I
a.:
, i.
John M. Lsterud
Pendergrass Diagnostic Research Laboratories Philadelphia. PA 19104-6086
Steven W. Sinton
Lockheed Palo A n 0 Research Laboratory 3251 Hanover St. Palo Alto, CA 94304
Gary P. Drobny Department of Chemistry University of Washington Seattle, WA 98195 Although the growth of magnetic resonance imaging in medicine has been rapid in recent years, there have been relatively few concerted steps toward the development of materials science applications. Almost all imaging systems are designed with the liquid state in mind, for use with large hiological subjects. Interest in microscopic and solid-state imaging is forming only now
ter the arena, and only a few now sur^ vive. The so-called sensitive point and sensitive line methods ( I ) consisted of some arrangement, using either nonlinear or time-varying gradients, that confined the source of signal at a particular point on the spectrum to a single volume element. This sensitive volume element would he moved, possibly by moving the object itself. These methods, although conceptually simple, have been largely supplanted by the two-dimensional techniques. The socalled spin warp imaging technique (2, 3 ) is a special case of two-dimensional NMR spectroscopy ( 4 ) and by far has
INSTRUMENTATON that the widespread, highly visible success of medical imaging has given materials researchers an inkling of the possibilities. NMR imaging depends on the use of linear magnetic field gradients to distinguish spatial position. Simply put, the Larmor frequency of a hydrogen nucleus is proportional to the local magnetic field. By introducing a linear magnetic field gradient, we index the spatial position in one dimension by frequency. By expressing the gradient as a ratio of frequency to distance, we obtain a proportionality constant hetween spatial and spectral positions. For protons, an isochromat (the signal from a small volume element of the sample) offset -112 cm in a gradient of 1G/cm (= 4.25 kHz/cm) will he shifted upfield by 2.12 kHz. Dimensional analysis highlights an important theme in NMR imaging: the mapping of time and frequency to wavenumher and spatial dimension, as shown in the box. The early history of imaging saw a number of contending techniques en0003-2700/89/036 1-023A/$01SO10 @ 1988 American Chemical Society
hecome the method of choice for medical imaging. Backprojection, the mathematical technique that supports X-ray computed tomography, has also been popular in the NMR literature (1, 5 ) and will receive special consideration here. Conceptually, the development of imaging has recapitulated the history of NMR spectroscopy. Just as with NMR itself, imaging began at low field on permanent magnet and resistive magnet systems, gravitating toward higher field superconductor systems.
ime t (s)
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The current effort is concentrated on hasic effects that determine the intensity and phase of signal, such as spinlattice and spin-spin relaxation, molecular diffusion, magnetic susceptibility, and chemical shift. The spectroscopist perusing this recent history will be reminded of the early history of NMR, as recounted by the older spectroscopy texts. The stimulated echo, pulsed gradient, and multiecho sequences recall the original Hahn spin-echo work ( 6 ) and the experiments separating spinspin and diffusion effects by Carr and Purcell ( 7 ) , Torrey (8), and Stejskal and Tanner (9). This article will provide an overview of the principal image reconstruction methods used in NMR today, andspecia1 attention will he given to hackprojection and spin warp (SW) imaging. We will also discuss so-called slice selection techniques, in which the image of a particular slice through an object can he obtained. Factors limiting spatial resolution in the images of liquids and solids will he considered, and in particular the effect of molecular diffusion on image intensity and resolution will he described. Finally, a number of applications of NMR imaging to the study of materials will be briefly reviewed. Future prospects for NMR imaging in materials sciences will he discussed. NMR image reconstruction techniques
As mentioned above, NMR imaging is based on the use of linear magnetic field gradients to frequency label spatial position. Any chemist who has been forced to contend with a poorly shimmed NMR magnet (Le., a magnet with poor field homogeneity) can im-
* wavenumber k
ANALYTICAL CHEMISTRY, VOL. 61. NO. 1. JANUARY 1.
