Upsala J Med Sci 91: 3 7 4 3 , 1986 Selective Properties of Rains of Real and Complex RF Micropulses in Slice-Selective Spin-Echo MRI B. Jung,' A. Ericsson,2 and A. Hemmingsson* 'Department of Hospital Physics, University Hospital, Uppsala and 'Department of Diagnostic Radiology, Uppsala University, Uppsala, Sweden ARST RACT The spin nutation properties o f frequency selective (space selective i n combination with a magnetic f i e l d gradient) trains o f radiofrequency micropulses were studied i n a numeric model. Two cases were considered, one simulating the 90 degrees excitation pulse in spin-echo MRI, the other t h e 180 degrees spin inversion pulse. Image reconstruction according t o the 2 - 0 Fourier transform technique requests t h a t the e f f e c t o f t h e 180 degree pulse i s independent o f t h e i n i t i a l phase o f t h e spin vector relative t o t h e radiofrequency field. I t was found t h a t with a t r a i n o f phase-stable radiofrequency micropulses with an envelope o f the traditional 'sinc' type the result o f the spin inversion pulse was depending on the i n i t i a l phase whereas this was not t h e case for a t r a i n o f phase-shifted (complex) micropulses. The complex t r a i n o f radiofrequency micropulses also had b e t t e r selective properties both f o r excitation and spin inversion than the r e a l one. INTRODUCTION In magnetic resonaiice imaging (MRI) (1, 3, 5, 6, 8) 'selective excitation' stands f o r a space selective process i n which a frequency-selective radiofrequency (rf) pulse is used t o nutate the spins in a defined slice o f t h e object under the influence o f a linear magnetic f i e l d gradient. In the imaging technique applying the 2 - 0 Fourier transform method ( 5 ) , t h e rf pulse and magnetic field gradient sequence may be a 90 degrees excitation pulse combined with ZGRAD. defocusing and phase-encoding during XGRAD+YGRAD, and a 180 degrees pulse combined with ZGRAD where ZGRAD stands f o r a magnetic field gradient along the major f i e l d axis and XtiRAD and YGRAD are mutually orthogonal gradients, orthagonal t o this' axis. A spin-echo recorded under XL;RAD i s then Fourier-transformed t o yield the distribution o f spins i n the X-direction. A number o f such sequences with proper Y-encoding (via a set o f properly chosen gradients in the Y-direction, YGRAD) i s recorded and a second Fourier transformation yields the two-dimensional 37 ZGRAO XGRAD - YljRAO __ ,A,. - A. R F T I M E Fig. 1: Simplified schematic timing diagram o f a spin-echo sequence i n 2-D Fourier transform MRI. spin-density i n the slice (in this example i n t h e XY-plane), defined by the two rf pulses applied under ZGRAD. , I n such a process the second rf pulse (180 degrees) o f necessity must handle, on an equal footing, elemental macroscopic magnetizations in the XY-plane, that have all possible phases in relation t o t h e rf pulse. When t r y i n g t o simulate numerically t h e behaviour o f a 2-D Fourier MRI system with respect t o slice definition, object, movement.s and spectrum of Larmor frequencies, an e r r a t i c behaviour w i t h sudden narrow drops i n spin-density sensitivity i n the selected slice was found. This motivated a closer look on the phase-dependence o f the space-selective processes. In t h e f i r s t simulations a selective phase-stable rf pulse, proposed by Locher (7), was applied, in l a t e r ones the complex selective rf pulse suggested by Silver e t al. (9, 10). Two situations were considered. In the f i r s t one t h e space selection and the resulting phase f o r nutation o f a macroscopic magnetization in thermal equilibrium, were studied. This case simulates the 90 degrees .excitation. I n the second situation t h e