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50 co 0! CO CD JC js h e •u o • -H O ST 4J WH-H T3 CD • C m oo o C «H u O &L, D. H co >4-i ce CD O 4-1 »-i c CD CD CD CO E •C i-4 -H 4-> 3 M D. CD 4-1 D. ^ O «W X N 1CD X 4J • i— 1 CD CD CD 3 CD a o. >. CO Ih CO CO u CD 00 CO .C c ^ CD O 9 "O 0) 3 • iH X! CD co o -u co 4J CO i-t U. C i-H >i 3 cd gi a E C T) •H O C O. U "O X C 3 c o <4-l -H H-l CD CO .c o w jj H c -a a. •H O CD • CD O v£> CO X t-H O -H CD u • CD 4-1 O * MNH •H 3 • bx en oo •H 4) -H CO l-i Lin

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51 o o o o o o o o o 01 M-l jr O 4-1 4J o co to 4-> -H 4-) 0) J-> to <-t 00 C 3 to o to Q. 0) c m u o G •H Vh G 4J jC •H 01 H T3 J3 C 4J • o ** o VH .-h ^ o rH N • to I 4-1 OC 4J «-H -H C G 3 Cn g ex, >> to E to V 01 — H X! c u o U CO o G a w i-H OJ C G o co x to 4J tH G iH U. C 3 SI o> O. G — E 5 1-f <4-l tO G Ih (-. CO JZ 0) 4-> O. G G X 0) JS u 0> U> 4-1 O 60 y-i 01 G u JC T3 G 4-1 H •U O. O C G • 00 3 U r^ t-h X i— t X) G .—i G • « C CT\ 00 E 1-1 •H -H to • fa4-1 4J 00 G.J3 -H

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52 that the optimal rf pulse corresponding to the 180 degree six zero sine function pulse has a higher power level than the optimal rf pulse corresponding to the 180 degree four zero sine function pulse if the pulse lengths are identical, or the optimal rf pulse corresponding to the 180 degree six zero sine function pulse has a longer pulse length than the optimal rf pulse corresponding to the 180 degree four zero sine function pulse if the power levels are identical. Thus, one may conclude that a higher slice quality requires a higher rf pulse power level in addition to a higher gradient or a longer rf pulse length. Even though the experimental results in Fig. 10 and Fig. 17 are very close to the calculation results, there are still some differences between the experiments and calculations. The reason for the difference between the simulations and the experiments is in the following: the rf amplifier in our instrument is not quite linear, especially in the regions near zero and maximum power level. Even though one can make some compensations for the nonlinear i ties , it is very hard to make the rf pulses in the experiments exactly as shown in Figs. 2, 11 and 14. Because of the nonlinearity of the rf power amplifier, the optimal rf pulse corresponding to a more zero sine function pulse will be more difficult to ;aake since it has a higher power level and a more complicated structure near the zero power. The second reason for the difference is that in our experiments one must input the rf pulse

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53 amplitude point by point; the pulse sequence compiler will allow at most about 90 points to make an experimental rf pulse. If there are more points, the CSI-II system will not compile the pulse sequence. However, 90 points are not enough to accurately make a experimental rf pulse. If there are some improvements in above two situations, the experimental results will be closer to the calculations. The author was required to input the experimental rf pulses point by point; however, a Fourier analysis of the calculated optimal rf pulses might be useful for some readers in defining the optimal rf pulses. The Fourier coefficients for the Fourier series representation of the optimal rf pulses in Fig. 2 and Fig. 14 are shown in the Appendices E and F, respectively. The Fortran program to obtain the Fourier coefficients of the Fourier series representation of the optimal rf pulses is in the Append ix G . The relation between the slice thickness and the gradient magnitude will be discussed in this paragraph. If the pulse sequence of Fig. 8 is applied, all three intervals of the z-direction magnetic field gradient will relate to the thickness of the slice. Since there is a quantitative relation between the dephasing interval and the refocusing interval, there are only two independent gradient magnitudes that will determine the slice thickness. The sample is a tube of water with a partition at the middle of the tube. The length of the tube is 100

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54 ram and the diameter of the tube is 10 mm. The thickness of the partition is about 1 mm. The experimental results is shown in Fig. 18. If one doubles the selective gradient magnitude while keeping other conditions unchanged, the slice thickness will be half with the same value of WTR , or vice versa, as shown in a), b) and c), d) of Fig. 18. From a) to b) of Fig. IS, the selective gradient magnitude is doubled, the frequency thickness and the spatial thicicness of the slice are half simutaneotisly. If one halves the refoc using gradient magnitude while keeping other conditions unchanged, the frequency slice thickness will be half with the same value of WTR, or vice versa, as shown in a) and c) of Fig. 18. In this case it can be emphasized that the spatial thickness of the slice is not changed even though the frequency thickness of the slice is half. If one doubles the refocusing gradient magnitude and the selective gradient magnitude simultaneously, the frequency slice thickness will not be changed and the spatial slice thickness will be half. Thus, if one doubles the selective gradient magnitude, the spatial slice thickness is always half, no natter the refocusing gradient magnitude is changed or not. It can be clear from the Fig. 18 that the following relation: dz = dv/(r*Gz) [63] is true only when Gz is the refocusing gradient and it is not true when Gz is the selective gradient, where dz is the spatial thickness of the slice.

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55 N X N O o o o o o r — o o o o o o o o o o 03 -C 4-1 C -H to 00 W 03 • 03 00 C -H u •H U-l -C o J-l 03 0) (J a c •H QJ iH 3 03 cr cu GJ 03 4J CU 03 C rH 03 3 03 Cl 4-> CU CU XT XI +J c e O 03 •h x: co r-l CU CU T3 Wi 3 U CU -H X C H oo CO • E 00 1-H 4J c 00 -H •h -a &H to oo 4-1 c CU •H -o CO u cu cox: 4-1 CU > 4-1 T3 U CU o t-l 4.) 0) 03 /-^ Xi • 0) •« j= u CJ 4J O > ••/— v RJ ^* O — O 03 C 4-1 -H 4-1 /~v QJ ^-s CO TJ (J 3 e 4J U O ~H O Ui c fc PC ""^ CO X) « E •a O CU 4J 4-> rH C. X) CU s-^ 3 -H CO o -c CO )-> 03 00 0) • 13 -O 3 01 4J •H -H rH C o. oou-i O. CO CJ CO E (H oo c •H CO 3 U o

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56 The the influence of the gradient nag nitudes on the signal magnitude will he discussed in this paragraph. If one increases the selective gradient magnitude, the signal magnitude will not be changed as shown in a) and b) of Fig. IT. If one halves the refocusing gradient magnitude, the signal magnitude will be double as shown in a) and c) of Fig. 18. Thus, only the refocusing gradient magnitude '..'ill have an effect on the signal magnitude.

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CHAPTER V CONJUGATE GRADIENT METHOD AND THE OPTIMAL 90 DEGREE SELECTIVE RF PULSES As we will see later, a truncated 90 degree sine function rf pulse is already a good 90 degree selective rf pulse, but one can still use the optimal control method and improve the rf pulse shape further to get better 90 degree slice selectivity. Since the 90 degree selective rf pulse design incorporates a refocusing gradient after the 90 degree rf pulse, the problem to design the 90 decree selective rf pulse is a little more complicated than that of the 18 degree pulse design. The time interval when there is a 90 degree selective rf pulse can be called the selective period. The interval when there is no 90 decree selective rf pulse and there is only refocusing magnetic field gradient can be called the refocusing period. As in the situation of the 180 degree selective rf pulse design one should decide what is the desired 90 degree magnetization, and then derive a procedure which can be programmed to find the optimal selective 90 degree rf pulse that causes the transverse magnetization at the end of the refocusing period to approach the desired magnetization as closely as possible. In the following discussion, we set U(t) = -r*Hl(t) as in the case of the design of the 180 degree selective rf pulse, in the 57

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58 refocusing period U ( t ) = 0. Ml, f 1 2 and 1 ! 3 represent x, y and z components of the magnetization in the selective period, and Ml, M2 and ii'3 represent x, y and z components of the magnetization in refocusing period, respectively. The bar, -, does not mean a vector here. Because there is no rf pulse in the refocusing period, the objective function, which is the measure of the difference between the desired transverse magnetization and the actual transverse magnetization, can be expressed as f tf J = \ dt S' [Ml(t,l,0)**2 + !'.2(t,l,0): *2-P(l*dv)**2]**2 ) tl 1 + S' Rl*[Ml(tf ,1,0)**2 + M2(tf ,1,0)**2-D(l*dw)**2]**2 1 T64] where tl and tf are the beginning and the end of the refocusing period, D ( 1 * d w ) is the desired transverse magnetization and can be expressed as w2 < l*dw MO -wl < l*dw < wl [65] I l*dw < -w2 where w2-wl is the width of the slice edge and wl+w2 is the thickness of the slice. The desired transverse magnetization, Eq . [65], can be graphically shown in Fig. 19. The length of the refocusing magnetic field gradient can be chosen to be identical with the 90 degree selective rf pulse length for convenience in the analysis. L(l*dw) =

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59 co CO 0) c 01 -* n o u •H 01 r— > 4J CO c 01 to J3 u 4J 4-> O 01 4-1 J= 4J 01 CM Ui a * o 01 VH — >• 01 i o c o 0) cj •H o 3 •H i-H 1—1 CO Q> CO « ,_ U. 0) o * 01 x: — 1 0) 4J rt CM * aou-i co 1 01 O -H -c x: ^> o 4-> PS O -O H •H s XS S w 0) M 01 o •H x: -h 0) Eh j-> Oi CO XI • u B
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60 Thus, the magnitude of the refocusing magnetic field gradient is half of the selective magnetic field gradient. Considering the problem in the rotating frame and assuming the selective rf pulse is applied along the x direction of the rotating frame on resonance, the I' loch equation for the selective period is d dt Ml l m: -I! Ml M2 M3 l*dw -l*dw u with the initial condition M1(0) = 0, K2(0) = 0, M3(0) = MO. The Eloch equation for the refocusing period is [66] [67] d dt Ml Ml M2 [6R] [69] -0.5*l*dw 0.5*l*dw J The initial condition for Eq. [6S] is Ml(tl) = Ml(tl), M2(tl) = M2(tl), M3(tl) = M3(tl). One can define a new variable, M 4 ( t , 1 , ) , by the equation dM4(t,l,0) = [Ml(t,l,0)**2 + M2(t,l,0)**2 D(l*dw)**2]**2 dt [70] with the initial condition M4(t 1,1,0) = 0. Substituting Eq. [70] into Eq. [64], the objective function becomes

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61 J = S 1 (M4(tf,l,0) + R1*[M1**2 + K2**2 D(l*dw)**2]**2] 1 Combining Eq. [ 6 c ] ] with E q . [7 0], the extended Bloch eauations for the refocusinp oeriod is dMl „ -0.5*l*dw*M2 dt dM2 0.5*l*dw*Kl dt d7l3 = dt dM4 = [ M 1 * # 2 + M2 ** 2 D(l*dw)**2]**2 dt with the initial condition Ml(tl) = Hl(tl), ~2(tl) = M2(tl), H3(tl) = M3(tl), M4(tl) = 0. Two cascaded systems, Eq. [66] and Eq. [72] can be generalized to Eq. [74] and Eq . [75], respectively. dMi(t,l,U) dt where i = 1, 2 and 3, and dMj(t,l,0) = fj(Ml(t,l,0) § ...,M4(t t 1.0)) dt [71] [72] [73] = fi(Ml(t,l,U),;i2(t,l,U),M3(t,l,U),U) [74] [75] where j = 1, 2, 3 and. 4. Because the variables in Eq. [75] are related to the variables in Lq. [74] by the initial

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62 condition, E q . [73], the objective function, E q . [711, can be generalized to J(U) = J(Ml[tf,l,Ml(tl,l,U),...,H3(tl,l,U)] M4[tf ,l,i:i(tl,l,U) K3(tl,l,U)]) [76] where 1 = 1 , 2 , . . . , L . L is chosen to be equal to 15?. One can start to derive the optimization procedure from the partial derivative of the objective function, Eq. [76], as we did in deriving the optimization procedure for designing the 180 degree selective rf pulse from the E q . [14], One obtains from E q . [76] dJ(U+qS) L 4 dj 3 dMj(tf) dMi( tl ,1 ,U+qS) = s' S 1 S' . dq 1 j dMj(tf) i dMi(tl) dq [77] Integrating Eq. [75], one gets Mj(t) Mj(tl) + ( fj(~l(v,l),...,~4(v,l))dv. [78] ) tl Differentiating both sides of the Eq. [73] v:ith respect to Mi(tl), one obtains dMj(t) dMj(tl) r t df j(Hl(v,l) M4(v,l)) = + \ dv dHi(tl) dMi(tl) J tl dMi(tl) d TT j ( 1 1 ) , t 4 df j d!7n ( v ) = + \ s' dv. [79] dMi(tl) J tl n diln(v) dMi(tl) Because of the initial condition, Eq. [73], one has dM j ( 1 1 ) = dij, 1 £ j < 3; dMi(tl) dM4(tl) = [80] dMi(tl)

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63 where d i j is the Kronecker delta. Differentiating both sides of Eq. [79], with respect to t , one obtains d dMj(t) 4 dlj dMn(t) = S ' . dt dMi(tl) n dMn(t) dMi(tl) In the matrix form, Eq. [81] can be written as [31] d dt d!il(t) dMi(tl) • d~4 ( t ) L dlii(tl) J df 1 dM 1 ( t ) d?4 L dTTl(t) df 1 dM 4 ( t ) df4 dU4 ( t ) dMl(t) dKi(tl) d!!4(t) L dMi(tl) J 82] According to Eq. [26], the solution of Eq. [82] is dKl(tl) d:ii(ti) dK 4 ( 1 1 ) dMi(tl) di;i(tf ) dMi(tl) * d ~4 ( t f ) L dlii(tl) (tf.tl) [83] where the matrix K(tf.tl) is the solution of the following equation : d?l dll dMl(t) dH4(t) dK(t,tl) dt K(t,tl) [84] df4 df4 L d~l(t) dH4(t) J with the initial condition K(tl,tl) = I. Because of Eq. [30], Eq. [83] can be written as dKj(tf) dh'i(tl) = Kji(tf.tl), 1 < j, i < 3;

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64 dH4(tf) =0 [85] dMi(tl) where Kji(tf,tl) is the jth row, ith column element of the matrix K(tf,tl). According to Eq. [29], one can have d'!i(tl ,l,U + qS) r 1 1 3 df n ( v , 1 , IJ + qS ) = \ dv S(v) S' Kin . [86] dq J tO n d'J Substituting Eq. [3 5] and Eq. [86] into E q . [77], one can get dJ(U+qS) dq 3 dJ 3 r tl 3 dfn(v,l,U+qS) = S* S' -S' Kji \ dv S(v) S' Kin 1 i dMj(tf) i ) tO n dU tl dfn(l) dJ dv S(v) S' S' S' --S' Kji(tf , t 1 )Kin ( t 1 , v ) tO In dU j dMj i t dv S(v)G(v,U+qS) [87] tO where E 3 dfn(l) 3 dJ 3 G(v,U+qS) = S' S' S' ----S' K ji ( tf , 1 1 )Kin( 1 1 , v ) In dU j dMj i L 3 dfn(v,l,U+qS) = S' S' Qn(v), [88] 1 n dU where 3 dJ 3 On(v) = S' --S* Kji(tf ,tl)Kin(tl,v). [89] J dMj i In the matrix form, Eq. [89] can be written as

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65 01 (v) Q2(v) Q3(v) Kll ... K31 K13 ... K33 Kll ... K 1 L K13 . . . K33 dJ dMl • dJ dM3 [90] or in the compact form 0(v) = K + (tl,v) Y+(tf ,tl)Jrn [91] where K""(tl,v) satisfies Eq. [55], Multiplying hoth sides of Eq. [55] by T "*"( t f , 1 1 ) Jm and according to the Eq. [91], one gets dQ(t) = -fn + Q(t) dt where r dfl dfl i [92] fm = dMl dM3 df3 df3 dMl dM3 J l*dw -l*dw -U(t) . [93] U(t) Substituting v = tl into Eq. [91], one derives the terminal condition for Eq. [92], Q(tl) = K + (tf ,tl)Jm. [94] Because Eq . [87] has exactly the same form as Eq. [30], the conjugate gradient method will minimize the objective function, Eq. [76], as it does for Eq. [14].