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1989
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Flgure 1. The basic principle of NMR imaging: the application of linear mag-
netlc field gradients to frequency-encode a free induction decay in one dimension with spatial information. Two water-filled bulbs are oriented along the axis parallelto the magnetic field direction. (a) if me magnetic field Is homogeneous. the NMR spectrum of the water in the bulbs consists of one line. (b) If a very large magnellc field gradient is appiled along the direction of me maQnetiCfield. me NMR apemum consists of two broad Ilnes. The separation between the lines is demndem an the size of the gradient
mediately appreciatethe principles underlying NMR imaging. In the NMR spectrum of a liquid, transition line shapes are idehlly Lorenzian, and typical lime widths are a few Hertz at most. In cases where the magnetic field is very inhomogeneous in the region occupied by the sample, the NMR line widths will increase. What is happening, of course, is that different regions of the samples are “seeing” different magnetic fields, and the nuclear spins in the sample are resonating at a variety of frequencies. This heterogeneous broadening of an NMR line by magnetic field inhomogeneity is an example of the frequency labeling of spatial position hy magnetic field gradients. We can get closer to an imaging experiment by constructing an NMR sample composed of two small bulbs of 24A
water, inserted ahout 1 cm apart in a conventional NMR tube, and placing this sample into a homogeneous magnetic field. The NMR spectrum of this sample (Figure la) consists of a single NMR line. Suppose that there is a gradient in the z direction (the magnetic field direction) of about 1 G/cm. For protons, this gradient would correspond in frequency units to about 4.25 kHz/cm. In Figure l b we see that the axis of the sample tube is parallel to the direction of the field gradient so that the spins in the two bulhs are in quite different magnetic fields. The NMR spectrum of this sample will consist not of one broad line but of two lines whose peaks are separated by ahout 4.25 kHz. The widths of these two lines will depend on the volume occupied by the water in each bulb. If the water in each bulb occupies a spherical volume about 4 mm across, each line will have a maximum width of about 1600 Hz. What we have obtained is a one-dimensional profile of the sample, an image projection. How images can be reconstructed from such NMR profiles is the subject of the next section.
(a) The free induction decay (FIO) signa is 58111pled immediately following excitetion of transvene magnetization with a * I 2 pulse. (b) A spinecho signal la formed by application of a T reto cusing pulse. me ac(Iuisition window Is centered about the echo
ProJectim-reconstructionimaging We will first discuss the way in which spatial information can be frequencyencoded in the projection-reconstruction (PR) experiment. Figure 2 shows two forms of the experiment. In Figure 2a, a single rf pulse excites a free induction decay (FID) signal after a gradient is turned on, and the FID is detected while the gradient is still on. The FID is encoded by the gradient with spatial information from the various resonance offsets from different locations in the sample. This form of the experiment has the advantage that the NMR response is detected immediately after the excitation, preserving maximum signal intensity, but has the disadvantage that the rf pulse must excite the entire range of frequencies produced by the gradient. A second version of this experiment is shown in Figure 2b. To avoid the necessity of applying rf pulses a t times when the gradients are on, a spin-echo pulse sequence is used (9). The sequence begins with a 90° pulse that generates magnetization transverse to the direction of the magnetic field. Assuming that spin-lattice relaxation is long compared with the echo pulse sequence, this magnetization will be confined to the xy plane, and an FID will be observed. Following the 90° pulse, a magnetic field gradient is turned on. This gradient is called the dephasing gradient because it causes the various spin isochromats a t different positions in the sample to precess a t different
rates; this causes the spins to lose phase coherence. After the dephasing gradient is turned off, a 180’ rf pulse is applied to the sample followed by a second gradient pulse with magnitude and direction identical to the first. This readout gradient pulse causes the spin isochromats to rephase to form a spin echo during the acquisition window. Use of either pulse sequence in Figure 2 produces an FID encoded with spatial information. The P R method of imaging requires that the experiment he repeated after either rotating the sample about an axis orthogonal to the direction of the gradient or by rotating the gradient vector itself. This can be accomplished electronically with two orthogonal gradient coils to give a resultant gradient whose form is given by GO= iG cos 8 j G sin 8. Mechanically rotating the sample and electronically rotating the gradient direction have identical effects. Regardless of the method used, the NMR experiment must be repeated until the rotation angle 8 has been varied from 0’ to 180°. To obtain a good quality image reconstruction, 8 should he varied in increments of one to two degrees a t most so that at the end of the experiment we will have accumulated a set of up to 180 FIDs, where each FID has been taken a t a different rotation angle 8. To understand how this set of FIDs can be used to reconstruct an image, we must turn to a simple mathematical analysis of the NMR imaging experi-
ANALYTICAL CHEMISTRY, VOL. 61, NO. 1, JANUARY 1, I989
Flgure 2. Projection4 pulse sequences.