space selection, t h e resulting magnetization i n the XY-plane and the resulting phase f o r inversion o f a macroscopic magnetization in the XY- plane were studied. In the l a t t e r case, simulating spin inversion, the calculations covered a l l i n i t i a l phases. METHOD A simplified timing diagram o f a spin-echo sequence i s given i n Fig. 1. The rf pulses were considered t o be composed o f 756 micropulses o f suitably selected strengths b u t o f constant amplitude and o f constant phase, with respect t o an assumed external clock, during their time period. They were applied in combination with the magnetic f i e l d gradient ZGRAD. The following XGRAD pulse defscuses the phases and i s applied t o c o r r e c t f o r the phase development du XGRAD pulse, under which the echo i s recorded. The YGRAO pulse, o f variable amplitude, encodes t h e echo in t h e Y-direction. he second The elements, BX and BY, o f a complex rf vector were assigned values according 38 002 0 01 000 -001 R F a 0010 - " 0.000 PULSE NUMBER -000: R F b t ' C . I I . O O l t oa01-l v w Fig. 2: Real pulse t r a i n (panel 2) according t o (7) and complex pulse t r a i n (pane! - b) according t o (9, 10). The amplitudes are given i n revolutions per rriicropulse period (Z=O) f o r a resulting nutation o f 90 degrees. In t h e complex case this applies for the real component only. Both pulse trains are centered around micropulses 128 and 129. Panel c shows the phase diagram o f the t h i r d quarter of the rf t r a i n and panel d the f o u r t h quarter. Arrows indicate micropulse number. t o the formulae given by (7) and by (9, 10). I n the former, real, case this yielded BX(I)=C*sinc(2*hI/p)*exp(-I*V(2*d*d) BY(I)=O (i.e. phase-stable) where 1=-128,128,1 p=128/4 d=9S*p/24* sinc(J)=siri(eI*J3/(PI*J) and PI-3.1 41 6.. . I n the l a t t e r , complex, case the formulae were BL(I)=C'*HEL~I *COS(HELP~) BY(I)=C'*HELPl "sin(HELP2) where HELP1 =2/(exp(I*INCR+5/256)+exp(-I*INCR+5/256)) HELPZ=S*ln(HELPI) and INCR=5/12Q~ The constants C and C' were varied in order t o obtain yaripus nutation angles. The calculations were performed in a coordinate system r o t a t i n g i n the appropriate direction in synchronism w i t h the Larmor frequencv o f t h e central disc of the slice (i.e. Z-0; no Z-gradient offset). The <-axis was pargllel t o the main magnetic field, BO. The X-axis was orthogonal t o t h e Z-axis and parallel t o the EX of the rf field, and t h e Y-axis orthogonal t o t h e other t w o axis and parallel t o the BY o f the rf field. The numerical representation o f a l l magnetic field vectors wa,s scaled t o give the precession angle during t h e time period o f one micropulse. The resultant o f the three components (in X-, Y - and Z-direction) was thus expressed as t h e I b C I PHASE OUT d 1 PHASE OUT Fig. 3: Resulting macroscopic magnetizations i n the Z-direction as a function o f distance from slice cent,re a f t e r nutation o f a (0,0,1) spin vector. Results with r e a l pulse t r a i n in panel a and with complex pulse t r a i n i n panel Q. The nutation angles i n panel 2 are 60 degrees (. . . .), 90 degrees (o o o 0) and 180 degrees (x x x x), and the r e l a t i v e strengths o f the complex t r a i n in panel b are 50 (. (. . . .), 70 (0 o o o), 140 (+ + + +) and 200 (x x x x) (see t h e t e x t f o r units). Panel - c shows the resulting phase a f t e r nutation with t h e r e a l pulse t r a i n as a function o f distance from slice centre arid the corresponding results w i t h the complex pulse t r a i n i s displayed in panel g. Nutation angles are in panel c 60 degrees (. . . .) and 180 degrees (x x x x) and i n panel d the pulse strengths are 50 (. . . .) and 200 (x x x x). Nutation angles or