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66 The optimization procedure will be the same as that for designing a 180 degree selective rf pulse. The only difference is the tormina! condition, E q . [94], which is different from Eq. [57]. To find the terminal condition, E q . [94], one can integrate E q . [84]. Because of Eq. [72], the coefficient matrix of Eq. [84] will be dll dll — — ™ — — — # m 9 — — — — — — dMl(t) dK4(t) df4 dH4(t) -0.5*l*dw 0.5*l*dw A(t) B(t) [95] df4 L dMl(t) where A(t)=2*[Ml(t,l)**2 + M2(t,l)**2 D(l*dw)**2]*2*Ml(t,l)**2 B(t)=2*[Ml(t,l)**2 + M2(t,l)**2 D( l*dw)**2 ]*2*M2( t , 1 )**2 . [96] The initial condition for Eq. [ ? 4 ] is dHl(tl) dHl(tl) dMl(tl) d~4 ( t 1 ) LdMl(tl) dM3(tl) d~4 ( 1 1 ) dM3(tl) 1 1 1 [97] One can notice that there is no conflict between Eq . [97] and K ( 1 1 , 1 1 ) = I because there is no need for the fourth column of Eq. [97] if Eq. [88] is used to obtain the gradient of the objective function, Eq. [88], Integrating Eq. [84] and substituting following Eq. [98] into Eq. [94],

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67 Jc = Rl*A(tf ) Rl*B(tf) 1 one can n et the terminal condition for ( t ) of P. q . [92], The gradient of the objective function can he found by E q . [99] as in the f o 1 1 o w i n g : [98] G(t.U) = S' (-i!3(t f l)02(t,l) + M2(t,l)Q(t,l)). [99] In summary, the procedure for designing the 9(5 degree selective rf pulses is in the following: 1 . Guessing an initial 90 degree rf pulse, for example, a 90 degree truncated sine function rf pulse, integrate • 7 q . [66] under the initial condition, E q . [67], and then integrate E q . [72] under the initial condition, Eg. [73], and find the first value of the objective function, J(0), by Eq. [71]. 2. Subtituting Eq . [9 5] whose terns, A(t) and B(t), are determined by the solution of Eq. [72] into E q . [84] and solving Eq. [84] under the initial condition, Eq. [97], one can find the terminal condition for Eq. [92] by Eq. [94], Integrate Eq. [92] backwards in time from tl to tO to get the vector function Q(t). 3 . Construct the gradient of the objective function by Eq. [99] using the solution of Eq. [66]. 4. Using Eq. [32] and Fq. [33], construct a new rf pulse, U(t). The numerical coefficient, q, is chosen to minimize

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68 the objective function J for given U ( i f t ) and S(i,t) and the minimizing q can be found by usinj; the Golden Section Search. The new rf pulse is used to find a new value of the objective function, J(n). Comparing the new value with the old value of the objective function, J(n-l), stop if J(n) > J(n-l), or construct a new gradient and continue by using Eq. [34] if J(n) < J(n-l). The four steps above can be inplenented by a !7 ortran program that is given in the Appendix H. The JOB program of Appendices B and H is given in the Appendix I. The parameter values used in the pro gran of Appendix H to obtain the 90 degree optimal rf pulses in Fig. 20, Fig. 25 and Fig. 2 S of Chapter VI are given in a), b) and c) of Appendix J .

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CHAPTER VI EXPERIMENTAL STUDY OF THE OPTIMAL 90 DEGREE SELECTIVE RF PULSES Before starting the computation, one should choose a desired transverse magnetization and an initial trial function. A three zero truncated sine function, as shown in Fig. 20 (dotted line), is chosen as an initial trial function for the conjugate gradient algorithm. The response to the 90 degree three zero sine function rf pulse is shown in Fig. 21 (dotted line). If the slice thickness of a desired response is identical with that of the response to the three zero sine function rf pulse and if one chooses WTR = 1/6, the corresponding optimal rf pulse is shown in Fig. 20 (solid line) and the response to the optimal rf pulse is shown in Fig. 21 (solid line). From the computer simulation results one can conclude that the optimal selective 90 degree rf pulse has better selectivity than the sine function selective 90 degree rf pulse. There are less artifacts on the slice top and less average artifacts outside of the slice. The above conclusion from the computer simulations has been proved by the experiments. The experiments are similar with that of the optimal 130 degree selective rf pulses. The instrument used in the experiment is the GE CSI-II 2T spectrometer/imager system. The sample is a tube 69

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70 epn)!|duiv •a|IB|»h CO U-i 0) o CO <4-i u CO O cu o CU rH c e w 3 0) ^ 4J O rH C > U CO -H 3 •H •nr 4J a. <4-l 4-1 x: 4-i u u u 4J C <4H V -C 3 l-l C i-l 0) 4J Vl_ o cu O -H I— 1 •H CQ •H > O CO 4J i-4 C E O i-i CO rH -H •H C CO CO CO 4-1 — s 3 E •a o Cto <4-l -H cy -h o o E 4-1 U 4-' U O Cl •H C 0) cu C O CO CU N sz e E •H OJ -o 4-1 CO C0T3 -H 0) c CL) c P O -H O CO 1_ •H Wi -o £-HXi co o c H 4-» 4-1 N O (11 X) a. • CO cu o CU CO *-^ rH 0) CU 01 0) 3 l-i o >-. u c o. eo 4-i J= U •H cu 4J O rH HH ~a « o u H CU -a c "y 0) c •HHON iJC rH CO Ih CC 4-1 O E 0) o cu n -h r 4-1 •a -o v-' 4J 4J (1) c C E o co CU o o co ON CO 4-1 (-. ^~v rH CU CO < CO 3 x: cu Q. e a, 4J co • -H c CU O rH <4H C O x: CN U iH CH a co co • • cu • vD 00 4-1 a; XI U CU "»«. •H 4-> u o co >—i b O CO cu T-\ •o tl 0£ 3 CO N-^ no 4J 4-1 D.-H

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71 o co o <4-l o o Q) tn c 4-1 •H l-l rH >N 3 •o o c 9 0) •H .-H 3 O a C CO • o w •H
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72 of water. The tube length is 100 mm and the tube diameter is 10 mm. A spin echo pulse sequence, as shown in Fig. 22 is used in the experiment. In the pulse sequence, the 90 degree rf pulse is frequency selective and the ISO degree refocussing rf pulse is not selective. In the experiment, the non-selective ICO degree pulse length is 330 us, the three zero sine function 90 degree selective rf pulse length is 2 ms, the five zero sine function selective 90 degree rf pulse length is 2 ms as well. The selective 90 degree rf pulse lengths can be reduced by increasing the output power level of the rf pulse power amplifier. The echo time, TE, is 30 ms and the acquisition data block size is 1024 points. If the sample axis is along the main magnetic field (z-direction), the chosen project direction is the z-direction. One can make the experimental rf pulses by inputing, point by point, the numerical data corresponding to the rf pulse shapes in Fig. 20. The experimental results are shown in Fig. 23 and Fig. 24. Comparing Fig. 23 and Fig. 24 with Fig. 21, one can see that the experimental results are very close to the computer simulation results. If the thickness of a desired slice is identical with that of the response to the three zero sine function rf pulse and if one reduces WTR, that is, one tries to get better slice quality, for example, one chooses WTR = 1/10, the optimal rf pulse obtained is shown in Fig. 25 (dotted line). The corresponding response to the dotted line rf

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73 ° r o OB c O c 3 cr d) u u-t CD c •H D. CU CO CO CO CD E jC -h H i-H • CO CN C CN O N O • CO 4J oc c o •H CD CD fa S tH •H GJ -O CD

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74 <1> -C 4J • o O CM 4-1 • 1o CO a> .h x: 3 r*. 4-> a. N X <*-l H-l ^ O U >» 4J 0) o .H Q) c 3 J* • 3 (7 CO 00 0) o • ^ -a w U. iH O CO Cs 4-1 c c 0) o E -H •H 4-> »h U a.' c O3 X U-( S> o (V c x: -h E-H to • o en u CN 0) N • oo ai •H CU u* u

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75 0) »4H -c o 4J 4-> O CO •u co s: •H 4J 0) en 4-> co C rH C3 O 3 a cc c CO CD O <3J U -H 1-1 4J 01 -H CD J3 T3 .c h c 4J O N • o • z «H O 0) w O CM h a co ca >« 4J . 4J JZ O rH CO C CO c a -h a> • co Pl, e a; 3 0) -H CO »-i >«H Ih i-l O 1) 3 U. iH Q. a. CO 01 X 4J CO CD <4H C rH U CD 3 CD E CX E CD h co -c U «4H CO 4-> 0) >-. CX 0) li X 0) J5 O 0) 0> 4J 4h t-i 0) 00 f-i 4-> -C oi 0) ex h"o -o «j n o • O 3 X •* O* CD CN TJ rH 0) CO • CO C CN ME-H •H -iH CO • (JL, 4J 4J 00 CXJZ-H O Oh

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76 m E E 4-1 OJ 4J U o c •w Q) rH i—i as S-. CC QJ to x OJ E C 4-1 U-i -rl -H o "4-1 4-1 1—1 cy o •H Q. ll 4-1 XI CO •H (U CO 0) x c 4J oi co 4-1 CO 4-1 •a c •H O o 5 CO T3 01 D. 4J XI CO n c o> J0> o Qj x Ih » «ih c i-i 0> 01 0) 3 I-. • X 1 X o. 0.0 S 4J 01 CN1 <4J t-l 0) 4-i U • CO o 0) 00 «H i-( C -H 3 4J CO -H &h QCC E rH X •H C 4-1 4-1 4J -O -H (-, C. -H X • O rH 03 I-t 4-1 o O CO CO iH — 1 0) CO E > -^ OJ 0) •H .— < 1• E 4J i— 1 oc a> co C_ CO II 01 i-l CO o u T3 -H •H K 4-1 01 t-t 4-1 H O O XI OJ c jj 0> U 4J x a; c 4-1 T3 4J O CO o -H 3 > 01 -H c X H <-> co co •h x H * • iH U CO 01 m en -h 4-1 CO c cni x C CO •H •c > a> a) — 1 • o> co c CO li CD 01 .* -c •H -H CO u u H [I* CO rH 0.-H rH 0) 3 0) X o •C C U 4J CO ©pnjildiuv ba|)B|»u

PAGE 85

77 pulse of Fig. 25 is shown in Fig. 26 (dotted line). One can notice thnt even though there are sharper edges on the slice, there are more artifacts on the top of the slice and outside of the slice. The above conclusion has been proved by the experimental result as shown in Fig. 27. Thus, one nay conclude that for a particular pulse length there is a minimum value of WTR , represented by WTRmin, at a certain slice thickness. If WTR > WTRmin, one can always find an rf pulse to which the response will be very close to the desired slice. If WTR < WTRmin, one can get a sharper slice edge, but the price is that there are nore artifacts on the top of the slice and outside of the slice. From the discussion above one may conclude that one cannot make a transverse magnetization slice as sharp as one wants with a single selective rf pulse at a certain slice thickness. All above conclusions are similar with that of the 130 degree selective rf pulses in the Chapter IV. If one tries to improve the slice quality for a particular pulse length, that is, to reduce the value of WTR, one must choose a desired slice that has a broader frequency range. For example, one can choose a desired slice that has the slice thickness which is identical with that of the response to a five zero sine funciton rf pulse and WTR = 1/10, the corresponding optimal rf pulse is shown in Fig. 28 (solid line). The response to the optimal rf pulse is shown in Fig. 29. The experimental results are shown in Fig. 30 and Fig. 31.

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78 CO o o E J-) c •H • a CT 4-1 CO o c w o u. 0) a. jC co j-i a Vh o ^ 4J o o CO ai cu co c C -H O .H Q. co x)
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79 o o o CO 1 m N CD 3 z X CO CO * w 4-1 (11 1h cC * <4-l c o >> O 0) o o X •H c 4J E4-1 • t-H •H 9 a • 3 • •o . co m c 01 CN o • " IL u u o; m • c H CJrt 1 • C -H « x 4-1 Cb 4-> CO c c ^ 0) <4-4 u . CU <4-4 O 01 x vE x H CO 4-1 o 01 CO _ o • c iH o . 00 (^ -H 0) o CM iH x u-i 1 4J [ • T3 4-1 eo oi i-i a. •H 4J Oi Oi [l< 4-1 o u o C X -o 3 0)

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80 epmndiuv •aj»»i»u CO O co y-i n-i (U o cu o ^ en c iH 0J ^: j-i c 3 > o ra o 3.-H H X! -H 4J £ 4J 4J • lw O 4J U O Wi CU X C — I tH a> 4-> 3 *««. c O T3 X -H -H rH w c H MO OJ o a) e N C • CO CU -H co /- 1-4 c oco o H Ul •H CU <4-l O tH <4-l -3 CU u »-i X! 0) •a o u CJ 0) •H r-H O ^ x r-l ro c CO 4J O E CU -H 0) co -h x! ro -a -o ^ • J-) 4J 4-J c Cu X) o ra CU o o o o co *-> *•> H JJ O < a> 3 X CU 4J c a. 4-> co • -H C P< CO t-t <4-l C O E-i CM U -H Q.J2 -a ra co • CU CU 4-> cu • oo -u CU Xi ^ O) •H 4J (h O CO b O oo cu i— i •o CU O X 3 V' •a 4J 4J a.

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81 ( o c m V o CO 4J c o to c-o CO c Q) O >-i 0) 01 x: Sh 4-J c <^ o o at so c J •H rH H 3 CO T3 CJ •H u O e CO c •H aj 4-1 X! (0 H i— 1 3 • g oo •H CN • m CO • CM M 00 0) H • jj tOO 3 •H a. 4-1 •o CN M •H rH • (U o CO JZ to •H 4_> fc a> o -C 4-J 4-1

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82 z >• o c a9 to 0) co ID JS x: H c *j o O 00 J-l CM o • w oo C -H O En n. co vi_( d) o 0) x: E CO •H in ai a) n x 3 a; a. 0) 4-1 O On C O 00 rH •H CO 3 4J fo > en ro •h bj:

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83 o o o CO o o o o 00 a; CD 4J OT c iH ta o S c K c to a< o QJ >H H 1-4 4-1 ai •H ^*t 0) Xi •o N J3 H c X 4-> o *+*r • u • H-l oc 0> >> O CN iH O. U cc to c 4J • 4J X! » i— 1 CO c tn 3 3 •H E C_ E OJ •H to Xi h 14-1 CO 4-1 <4-J a) CO |H 4-1 J5 0) 0) o. H o •a a) c u • c 3 X t— i C-* 01 oo •o r-i qj m • cc C CN 00 E •H •H •H CO • tn 4-1 4-1 00 O. Xi -H o O Ua

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84 Fron the computer simulation results and the experimental results one may notice that the improvements for the selectivity of the 90 degree selective rf pulse after the optimization procedure are not so obvious as for that of the 180 degree selective rf pulse. The reason may he attributed to the nature of the magnetic resonance spin system, hut not the optimization procedure. For the convenience for some readers, portions of the Fourier coefficients of the Fourier series for the 90 degree optimal rf pulses in Fig. 20 and Fig. 28 are given in the Appendices K and L. The first 20 coefficients of both cosine and sine columns in the Appendices K and L will reconstruct the rf pulses in Fig. 20 and Fig. 2S, respectively . The discussion of the influence of the gradient magnitude on the slice thickness is similar as that of Chapter IV. The pulse sequence used in the experiment is shown in Fig. 22. The sample is a tube of water with a 1 mm thick bloc!; in the middle of the tube. The experimental result is shown in Fig. 32. One can see that Fig. 18 and Fig. 32 are quite similar; thus, the conclusions from Fig. 18 and Fig. 32 should be the same and it is already presented in Chapter IV.