+
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ment. The basic equation describing the evolution of transverse magnetization in the ry plane from a single isochromat after a 90° (or the nI2) pulse is: d -M =?n x M -M/T, (1) dt where M is a magnetization vector, n is a vector describing the effects of chemical shift and resonance offset terms such as the gradient, y is the nuclear magnetogyric ratio, and T2 is the spinspin relaxation time. Equation 1 haa the form of a classical torque equation, where as a result of the crow product between n and M,a precession of the magnetization occurs a t the Larmor frequency. The second term on the right of Equation 1is essentially a dissipative or damping term, and 1/T2 is therefore analogous to a damping rate or a rate of transverse relaxation. Equation 1 can be written as a scalar equation by setting m = M, iMy and by recognizing that in the absence of rf pulses, 0 is a scalar:
been divided into two parts. The first part arises from chemical shift effects; accordingly, HOis the static field and u is the chemical shift. The second term arises from the interaction of the magnetic field gradient Gg and the spin isocbromat whose location is designated by the vedor R. The general solution to Equation 2 is
m(t!)= Moe-t‘/T2e-i$(t)
where MOis the amplitude of the magnetization a t t = 0. The phase @ is the time integral of 0:
@(t’)= y Jt’(H0u
dt
+
+ R * G,)dt
(4)
What is the meaning of this phase term @, and how is it relevant to our discussion of NMR imaging? The phase term arises as a result of the torque exerted by the field n on the magnetization M, and within this phase is stored information about the interactions that the spins are experiencing. In high-resolution NMR, for example, the dominant interaction is the chemical shift u, thus the Hou term in the integrand of Equation 4 will dominate @. Fourier transforming
+
- m = -y(Hou R . G J m
(3)
- m/T2 (2)
where the magnetic field vedor Q has
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ANALYTICAL CHEMISTRY, VOL. 81, NO. 1, JANUARY 1, 1989
Equation 3 will yield the NMR absorption spectrum with the NMR transition frequencies marked by the chemical shift. But suppose that very large, linear magnetic field gradients occur acrow the sample volume. Then the R Go term in the integrand of Equation 4 dominates 6, and this term contains spatial information. Under such circumstances, Fourier transformation of Equation 3 will yield the type of spectrum schematized in Figure 1, where, because of the dominance of the gradient term in the phase @, NMR frequencies reflect spatial position. Therefore, the game that the NMR imager plays is how to design NMR experiments that will yield spatial information in the phase term @ and how to use this information to reconstruct an image. To understand imaging experiments, we must understand how to inspect a pulse sequence and extract the phase using Equation 4. Equation 4 seems very complicated but really is not. All of the terms in the integrand are either time independent or have a simple time dependence. For the purposes of this article, the phase integral can be evaluated simply by multiplying the terms in the integrand by the time duration of interest. With this easy recipe in mind, we can calculate the phase @ of a spin isochromat located within a sample that has been subjected to the spinecho pulse sequence in Figure 2b. The phase @ just before the 180° pulse, also called a T pulse, is calculated from Equation 4 as
.
d t , ) = r(R * Gg)t, + YHout, (5) where t , is the time just before the T pulse and to is the duration of the gradient pulse. The effect of the T pulse is to negate the phase of the magnetization rn, making @ = -@(tJ just after the T pulse. After the T pulse, the readout gradient is turned on, and a t any time t, after the readout gradient is turned on, the phase @ i sgiven by
.
