Fourier Analysis and Structure Determination Part II: Pulse NMR and NMR Imaging John P. Chesick Haverford College, Haverford. PA 19041 In this paper we seek to extend and amplify the two previous articles by Rabenstein ( I ) and Macomber (2) dealing with pulse NMR with extension in more detail t o the application of NMR imaging. As was noted in the preamble to Part I(3) of this three-part series, NMRimagingis assuming considerable clinical importance in the medical world and has a similarity in its theoretical analysis to that of single crystal X-ray crystal structure determination. We will here consider the simplest pulse NMR experiments using the background of our discussion of the Fourier transform in Part I and then move to consideration of the generation of snin echoes as the basis for the imaeine ~rocedure. " We purposely avoid providing a review of options and variations in NMR imagingpractice. One method, the threedimensional Fourier transform of spin echo data, was chosen for illustration here. This is of current practical use in a number of imaging systems, provides adiantages in signalto-noise ratio, permits separation and control of the weighting of the images by the TI and Tz relaxation time mappings, and probably is the clearest in analogy to the formalism and methods of X-ray crystallography, which will be discussed in Part 111of our presentation. Other imaging methods using different pulse and gradient sequence methods can be relatively easily described and explained as special cases of the one discussed here. Sources for more discussions of NMR imaging are the pioneering paper on the Fourier transform method by Kumar, Welti, and Ernst (4a), some more general expositions (46-d), and a review of NMR imaging methods by King and Moran (4e).
used to identify a frequency in radianslsecond, an angular velocity. Then f,r = wrf/(2r). I t is useful, however, to define a rotating coordinate system with the axes x' and y' rotating around the z axis a t the angular frequency of the rf radiation, w,f. The rf field vector HI is then stationary in the x'y' plane of this rotating coordinate system. The Bloch equations take the following form in this coordinate system rotating about the z axis dMJdt = rM,H, dM,./dt
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Rotatlng Frame and the Bloch Equatlons The proton, the hydrogen nucleus, in a magnetic field of strength Ho shows two possible components of magnetization in the field direction; the corresponding nuclear spin states differ in enerw bv AJ3 = rhH.. Here r is the nroton magnetogyric ratio. A t room temperature and using almost anv ~racticalfield streneth A E w,r, or 6wz > 0. Figure 5a shows the transverse magnetization components for wl < 0, corresponding to rotation in the opposite direction in the x'-y' plane. Figure 5b shows a magnitude and phase angle plot that is equivalent to the component plot of Figure 5a, illustrating both the rotation of the transverse maanetization vector in the x'-v'- olane and . the exponential decay characterized by the Tz time constant. Equations 3 and 4 (or 5) are summed over both t w e s of 6a protons to obtain the actual observed S,, and Sf. shows the algebraicaddition of the components of Figures 4a and 5a with a 211 initial amplitude ratio. This gives the complex signal to be observed for this two-spin type system.
r re
Figure 5. (a) Transverse magnetization in rotating frame with f = ( r h - w,# (2r)< 0: - M y , , . . . . M,,. (b) Display of (a) as ampliude and phase.
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Figure 6. (a) Transverse magnetization from mixture of two spin types; My,,. . . . . M.,. (b) Fourier bansformof (a), frequency mearurec as shin f from the resonant frequency f"; -absorption, . . . . . dispersion. As before, a frequency analysis (Fourier transform) of the complete FID signal in Figure 6a yields the conventional frequency spectrum. Our example with two types of protons gives the expected spectrum in Figure 6b showing the two frequencies present with relative absorption amplitudes MY,1(0)and Mydo). Real Magnets and Effective T2 The decay factor exp (+ITz) in eqs 3-5 defines the envel o ~ eaivina the ideal or maximum Dossible value for the m&n&izaiion components MJt) and My@) a t any time. This exponential decay factor is determined by the irreversible loss of the transverse spin magnetization components, and no trick will retrieve this lost amplitude. The transverse relaxation time constant Tz is typically in the range of 502000 ms for protons in aqueous biological systems. I t is observed, however, that the FID decays much more rapidly than predicted by TZfor magnetic fields of the homogeneity practical for all NMR imaging applications and for most analytical NMR spectrometers. The lines in the NMR spectrum have widths correspondingly greater than the "natural" value of ll(sT2) for the protons observed. The spread in magnetic field due to the inhomogeneity of any real magnet system gives a proportionate spread in proton resonance frequency and appears as an increase in the spectral line width. This also results in aspreading or smearing out of the magnetization vectors in the x'y' plane during the course of a free induction decay experiment. Cancellation of the resulting sinusoidal signals of slightly different frequencies tends Volume 66 Number 4
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to occur as time increases following the initial 90' rf pulse. The ensemble average of the signals from all parts of the sample goes to zero much faster than predicted by Tz. The envelope describing the decay of the actual signal amplitude is given by M$O) exp (-tlT;), where the effective decay constant 1/Tn = 1/T? r6Hn. 6Ho is the effective variation in field strength a i o s i the saLple:~he Fourier transform of the FID obtained with a real maenet then eives the absorvtion spectrum with l/(rT;) ~ [ a sthe instrument dependent line width, always larger than the natural or intrinsic line width determined by Tz. Figure 7 shows schematically the effects of real magnet inhomogeneity in the decay of the FID. The difference between the two decay curves with the time constants Tz and T?' represents the "y' magnetization signal that apparently disappeared but that is in l'act recoverable by thegeneration of a "spin echo", which will be described in the next section. In this description of the source and analysis of the FID sienal. we have the ohvsiral basisof the FT N M R soectrome t k oberation. or-suih a spectrometer, the magnetic field should be hiehlv - .uniform over a homoeeneous sample. . . and the frequency spectrum, one component of the Fourier transform of the FID transient. shows the different frequencies of absorption due to varying chemical shifts and ;pinspin cou~line . - effects.Benn and Cunther (fi) provide a convenient review of pulse methods in NMR speciroscopy.