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85 N CD -C 4J •U C CO C T3 -H -H C X) CO CN CO CN U CD co cox: co • jj (Li CO CU C -H > J* fc, -H ^-v U 4-> T3 •H 14-1 O J= O QJ O 0J CD cd u co o c •H OJ CD -I H£ co cr +J CD CO » *J CD "O (0 C i-H O Q) 3 4J (Li a. 5 /-> 4-1 CD U CD JZ o c c O CD /-n •HX ^ CO O HOW o -a M 3 /-n 4J CO CD -H JC C E HMO CO Vj • E En CN CO 4J . C T3 • CD CD CO -H -H •H -O t-l hoc. eo co .n • -a o > 1— I •/-> CO u x: o co •u -H -^ CD to ~a 3 E 4J O -H SC fc co co • E CD 4-1 iH C X2 CD 3 -H O T3 a co u CO 00 •H 00 CD C T3 -H 3 CO 4J 3 •H O C O O0<4-( CO CD E to

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CHAPTER VII CONCLUSION There are more than 30 articles devoting to the selective rf pulse design problem, except the articles which utilize the optimal control method, since the existence of the magnetic resonance imaging. However, none of them completely solved this problem. From the discussion above, one can see that the conjugate gradient method has completely solved the problem of designing a selective rf pulse. The conjugate gradient method is a established method and has been utilized to solve the optimal control problem since 1967 (10). Unfortunately, this powerful method is just recently introduced into MRI and used to solve the pulse shape design. This fact has demonstrated the importance again that the researchers in different fields should understand each other and learn each other. 86

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APPENDIX A FOURTH ORDER RUNGE-KUTTA METHOD Because the fourth order Runge-Kutta method in a standard textbook, for example in (17), is hard to nrogram for our purpose, we rearrange it as in the following: if a differential equation is x'(t) = fl(t,x,y,z,w) y'(t) = f2(t,x,y,z,w) [ Al ] z'(t) = f3(t,x,y,z,w) w'(t) = f4(t,x,y,z,w) where the prime, ', means differentiate with respect to the time variable, t, the equivalent difference equation to above equation for the numerical iteration calculation is x(n+l) = x(n)+(Al+2*Bl+2*Cl+Dl)/6 y(n+l) = y(n)+(A2+2*B2+2*C2+D2)/6 [ A2 ] z(n+l) = z(n)+(A3+2*B3+2*C3+D3)/6 w(n+l) = w(n)+(AA+2*B4+2*C4+D4)/6 where the asterisk, *, means multiplication and Ai=dt*fi(t(n),x(n),y(n),w(n)) Bi=dt*fi(t(n)+dt/2,x(n)+Al/2,y(n)+A2/2,z(n)+A3/2,w(n)+A4/2) Ci=dt*fi(t(n)+dt/2,x(n)+Bl/2,y(n)+B2/2,z(n)+B3/2,w(n)+B4/2) Di=dt*fi(t(n)+dt,x(n)+Cl,y(n)+C2,z(n)+C3,w(n)+C4) where dt is the time increment and i = 1, 2, 3 and 4. 87

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FORTRAN PROGRAI APPENDIX B FOR 180 DEGREE PULSE DESIGN C SEPT 29 19 REAL J DIMENS DIMENS DIMENS DIMENS DIMENS DIMENS DIMENS DIMENS DIMENS DD=f ML-f D=bl 11,1 DRTA RFA = COE= COEF IDD= IDE= C IDF= C IDG = C CD f C c c c c c c c c c req . l req . s ank; P 2,13, G ,DRTA1 sectio propor =renew ID. of 1 fron 1 for 2 no o orces READ(1 READ(1 READ(1 READ(1 RAED(1 READ(1 READ(1 PRINT PRINT PRINT PRINT PRINT PRINT PRINT 86 l.J ION ION ION ION ION ION ION ION ION ION ncr tep = pa I d = co n 1 tio ed de si 180 pti pro 1-* 1 1 1 1 2,JJ,J,HUA,JA,JB,JC,JD X(400),Y(400),Z(400), RF1A(400),RF1R(400),R RF2A(400),RF2B(400),R RF0A(400),RF0B(400) ,R XX(400),YY(400),ZZ(40 0MAA(400),0MBB(400),0 W(400) ,UW(400) ,TW(200 X1(200),Y1(200),Z1(20 FILOUl(1200),FILOU2(l FIL0U4(800),FIL0U5(12 enent; DT=tirae increme No.; 0M=func. amplitu rameter of sine func. ; etermine desired Mz; nvergence tolarence , DR ength of golden sectio n factor for integral, section length, C0EF=0 sired prof ile , ( 1 , 2 ) ; nc func.,IDE=2 from sa degree, IDF=3 for 90 mization; KD=12~20,I4= ,JQ,L1,L2,L3 S1(400),FILI F1C(400),DES F2C(400),DES F0C(400),S(4 0),G(400),GG MCC(400) ),TZ(200),Z2 0),X2(200),Y 200),FIL0U3( 00),FIL0U6(1 nt; H=initia de; M=ID. of NL=time ste ,L4,II NP(400) FIL(SOO) (200) 00) (400) (200) 2(200) 1200) 200) 1 Mz; func . ; p No.; run KD times, each TA=10 50,DRTA1=1 10; n,RFA=0.05"0.1E-03; C0E=0.l"l.0E-05; .l'l.OE-03; ved func . ; degree ; 5~10, time 14 iterations; DD,DT,H,ML 0M,M,D,P,NL II, 12, 13, 14, GI DRTA , RFA , C0E , COEF , DR IDD, IDE, IDF, IDG RX1,RX2,RY1,RY2,RZ1, KC, RATIO, KD DD,DT,H,ML 0M,M,D,P,NL II, 12, 13, 14, GI DRTA , RFA , C0E , COEF , DRTA IDD, IDE, IDF, IDG RX1,RX2,RY1,RY2,RZ1,RZ KC, RATIO, KD TA1 RZ2 C rf p N3=NL-1 M2=3*N3 ulse from saved func IF (IDE.EQ.2) THEN 88

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89 10 REAP (10,*) (IDUM,FILINP(L),L=1 DO 10 1=1, N3 RFOA(I)«FILINP(I) RF0B(I)=FILINP(I+N3) RF0C(I)-FILINP(I+2*N3) CONTINUE ENDIF M2) Nl=(NL-l)/2+l N2-N1-1 na=:u-2 RH1-N1 Ml=(KL+l)/2 RM1-M1 desired Hz DO 11 1-1,11 DESFIL(I)=H 11 CONTINUE IF (IDD.EQ.l) THEN DESFIL(I1+1)=0.0 DESFIL(Il+I2+2)=0.0 DO 12 I-.I2 DESFIL(I+I1+1)=CI 12 CONTINUE DO 13 1=1,13 DESFIL(I+I1+I2+2)=H 13 CONTINUE ENDIF 16 17 IF (IPD.EQ.2) THEN DESFIL(Il+l)=GI/2 PESFIL(Il+2)-0.0 DESFIL(Il+3)»GI/2 DESFIL(Il+3+I2+l)=GI/2 DESFIL(Il+3+I2+2)=0.0 D3SFIL( I 1+3+1 2+3) -GI/2 DO 16 1=1,12 DESFIL(Il+3+I)=-GI CONTINUE DO 17 1-1,13 DESFIL(Il+3+I)=H CONTINUE ENDIF DO 9 I=ML+1,4*ML DESFIL(I)=0.0 9 CONTINUE WRITE(18, 1200) (L.DESFIL(L) ,L=1 ,4*HL) 1200 F0RHAT(I5,E14.6) DO 14 1=1, ML DES(I)=DESFIL(I) 14 CONTINUE ! I X = I D . of gradient change times, related to KD

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90 IK=1 112 BETA-0.0 C K = I D . of one-dimen. optini. iter, times, rela. to 14 K=l DO 15 N=1,NL S(N)=0.0 S1(N)=0.0 15 CONTINUE C 111 DO 100 L=1,ML RL=L D0M=(RL-RM1)*DD X(1)=0.0 Y(1)=0.0 Z(1)=H U(1)=0.0 c DO 101 N-1.N3 EN=N IF (K.EQ.l) THEN C IF (IDE.EQ.l) THEN C sine func. IF (rI.EQ.3) THEN IF (N.E0.N1) THEN OMA-OM ELSE EXA=P*(RN-RN1)*DT OMA-OM*SIN ( EX A ) / ( EXA ) ENDIF EXB=P*((RN-RNl)*DT+DT/2) 0MB=0M*SIN(EXB) / ( EXB ) IF (N.EQ.N2) THEN OMC-OM ELSE EXC=P*((RN-RN1)*DT+DT) OHC-OM*SIN ( EXC ) / ( EXC ) ENDIF C ENDIF C ELSEIF (IDE.E0.2) THEN C a saved func. as initial trial func. 0MA=RF0A(N) 0MB=RF0MN) OMC-RFOC(N) ENDIF C OMAA(K)-OMA OMBB(N)-OMB 0:iCC(N)=0riC C ELSELF (K.NE.l) THEN 0MA»0MAA(N)

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91 OMB=OMBB(N) OMC=OMCC(N) ENDIF IF (K.EQ.l) THEN RF1A(N)-0MA RF1B(N)=0MB KFlC(N)=0'iC ENDIF A1=DT*D0M*Y(K) B1=DT*(-D0M*X(N)-0MA*Z(N)) C1=DT*0MA*Y(N) D1=DT*(Z(N)-DES(L))**2 A2=DT*D0M*(Y(N)+Bl/2) B2=DT* ( -DOM* ( X ( N ) +A 1 / 2 ) -OMB* (Z(N )+Cl /2 ) ) C2=DT*0MB*(Y(N)+Bl/2) D2=DT*(Z(N)+Cl/2-DES(L))**2 A3=DT*D0H*(Y(N)+B2/2) B3=DT*(-D0M*(X(N)+A2/2)-0ME*(Z(N)+C2/2)) C3=DT*0MB*(Y(N)+B2/2) D3=DT*(Z(N)+C2/2-DES(L))**2 A4=DT*D0M*(Y(N)+B3) B4=DT* ( -DOM* ( X ( N ) +A3 ) -0!'C* ( Z ( N ) + C3 ) ) C4=DT*0MC*(Y(N)+B3) D4=DT*(Z(N)+C3-DES(L))**2 X(N+l)=X(N.) + (Al+2*A2+2*A3+A4)/6 Y(N+l)=Y(N)+(Bl+2*B2+2*B3+B4)/6 Z(N+l)=Z(N)+(Cl+2*C2+2*C3+C4)/6 W(N+l)=W(N)+(Dl+2*D2+2*D3+D4)/6 101 CONTINUE IF (K.EQ.l) THEN IF (L.EQ.H1) THEN PRINT *, Z(l) ,Z(N1),Z(NL) ENDIF ENDIF IF (X.EO.l) THEN X1(L)=X(NL) Y1(L)=Y(NL) Z1(L)=Z(NL) ENDIF TZ(L)«Z(NL) TW(L)=U(NL) 100 CONTINUE J0=0.0 JA=0.0 DO 21 L=1,ML J=RZ2*(TZ(L)-DES(L))**2+RZl*T T v.'(L) JO=JO+J J*RZ2*(TZ(L)-DES(L))**2 JA=JA+J 21 CONTINUE

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92 C IF (K.EQ.l) THEN PRINT *, JO, J A ENDIF C no optimi. goto the end IF (IDG.E0.2) THEN GOTO 402 ENDIF C DO 200 L=1,ML RL=L do;:=(rl-r;h)*dd X(1)=0.0 Y(1)=0.0 Z(l)-H W(1)=0.0 c DO 201 N=1,N3 RN-R IF (K.EQ.l) THEN C C c c c IF (IDE.EQ.l) THEN IF (M.EQ.3) THEN IF (N.EQ.N1) THEN OMA-OM ELSE EXA=P*(RN-RN1)*DT OMA=OM*SIN(EXA) / ( EXA ) ENDIF EXB=P*((RN-RNl)*DT+DT/2) OMB=OK*SIK(EXB)/(EXB) IF (N.EQ.N2) THEN OMC=OM ELSE EXC=P*((RN-RN1)*DT+DT) OMC=OM*SIN ( EXC ) / ( EXC ) ENDIF ENDIF ELSEIF (IDE.E0.2) THEN OMA=RFOA(N) OM'B=RFOB(N) OMC=RFOC(N) ENDIF OMAA(N)-OMA OMBB(N)-OMB OMCC(N)=OMC ELSELF (K.NE.l) THEN OMA=OMAA(N)

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OMB=OMBB(N) OMC=OMCC(N) ENDIF 93 IF (K.EQ.l) RF1A(N)=0MA RF1B(N)=0MB RF1C(N)=0MC ENDIF THEN AUDT' E1=DT : C1 = DTD1=DT* A2 = DT : B2=DTC2 = DT J D2=DT S A3 = DT* B3=DT< C3 = DT ! D3=BT S A4=DTB4 = DT ; C4=DT" D4=DT> DOH*Y(N) (-DOM*X(N)-OMA*Z(N)) OMA*Y(N) (Z(N)-DES(L))**2 D0M*(Y(N)+Bl/2) ( -DOM* ( X ( N ) +A 1 / 2 ) -OHB* ( Z ( N ) +C 1 / 2 ) ) OMB*(Y(N)+Bl/2) (Z(N)+Cl/2-DES(L))**2 DOM*(Y(N)+B2/2) (-DOM*(X(N)+A2/2)-OMB*(Z(N)+C2/2)) OMB*(Y(N)+B2/2) (Z(N)+C2/2-DES(L))**2 DOM*(Y(K)+B3) ( -DOM* ( X ( N ) + A3 ) -OMC* ( Z ( N ) +C3 ) ) OMC*(Y(N)+B3) (Z(N)+C3-DES(L))**2 )+(Al+2*A2+2*A3+A4)/6 )+(Bl+2*B2+2*E3+B4)/6 )+(Cl+2*C2+2*C3+C4)/6 )+(Dl+2*D2+2*D3+D4)/6 X(N+1)=X(N Y(N+1)=Y(N Z(N+1)=Z(N W(N+1)=W(N 201 CONTINUE XX(1)=0.0 YY(1)=0.0 ZZ(l) = 2*RZ2*( r fZ(L)-DES(L) ) WU(1)=RZ1 C in C BO 202 N=l f N3 RN = N tegrating adjoint DDT=-DT enua. back wards in tine IF (K.EQ.l) THEN IF (IDE. EG. 1) THEN IF (M. EQ.3) THEN IF (N. EQ.N1) THEN o:iA=Oi! ELSE EXA=P* ! (RN-RN1)*DT 0MA=0I I*SIN(EXA)/(EXA) ENDIF EXB=P* : ((RN-RNl)*DT+DT/2) OMB=OM*SIN ( EXB) / ( EXB )