d t ’ ) = yR G,(t‘ - t, yHou(t’ - 2t,)
- t,) + (6)
Note that at a single unique time TE = t , tg, the phase shift proportional to position is identically zero. The signal a t this time TE is called the gradient echo. At time t’ = 2t, the phase shift due to the chemical shift (the term containing u in Equation 6) is also zero. In most imaging experiments, the gradient is the dominant dephasing mechanism and the echo a t TE can he thought of as modulated by the envelope of the spin echo a t 2t, (IO). In the spin-echo sequence shown in Figure 2b, the acquisition window is
+
INSTRUMENTATION centered on the gradient echo. By offsetting time hy t = t' - T E to reflect this, we can rewrite Equation 6 as +(t
+ TE) = y[R + (Hoo/Go)l].Got+ yH,o(TE
- 2t,)
(7)
where l is a unit vector in the direction of GO.This formula exhibits two important effects of chemical shift, or any constant offset to frequency. There is a spatial "misregistration" because of chemical shift evolution during the acquisition that depends on the ratio HoalGo, and a constant phase shift because of the difference in echo times. Unless the gradient magnitude effectively dominates the chemical shift, the image will contain distortions that might be incorrectly interpreted as true spatial features. The Dhase term eiven in Eauation I corresponds to a single spin isdchromat located at agivenspatialposition in the sample. To obtain an expression for the FlD signal S ( f )that has been encoded withspatialinformation, wemustintegrate over the spatial extent of the object. Neglecting chemical shift effects:
where p ( x , y) contains information on the spatial distribution of spin density and the effects of Tz relaxation. To form an image, we need to recover p ( x , y ) by inversion of Equation 8. To show how this is done in the hackpro-
jection method, we introduce the concept of the wavenumher by the definition k = (kz,k,) = (yG,t, yG,t), which yields Equation 8 in the form:
S(k,, k,) =
P(&
J J S(k,, k,)dbse*"dk,d&, (lo)
When an imaging experiment is per. formed PR, the F1D s ,(~ ~~. ir,,Rr. -. quired as a fhction of the rotation angle 0. Thus the data set can be thought of as a two-dimensional data matrix with the horizontal dimension laheled by wavenumber and the vertical dimension laheled by 8. We can take advantage of the actual form of the data (S(K, 0 ) ) by transforming Equation 10 into polar coordinates to give: I
~
~~~
~
(X,Y).
The factor of IKI arises from the transformation polar coordinates andfrom the Cartesian chance intosign of
Y)e-".'dxdY
Ip(X,y)e''b"e-Jk"drd) (9) From Equation 9 it can be seen that the FID S ( t ) gives the two-dimensional spatial Fourier transform of the spin density p ( x , y) along some ray defined by 0 through the origin of the two-dimensional spectrum. The wavenumber origin coincides with the echo peak. Equation 9 also implies that p can he recovered by a two-dimensional inverse transformation of S(k,, k,) with respect to k, and k,: p(x, y) =
+
where r = x cos 0 y sin 0 is the displacement along the gradient direction from the gradient origin of the point
theargument of theexpon;ntial. Equation 11 means that the image can he reconstructed from the FID by performing a one-dimensional Fourier transform of lKlS with respect t o K followed by an integration over 8 from 0 to x . In practice, the f i s t step is done by Fourier transformation of each row of the two-dimensional data set followed by a convolution with a filter function that ap roximates the Fourier transform of?Kl (11). The final step is handled by placing each transformed and convoluted row a t an angle 0 to the x axis of a Cartesian coordinate svstem. which we call the image buffer (Figure 3, left). Intensities in the image buffer along all rays perpendicular to this rotated row are incremented by the value in the row at the intersection of each ray and the row. This backprojection of intensity information is the step for which the algorithm is named. Figure 3 (right) shows a series of four frames representing the reconstruction of a circular image by backprojection. Beginning with the upper left-hand frame, a single convoluted spectrum, corresponding to 0 = 0, is backprojected onto the image buffer, resulting in an intense band running vertically through the buffer. The upper righthand frame corresponds to backprojec-
..gum 3. The reconstruction of images by backprojectic..-. .JMR data. An NMR data lid Is ecwmuiated 85 a lunction Of the sample rotation angle 8. lien)The backprojection step is handled by laying the NMR spectrum conesponding to a rotation angle 8 Mo Um Image buffer a i an angle 8 to hxaxis. The imensity a1 a p i n t (r, e) contributes to a point (x, ,q il (x, ,q iles on a ray that 16 perpenilne. (risM)The remnsbucllon 01 a Circular Dhantm by. backc-rolection. The lour h a m show lormation 01 me imam as NMR data are dicular 10 Um ~)wctrai . . backprojectedmrough. '0 45'. 90'. and 180'
