Relaxation-Selective Magnetic Resonance Imaging

1 October 1999

Chemical Physics Letters 311Ž. 1999 379±384 www.elsevier.nlrlocatercplett

Relaxation-selective magnetic resonance imaging

Seth D. Bush a, David E. Rourke b,), Lana G. Kaiser a, Alexander Pines a a Materials Sciences DiÕision, Lawrence Berkeley National Laboratory and Department of Chemistry, UniÕersity of California, Berkeley, CA 94720, USA b Magnetic Resonance Centre, School of Physics and Astronomy, Nottingham UniÕersity, Nottingham NG7 2RD, UK Received 7 December 1998; in final form 7 July 1999

Abstract

The Bloch equations with T2 relaxation can be inverted in closed form with respect to T2, using inverse scattering theory. Hence, radio frequency pulses can be calculated that cause a final magnetization response that is any desired function of T2, provided that function is physically realizableŽ. however, there are strong constraints on what is physically realizable . A useful subclass of such pulses are `dressing' pulses, which store the magnetization on the z-axis, with magnitude a given function of T2. This enables spins to be selectively nulled according to their T2 ± this is demonstrated by obtaining a relaxation-selective image of a phantom. q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction space of allowed pulses, calculating the magnetization response for each trial pulse. Typical algorithms When radio frequencyŽ. rf pulses are applied to a used are gradient-based minimizationwx 1±3 and sim- system made up of spins that are distinguished by ulated annealingwx 4±6 . one or more parametersŽ such as free-precession For sufficiently restricted spaces of allowed frequency, or T2 relaxation time. , the final state of pulses, the determination of the best pulse shape can the spins will be a function of those parameters. The be done more simply. For example, rf pulses that rf pulse design problem involves finding that rf pulse give a desired magnetization response as a function which leaves the spin system in a given desired final of T22relaxation timeŽ. `T -selective pulses' have state, or as close to it as possible. been obtained by calculating the response of a rect- This problem is most commonly approached in angular soft pulsewx 7 . This response can be given in the `forward' direction. For example, rf pulses that closed form ± it is then easy to find rectangular give a desired magnetization response as a function pulses that suppress spins with either long or short of resonance offsetŽ. `frequency-selective pulses' are T21times. In a similar manner, T -contrast enhance- often designed by numerically searching through a ment has been obtained by calculating the response of a train of hard pulseswx 8 . This Letter is concerned with the calculation of

T2-selective pulses via an inÕerse method. Firstly, ) Corresponding author. E-mail: constraints on the total space of allowed magnetiza- [email protected] tion responses, as functions of T2 , are obtained. It is

0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S0009-2614Ž. 99 00890-8 380 S.D. Bush et al.rChemical Physics Letters 311() 1999 379±384 important to do this, as the space is quite restrictive. Given an initial specification of F as a function For example, it is not possible for truly `notch filter' of j , together with a `physically realizable' final pulses to exist, as they do with frequency-selective FjŽ., inverse scattering theory allows the determi- wx pulses. nation of VtŽ. 11 . The Bloch equations with T2 Secondly, inverse scattering theory allows the relaxation can be written in the form of Eq.Ž. 1 , with s wx exact calculation of the rf pulse that will give any n 3 10 . In this case, j corresponds to the T2 wx s r allowable response 9,10 . This Letter concentrates relaxation rate, G221 T . F corresponds to the on an important subset of allowable responses. It is magnetization vector, and V to the rf pulse. important because the corresponding pulses are able Due to the particular form of the Bloch equations, to selectively null spins with particular T2 values, there are constraints on the allowable final magneti- and also because the pulses can be calculated more zation, Ž.m xyz,m ,m , as a function of G2 after an rf simply than in the general case, using the `dressing pulse. The most important constraints are that the wx method' ± a tool from inverse scattering theory 10 . final longitudinal magnetization, m z , is analytic in wx Such pulses are therefore `notch filter' pulses, to the right half complex G2 plane 9,10 , and that ™ <<™ within the constraints described below. This comple- m z 1as G2 `. It is also bounded in magni- <<( ments the use of a 908 pulse followed by a wait tude, m z 1, when G2 is imaginary.Ž The previous period, which would constitute a high pass T2 filter, two statements assume that the magnetization is and the square pulses described in Ref.wx 7 , which normalized so its magnitude equals 1 at equilibrium.. were used as low pass T2 filters. Standard results from complex analysis can be A pulse was calculated using this method to null used to obtain some physically obvious results on the g qq magnetization of spins with a particular T2 value, bounds of m z for G2 R , where R is the closed ( ( leaving other spins as unaffected as possible. This positive real axis, 0 G2 `. Apply the conformal X s y r q same pulseŽ. after scaling its duration and amplitude map G22Ž.Ž.G 1 G 21 , to map the right half < X <( could be used to null spins with different T2 values complex G22plane to the unit disk, G 1, with to that originally specified. It was tested experimen- the imaginary axis mapped to the disk's boundary. <<( tally as part of imaging a two-component system, Then, from above, m zz1 on the boundary, and m consisting of a short T2 spin species, physically is analytic inside the disk. Hence, the maximum wx <<- separated from a long T2 species. By nulling first the modulus theorem 12 shows that either m z 1 <