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94 C C C IF (N.EQ.N2) THEN OMC-OM ELSE EXC=P* ( ( RN-RH 1 )*DT+DT) OIiC=OM*SIN(EXC)/(EXC) ENDIF ENDIF ELSEIF (IDE. EC). 2) THEN OMA-RFOA(N) OMB=RFOE(N) OMC-RFOC(K) ENDIF C ELSELF (K.NE.l) THEN OMA-OMAA(N) OMB-OKBB(N) OMC-OMCC(K) ENDIF C ZD-(Z(N)+Z(N+l))/2 Al-DDT*DOM*YY(N) 3 1=DDT* ( -DOM* XX ( N ) -OHA*ZZ ( N ) ) C 1 = DPT* ( OMA*YY ( N ) -2* ( Z ( N ) -DES ( L ) ) *WW (N ) ) Dl-0.0 A2-DDT*DOM* ( YY ( N ) +E 1 / 2 ) B2=DDT* (-DOM* ( XX ( N ) +A 1 / 2 ) -OHB* (ZZ( N ) +C 1 / 2 ) ) C2=DDT*(OMB*(YY(N)+Bl/2)-2*(ZD-DES(L))*(WW(N)+Dl/2)) D2=0.0 A3-DDT*DOM* ( YY(N )+B2/2) B3=BDT*(-DOK*(XX(N)+A2/2)-OMB*(ZZ(N)+C2/2)) C3=DDT*(OMB*(YY(N)+B2/2)-2*(ZD-DES(L))*(WW(N)+D2/2)) D3=0.0 A4=DDT*D0M*(YY(N)+B3) B4-DDT* ( -DOM* ( XX( N) +A3 )-OMC*( ZZ (N )+C3 ) ) C4=DDT*(OMC*(YY(N)+B3)-2*(Z(N+l)-DES(L))*(WW(N)+D3)) D4=0.0 C XX(N+l)=XX(N)+(Al+2*A2+2*A3+A4)/6 YY(N+l)=YY(N)+(Bl+2*B2+2*B3+B4)/6 ZZ(N+l)=ZZ(N)+(Cl+2*C2+2*C3+C4)/6 WW(N+l)=WW(N)+(Dl+2*D2+2*D3+D4)/6 202 CONTINUE C G(N)=gradient of objective func. DO 23 N-l.NL G(N)=-Z(N)*YY(NL+1-N)+Y(N)*ZZ(NL+1-N) 2 3 CONTINUE IF (L.EQ.l) THEN DO 24 N=1,NL GG(N)=0.0 24 CONTINUE ENDIF

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95 DO 25 N-l.NL GG(N)-G6(N)+G(N) 2 5 CONTINUE C 200 CONTINUE C SC1=0.0 DO 31 1=1, N2 SG1=SG1+(C0E*GG(2*I-1))**2 31 CONTINUE SG2=0.0 DO 32 I-1.K4 SG2=SG2+(C0E*GG(2*I))**2 32 CONTINUE DOT2=((COE*GG(l))**2+(COE*GG(NL))**2+4*SGl+2*SG2)*DT/3 IF (K.NE.l) THEN BETA=D0T2/D0T1 END IF D0T1=D0T2 C DO 33 N=1,NL S ( N ) =-GG ( N)+BETA*S ( N ) 5 3 CONTINUE C initial setting for golden section search HUA=0. 618034 RFA1 = RFA*H1JA RFA2=RFA L1=RFA*KUA**2 L2=RFA*HUA L3=0.0 L4=RFA RLN=L1-L3 RLM=L4-L2 C after KC times G(N), setting is changed to keep converge IF (K.NE.l) THEN IF (K.GT.KC) THEN C0EF=RATI0*C0EF ENDIF RFA1-C0EF*RFA*HUA RFA2=COEF*RFA L1-C0EF*RFA*HUA**2 L2=COEF*RFA*HUA L3=0.0 L4«C0EF*RFA RLN=L1-L3 RLN=L4-L2 ENDIF C ID=0 1 = C 333 DO 300 L=1,KL RL«L D0M=(RL-RM1)*DD

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c c 96 X(1)=0.0 Y(l)=U.O Z(l)-H ','(1) = 0.0 DO 301 K-1,N3 RN=N IF (K.EQ.l) THEN IF (IDE.EQ.l) THEM IF (K.EQ.3) THEN IF (N.EQ.N1) THEN 0MA=0K+RFA1*S(N) ELSE EXA=P*(RN-RN1)*DT OMA-OM*SIM ( EXA) / ( EXA ) +RFA1*S( N ) ENDIF EXB=P*((RN-RNl)*DT+DT/2) 0MB=0M*SIN(EXB)/(EXB)+RFAl*(S(N)+S(N+l))/2 IF (N.E0.N2) THEN OMOOM ELSE EXC=P* ( ( RN-RN 1 ) *DT+DT ) 0MC=0M*SIN(EXC)/(EXC)+RFA1*S(N+1) ENDIF ENDIF ELSEIF (IDE.EQ.2) THEN OH A =RFO A ( N ) +RF A 1 * S ( N ) 01 iB=RFOB ( N ) + RFA 1* ( S ( N )+S ( N+l ) ) /2 0MC-RF0C(N)+RFA1*S(N+1) ENDIF ELSELF (K.NE.l) THEN OHA-OMAA ( N ) +RFA 1*S (N ) OMB-OMBB( N ) +F.FA 1 * ( S ( N ) +S ( N + 1 ) ) / 2 0MC-OMCC( N ) +RF A 1*S ( N+l ) ENDIF ONAA(N)-OMA OMBB(K)-OMB OMCC(K)«OMC RF2A(N)=0MA RF2B(N)«0KB RF2C(N)»0MC ENDIF A1=DT*D0M*Y(N) B1=DT*(-D0M*X(N)'-0KA*Z(N)) C1=DT*0MA*Y(N) D1=DT*(Z(N)-DES(L))**2

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c c 97 A2 = DT*D0.I*(Y(N) + 31/2) B2=DT* (-DOM* ( X ( K ) + A 1 / 2 ) -03 IB* ( Z ( N ) +C 1 / 2 ) ) C2=DT*0MB* ( Y ( N ) +E 1 / 2 ) D2=DT* ( Z ( N ) +C 1 / 2-DES ( L ) ) ** 2 A3=DT*D0M*(Y(N)+B2/2) B3=DT* ( -DOM* ( X (N )+A2/2 ) -OMB* ( Z ( II ) +C2 / 2 ) ) C3=DT*0MB*(Y(N)+B2/2) D3=DT*(Z(N)+C2/2-DES(L))**2 A4 = DT*D0ii*(Y(N) + B3) B4=DT* (-DOM* ( X ( N ) +A3 ) -OMC* ( Z ( N ) +C3 ) ) C4=DT*0MC*(Y(N)+B3)" D4=DT*(Z(N)+C3-DES(L))**2 X(N+l)=X(N)+(Al+2*A2+2*A3+A4)/6 Y (N+l ) =Y ( N )+(Bl+2*B2+2*B3+B4 ) /6 Z(N+l)=Z(N)+(Cl+2*C2+2*C3+C4)/6 \\ T (N+l)=W(N) + (Dl+2*D2+2*D3+D4)/6 301 CONTINUE X2(1)=X(NL) Y2(1)=Y(NL) Z2(1)=Z(KL) TW(1)=W(NL) 300 CONTINUE JJ=0.0 J3=0.0 DO 41 L-l.ML J=RZ2*(Z2(L ) -DES ( L ) ) **2+RZl *TW(L ) JJ=JJ+J J=RZ2* ( Z2 ( L ) -DES (L) )**2 JB=JB+J 41 CONTINUE 1 = 1 + 1 IF (I.EQ.l) THEN J2=JJ RFA1=RFA1*HUA GOTO 333 ELSEIF (I.EQ.2) TEEN J1=JJ ENDIF IF (ID.EQ.l) THEN J1=JJ ELSEIF (ID.E0.2) THEN J 2= J J ENDIF IF (K.GT.l) THEN IF (Jl.GT.JC) THEN ID0=1 GOTO 400 ELSEIF (JB.GT.JD) THEN ID0=1 GOTO 400

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98 EN DIP EM DIP C IF (K.EQ.I4) THEN IB0=1 GOTO A 00 EKDIF C p, olden section search is working IF (J2.LT.J1) THEN RFA2=RFA2-RLN ID=2 L3=L1 L1=L2 J1 = J2 L2=L3+HUA*RFA2 RLM-L4-L2 RLN=L1-L3 EFA1=L2 GOTO 333 C ELSE IF (J2.GT.J1+DRTA) THEN RFA2-RFA2-RLH ID=1 L4=L2 L2 = L1 J2=J1 L1=L4-HUA*RFA2 RLH=L4-L2 RLN=L1-L3 RFA1=L1 GOTO 333 ELSE JC=J1 JD-JIi PRINT *, Jl.JA.JE EKDIF DO 43 N=1,NL S1(N)=S1(N)+RFA1*S(K) 43 CONTINUE K=K+1 IF (J2.LT.J1+DRTA1) THEN ID0=1 GOTO 400 ' ELSE ID0=2 GOTO 111 ENDIF 400 CONTINUE IF (IDE.EQ.2) THEN IF (IK.LT.KD) THEN DO 401 1=1, N3 RF0A(I)=RF2A(I)

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99 i?FOB(I FOC(I )=I!F2:.( T ) )=RF2C(I) 401 CONTINUE IK=IK+1 PRINT *,IK GOTO 112 ENDIF END IF 40 2 DO 4 4 FIL0U1 FIL0U1 FIL0U1 44 CONTIN DO 4 5 FIL0U2 F I LOU 2 FIL0U2 45 CONTIN DO 46 FIL0U3 FIL0U3 FIL0U3 SQ=X1( FIL0U3 4 6 CONTIN DO 4 7 FIL0U4 FIL0U4 FIL0U4 S0=X2( F I LOU 4 47 CONTIN M3=4*M DO 48 FIL0U5 4 3 CONTIN DO 49 F I LOU 6 49 CONTIN WRITE( WRITE( WRITE ( V.T.ITE( WRITE ( WRITE ( 1100 FORMAT STOP END KK=1,N3 (KK)-RFIA (KJC+N3)=R (D'+2*N3) UE K?:=1,N3 (KK)=RF2A (KK+iJ3)=S (K +2*;'3) UE KK-l.ML (KK)=Z1(K (KK+HL)=X (KK+2*I5L) KK)**2+Y1 (KK+3*ML) UE J\ K = i 9 V1U (KK)=Z2(K (KK+ML)-X (KK+2*ML) KK)**2+Y2 (KK+3*ML) UE T KK=1,M2 (XK)-O.O UE KK-1.K3 (KK)=0.0 UE 22,1100) 13,1100) 14,1100) 15,1100) 16,1100) 17,1100) (I5.F14.6 (KS) FIB(KK) =RF1C( KK) (KK) F2B(KK) =RF2C(KK) K) 1(KK) =Y1(KK) (KK)**2 =SQRT(SQ) 2(KK) =Y2(KK) (KK)**2 »S0RT(SQ) (L,FIL0U1(L),L=1,M2) (L,FIL0U2(L) ,L-1,M2) (L,FIL0U3(L),L=1,M3) (L,FIL0U4(L),L=1,M3) (L,FILOU5(L),L=l,M2) (L,FIL0U6(L),L=1,M3) )

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APPENDIX C PARAMETER VALUES OF THE PEOAGRAM IN THE APPENDIX B TO OBTAIN THE OPTIMAL RF PULSES OF FIG. 2, FIG. 11 AND FIG. 14 a) The parameter values for the optimal rf pulse of Fig. 2 2111.0, 1. 11111 1E-05, 2000, 159 -26911.0, 3, 0.7, 25300.0, 91 69, 15, 69, 5, 2000 50.0, 0.5E-03, 0.5E-05, 0.5E-03, 10.0 2, 1, 1, 1, 1, 1, 1, 1, 60, 0.7, 20 b) The parameter values for the optimal rf pulse of Fig. 11 2111.0, 1.11111 1E-05, 2000, 159 -26911.0, 3, 0.7, 25300.0, 91 70, 17, 70, 5, 2000 50.0, 0.5E-03, 0.5E-05, 0.5E-03, 10.0 2, 1, 1, 1, 1, 1, 1, 1, 60, 0.7, 20 c) The parameter values for the optimal rf pulse of Fig. 14 2111.0, 1.111111E-05, 2000, 159 -39615.0, 3. 0.7, 3S350.0, 91 64, 25, 64, 5, 2000 50.0, 0.5E-03, 0.5E-05, 0.5E-03, 10.0 2, 1, 1, 1, 1, 1, 1, 1, 60, 0.7, 15 100

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APPENDIX D FORTRAN PROGRAM FOR REFOCUSING PULSE STUDY C JAN'. 8. 1987 C program SELE53 DIENSIOH X ( 400 ), Y ( 400 ), Z ( 400 ),FILINP( 1200) DIMENSION RF1A(400) ,RF1B(400) ,RF1C(400) DIMENSION RF0A(400) ,RF0B(400) ,RF0C(400) DIMENSION XO(IOOO) ,YO(1000) ,Z0(1000) DIMENSION X 1 ( 1 000 ) , Yl ( 1 000 ) , Z 1 ( 1 000 ) DIMENSION XA( 1000 ), YA ( 1 000 ),ZA( 1000) DIMENSION XB ( 1 000 ), YB ( 1 000 ),ZB( 1000) DIMENSION XYl(lOOO) ,XYA(1000) .XYB(IOOO) DIMENSION XY0( 1000), XYR( 1000) DIMENSION FILOU1(1200), FIL0U2( 1200) , FIL0U3( 1C00) DIMENSION FIL0U4(1000), FIL0U5( 1 200) , FIL0U6( 1000) C IDD desired profile (1,2,3), IDE run(l) run(2) C IDH for different rotation axis READ (11,*) DD,DT,H,ML,IHW RLAD(11,*) OM,M,D,P,NL READ(11,*) II, 12, 13, 14, GI READ (11,*) IDD, IDE, IDF, IDG, IDH PRINT *, DD,DT,H,ML,IKW PRINT *, OM,M,D,P,NL PRINT *, II, 12, 13, 14, GI PRINT *, IDD, IDE, IDF, IDG, IDH C N3=NL-1 M2=3*N3 IF (IDE.EQ.2) THEN READ (10,*) (IDUM,FILINP(L),L=1,M2) DO 10 1=1, N3 RF0A(I)=FILINP(I) RF0B(I)=FILINP(I+N3) RF0C(I)=FILINP(I+2*N3) 10 CONTINUE ENDIF C X(1)=0.0 Y(l)-H Z(1)=0.0 c DO 100 L=1,ML RL=L D0M=(RL-RM1)*DD . C DO 101 N=1,N3 RN = N 101