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INSTRL tion of data rrom Oo through 45O, and the lower left-hand frame corresponds to backprojection through 90'. The lower right-hand frame shows the complete reconstruction of the image following backprojection through 180". Spin warp imaging
The second principal NMR imaging technique is known as spin warp imaging. This technique, based on the twodimensional zeugmatography concept of Kumar, Welti, and Ernst (2) and first proposed by Edelstein et al. (3), has been most widely applied in medicine. A version of thii experiment is shown in Figure 4. The effects of the rf pulses and the G, gradient are the same as thoae found above in the discussion of PR imaging. Instead of rotating the gradient or the sample through different angles, a second orthogonal gradient G, is applied during the first half of the pulse sequence. The phase shift induced in a single isochromat at y from this gradient is given by (neglecting chemical shift effects):
or, using the wavenumber notation as introduced in the last section,
S(k,,k,) =
ss
p ( x , y)e-ik*e-ik.ydxdy
(13b) In Equations 13a and 13b, t , is the acquisition time and ty is the duration of they gradient. Note that unlike the PR experiment in which data are gathered as a function of a polar coordinate, the data matrix resulting from an SW experiment is in a form that will yield the image p in Cartesian coordinates directly by two-dimensional Fourier transformation. The phase encoding proceeds by holding tyconstant and incrementing the amplitude of G,, and the increment in G, and the duration t, are chosen to give the same field of view (FOV) in the x and y directions:
plied to the sample simultaneously with a gradient field. The bottom half of Figure 5 shows the amplitude of the rf field plotted against frequency in the rotating reference frame (a reference frame that is rotating a t the Larmor frequency about the magnetic field direction). At the same time, the resonance frequencies of the spins along the gradient direction G, are spread out by the gradient. This is diagrammed in the upper half of Figure 5. The only spins that will be excited are those that resonate within the bandwidth Au of the rf pulse; thus, only those spins that lie within the slice defined by AZ are affected hy the rf. From Figure 5 it is obvious that the adjustable parameters Aw and G, control the width of the selected slice, and we therefore require either a narrow-band rf irradiation or a strong gradient G, to obtain a thin slice. The location of the slice is controlled by the offset frequency of the rf transmitter. A number of schemes have been devised t o produce narrow-band rf . ....... .. ..
...
(14) The amplitude of this gradient is incremented systematically for each row of data collected. Thus the two-dimensional data matrix is indexed by yG,ty for each row and yG.t, for each column. The overall echo signal is:
Figure 4. Spin warp pulse sequf using spin-echo refocusing. The rl and 0. pulse timing are the s ~ m as e in the projection-reconstructionspln-ecb sequence.
An Bddlli~nalgradient 0, Is applied between ths rl pulses to phase encod8 the echo signal wllh spallal information In the ydlmction. The ampinude 01 this gradient Io lncrenmnted lor each new row
collected
Cy is known as the phase-encoding gradient to reflect the fact that this gradient encodes the echo with spatial information through its phase. Slice selection in NMR imaging
The imaging methods that we have presented thus far each produce a twodimensional image of an object without spatial resolution along an axis perpendicular to the image plane. The image in this case is a projection of spin density along this third orthogonal direction. In the case of SW imaging, resolution in this third dimension could he obtained by adding a third gradient and collecting a full three-dimensional image matrix of encoded NMR signals through a straightforward extension of the pulse sequence shown in Figure 4. Often, however, only images of selected planes through an object are desired, and acquisition of a complete threedimensional data set would be unnecessary as well as time consuming. An alternative to a full three-dimensional experiment has been devised that restricts rf irradiation of spins to welldefined slices through the object. This method of slice-selective imaging, which has been extensively applied in medical imaging, can be extended to the imaging of solids with some limitations. Before discussing these limitations we present the basic concepts behind slice-selective imaging. Figure 5 shows how slice selection works. A narrow-band rf field is ap-
Flgure 5. The basic slice selection cor cept in NMR imaging. The bonom diagram shows the amplitude of the rl field 9, as a lunction 01 frequency in the rotating frame. The rf field is zero everywhere omide the region A*. The top diagram illuslrates the connection between frequency offsets produced by a linear gradient In the z direction and the m r e spondlng +&xis coordinate. The doned lines Indlcab how the rl profile will attect only those isocbomats whose wordinales lie withln the sllce defined by Az.