Inverse scattering theory allows the `inversion' of been moved off the z axis, unless it has no T2 systems of the form relaxationŽ or infinitely fast T2 relaxation ± in which EF case it can never move off the z axis. . s ij JqVtŽ.Fj Ž,t .,1 Ž. Suppose it is desired to construct an rf pulse that Et s g will cause m z 0 for some value, say g 22,of G where FjŽ.,t , J, and Vt Ž.are n=n matrices, and j R q. Such a pulse would enable the nulling of the s r is a scalarŽ. the scattering parameter . It is possible, magnetization of spins with T221 g . Without loss and often convenient, to make all quantities in the of generality, g 2 can be taken equal to 1. Then the above equation dimensionlessŽ e.g., the time t is function m zŽ.G2 is mapped by the above conformal X s expressed as a multiple of some convenient charac- transformation so that m z is zero at G2 0. Then wx < X <<( X < teristic time. . This has been done throughout this the Schwarz lemma 12 implies that m zŽG22. G s s section. within the unit disk. Hence if m z 0at G2 1, then S.D. Bush et al.rChemical Physics Letters 311() 1999 379±384 381

<<<( y r q < m z Ž.Ž.G221 G 1 for all G 2in the right half complex plane. In general, if m z is required to s equal 0 for G22g , with g 2not necessarily equal to 1, then m z must obey the constraint y G22g <

Fig. 2.Ž. a Pulse shapes for a square pulse, a sech pulse, which is the first-order dressing pulse, and a third-order dressing pulse.Ž. b and Ž. c r show the longitudinal and transverse magnetization responses versus 1 T2 for spins on resonance, obtained by numerical integration of the Bloch equations for the given pulse shapes. The transverse responses of the dressing pulses are close to zero. They are not identically zero s as the pulses have been truncated.Ž. d shows the mz responses versus resonance offset for spins with T2 1 ms after using the three pulse shapes. In all graphs, the dashed, solid, and dotted lines refer to the square, sech, and third-order dressing pulses, respectively.

s r This pulse therefore nulls spins with T2 1 g. Fur- is less resonance-offset sensitive than the sech pulse, thermore, by comparison with Eq.Ž. 2 , it is the pulse and might therefore be a better choice as a relax- that allows maximum contrast between these spins ation-selective pulse in many cases. See also Fig. 2, and spins with a different T2 time. which compares the sech pulse, the third-order dress- One problem with this pulse, however, is that it is ing pulse, and a square pulse designed to null spins s quite sensitive to resonance offsetŽ when pulses are with T2 1 ms. The third-order dressing pulse corre- calculated with inverse scattering theory to have a sponds to that shown in Section IVC of Ref.wx 10 . specified T2 response, it is assumed that all spins are The figure shows the pulse shapes, the longitudi- on resonance. . It is possible, just as with the square nal and transverse magnetization responses as func- wx 8 pulses of Ref. 7 , to use hard 180 pulses within the tions of G2 Ž.for spins on resonance , and the m z pulse to decrease this sensitivity. However, we found, responses as functions of resonance offsetŽ for spins s by numerical simulation, that the third-order dressing with T2 1 ms. . All responses were obtained via pulse with m z response numerical integration of the Bloch equations for y ywxq r y wxy r Ž.G221 Ž.Ž.G 1 i3''2 G 21 i3 2 pulse durations as shown in the figure. m s z q qwxq r q wxy r Note that the transverse responses for the two Ž.G221 Ž.Ž.G 1 i3''2 G 21 i 3 2 dressing pulses are very close to zero for all T rates, Ž.Ž.G y1 wxG 2y G q1 2 s 222 Ž.6 whereas that for the square pulse is appreciableŽ in G q wxG 2q G q Ž.Ž.2221 1 principle, the transverse responses for the dressing S.D. Bush et al.rChemical Physics Letters 311() 1999 379±384 383

pulses should be zero ± their being non-zero is a of two concentric water samples, doped with MnCl2 , consequence of the pulses being truncated. . Hence, so that the separate components had different T2 dressing pulses do truly null the magnetization at times, but similar resonance offsetsŽ MnCl2 was specified T2 values, whereas other pulses null just used as a doping agent, as it has very little effect on the longitudinal response. the resonance offset of the water. . Combined with

It is interesting that the m z response of the square the homogeneity of the fieldŽ. 10 Hz broadening , all pulse is very similar to the response of the sech spins in both compartments lay within the range of << pulse, although m z for the latter is always greater resonance offsets over which the third-order dressing than or equal to that of the square pulseŽ as follows pulse works. from the previous section. . The resonance offset The two components had different T1 times. The behaviour of the third-order dressing pulse is approx- outer component's T1 was sufficiently long to be imately the same as would be obtained from the taken as infinite. The inner component had a T1 of square pulse shown if two 1808 hard pulses were 12 ms. Although the dressing pulses are designed r r wx added to the latter at times T 3 and 2T 3, where T assuming no T1 relaxation, it can be shown 10 that is the square pulse's duration. odd-order dressing pulses work with T1 present. As a simple test of using a dressing pulse to Finite T12has the effect of decreasing the T nulled selectively null spins with a specified T2 , a phantom by the pulseŽ to what value it is decreased to must be as shown in Fig. 3a was constructed. This consisted determined by numerical simulation. .