pulse strengths between those given i n panels c and 4 yield results in between the two curves i n the digrams. The 2-distance is measured i n 2-offset revolutions per micropulse period. precession angle during the time period o f one micropulse. The simulation procedure was then merely t o multiply the magnetization vector, currently present, with t h e appropriate r o t a t i o n natrix and t o repeat this procedure 256 times. The nutation o f the c e n t r a l ( i s . Z-0) elemental magnetization, when subjected t o the r e a l component o f a micropulse train, i s regarded as t h e nominal f l i p angle o f the pulse. Two cases o f i n i t i a l macroscopic magnetization were considered. I n the f i r s t one the i n i t i a l magnetization was the one a t thermal equilibrium, i.e. parallel t o t h e main magnetic f i e l d (BO), in vector notation (O,O,l). I n t h e other case t h e macroscopic magnetization vector was supposed t o initially be nutated t o t h e XY-plane, i n vector notation (sin(v),cos(v),O). (The vector notation is (x,y,z) w i t h x=component o f the macroscopic magnetization in t h e X-direction etc. and v-phase angle relative t o the X-axis i n the XY-plane immediately before the f i r s t micropulse o f the simulated rf pulse train.) Relaxation processes were ignored in the simulations which were performed on a PDP 11/40 computer with programs written in Fortran IV. Source codes are available on request. 40 d '5LpHA5E OUT 1 X Y - V E C T O R a L 1 0 PHASE I N ooOO 1 XY-VECTOR b OO0.0 - I P H A S E IN f PHASE OUT C 00 05 1 0 PHASE I N I . t 0.0 05 1.0 P H A S IN Fig. 4 : Macroscopic magnetizations i n t h e XV-plane as a function o f i n i t i a l phase aft,er inversion of (sin(v),cos(v),O) spin vectors in the XY-plane. Results w i t h r e a l pulse t r a i n in panel 2 and w i t h complex pulse t r a i n in panel Q. Z=O (central plane o f the slice, --------- ), Z-0.01 (panel a) and Z=0.025 (panel 4) (. . . .), Z=0.025 (panel - a) and Z=O.O3O (Panel b) (0 0 0 0). The resulting phases o f t h e macroscopic magnetizations as a function o f i n i t i a l phase a f t e r the same inversion i s shown i n panel c f o r the r e a l pulse t r a i n and in panel d f o r t h e complex pulse train. Z=O (central plane o f t h e slice, -------- ), 2=0.010 in panel c and Z=0.025 in panel r! (. . . .), Z=0.020 in panel c and Z=O.O28 i n panel d (0 o o oj, Z=0.025 i n panel - c and Z=O.032 in panel d (x x x x). In panel 24J.030 (+ + + +). A l l phar .,es are measured in revolutions and t h e Z-distance i s given as 2 - o f f s e t precession revolutions per micropulse period. RESULTS The rf pulse trains are illustrated i n Fig. 2. The resulting macroscopic magnetization in t h e 2-direction a f t e r an rf pulse t r a i n applied t o a (0,0,1) macroscopic magnetization vector is given in Fig. 3 as a function o f distance from the central plane (Z=O). Distance i s measured as precession revolutions in the rotating coordinate system. For symmetry reasons only half o f the selected slice is given i n the illustration. The resulting phase a f t e r this procedure i s also given i n Fig. 3. The macroscopic magnetization remaining in the XY-plane a f t e r nutation o f a (sin(v),cos(v),O) macroscopic magnetization is given in Fig. 4 as a function o f the i n i t i a l phase v and the resulting phase a f t e r this procedure i s also given i n Fig. 4. Only p a r t o f the results o f the calculations are displayed in the figures. DISCUSSION The space selectivity o f both pulse trains i s rather good f o r 90 degrees pulses as illustrated in Figs. 3 and 4 . The complex pulse t r a i n yields a somewhat more uniform magnetization along t h e Z-axis than does the r e a l one. This e f f e c t