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ANALYTICAL CHEMISTRY, VOL. 61, NO. 1, JANUARY 1, 1989
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pulses. Figure 6 shows one commonly used method. The amplitude of the rf pulse is modulated while a slice-selection gradient is applied. The rf modulation is chosen so that the frequency profile of the pulse is as square as possible, thus affording as selective an irradiation as possible. A sin(r/t')/rt' (or sinc) function is typically used, although other shapes have been implemented (12, 13). The duration of the pulse t defines the bandwidth Au: Au = 2rn& where n depends on the number of cycles in the sinc function. The amplitude of the rf pulse is adjusted to give either u12 rotation of m, in the case of selective excitation or T rotation of transverse magnetization for selective spin-echo refocusing. For selective excitation (Figure 6a) the slice selection gradient is reversed followingthe rfpulse. This reverses the precession of the spin isochromats within the slice that have dephased during the rf pulse, and as a result all spin isochromats will have the same phase at the end of the gradient pulse. If this selective sequence replaces the 90' pulse in Figure 2b and Figure 4, only those spins within the selected slice will contribute to the echo signal and hence to the image. The refocusing r pulse in the spin-echo pulse sequences can be replaced by the sequence shown in Figure 6b. In this case, the slice selection gradient does not need to be reversed because the r rotaon of iagnetizatio Nady ne-
gates the phase of each of the isochromats within the slice. If only one slice is imaged, only one of the rf pulses in the echo sequence needs to be selective. Multislice imaging techniques employ both selective excitation and selective refocusing pulses to avoid perturbation of spins outside the slice. In this way data from one slice can he collected while z magnetization from the other slices is restored through spin-lattice relaxation.
Factom limiting spatlal d u t h We have discussed the two principal methods whereby images may be reconstructed from NMR data: projection reconstruction and spin warp. We have also explained how, with the controlled application of one or more linear magnetic field gradients, the FID can be encoded with information on the spatial distribution of spin density, and how NMR spectral information can he manipulated to yield an image. Several important and practical questions remain. How are gradient strength, Fourier transform size, and spectral width quantitatively related to the spatial resolution of the image, and what are the factors limiting spatial resolution in solids and liquids? To obtain the relationship between gradient strength, spatial width, and Fourier transform size we begin with the basic relationship between the time
resolution of the FID, also called the dwell time D W, and the spectral width
Sw: DW.SW=l (15) If N is the number of points in the digital Fourier transform (assumed also to be equal to the number of points in the FID), and if W is the spectral resolution in units of Hertz per point
C-M" N.W.DW=W.AT=l
(16)
where AT is the total acquisition time. Suppose G is the gradient strength in units of Gauss cm-1 and AX is the spatial resolution in units of cmlpt. Then
yG. AX. AT/2u
=1
(17) Equation 17 means that the product of the gradient strength yG, the spatial resolution AX, and the acquisition time AT equals a constant. The relationship between gradient strength, spatial resolution, and acquisition time is shown in Figure 7 as a log-log plot of gradient strength on the vertical axis versus acquisition time on the horizontal axis. Diagonal lines are l i e s of equal spatial resolution. From Figure 7 we see that to attain a spatial resolution of 0.24 mm/pt we might use a gradient strength of 10 Glcm and an acquisition time of 1 ms. If the acquisition time is held constant a t 1ms and
1-
\ 10
5.0
Figure 6. Selective sequences consisting of amplitude-modulated rf pulses and sllce gradients. (a) Selecllve exoltstlon schwm used lo wnvell m, to -m, wlmln a sllca and (b) seIBc1Ive relowing pulsa mal corn (m,, q) to (m, -my) whhln a sIIce.