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102 IF (IDE.EQ.l) THEN IF (M.EQ.l) THE:i OMA-OM OMB=OM OMC-OM ELSEIF (M.EQ.3) THEN IF (N.EQ.N1) THEN 0MA=OM ELSE EXA=P*(RN-RN1)*DT 0MA=0M*SIN ( EXA ) / ( EXA ) ENDIF EXB=P*((RN-RNl)*DT+DT/2) 0;iB=OK*S IN ( EXB ) / ( EXB ) IF (N.EQ.22) THEN OMC-OM ELSE EXC-P*((RN-RN1)*DT+DT) OMC=OM*SIN(EXC)/(EXC) ENDIF ENDIF ELSEIF (IDE.EQ.2) THEN OMA-RFOA(N) OMB-RFOBNN) OMC-RFOC(N) ENDIF RF1A(N)-0MA RF1B(N)-0MB RF1C(N)=0MC IF (IDII.EQ.l) THEN A1=DT*D0M*Y(N) B 1 = DT* ( -DOM*X ( N ) -0MA*Z( N ) ) C1=DT*0MA*Y(N) A2«DT*DOM*(Y(N)+Bl/2) B2=DT*(-DOM*(X(N)+Al/2)-OMB*(Z(N)+Cl/2)) C2=DT*OMB*(Y(N)+Bl/2) A3-DT*D0M*(Y(N)+B2/2) B3=DT*(-DOM*(X(K)+A2/2)-OMB*(Z(N)+C2/2)) C3=DT*0MB*(Y(N)+B2/2) AA=DT*D0M*(Y(N)+B3) B4-DT*(-DOM*(X(N)+A3)-OKC*(Z(N)+C3)) C4-DT*0MC*(Y(N)+B3) ELSEIF (IDH.EQ.2) THEN A1=DT*(D0M*Y(N)-0MA*Z(N)) B1-DT*(-D0M*X(N)) C1=DT*0MA*X(N) A2-DT*(D0M*(Y(N)+Bl/2)-0MB*(Z(N)+Cl/2)) B2-DT*(-DOM*(X(N)+A1*0.5)) C2»DT*0MB*(X(N)+Al/2) A3-DT*(D0M*(Y(N)+B2/2)-0MB*(Z(N)+C2*0.5)) B3=DT*(-D0M*(X(N)+A2*0.5))

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103 C3=DT*OMB*(X(N)+A2/2) A4=DT*( DOM* ( Y ( N )+B3 )-OMC* ( Z ( N ) +C3 ) ) B4=DT*(-D0M*(X(N)+A3)) C4=DT*0HC*(X(N)+A3) ELSEIF (IDH.EQ.3) THEN A1=DT*(-0MA*Z(N)) B1=DT*D0M*Z(N) C 1 =DT* ( OMA*X( N ) -POM* Y ( N ) ) A2«=DT* ( -OMB* ( Z ( N ) +C 1 / 2 ) ) B2=DT*D0M*(Z(N)+Cl/2) C 2 = DT* ( OMB* ( X ( N ) +A 1 / 2 ) -DOM* ( Y ( N ) +E 1 / 2 ) ) A3=DT*(-0MB*(Z(N)+C2/2)) B3=DT*D0M*(Z(N)+C2/2) C3=DT*(0MB*(X(N)+A2/2)-D0H*(Y(N)+B2/2)) A4=DT*(-0MC*(Z(N)+C3)) B4=DT*D0M*(Z(K)+C3) C4=DT* ( OMC* ( X (N ) +A3 ) -DOM* ( Y ( 11 ) +B3 ) ) ENDIF X(N+l)=X(N)+(Al+2*A2+2*A3+A4)/6 Y(N+l)=Y(N)+(Bl+2*E2+2*33+B4)/6 Z(N+l)=Z(N)+(Cl+2*C2+2*C3+C4)/6 101 CONTINUE IF (L.EQ.H1) THEN PRINT *, X(1),X(N1),X(L) PRINT *, Y(1),Y(N1),Y(NL) PRINT *, Z(1),Z(N1),Z(NL) ENDIF Z1(L)=Z(NL) xi(l)=x(i;l) Y1(L)=Y(NL) X Y 1 ( L )=SQRT ( XI ( L ) **2+Y 1 (L ) **2 ) 100 CONTINUE DO 44 KK=1,N3 FIL0U1(KK)=RF1A(KK) FILOU 1 ( KK+N3 ) =RF1B ( KK ) FIL0U1(KK+2*N3)=RF1C(KX) 44 CONTINUE WRITE(22,1100) (L,FIL0II1(L) ,L=1,3*N3) WRITE(17,1100) (L, XI (L), Lei, ML) VRITE(13,1100) (L,Y1(L),L=1,ML) WRITE(19,1100) (L,Z1(L),L=1,ML) URITE(20,1100) (L,XY1(L) ,L-1,ML) 1100 F0RMAT(I5,E14.6) STOP END

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APPENDIX E FOURIER COEFFICIENTS FOR THE FOURIER SERIES OF THE OPTIMAL RF PULSE IN FIG. 2 REPRESEKTION 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Cosine 0, .296246E+04 0, .602884E+04 0. .603475E+04 0. , 623589E+04 0, .382474E+04 0. .934820E+04 0, .690043E+04 0. .359159E+04 0. .256553E+04 0. , 160030E+04 0, .122225E+04 0, .771072E+03 0. 599388E+03 0. .376579E+03 0. .2S8086E+03 0. .177351E+03 0, .12S165E+03 0. .758214E+02 0, .546459E+02 0. .213661E+02 0. , 183438E+02 0. .924536E+00 0, .111633E+02 0, .225646E+01 0, •673807E+01 0, .609533E+01 0, .504489E+01 0, .421767E+01 0, .760243E+01 0, .255727E+01 0, .457754E+01 0, .672835E+01 0, .353689E+01 0. .637534E+00 0, .101881E+02 0. .218592E+01 0, .753347E+01 0. .971475E+00 0, .833246E+01 0, , 126060E+00 Sine O.OOOOOOE+00 0.374348E+03 0.776027E+03 0.118075E+04 0.173430E+04 0.310588E+04 0.246532E+04 0.170867E+04 0.136224E+04 0.104353E+04 0.863594E+03 0.667074E+03 0.543747E+03 •0.424560E+03 0.329281E+03 -0.267328E+03 0.183564E+03 0.157226E+03 0.919727E+02 0.751788E+02 629E+02 716E+02 393E+02 127E+02 695E+02 642E+02 206E+01 300E+01 217E+01 141E+01 544E+01 982E+01 891E+01 0.383432E+01 157 919 751 0.329 -0.357 0.210 0.259 0.109 0.118 0.208 0.435 -0.697 0.506 0.269 0.547 0.132 18 0.101 0.842 •0.556 0.392 0.399 0.204 133E+02 952E+01 266E+01 170E+01 435E+01 690E+01 104

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APPENDIX F OURIER COEFFICIENTS FOE THE FOURIER SERIES REPRESENTATION OF THE OPTIMAL RF PULSE IN FIG. 14 Sine O.OOOOOOE+OO -0.412471E+03 0.907366E+03 -0.171388E+04 0.125197E+04 -0.616422E+03 0.433608E+03 -0.248337E+03 0.155689E+03 -0.923386E+02 0.571649E+02 -0.273676E+02 0.2510R7E+02 -0.237399E+01 0.108717E+02 0.189106E+01 0.865308E+01 0.412969E+01 0.352519E+01 0.347007E+01 0.824660E+01 0.281181E+00 0.252998E+01 O.S98227E+01 -0.1S4282E+01 0.509570E+01 0.202300E+01 0.351510E+01 0.183202E+01 0.868455E+00 0.426226E+01 0.137229E+01 -0.199483E+00 0.457037E+01 -0.280655E+00 0.121986E+01 0.143490E+01 0.135804E+01 0.359790E+00 0.413855E+00 105 Cosine 1 -0. 291488E+04 2 0. 596038E+04 3 -0. 647590E+04 4 0. 803627E+04 5 -0. 456489E+04 6 0. 165962E+04 7 -0. 997046E+03 8 0. 451111E+03 9 -0. 25991 c E+03 10 0. 118195E+03 11 -0. 748929E+02 12 0. 23O019E+02 13 -0. 275648E+02 14 -0. 214526E+01 15 -0. 106583E+02 16 -0. 416702E+01 17 -0. 679630E+01 18 -0. 417355E+01 19 -0. 331207E+01 20 -0. 269037E+01 21 -0. 352532E+C1 22 -0. 15184°E+01 23 -0. 142062E+01 24 -0 143621E+01 25 -0. 130772E+01 26 -0. 287690E+00 27 -0. 962 701E+00 28 0. 378960E-01 29 0. 641276E-01 30 -0. 882932E+00 31 0. 143774E+01 32 0. 274126E+00 33 -0 12476SE+01 34 0. 308371E+01 35 -0 692546E+00 36 0. 518338E+00 37 109133E+01 38 121201E+01 39 676590E+00 40 -0 .377214E+00

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APPENDIX G FORTRAN PROGRAM FOR FOURIER COEFFICIENTS C NOV. 23RD 1986 DIMENSION A(400),B(400),F(606),FILINP(606),FS(600) DIMENSION SG1(400) ,SG3(400) , SG4(400) , SG2(400) ,S(400) READ(11,*) DD,DT,H,ML,ID READ ( 11,*) OH , M , D , P , NL , N J PRINT *, DD,DT,H,ML,ID PRINT *, OM,M,D,P,NL,NJ N3-NL-1 READ (10,*) (IDUM,FILINP(L),L=1,N3) DO 10 1=1, N3 F(I)=FILINP(I) 10 CONTINUE TPI=6. 2831853 A(1)=0.0 B(1)=0.0 S(1)=0.0 DO 12 N=1,NL DO 11 K=1,N3 A ( N )=A( N )+F( K ) *COS (TPI* ( N-l )*K/N3) B(N)=B(N)-F(K)*SIN(TPI*(N-1)*K/N3) 11 CONTINUE A(N)«2*A(N)/N3 E(N)=2*B(N)/N3 12 CONTINUE A(l)=A(l)/2 C DO 22 K=1,N3 DO 21 N=1,NJ SS=B(N)*SIN(TP1*(N-1)*K/N3) S(K)=S(K)+A(N)*C0S(TPI*(N-1**K/N3)-SS 21 CONTINUE FS(K)=S(K) FS(K+N3)=S(K) FS(K+2*N3)=S(K) 22 CONTINUE WRITE(14,1100) (L,A(L),L=1,NL) WRITE(15,1100) (L,B(L),L=1,NL) WRITE ( 1 6 , 1 1 00 ) ( L , FS ( L ) , L= 1 , 3*N3 ) 1100 F0RMAT(I5,E14.6) STOP END 106

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APPENDIX II FORTRAN PROGRAM FOR 90 DEGREE PULSE DESIGN C Sept. 29 1986 C program SELE52 REAL Jl, J2, J, JJ.HUA.II, JA, JE, JC , JD , JO , LI , L2 , L3 , L4 C C C c IDD IDF IDO DIMEN DIMEN DIMEN DIMEN DIMEN DIMEN DIMEN DIMEN DIMEN DIMEN DIMEN DIMEN desir 180 d Dz(l) EEAD( READ( READ( READ( READ( SIGN X(201),Y(201),Z(201),S1(201),FILINP(606) SION XA(201),YA(201),ZA(201),U3(160),Z2(160) SION XB(201),YB(201),ZB(201),WB(201) SION RF1A(201),RF1B(201),RF1C(201),DESFIL(640) SION RF2A(201),RF2B(201),RF2C(201),DXY(160) SION RF0A( 201 ) , RF0B( 201 ) , RFOC( 201 ) , S( 201 ) SION OMAA(201),OMBB(201),OMCC(201),TZ(160) SION W( 201 ) , WW( 201 ) , TV.'( 1 60) , TX ( 1 60 ) , TY ( 1 60) SION X1(160),Y1(160),Z1(160),X2(160),Y2(160) SION X3(160),Y3(160),Z3(160),X4(160),Y4(160) SION X0(160),Y0(160),Z0(160),Z4(160) SION FIL0U1(606), FIL0U2 ( 606 ) , FIL0U3( 730) ed Mxy ( 1 , 2 , 3 ) , IDE=1 from calcu. curve; =2 saved egree(l), 90 degree(3); IDG=2 no optimization; , Dxy(2); ID1I = 4 taking half of the 180 as 90; 11,*) DD.DT.II.ML.IDP 0M,M,C0C,P,NL 11,12,13, 14, GI.SIKC DRTA , RFA , COE , COEF , DRTA 1 IDD, IDE, IDF, IDG, IDH, IDO 11 11 11 11 READ(11 PRINT * PRINT * PRINT * PRINT * PRINT * PRINT * PRINT * *) *) *) *) *) KC.P.ATIO.KD DD,DT,H,ML,IDP OM,M,COC,P,NL I1,I2,I3,I4,GI,SINC DRTA , RFA , COE , COEF , DRTA 1 IDD .IDE, IDF, IDG, IDH, IDO RXY1,RXY2,RY1,RY2,RZ1,RZ2 KC, RATIO, KB N3 = N ;;2=3 if ( READ IF ( DO 7 RFOA RFOB RFOC L-l *N3 IDE. (10, IDH. 1 = (D = (D = (D = EQ.2) THEN *) (IDUM.FILINP(L) EQ.4) THEN 1.N3 FILINP(I)/2 FILINP(I+N3)/2 FILINP(I+2*N3)/2 L=1,M2) 107

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108 70 CONTINUE ELSE DO 71 1=1, N3 RFOA(I)=FILINP(I) RF0B(I)=FILINP(I+N3) RF0C(I)=FILINP(I+2*N3) 71 CONTINUE ENDIF ENDIF :a = (NL-i)/2+i N2=N1-1 N4=Nl-2 RN1=N1 Iil = (HL+l)/2 RM1=M1 IF (IDO.EQ.l) THEN DO 72 1=1,11 DESFIL(I)=H 72 CONTINUE IF (IDD.EQ.l) THEN IF (IDF.E0.3) THEN DESFIL(I1+1)=0.5*H DESFIL(I1+I2+2)=0.5*H ELSE DESFIL(I1+1==0.0 DESFIL(Il+I2+2)=0.0 ENDIF DO 73 1=1,12 DESFIL(I+I1+1)=-GI 73 CONTINUE DO 74 1=1,13 DESFIL(I+I1+I2+2)=H 74 CONTINUE ENDIF IF (IDD.EQ.2) THEN IF (IDF.EQ.3) THEN DESFIL(Il+l)=3*H/4 DESFIL(Il + 2)=II/2 DESFIL(Il+3)=H/4 DESFIL(Il+3+I2+l)=H/4 DESFIL(Il+3+I2+2)=H/2 DESFIL( I 1+3+12+3 )=3*H/4 ELSE DESFIL(Il+l)=H/2 DESFIL(Il+2)=0.0 DESFIL(Il+3)=-GI/2 DESFIL( I 1+3+12+1 )=-GI/2 DESFIL(Il+3+I2+2)=0.0 DESFIL(Il+3+I2+3)=H/2 ENDIF DO 75 1=1,12

