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Relaxation-Selective Magnetic Resonance Imaging

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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 magnetiza- tion 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 <<s short T22speciesŽ. leaving the long T species alone , everywhere inside the disk, or that m z 1 every- <<( and then nulling the long T2 speciesŽ leaving the whereŽ. including the boundary . Hence m z 1 for g q <<s short T2 species alone. , selective images of the G2 R , with the proviso that if m z 1 for any q - - different T2 species were obtained. G22in the open interval R Ž.i.e., 0 G ` , then <<s g q m z 1 for all G2 R . This is basically proving that the magnetization vector cannot ever increase in 2. Theory length above its equilibrium value, and that its length will be less than its equilibrium value once it has 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 <<m ( 2 z q Ž. G22g for G2 in the right half complex plane. This result means that rf pulses cannot be used to obtain arbitrarily sharp notch filter m z responses as functions of G2 . For example, consider the notch filter function: ° - 1 for G2 0.5, s~ -G - m z Ž.G2 0 for 0.5 2 1.5, Ž.3 ¢ ) 1 for G2 1.5. It would not be possible to obtain a response that Fig. 1. The evolution of the magnetization, for a series of spins were arbitrarily close to this desired response. If a with different T2 values, during a third-order dressing pulse G s s designed to null species with a T2 of 1. All magnetizations start at pulse were designed that gave m zŽ.2 1 0, then Ž.0,0,1 and move to a position on the z axis dependent on their Ž.G s s the maximum achievable value of m z 2 1.5 is T22.Ž The path of the T ` species is not clear; it overshoots the z 0.5r2.5s0.2. This is in contrast to pulses used for axis, but then retraces its path to the axis.
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