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Spln Echo Formation The spin echo formation experiment provides the framework for the NMR imaging and should first be considered for a uniform sample. I t must be emphasized that the following discussion of spin echo formation assumes only the interaction of the spin magnetization vectors with the magnetic field and neglects spin-spin coupling effects. The spin echo description for operation of a high resolution, high field analvtical NMR apparatus would require a much more detailed treatment. statements about ideal refocus of precessing transverse spin components following .rf -pulse induced 1800 rotations are incorrect for significant spin-spin coupling effects. However, it is valid to neglect such spin-spin coupling effects in the imaging application to be considered here. Fieure 8 orovides a diamammatic summanr of the spin echo-formacon process. ~ & e8g shows somebf the events on a time line, and the other parts of the figure depict the magnetization vectors for two types of protons a t various staees in the spin echo sequence. These two types are picked wr,-which to show resonance angdar frequencies wl b r a ~ k ethe t transmitter angular frequency w,f. Honow represents the average field strength over the sample for aslightly inhomogeneous field. Figure 8a shows the initial equilibrium maenetization. Fieure 8b deoicts the result of the short rf puge from the trksmitter ciil producing the M,4O) magnetization. As time passes the magnetization vectors evolve as
Flgure 7. Effect of field inhomogeneity on FID component.
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described by eqs 1 and 2 for each proton type to produce Figure 8c. Figure 8g shows one component of the resulting FID signal decaying with time constant T;. The two spin types aremoving away in opposite directions from the y'axis &the rotating c ~ o r d i n a t e ~ s ~ s t eAtma. time r the phase of the rftransmitterisadiustedand therf transmitter is turned on again long enough to rotate all magnetization vectors by 180° about the y'axis. This flips the spins from the arrangement in Fieure 8c to that shown in Fieure 8d. The laeeine and l e a d i 6 spin types have traded iositions in the-dy' dane. With this tortoise-hare interchanee. - . another evolution time interval r brings the spins back together as shown in Fieure 8eat thecumulative time of 27. In further time the vec& spread again as shown in Figure 8f. As the spins reconvene into more uniform phase the My value grows and a signal appears with maximum amplitude a t the time 27. This refocus a t time 27 occurs regardless of whether field inhomogeneities (variation of Hoin space) or chemical shift differences (different values of a) caused the different 6w values. However this refocus will not occur perfectly if spinspin coupling effects are significant. The convergence of the spreading spin moment vectors along they' axis and corresponding creation of a maximum amplitude signal is referred to as a spin echo, first observed by Hahn (7). The spin amplitude lost due to the effects of field inhomogeneity is maximally recovered a t time 2r and is the source of the signal. The phase sensitive detector keeps track of the transmitter frequency and phase so that the subtraction of the rf frequency w,f with proper phase can be made to give the audio frequency signals with consistent phase relationships
Figure 8. Magnetization vectffs fw spin echo experiment.(a)initial condition. (b) After 90- rotation induced by strong rt pulse. (c)Afterevolution time 7 . (d) After 180" rotation induced by strong rt pulse. (8)After evolution time r. If) After funher evolution time. (g) M,.(q showing spin echo maximum at time t = 2,.
even when the transmitter is not feeding rf power to the transmitter coil around the sample. The superposition of the sign& from spins of a number of different frequencies will give much more complicated FID and spin echo transients, but the spectral analysis of the FID, or of either half of the spin echo, will give the frequency spectrum of the sample. Again "spectral analysis" means mathematically the calculation of the Fourier transform of the transient signal. The formation and decay of the maximum amplitude of the spin echo will he descrihed hy the same Tz* time constant as was seen for the initial FID. Another rf pulse may be applied a t time 3r to rotate again all magnetic moment vectors by 180° about the y' axis. This repeats the reversal of the spins in the x'y' plane, and a second spin echo maximum appears a t time 47. This process may he repeated to make rnor~kpinechoesuntilthe iireversihle decay descrihed by 'I;annihilates all transverse maanetization. Imager Local Field Control Let us assume now that we wish to make images of an inhomogeneous sample. We will define an image as a threedimensional proton density map. This will he shown as a set of narallel slices. with the contents of each slice . oroiected . onto a viewing plane with displayed intensity proportional to the ~roiectedcontents of the slice a t each point in the plane. ~ h d c h o i c of e slice planes and hence the appearance of an individual slice is arhitrary. An ideal imager would produce a slice in the plane of a uniform test bodythat would he of uniform intensity and of a shapesimilar to that of the wst body. The magnetic resonance imaging process, in contrast to the analytical NMR spectrometer, operates most simply when the field is low enough that all protons may he considered to have the same chemical shifts. Differences in chemical shifts do in fact appear in magnetic resonance imaging a t the highest fields employed, but these appear as "chemical
shift artifacts", formally equivalent to magnetic field inhomogeneity. These effects serve to distort the images when using the highest possible magnetic field strengths. Spinspin coupling effects also can generally be neglected. The sample of usual interest is now spatially inhomogeneous, not only in the proton density but also in the relaxation times TI and Tz associated with each volume element (voxel) in the sample. Spatial codins in three dimensions will now he achikved through cont~olledthree-dimensional variation in the magnetic field. The spatial dependence of T, and T?will now heexpected to play role in