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109 DESFIL(Il+3+I)=-GI 7 5 CONTINUE DO 76 1=1,13 DESFIL(Il + I2 + 6 + I)=II 7 6 CONTINUE ENDIF IF (IDD.EQ.3) THEN DESFIL(Il+l)=3*GI/4 DESFIL(Il+2)=GI/2 DESFIL(Il+3)=GI/4 DESFIL(Il+4)=0.0 DESFIL(Il+5)=-GI/4 DESFIL(Il+6)=-GI/2 DESFIL(Il+7)=-3*GI/4 DESFIL(Il+7+I2+l)=-3*GI/4 DESFIL(Il+7+I2+2)=-GI/2 DESFIL( I 1+7+12+3 )=-GI/4 DESFIL(Il+7+I2+4)=0.0 DESFIL(Il+7+I2+5)=GI/4 DESFIL(Il+7+I2+6)=GI/2 DESFIL( I 1+7+12+7 )=3*GI/4 DO 77 1=1,12 DESFL(Il+7+I)=-GI 7 7 CONTINUE DO 78 1=1,13 DESFIL( I 1+7+1 2+7+1 )=GI 78 CONTINUE ENDIF ENDIF C IF (ID0.EQ.2) THEN DO 79 1=1,11 DESFIL(I+3*HL)=0.0 79 CONTINUE C desired profile is Mxy IF (IDD.EQ.l) THEN DESFIL(I1+1+3*ML)=0.5*H DESFIL(Il+I2+2+3*ML)=0.5*H DO 80 1=1,12 DESFIL(I1+1+I+3*ML)=H 80 CONTINUE DO 81 1=1,13 DESFIL(Il+I2+2+I+3*ML)=0.0 81 CONTINUE ELSEIF (IDD.EQ.2) THEN DESFIL(1+I1+3*ML)=0.25*H DESFIL(2+I1+3*KL)=0.5*H DESFIL(3+I1+3*ML)=0.75*H DESFIL(l+Il+I2+3+3*ML)=0.7 5*H DESFIL(2+Il+I2+3+3*HL)=0.5*H DESFIL(3+Il+I2+3+3*ML)=0.25*H DO 32 1=1,12 DESFIL(3+I1+I+3*KL)=H 82 CONTINUE

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110 DO 83 1=1,13 DESFIL(I+6+Il+I2+3*ML)=0.0 83 CONTINUE ENDIF i ENDIF DO 84 I = l,3*iIL DESFIL(I)=0.0 84 CONTINUE URITE(1S,1200) (L,PESFIL(L),L=1,4*KL) 1200 F0RiIAT(I5,E14.6) DO 35 1=1, ML DXY(I)=DESFIL(I+3*ML) 8 5 CONTINUE i IK = 1 112 BETA=0.0 K = l DO 86 N=1,NL S(N)=0.0 S1(N)=0.0 86 CONTINUE 111 DO 100 L-l.ML RL=L IF (L.LE.M1) THEN D0M-(RL-RM1)*DD ELSEIF (L.GT.M1) THEM D0M=(RL-RM1)*DD ENDIF X(1)=0.0 Y(1)=0.0 Z(1)=H DO 101 N=1,N3 rn=n IF (K.EQ.l) THEN IF (IDE.EQ.l) THEN IF (M.EQ.l) THEN 0MA=0M OME=OM OMC=OM elseif (m.eq.2) then exa=((rn-rn1)*dt)**2/(2*d**2) o;;a=om*exp(-exa) EXB=((RN-RNl)*DT+DT/2)**2/(2*D**2) 0MB=0M*EXP(-EXB) EXC=((RN-PN1)*DT+PT)**2/(2*D**2) OMC=OM*EXP(-EXC) ELSEIF (M.EQ.3) THEN IF (N.EQ.N1) THEN OHA=OM ELSE exa=p*(rn-rn1)*dt oi;a=oh*sin(exa)/(exa)

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Ill ENDIF EXB=P*((RN:Nl)*DT+DT/2) OMB-OM*SI ( :XB)/(EXB) if (N.EQ.N2) the;; o:;c=om ELSE exc=p*((en-rni)*dt+dt) omc=om*sin(exc)/(exc) ENDIF ENDIF ELSEIF (IDE.EQ.2) THEN OMA=RFOA(N) OMB=RFOB(N) OMC=RFOC(N) ENDIF OMAA(N)-OMA OMBB(N)=OMB OMCC(N)-OMC ELSEIF (K.NE.l) THEN oi;a=omaa(m) omb=ombb(n) omc-omcc(n) ENDIF IF (K.EQ.l) THEN RF1A(N)=0HA RF1B(N)=0MB RF1C(N)=0MC ENDIF Al=DT*DOM*Y(N) Bl=DT*(-DOM*X(N)-OMA*Z(N)) Cl=DT*OMA*Y(N) A2=DT*DON*(Y(N)+Bl/2) B2=DT*(-DOM*(X(N)+Al/2)-OKB*(Z(N)+Cl/2)) C2=DT*OMB* ( Y ( N ) +B1 / 2 ) A3=DT*DOH*(Y(N)+B2/2) B3=DT*(-DOM*(X(N)+A2/2)-OMB*(Z(N)+C2/2)) C3=DT*0;iE*(Y(N) + B2/2) A4=DT*DOM*(Y(N)+B3) B4=DT*(-DOM*(X(N)+A3)-OMC*(Z(N)+C3)) C4=DT*0MC*(Y(N)+B3) X(N+l)=X(N)+(Al+2*A2+2*A3+A4)/6 Y(N+l)-Y(N)+(Bl+2*B2+2*B3+B4)/6 Z(N+l)-Z(N)+(Cl+2*C2+2*C3+C4)/6 101 CONTINUE IF (K.EQ.l) THEN IF (L.E0.M1) THEN PRINT *, Z(1),Z(N1),Z(NL) ENDIF ENDIF ZO(L)=Z(NL) XO(L)=X(NL)

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112 YO(L)=Y(NL) 100 CONTINUE IF (IDF.EQ.3) THEN DO 102 L=1,ML RL = L D0M=-(RL-P.M1)*DD*0.5 X(1)=X0(L) Y(D = YO(L) Z(1)=Z0(L) W(1)=0.0 DO 103 N-1.N3 OIIA = 0.0 Oi!B=0.0 0MC=0.0 A1=DT*D0M*Y(N) B 1=DT* ( -DOM*X ( N ) -OMA*Z ( N ) ) C1=DT*0MA*Y(N) D1=D1*(X(N)**2+Y(N)**2-DXY(L)**2)**2 A2=DT*D0M*(Y(N)+Bl/2) B2=DT*(-D0M*(X(N)+Al/2)-0MB*(Z(N)+Cl/2)) C2=DT*OMB*(Y(N)+Bl/2) DD= ( Y ( N )+B 1*0.5) **2-DXY ( L) **2 D2=DT*((X(N)+A1*0.5)**2+DD)**2 A3=DT*DOM*(Y(N)+B2/2) B3=DT*(-D0M*(X(N)+A2/2)-0MB*(Z(N)+C2/2)) C3=DT*0MB* ( Y( N )+B2/2 ) DD=(Y(N)+B2*0.5)**2-DXY(L)**2 D3-DT*((X(N)+A2*0.5)**2+DD)**2 A4=DT*D0M*(Y(N)+B3) B4=DT*(-D0M*(X(N)+A3)-0MC*(Z(N)+C3)) C4=DT*0MC*(Y(N)+B3) DD=(Y(N)+B3)**2-DXY(L)**2 D4=DT*((X(N)+A3)**2+DD)**2 X(N+l)«X(N)+(Al+2*A2+2*A3+A4)/6 Y(N+l)=Y(N)+(Bl+2*B2+2*B3+B4)/6 Z(N+l)=Z(N)+(Cl+2*C2+2*C3+C4)/6 W(N+l)=W(N)+(Dl+2*D2+2*D3+D4)/6 103 CONTINUE IF (K.EO.l) THEN Z1(L)=Z(NL) X1(L)=X(NL) Y1(L)=Y(NL) ENDIF TX(L)=X(NL) TY(L)=Y(NL) TW(L)=W(NL) 102 CONTINUE ENDIF J0=0.0 JA=0.0 DO 21 L=1,KL

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c c 113 J=RXY2*(TX(L)**2+TY(L)**2-DXY(L)**2)**2+TW(L) JO=JO+J J=RXY2*(TX(L)**2+TY(L)**2-DXY(L)**2)**2 JA=JA+J 21 CONTINUE IF (K.EQ.l) THEN PRINT *, JO.JA ENDIF no optimization, goto to the end IF (IDG.EQ.2) THEN GOTO 402 ENDIF DO 200 L=1,ML RL=L D0M=(RL-RH1)*DD XA(1)=0.0 YA(1)=0.0 ZA(1)=H DO 201 N=1,N3 RK-N IF (K.EQ.l) THEN IF (IDE.EQ.l) THEN IF (M.EQ.l) THEN 0MA=0H OMB-OH 0MC=0H ELSEIF (M.EQ.2) THEN EXA=((RN-RN1)*DT)**2/(2*D**2) OMA=OM*EXP(-EXA) EXB-((RN-RNl)*DT+DT/2)**2/(2*D**2) 0MB=0h*EXP(-EXB) exc=((rn-rn1)*dt+dt)**2/(2*d**2) o;jc=om*exp(-exc) elseif (m.eq.3) then if (n.eq.n1) then oka=om ELSE EXA=P*(RN-RN1)*DT OMA=OM*SIN(EXA)/(EXA) ENDIF EXB=P*((RN-RNl)*DT+DT/2) OMB=OM*SIN(EXB)/(EXB) IF (N.EQ.N2) THEN OMC=OM ELSE EXC=P*((RN-RN1)*DTDDT) OMC=OM*SIN(EXC)/(EXC) ENDIF ENDIF ELSEIF (IDE.EQ.2) THEN 0MA=RFOA(N)

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114 OMB o;:c END ELS OMA OMB OMC END integra Al = Bl = Cl = A2 = B2 = C2 = A3 = 33 = C3 = A 4 = B4= C4 = XA( YA( ZA( CON RL= DOM XB( YB( ZB( W(l integra DO OMA !01 =RFOB(N) -RFOC(N) IF EIF (K.NE. =OMAA(N) =OMBB(N) =OMCC(N) IF te Eq.[66] DT*D0i:*YA( DT*(-DOM*X DT*OMA*YA( DT*DOM*(YA DT*(-DOM*( DT*OMB*(YA DT*DOM*(YA DT*(-DOM*( DT*OMB*(YA DT*DOM*(YA DT*(-DOM*( DT*OMC*(YA N+1)=XA(N) N+1)=YA(N) N+1)=ZA(N) TINUE L =-(RL-RMl) 1)=X0(L) 1)«Y0(L) 1)=Z0(L) ) = 0.0 te Eq.[72] 211 N-1.N3 = 0.0 OMB=0.0 OMC=0.0 A1=DT*D0M*YB( B1=DT*(-D0M*X C1=DT*0MA*YB( B1=DT*(XB(N)* A2»DT*D0K*(YB 32=DT*(-D0M*( C2=dt*o;:b*(yb DD=(YB(N)+B1* D2=DT*((XB(N) 3=DT*D0M*(YB( B3=DT*(-D0M*( C3=DT*0MB*(YB DD=(YB(N)+B2* DB=D?*((XE(N) A4=DT*D0K*(YB B4=DT*(-D0M*( C4=DT*0HC*(YE I)D=(YB(N)+B3) 1) THEN of Chap N) A(N)-OKA N) (TO+B1/2 XA(N)+A1 (IJ) + Bl/2 (N)+B2/2 XA(N)+A2 (N)+B2/2 (N)+B3) XA(N)+A3 (N)+B3) +(A1+2*A +(B1+2*B +(C1+2*C *DB*0.5 ter V *ZA(N)) ) /2)-0MB*(ZA(N)+Cl/2)) ) ) /2)-OMB*(ZA(N)+C2/2)) ) )-0MC*(ZA(N)+C3)) 2+2*A3+A4)/6 2+2*B3+B4)/6 2+2*C3+C4)/6 of Chapter V N) B(N)-OMA N) *2+YB(M) (N)+Bl/2 XBfN)+Al (IO + B1/2 0.5)**2+A1*0.5) N)+B2/2) XE(K)+A2 (N)+B2/2 0.5)**2+A2*0.5) (N)+B3) XB(N)+A3 (N)+B3) **2-DXY( *ZB(N)) **2-DXY(L)**2)**2 ) /2)-0MB*(ZB(N)+Cl/2)) ) DXY(L)**2 **2+DD)**2 /2)-OMB*(ZB(N)+C2/2)) ) DXY(L)**2 **2+DD)**2 )-0MC*(ZE(N)+C3)) L)**2

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115 D4=DT*((B(N)+A3)**2+DD)**2 XE(N+l)»XB(N)+(Al+2*A2+2*A3+A4)/6 YB(N+l)=»YB(N) + (Bl+2*E2+2*B3+B4)/6 ZB(N+l)=ZB(N)+(Cl+2*C2+2*C3+C4)/6 W(N+l)=!v'(N) + (Dl+2*D2+2*D3+D4)/6 211 CONTINUE C integrate Eq.[84] of Chapter V X ( 1 ) = 1 . Y(1)=0.0 Z(1)=0.0 U(1)=0.0 BO 202 N=1,N3 O!!A=0.0 OMB-0.0 o;ic=o.o XD=(XE(N)+XB(N+l))/2 YD=(YB(N)+YB(N+l))/2 Al=DT*DOH*Y(N) B1=DT*(-D0M*X(N)-0MA*Z(N)) C1=DT*0MA*Y(N) DA=2*(XB(N)**2+YB( N ) **2-DXY ( L ) **2 ) *2*XB ( N ) *X ( H ) DB=2*(XB(N)**2+YB(N)**2-BXY(L)**2)*2*YB(N)*Y(N) D1=DT*(DA+DB) A2-DT*DOM*(Y(N)+Bl/2) B2=DT*(-D0M*(X(N)+Al/2)-0MB*(Z(N)+Cl/2)) C2-DT*OHB*(Y(N)+Bl/2) DA=2*(XD**2+YD**2-DXY(L)**2)*2*XD*(X(N)+A1*0.5) DB=2*(XD**2+YD**2-DXY(L)**2)*'2*YD*(Y(N)+B1*0.5) D2=DT*(DA+DB) A3=DT*DOM*(Y(N)+B2/2) B3=DT* ( -DOM* (X(N)+A2/2 ) -OKB* ( Z( N )+C2/2 ) ) C3=DT*OMB*(Y(N)+E2/2) DA=2*(XD**2+YD**2-DXY(L)**2)*2*XD*(X(N)+A2*0.5) DB=2* ( XD**2+YD**2-DXY( L ) **2 )*2*YD* ( Y ( N )+B2*0 . 5 ) D3 = DT*(1)A + DB) A4-DT*D0H*(Y(N)+B3) B4-DT* ( -DOM* ( X ( N ) +A3 ) -CMC* ( 7, ( N ) +C3 ) ) C4=DT*0MC*(Y(N)+B3) AA=2*XB( N+l ) * ( X (N) + A3 ) BB=2* YB ( N+l ) * ( Y( N ) +B3 ) DA=2*(XB(N+1)**2+YB(N+1)**2-DXY(L)**2)*AA DB=2* (XB(N+1 ) **2+YB (N+l )**2-DXY (L)**2) *BB D4oDT*(DA+DB) X(N+l)=X(N)+(Al+2*A2+2*A3+A4)/6 Y(K+l)=Y(N)+(Bl+2*B2+2*B3+B4)/6 Z(N + l)=Z(lI) + (Cl + 2*C2+2*C3 + C4)/6 W(N+l)=W(N) + (Dl+2*D"2+2*D3+D4)/6 202 CONTINUE Z3(L)=Z(NL) X3(L)=X(NE) Y3(L)=Y(NL) W3(L)«W(NL) DZ=X3(L)*(TX(L)**2+TY(L)**2-DXY(L)**2)*2*TX(L) EZ=Y3(L)*(TX(L)**2+TY(L)**2-DXY(L)**2)*2*TY(L)