i s accentuated when the strengths o f t h e pulse trains are increased t o approach a nutation angle o f 180 degrees. The mysterious property of the complex t r a i n 41 t o a p p r o a c h 180 d e g r e e s inversion a s y m p t o t i c a l l y f o r high p u l s e s t r e n g t h - was n o t e d by (9, 10) a n d i s e v i d e n t in Fig. 3b. For b o t h t y p e s of YO d e g r e e s pulse t r a i n s t h e r e s u l t i n g p h a s e was almost l i n e a r l y depending on p o s i t i o n (Fig. 3c a n d d). As s t a t e d e a r l i e r (7, Y , 10) t h i s seems t o b e a p r e r e q u i s i t e f o r p r o p e r r e f o c u s i n g o f t h e s p i n s b e f o r e t h e r e c o r d i n g o f t h e e c h o . The b r e a k s in t h e l i n e a r f u n c t i o n s o c c u r well o u t s i d e t h e s p a c e - s e l e c t i v e r e g i o n s . ’ The e f f e c t of t h e spin-inversion p u l s e t r a i n s i s e v i d e n t l y depending on t h e i n i t i a l p h a s e (Fig. !+). This e f f e c t is, however, n o t v e r y a c c e n t u a t e d f o r t h e complex t r a i n u n t i l v e r y c l o s e t o t h e s l i c e b o r d e r . For t h e real p u l s e t r a i n t h e e f f e c t o c c u r s c l o s e r t o Z=O, a n d , in f a c t , Z=O.O25 in Fig. 4a r e p r e s e n t s a case when a minimum in t h e XY-vector h a s a l r e a d y o c c u r r e d at, a smaller Z value. Also t h e r e s u l t i n g s p i n p h a s e i s l i n e a r l y d e p e n d i n g o n t h e incoming p h a s e u p t o v e r y c l o s e t o t h e s l i c e b o r d e r f o r t h e complex r f t r a i n , w h e r e a s t h e r e a l t r a i n g i v e s d i s t o r s i o n s r a t h e r c l o s e t o t h e s l i c e c e n t r e (Fig. 4b a n d c). This d i s t o r s i o n r e s u l t s in a bunching o f t h e r e s u l t i n g p h a s e s a r o u n d two p r e f e r r e d values. I t i s believed t h a t t h i s bunching may s e r i o u s l y a f f e c t t h e p e r f o r m a n c e of t h e imaging s y s t e m . In simulations with o t h e r p h a s e - s t a b l e pulse t r a i n s , n o t f u r t h e r d e t a i l e d h e r e , t h e bunching o f p h a s e s was f o u n d t o b e a s s o c i a t e d with t h e o c c u r r e n c e of zeroes f o r some initial p h a s e s in t h e m a c r o s c o p i c magnetization in t h e XY-plane a f t e r inversion. An i n v e r s e p r o p o r t i o n a l i t y of r e s u l t i n g p h a s e on i n i t i a l p h a s e was f o u n d f o r Z v a l u e s l a r g e r t h a n t h e smallest o n e giving a z e r o ( f o r some i n i t i a l p h a s e s ) in t h e r e s u l t i n g XY-vector. Such a b e h a v i o u r i s i n d i c a t e d i n Figs. 3 a n d 4 . I t t h u s seems d i f f i c u l t t o h a v e a good s e l e c t i v e s p i n i n v e r s i o n in a n MRI s y s t e m provided t h e r f p u l s e i s r e s t r i c t e d t o b e r e a l ( p h a s e - s t a b l e ) . Simulation e x p e r i m e n t s by (2, 4 ) s u p p o r t t h i s s t a t e m e n t . U n r e p o r t e d r e s u l t s o b t a i n e d by u s s t r e n g t h e n s t h i s conclusion s i n c e i t was invariably f o u n d t h a t spin bunching a n d low i n v e r s i o n e f f i c i e n c y o c c u r r e d b e f o r e t h e b o r d e r s of t h e s e l e c t e d s l i c e . On t h e o t h e r h a n d a f i r s t t r i a l with a t r a i n of complex r f micropulses g a v e a r e s u l t f r e e of most of t h e s h o r t - c o m i n g s of t h e r e a l p u l s e t r a i n . A complex r f pulse would a d d t o t h e complexity of a n MRI s y s t e m b u t t h e r e should b e no a b s o l u t e t e c h n i c a l o b s t a c l e s t o s u c h a s o l u t i o n . 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