Fig. 3.Ž. a The doped water phantom used to demonstrate the T2 selectivity of the third-order dressing pulse. It consisted of two concentric tubes, filled with MnCl22 with concentrations shown. The T times of the outer and inner solutions were 3.79 and 0.45 ms, respectively. The T1 relaxation of the outer solution was long enough to be taken as infinite, the T1 time of the inner solution was 12 ms.Ž.Ž. b , c , and Ž. d show three images obtained from the phantom. InŽ. b , all the spins were excited with a 908 hard pulse prior to imaging. In Ž. c , the fast-relaxing spins were selectively nulled prior to the excitation pulse. InŽ. d , the slow-relaxing spins were selectively nulled prior to the excitation pulse. All images have their intensities to the same scale. 384 S.D. Bush et al.rChemical Physics Letters 311() 1999 379±384

Fig. 3b shows an initial image of both compo- scheme used in Ref.wx 7 works for dressing pulses. . nents of the phantom, obtained at 4.2 T using a hard However, it is not possible, even in principle, to 908 pulse, followed by an imaging sequence. sharpen the discriminatory ability of rf pulses be-

Subsequent images were obtained by preceding tween species with different T2 times above a certain the above sequence with the third-order dressing limit. pulse described above, and shown in Fig. 2a. By scaling the pulse in duration and amplitude, it could be used to selectively null either the fast or the slow Acknowledgements relaxing speciesŽ e.g., doubling the pulse duration D.E.R. thanks the UK Medical Research Council and halving its amplitude will double the T2 nulled by the pulse. . These images are shown in Fig. 3c,d, for financial support. This work was supported by confirming the ability of this pulse to act as a notch the Director, Office of Energy Research, Office of T filter. Basic Energy Science, Materials Sciences Division, 2 of the US Department of energy under Contract No. DEAC03-76SF00098. 4. Conclusions References Relaxation-selective pulses are a useful comple- ment to frequency-selective pulses. They have previ- wx1 S. Conolly, D. Nishimura, A. Macovski, IEEE Trans. Med. ously been used to create contrast between spin Imag. 5Ž. 1986 106. species with different relaxation timeswx 14 , and to wx2 J.W. Carlson, J. Magn. Reson. 67Ž. 1986 551. wx selectively null magnetizationwx 7 . They have also 3 J.T. Ngo, P.G. Morris, Magn. Reson. Med. 5Ž. 1987 217. wx wx4 S. Kirkpatrick, C.D. Gelatt Jr., M.P. Vecchi, Science 220 been used in magnetization transfer experiments 15 . Ž.1983 671. In contrast to previous work, we have found wx5 C.J. Hardy, P.A. Bottomley, M. O'Donnell, P. Roemer, J. strong constraints on the physically allowable T2 Magn. Reson 77Ž. 1988 233. responses, and determined how to invert any such wx6 H. Geen, S. Wimperis, R. Freeman, J. Magn. Reson. 85 response. We have concentrated on `dressing' pulses Ž.1989 620. wx7 M.S. Sussman, J.M. Pauly, G.A. Wright, Magn. Reson. Med. as they are relatively easy to calculate, and they 40Ž. 1998 890. work well in nulling the magnetization of one or wx8 N.K. Andreev, A.M. Khakimov, D.I. Idiyatullin, Instrum. more spin species distinguished by T2 ± although the Exp. Tech. 41Ž. 1998 251. constraints on the allowable responses mean that wx9 S.V. Manakov, Sov. Phys. JETP 38Ž. 1974 248. wx pulses that null multiple T values will work well 10 D.E. Rourke, S.D. Bush, Phys. Rev. E 57Ž. 1998 7216. 2 wx11 P.J. Caudrey, Physica D 6Ž. 1982 51. only if the values are well separated. wx12 J.E. Marsden, M.J. Hoffman, Basic Complex Analysis, 2nd For example, we have demonstrated the selective edn., W.H. Freeman, New York, 1987, Chap. 2. nulling of spins in a two-component system, where wx13 V.E. Zakharov, A.B. Shabat, Funct. Anal. Appl. 13Ž. 1980 the T times are different by about a factor of 8. We 166. 2 wx also chose the system so that all spins were reason- 14 J.H. Brittain, B.S. Hu, G.A. Wright, C.H. Meyer, A. Macov- ski, D.G. Nishimura, Magn. Reson. Med. 33Ž. 1995 689. ably close in resonance offset. For a system with a wx15 N.P. Davies, W. Vennart, Abstracts of the British Chapter of wide range of resonance offsets, a bandwidth-broad- the International Society of Magnetic Resonance in Medicine, ening scheme would need to be consideredŽ the Nottingham, 1998, p. 35.