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116 XX C i n t e g r X( Y( Z( W( do OH OH O'l XD YD Al Bl CI DA DB Dl A2 E2 C2 DA DB D2 A3 B3 C3 DA DD D3 A A FA C4 A A BB DA DB DA X( (1)-DZ+EZ+W ate Eq.[84] 3(L) of Chapter V 212 Y( Z( W( CO Z3 X3 Y3 DZ 1)=0.0 1)-1.0 1)=0.0 l)=0.0 212 M=1,K3 A = 0.0 B=0.0 C=0.0 =(XB(N)+XB =(YB(N)+YB =DT*D0M*Y( =DT*(-DOH* =DT*0MA*Y( =2*(XB(N)* =2*(XE(N)* =DT*(DA+DB «DT*DOM*(Y -DT*(-DOM* =DT*OMB*(Y =2*(XD**2+ =2*(XD**2+ =DT*(DA+DB =DT*DOM*(Y =DT*(-DOM* =DT*OMB*(Y =2*(XD**2+ «2*(XD**2+ =DT*(DA+DE =DT*DOM*(Y -DT*(-DOM* =DT*OMC*(Y -2*XB(K+1) =2*YB(N+1) =2*(XB(N+1 = 2*(XB(N T +1 =DT*(DA+DB M+1)=X(N)+ N+1)-Y(N)+ N+1)=Z(N)+ N+1)-W(N)+ NTINUE (L)-Z(NL) (L)-X(NL) (L)-Y(NL) (L)-W(NL) =X3(L)*(TX( N+l))/2 N+l))/2 ) (N)-OMA*Z(N)) ) 2+YB ( N ) **2-DX Y ( L) **2 )*2*XB( H ) *X ( N ) 2+YB(M ) **2-DXY (L)**2) *2*YB(N)*Y(N ) N)+Bl/2) X(N)+Al/2)-0MB*(Z(X)+Cl/2)) N)+Bl/2) D**2-DXY( L ) ** 2 ) *2*XD* (X (N )+A 1 *0 . 5 ) D**2-DXY( L) **2 ) *2*YB* ( Y(N )+Bl*0 . 5 ) N)+52/2) X ( N ) +A2/ 2 ) -OMB* (Z(N )+C2/2 ) ) N)+B2/2) D**2-DXY(L)**2)*2*XD*(X(N)+A2*0.5) D**2-DXY ( L ) * * 2 ) * 2 * YD* ( Y ( N ) +B2*0 . 5 ) N)+B3) X ( N ) +A3 ) -OMC* (Z (N )+C3) ) N)+B3) (X(N)+A3) (Y(N)+B3) **2+YB( N+ 1 ) **2-DXY ( L ) **2 ) * A A **2+YB(N+l)**2-DXY(L)**2)*BB A1+2*A2+2*A3+A4)/G Bl+2*B2+2*B3+B4)/6 Cl+2*C2+2*C3+C4)/6 Dl+2*D2+2*D3+D4)/6 YY C integr L)**2TY(L)**2-DXY(L)**2)*2*TX(L) EZ=Y3(L)*(TX(L)**2+TY(L)**2-DXY(L)**2)*2*TY(L) 3(L) of Chapter V (1)=DZ+EZ+W; ate Eq.fSA] X(l)-0.6 Y(1)=0.0 Z(l)=1.0

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117 222 C C W(l)DO 22 o:;a=o OMB=0 o;;c=o XD = (X YD=(Y A1 = DT B1=DT C1 = DT DA = 2* DB=2* D1=DT A2=DT B2=DT C2 = DT DA=2* DB=2 J i D2=DT A3=DT B3 = DT C3=DT DA=2* DB=2* D3 = DT A4 = DT B4=DT C4 = DT AA=2* BB=2* DA-2* DB=2* D4=D* X(N+1 Y(N+1 Z(N+1 w(n+i CONTI Z3(L) X3(L) Y3(L) W3(L) DZ=X3 EZ=Y3 ZZ(1) 0.0 2 K-l.N .0 .0 .0 B(N)+XB B(N)+YB *DOM*Y( *(-DOM* *OMA*Y( (XB(N)* (XB(K)* *(DA+DE *DOM*(Y *(-DOM* *OMB*(Y (XD**2+ (XD**2+ *(DA+DB *DOM*(Y *(-dom* *OMB*(Y (XD**2+ (XD**2+ *(DA+DB *DOM*(Y *(-DOM* *OMC*(Y XB(N+1) YB(N+1) (X3(: +i (XB(N+1 (DA+DB) )=X(N)+ )=Y(N)+ )=Z(N)+ )=W(N)+ MI1T7 11 U Jj -Z(NL) =X(NL) =Y(NL) =W(NL) (L)*(TX (L)*(TX =DZ+EZ+ N+l))/2 N+l))/2 ) (K)-OMA*Z(N)) ) ! 2+YB(N)**2-DXY(L) : ! 2+YB(N)**2-DXY(L) ! *2)*2*XB(N)*X(N) *2)*2*YB(N)*Y(N) ;;)+bi/2) X(N)+Al/2)-OMB*(Z(N)+Cl/2) ) N) + 1:1/2) D**2-DXY(L)**2)*2 D**2-DXY(L)**2)*2 *XD*(X(N)+A1*0.5) *YD*(Y(N)+B1*0.5) )+E2/2) X ( N ) + A 2 / 2 ) -OMB* ( Z ( N ) +C2 / 2 ) ) N)+B2/2) D**2-DXY( L ) **2 ) * 2 D**2-DXY(L)**2)*2 *XD*(X(N)+A2*0.5) *YD*(Y(N)+B2*0.5) ,:)+B3) X(N)+A3)-0KC*(Z(N)+C3)) N)+B3) (X(N)+A3) (Y(N)+B3) **2+YE(N+l)**2-DX **2+YB(N+l)**2-DX (A1+2*A2+2*A3+A4)/ (B1+2*B2+2*B3+B4)/ (C1+2*C2+2*C3+C4)/ (D1+2*D2+2*D3+D4)/ Y(L)**2)*AA Y(L)**2)*BB 6 fc> 6 6 (L)**2+TY(L)**2-DXY(L)**2)*2*TX(L) ( L ) **2+TY ( L ) **2-DXY ( L ) **2 )*2*TY( L ) W3(L) 1)0M=-2*D0M DO 203 N = -1, K3 RN = -.11 DD1 :=-dt IF (K.EQ .1) TI IEN IF (IDE.] ::o. 15 THEN IF (H.EO i) TI IEN

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118 OMA-OM OMB-OM OKC-OM ELSEIF (M.EQ.2) THEN EXA=((RN-RN1)*DT)**2/(2*D**2) 0MA=0M*EXP(-EXA) EXB=((RN-RNl)*DT+DT/2)**2/(2*D**2) OMB=OM*EXP(-EXB) EXC=((RN-RN1)*DT+DT)**2/(2*D**2) OMC=OM*EXP(-EXC) ELSEIF (M.EQ.3) THEN IF (N.EQ.N1) THEN OMA-OM ELSE EXA=P*(RN-RN1)*DT OMA=OM*SIN(EXA)/(EXA) END IF EXE=P*((RN-RNl)*DT+DT/2) OMB-OM*SIN ( EXB ) / ( EXB ) IF (N.EQ.N2) THEN OMC-OM ELSE EXC=P*((RN-RN1)*DT+DT) OMC-OM* S I N ( EXC ) / ( EXC ) ENDIF ENDIF ELSEIF (IDE.EQ.2) THEN OMA-RFOA(N) OMB-RFOB(N) OMC-RFOC(N) ENDIF ELSEIF (K.NE.l) THEN OMA-OMAA(N) OMB-OMBB(N) OMC-OMCC(N) ENDIF integrate Eq.[92] backwards in time Al=DDT*DOM*YY(N) Bl»DDT*(-DOM*XX(N)-OMA*ZZ(N)) C1=DBT*0MA*YY(N) A2=DDT*DOM*(YY(N)+Bl/2) B2-DDT*(-D0M*(XX(N)+Al/2)-OHB*(ZZ(N)+Cl/2)) C2-DDT*OIIB* ( YY ( K ) +B 1 / 2 ) A3=DUT*DOM*(YY(N)+B2/2) B3=DDT*(-DOM*(XX(N)+A2/2)-OMB*(ZZ(N)+C2/2)) C3-DDT*0MB*(YY(N)+B2/2) A4=DDT*D0M*(YY(N)+B3) B4=DDT*(-D0I1*(XX(N) + A3)-0MC*(ZZ(N)+C3)) C4=DDT*OMC*(YY(N)+B3) XX(N+l)=XX(N)+(Al+2*A2+2*A3+A4)/6 YY(N+l)=YY(N)+(Bl+2*B2+2*B3+B4)/6

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c 119 ZZ(N+l)«ZZ(N)+(Cl+2*C2+2*C3+C4)/6 203 CONTINUE DO 2 3 K-l.NL G( M) =-ZA (N ) *YY ( ML+ 1-N ) + Y A ( N ) *ZZ( NL+ 1-N) 23 CONTINUE IF (L.EQ.l) THEN DO 2 4 N-l.NL GG(N)-0.0 24 CONTINUE ENDIF DO 25 N-l.KL GG(N)=GG(N)+G(N) 25 CONTINUE 200 CONTINUE SG1-0.0 DO 31 1=1, N2 SG1=SG1+(C0E*GG(2*I-1))**2 31 CONTINUE SG2=0.0 DO 32 1=1, N4 SG2=SG2+(C0E*GG(2*I))**2 32 CONTINUE SG= ( C0E*GG( 1 ) ) **2+( COE*GG ( NL ) ) **2 D0T2=(SG+4*SGl+2*SG2)*DT/3 IF (K.NE.l) THEN BETA=D0T2/D0T1 ENDIF D0T1=D0T2 DO 33 N=1,NL S ( N ) =-GG ( N ) +BETA*S ( N ) 3 3 CONTINUE PRINT *, GG(1),GG(N1),GC(NL),K PRINT *, S(1),S(N1),S(NL),BETA HUA=G. 613034 RFA1=RFA*HA RFA2=RFA L1=RFA*HUA**2 L2=RFA*HUA L3=0.0 L4=RFA RLN-L1-L3 RLM=L4-L2 IF (K.NE.l) THEN IF (K.GT.IDP) THEM RFA=RFA*C0C ENDIF RFA1=C0EF*RFA*HUA RFA2=C0EF*RFA L1=C0EF*RFA*HUA**2 L2=C0EF*RFA*HUA L3=0.0

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120 333 L4=C0EF*RFA RLN=L1L3 RLM=L4-L2 ENDIF ID = 1 = DO 300 L-l.ML RL=L D0M=(RL-RM1)*DD X(1)=0. Y(1)=0. Z(l)-H DO 301 N=l ,N3 RN=N IF (K.EQ.l) THEN IF (IDE.EQ.l) THEN IF (M.EQ.l) THEN 0MA=0M+RFA1*S(N) 0;iB=0M+RFAl*(S(N) + S(N + l))/2 0MC=0M+RFA1*S(N+1) ELSEIF (M.EQ.2) THEN EXA=((RN-RN1)*DT)**2/(2*D**2) ONA=OH*EXP(-EXA)+RFA1*S(N) EXB=((RN-RNl)*DT+DT/2)**2/(2*D**2) 0MB=0M*EXP(-EXB)+RFAl*(S(N)+S(N+l))/2 EXC=((RN-RN1)*DT+DT)**2/(2*D**2) 0iiC = 0H*EXP(-EXC)+RFAl*S(N + l) ELSEIF (M.EQ.3) THEN IF (N.EQ.N1) THEN 0MA=0M+RFA1*S(N) ELSE EXA=P*(RN-RN1)*DT 0MA=0M*SIN(EXA)/(EXA)+RFA1*S(N) ENDIF EXB=P*((RN-RNl)*DT+DT/2) 0ME=0K*SIN(EXB)/(EXB)+RFAl*(S(N)+S(N+l))/2 IF (N.EQ.N2) THEN 0MC=0M ELSE EXC=P*((RN-RN1)*DT+DT) 0MC=0M*SIN(EXC)/(EXC)+RFA1*S(N+1) ENDIF ENDIF ELSEIF (IDE.EQ.2) THEN 0KA=RF0A(N)+RFA1*S(N) OMB=RF0B(N)+RFAl*(S(N)+S(++l))/2 0MC=RF0C(N)+RFA1*S(N+1) ENDIF

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121 ELSEIF (K.NE. 0MA=0ilAA(N)+R OMB=OI;BB(N) + R OMC-OKCC(N)+R ENDIF integrate Eq . [ 66 ] o;:aa(n)=oma ombb(n)»omb ohcc(n)=omc RF2A(N)=0MA RF2B(N)=0MB RF2C(N)=0MC A1»DT*D0M*Y(N B1=DT*(-D0M*X C1=DT*0HA*Y(N A2=DT*D0M*(Y( B2*DT*(-D0M*( C2=DT*0MB*(Y( A3=DT*D0M*(Y( B3-DT*(-D0M*( C3=DT*0KB*(Y( AA=DT*DOM*(Y( B4*=DT*(-D0M*( C4=DT*0HC*(Y( X(N+l)=X(N)+( Y(N+l)=Y(N)+( Z(N+l)=Z(N)+( 301 CONTINUE ZO(L)=Z(NL) XO(L)=X(NL) YO(L)=Y(NL) 300 CONTINUE 1) THEN FA1*S(N) FAl*(S(N)+S(N+l))/2 FA1*S(N+1) of Chapter V ) (N)-0MA*Z(N)) ) N) X( N) N) X( N) N) X( N) Al El CI + B1 N) + + B1 + B2 R) + + 32 + B3 N) + + B3 + 2* +2* + 2* /2) Al/2)-0MB*(Z(N)+Cl/2)) /2) /2) A2/2)-OMB*(Z(N)+C2/2)) /2) ) A3)-0MC*(Z(N)+C3)) ) A2+2*A3+A4)/6 B2+2*B3+B4)/6 C2+2*C3+C4)/6 IF (IDF.EQ.3) THEN DO 302 L-l.ML RL=L D0M=-(RL-RM1)*DD*0.5 X(1)=X0(L) Y(1)=Y0(L) Z(1)=Z0(L) W(1)=0.0 integrate E q . [ 7 2 ] of Chapter V DO 303 N=1,N3 01 i A = 0.0 0MB=0.0 onc=o.o Al=DT*DOM*Y(N Bl=DT*(-DOM*X Cl-DT*OMA*Y(N D1=D1*RXY1*(X A2«DT*D0M*(Y( B2=DT*(-D0M*( C2=DT*OMB*(Y( DD=(Y(N)+B1*0 ) (N)-0MA*Z(N)) ) (N)**2+Y(N)**2-DXY(L)**2)**2 N)+Bl/2) X(N) + Al/2)-0JIB*(Z(N)+Cl/2)) IO + B1/2) ,5)**2-DXY(L)**2

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122 D2=DT*RXY1*((X(N)+A1*0.5)**2+DD)**2 A3=DT*D0M*(Y(N)+B2/2) B3=DT*(-DOM*(X(N)+A2/2)-OMB*(Z(N)+C2/2)) C3=DT*0MB*(Y(N)+32/2) DD= ( Y( N )+B2*0 . 5 ) #*2-DXY(L)**2 H3»DT*RXY1*((X(N)+A2*0.5)**2+DD)**2 A4=DT*D0M*(Y(N)+B3) B4=BT* ( -DOM* ( X ( N ) +A3 ) -OMC* ( Z ( N ) +C3 ) ) C4=DT*0MC*(Y(N)+B3) DD=(Y(N)+B3)**2-DXY(L)**2 D4=DT*RXY1 * ( ( X ( N ) +A3 ) ** 2 + DD ) **2 X(N+l)«X(N)+(Al+2*A2+2*A3+A4)/6 Y(N+l)=Y(N)+(Bl+2*B2+2*B3+B4)/6 Z(N+l)=Z(N)+(Cl+2*C2+2*C3+C4)/6 V(N+l)=W(N)+(Dl+2*D2+2*D3+D4)/6 303 CONTINUE Z2(L)=Z(NL) X2(L)=X(IIL) Y2(L)=Y(NL) TW(L)-W(NL) TX(L)=X(NL) TY(L)-Y(NL) 302 CONTINUE ENDIF JJ=0.0 JB=0.0 DO 41 L=1,HL J*=RXY2*(TX(L)**2+TY(L)**2-DXY(L)**2)**2+TW(L) JJ=JJ+J J = RXY 2* ( TX ( L )**2+TY ( L ) **2-DXY (L) **2 )**2 JB=JB+J 41 CONTINUE 1 = 1 + 1 IF (I.EQ.l) THEN J2=JJ RFA1-RFA1*HUA GOTO 333 ELSEIF (I.EQ.2) THEN J1=JJ ENDIF IF (ID.EQ.l) THEN J1=JJ ELSEIF (ID.EQ.2) THEN J2=JJ ENDIF IF (K.GT.l) THEN IF (Jl.GT.JC) THEN IDO-1 GOTO 400 ELSEIF (JB.GT.JD) THEN ID0=1 GOTO 400

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123 EX DIP ENDIF IF (K.EQ.I4) THEN ID0=1 GOTO 400 ENDIF IF (J2.LT.J1) TEEN RFA2=RFA2-RLN ID=2 L3=L1 L1=L2 J1=J2 L2=L3+HUA*RFA2 RLM-L4-L2 RLN=L1-L3 EFA1=L2 GOTO 333 ELSEIF (J2.GT.J1+DRTA) THEN RFA2=RFA2-RLH ID=1 L4=L2 L2=L1 J2=J1 L1=L4-HUA*RFA2 RLM=L4-L2 RLN=L1-L3 RFA1=L1 GOTO 333 ELSE JC=J1 JD=JB PRINT *, 0,J1,JA,JB ENDIF DO 43 N=1,LL S1(N)=S1(N)+RFA1*S(N) 4 3 CONTINUE K = K+1 IF (J2.LT.J1+DRTA1) THEN ID0=1 GOTO 400 ELSE ID0=2 GOTO 111 ENDIF 400 CONTINUE IF (IDE.EQ.2) THEN output as input, run again IF (IK.LT.KD) THEN DO 401 1=1, N3 RF0A(I)=RF2A(I)

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124 RF0B(I)=RF2B(I) RF0C(I)=RF2C(I) 401 CONTINUE IK=IK+1 PRINT *,IK GOTO 112 ENDIF ENDIF i 402 DO 44 KK=1,N3 FILGU1(KK)=RF1A(KK) FIL0U1 (KK+N3 ) =RF1E ( KK ) FIL0U1(KK+2*N3)=RF1C(KK) 44 CONTINUE DO 45 KK-1.N3 FILOU2(KK)=RF2A(KK) FILOU2(KK+N3)=RF2B(KK) FIL0U2(KK+2*N3)=RF2C(KK) 4 5 CONTINUE DO 46 KK=1,ML FIL0U3(KK)=Z1(KK) FIL0U3 ( KK+ML ) =X 1 ( KK ) FIL0U3(KK+2*ML)=Y1(KK) SQ=X1(KK)**2+Y1(KK)**2 FIL0U3(KK+3*ML)=SQRT(SQ) 46 CONTINUE DO 47 KK11.ML FIL0U4(KX)=Z2(KK) FIL0U4(KK+ML)=X2(KK) FILOU4(KK+2*ML)=Y2(KK) SQ=X2(NK)**2 + Y2(KK)' : *2 FIL0U4(KK+3*HL)=SQRT(SQ) 47 CONTINUE M3-4*ML DO 48 KK=1,M2 FIL0U5(KK)=0.0 48 CONTINUE DO 49 KK=1,K3 FILOU6(KK)=0.0 49 CONTINUE WRITE(22,1100) (L,FIL0U1(L),L=1,M2) WRITE (2 1,1100) (L,FILOU2(L),L=l,!I2) WRITE( 14 , 1 100) ( L , FIL0U3 ( L) , L=l , M3 ) WRITE (15, 11 00) (L,FIL0U4(L),L= 1,113) V;RITE(16,1100) (L,FIL0U5(L),L=1,M2) WRITE(17,1100) (L,FILOU6(L),L=l,M3) C suppress the floating point underflow CALL ERRSET(208, 256,-1, 1) 1100 FORMAT (1 5, E44. 6) STOP END

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APPENDIX I JOB PROGRAM FOR APPENDICES B AND H THE JOB FOR APPENDIX E: //MAOl JOB (2003,5001, 100, 7, 999), MAO, CLASS=1 // EXEC FORTVCLE /INCLUDE SELE51 FORTRAN //G0.FT18F001 DD SYSOUT=C //G0.FT10F001 DD * /INCLUDE FILINP DATA //G0.FT10F001 DD * /INCLUDE INPI //G0.FT22F001 //GO.FT13F001 //G0.FT14F001 //G0.FT15F001 //G0.FT16F001 //G0.FT17F001 //GO.SYSIN DD THE JOB FOR APPENDIX H: //MA02 JOB (2003,5001, 100, 7, 999), MAO, CLASS-1 // EXEC FORTVCLE /INCLUDE SELE52 FORTRAN //G0.FT18F001 DD SYSOUT=C //G0.FT10F001 DD /INCLUDE FILINP DATA //G0.FT10F001 DD /INCLUDE INPI //G0.FT22F001 //G0.FT13F001 //GO.FT14F001 //G0.FT15F001 //G0.FT16F001 //G0.FT17F001 //GO.SYSIN DD TF DATA DD SYSOUT=J DD SYSOUT-E DD SYSOUT=F DD SYSOUT=G DD SYSOUT-K DD SYSOUT-I * TF DD DATA SYSOUT=J DD SYSOUT=E DD SYSOUT=F DD SY30UT=G DD SYSOUT=H DD SYSOUT=I 125

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APPENDIX J PARAMETER VALUES OF THE PROGRAM IN APPENDIX H TO OBTAIN TEE OPTIMAL RF PULSES OF FIG. 20, FIG. 25 AND FIG. 28 a) The parameter values for the optical rf pulse of Fig. 20 1570.0, 1.11111 1E-05 , IOC, 159, 2 -9065.0, 0.7, 19300.0, 91 66, 21, 66, 5, o, 1 50.0, 0.1E-03, 0.5E-05, 0.1E-03, 10.0 2, 1, 3, 0, 5, 2 1, 1, 1, 1, 1, 1 20, 0.7, 15 b ) The parameter values for the optimal rf pulse of Fig. 25 1570.0, 1.1] 11111E-05, 100, 159, 1 -9065.0, 0.7, 19300.0, 91 67, 23 67, 5, o, 1 50.0, 0.1E-03, 0.5E-05, 0.1E-03, 10.0 1, 2, 3, o, 5, 2 1, 1, 1, 1, 1, 1 20, 0.7, 12 c) The parameter values for the optimal rf pulse of Fig. 28 1570.0, 1, . 11111 1E-05, 100, 159, 10 -15396.0, 3, 0.7, 31975.0, 91 58, 37, 53, 5, o, 1 50.0, 0.1E-03, 0.5E-05, 0.1E-03, 10.0 2, 1, 3, 0, 5, 2 1, 1, 1, 1, 1, 1 20, 0.7, 15 126

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APPENDIX K FOURIER COEFFICIENTS OF THE FOURIER SERIES REPRESENTATION OF THE OPTIMAL RF PULSE IN FIG. 20 Sine O.OOOOOOE+OO -0.201899E+03 0.507699E+03 -0.337984E+03 -0.383514E+00 -0.579742E+02 -0.342352E+G2 -0.407180E+02 -0.337707E+02 -0.316671E+02 -0.279499E+02 -0.251290E+02 -0.233887E+02 -0.208346E+02 -0.192330E+02 -0.182061E+02 -0.155616E+02 -0.1586S3E+02 -0.134981E+02 -0.130264E+02 -0.125601E+02 -0.106989E+02 -0.109925E+02 -0.951433E+01 -0.912414E+01 -0.867219E+01 -0.771478E+01 -0.740869E+01 -0.689631E+01 -0.605017E+01 -0.626775E+01 -0.489746E+01 -0.523200E+01 -0.427739E+01 -0.39S430E+01 -0.390304E+01 -0.303956E+01 -0.304002E+01 -0.249087E+01 -0.203356E+01 127 Cosine 1 -0. , 169372E+04 2 0. .288795E+04 3 -0, .361280E+04 4 0. .159014E+04 5 0. ,141494E+01 6 0. .159420E+03 7 0. .769519E+02 8 0. , 765642E+02 9 0. , 540168E+02 10 0. 435509E+02 11 0. .332784E+02 12 0. , 259571E+02 13 0. .210232E+02 14 0. .162115E+02 15 c. •129513E+02 16 0. .104777E+02 17 0. .755129E+01 18 0. .633576E+01 19 0. .434571E+01 20 0. .325267E+01 21 0. .216887E+01 22 0, •110160E+01 23 0. •369705E+00 24 -0, .350803E+00 25 -0, .974517E+00 26 -0. .156207E+01 27 -0. .193531E+01 28 -0. .242C71E+01 29 -0, .278906E+01 30 -0. •298010E+01 31 -0. .364296E+01 32 -0. .332352E+01 33 -0, .415392E+01 34 -0, .383264E+01 35 -0, .415360E+01 36 -0, .464790E+01 37 -0, .422144E+01 38 -0, .486577E+01 39 -0, .469502E+01 40 -0, .457746E+01

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APPENDIX L FOURIER COEFFICIENTS OF THE FOURIER SERIES REPRESENTATION OF THE OPTIMAL RF PULSE IN FIG. 28 Sine O.OOOOOOE+OO -0.199309E+03 0.447691E+G3 -0.616315E+03 0.103004E+04 -0.620669E+04 0.413190E+02 -0.114660E+03 -0.552481E+02 -0.567666E+02 -0.403002E+02 -0.386490E+02 -0.310853E+02 -0.291230E+02 -O.257630E+02 -0.230600F+02 -0.209952E+02 -0.190140E+02 -0.177844E+02 -0. 162040E+02 -0.151512E+02 -0.139326E+02 -0.123477E+02 -0.121618E+02 -0.111424E+02 -0.10496&E+02 -0.955069E+01 -0.899339E+01 -0.828694E+01 -0.757017E+01 -0.727276E+01 -0.634151E+01 -0.602479E+01 -0.542441E+01 -0.500013E+01 -0.437506E+01 -0.333065E+01 -0.357354E+01 -0.292428E+01 -0.264080E+01 128 Cosine 1 -0. 158867E+04 2 0. 285033E+04 3 -0, 318531E+04 4 0. 290026E+04 5 -0. 359238E+04 6 0. 170538E+04 7 -0. 934551E+02 8 0. 214829E+03 9 0. 830845E+02 10 0. 778256E+02 11 0. 47S703E+02 12 0. 398490E+02 13 0. 27891 1E+02 14 0. 226311E+02 15 0. 173047E+02 16 0. 132494E+02 17 0. 101745E+02 18 0. 762904E+01 10 0. 571372E+01 2 0. 402426E+01 21 0. 262465E+01 22 0. 144144E+01 23 0. 424740E+00 24 -0. 445334E+00 25 -0. 120196E+01 26 -0. 190271E+01 27 -0. 240136E+01 28 -0 295330E+01 29 -0. 336101E+01 30 -0. 370644E+01 31 -0 . 422037E+01 32 -0. 430422E+01 33 -0. 473654E+01 34 -0. 434606E+01 35 -0. 518969E+01 36 -0 522512E+01 37 -0 531313E+01 38 -0 573393E+01 39 -0 549935E+01 40 -0 596654E+01

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REFERENCES 1 G. A. Morris and R. Freeman, J. Magn. Reson. 29, 433 (1978). 2 I). I. Moult, J. Hagn. Reson. 35, 69 (1979). 3 W. S. Warren, J. Chem. Phys. 31(12), Pt. 1, 5437 (1984). 4 D. G. Nishimura, Med. Phys. 12(4), 413 1985. 5 Hong Yan and J. C. Gore, J. Magn. r :eson. 71, 116 (1987). 6 M. S. Silver, R. I. Joseph and !>. I. Moult, J. Ma?n. Reson. 59, 347 (1984). 7 A. H . Lent and M. P. . Kritzer, Book of Abstracts, P1015. Society of Magnetic Resonance in Medicine, Fourth Annual Meeting, London, UK, August 1985. 10 S. Conolly, D. G. Nishimura and A. Macovski, IEEE Trans. Med. Imaging MI-5, 106 (1986). M. O'Donnell and U. J. Adams, Magn. Reson. Imaging 3, 377 (1935). L. S. La son, S. K . Mitter and A. D. War en, IEEE Trans. Autom. Control AC-12, 132 (1967). 11 D. G. Nishimura, A, Macovski, J. M. Pauly and S. M. Conolly, Magn. Reson. Med. 4, 193 (1987). 12 Jintong Mao, T. H. Mareci, K. N. Scott and E. R. Andrew, J. Magn. Reson. 70, 310 (1986). 13 L. liasdorff, Gradient Optimization and Nonlinear Control, Wiley, New York, 1976. 14 Jintong Mao, T. H. Mareci and E. R. Andrew, Experimental Study of The Optimized Selective 180 Degree Padiof requency Pulses, (manuscript submitted for publication) (1987). 15 Jintong Mao, T. II. Mareci and E. P . Andrew, Selective 90 Degree Radiof requency Pulses by Optimal Control, (manuscript submitted for publication) (1987). 129

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130 16 D . G . Luenberger, Linear and Nonlinear Programming, 2nd Ed., A ddi son-Wesley , Reading, Massachusetts, 1984, 17 B. Carnahan, I!. A. Luther and J. 0. Wilkes, Applied Numerical Method, Wiley, New York, 1969

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BIOGRAPHICAL SKETCH Jintong ?>ao was born in 1944, Shanghai, China. He received Bachelor of Mechanical Engineering degree from the liuazhong Institue of Technology, Wuhan, China, with major in thermal energy engineering in 1967. After ten years as an electronics and control technician and engineer in !;uhan Iron and Steel Company, Wuhan, China, he entered the Graduate School of the University of Technology and Science of China, Beijing, in 1978 and received the Master of Electrical Engineering degree with major in automatic control in 1981. Because of his attraction to the sunny weather in Florida, he became a graduate student in the Physics Department of the University of Florida in 1981, at the age of 37. He is presently pursuing a Doctor of Philosophy with emphasis on magnetic resonance imaging. He is married and has a daughter who is fourteen years old. 131

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. 6 K LvvilW-U^ E. R. Andrew, Chairman Graduate Research Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. VaJJW. S~T B~r Prof esso Chemistry I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. N. S. Sullivan Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. K.U S>c3# K. N. Scott Associate Professor of Nuclear Engineering Sciences

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I certify that I have read this study and that in ray opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. /J.^ eZ>CeZ-6^ T. ii . llareci Assistant Professor of Physics This dissertation was submitted to the Graduate Faculty of the Department of Physics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philsophy. April 1987 Dean, Graduate School

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UNIVERSITY OF FLORIDA 3 1262 08554 1620