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Figure 7. A log-log plot of G ..T with the diagonal lines definingspatial resoiution. The relationship between gradient strength G,acquisition time AT, and spatial resolution Axis given by the equation yG. A X - A V 2 a = 1. The red m e 8 ahow l l n s d amnuallon I& 2 5 X 10-5 cm.2 sc'.
ANALYTICAL CHEMISTRY, VOL. 61, NO. 1. JANUARY 1, 1989
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the gradient strength is increased to 100 Glcm, the spatial resolution becomes 0.024 mmlpt. It seems too good to be true. Equation 17 implies that to increase spatial resolution, all we need do is go to higher and higher gradient strengths. This discussion begs the question: What are the fadors, if any, that limit spatial resolution in solids and liquids? It is tempting to speculate that very high resolution images of liquids will be readily attainable, and certainly nothing in our discussion thus far suggests that anything but gradient strength limits the spatial resolution of NMR images of liquids. Such, however, is not the cme, because our discussion of image reconstruction has thus far omitted any mention of molecular motion. If we assume that the molecules in a sample are undergoing Brownian motions with diffusion coefficient D,what is the effect of this motion on the NMR signal that we detect during the acquisition window of the pulse sequence in Figure 2b? If no free precession periods are included between gradient pulses, then the gradient echo time TE equals the acquisition time AT, and TE = AT = Zt,, where 2t, is the duration of the readout gradient. The ratio of the spinecho intensity in the presence of diffusion ID to the intensity of the spin echo in the hypothetical case of no diffusion Is is given by:
tio of ID& = 0.86. If, however, the gradient is increased to somewhat over 80 Glen, theoretically yielding a spatial resolution of better than 1rmlpt, ID& decreases to about 0.009.This means that higher spatial resolution can be realized by using ever-increasing gradient strengths, but only a t a severe cost in signal-to-noise ratio. These expectations are borne out by experimentation. Figure 8 shows a series of four image frames consisting of two juxtaposed glass bulbs (diameter ofeachbulbis4mm). Ineachframe the upper right-hand bulb is filled with glycerol, and the lower left-hand bulb is filled with water. The four images were obtained using the PR sequence shown in Figure 2b. We will defer discussion on the experimental details of PR imaging until the next section and simply discuss the results. This series of imaging experiments was performed as a function of gradient strength, with the first (upper left) image obtained with a gradient of 5.25 Glcm, the second (upper right) with a gradient of 8.75 Glen, the third (lower left) with a gradient of 18.90 Glcm, and the fourth (lowerright) withagradient of 33.25 GI em. The imaging experiments were carried out a t constant FOV by decreasing the dwell time to compensate for the increase in gradient (see Equation 14). Therefore all four frames reconstruct with their image dimensions identical. The intensity ratios of these four images relative to the intensity of the wa-
ter signal following a 90' pulse are 0.91, 0.77,0.30, and 0.02. Note that the water image almost vanishes in the fourth frame. The glycerol image is attenuated hardly a t all because the diffusion rate of glycerol is much lower than that of water. The good news seems to be, however, that high spatial resolution may be achievable in solids where molecular diffusion is orders of magnitude slower or is absent altogether. The bad news is that other factors come into play that hinder our attempts to achieve high spatial resolution in solids. The foremost of these obstacles is the large NMR line widths observed in solids. We mentioned earlier that in liquids, typial NMR line widths are a few Hertz at most, but the NMR spectra of solid materials are dominated by a variety of anisotropic interactions that are not manifested in the spectra of liquids because of the averaging effect of rapid and isotropic molecular tumbling. For example, the NMR spectrum of protons in a polycrystalline powder is dominated by the direct nuclear dipole-dipole coupling; spectral line widths may be 20,000 Hz or more. In terms of imaging resolution, this means that if we apply a 1Glcm gradient to a solid sample, two spin isochromats separated by 5 mm will be offset by 2.12 kHz. If the NMR line width is 30 kHz, no appreciable spatial information will be stored in the FID. To resolve isochromats 5 mm apart, gradients of
If, on the other hand, TE greatly exceeds %,(AT = %,), then considerable diffusion takes place between the gradient pulses. In this case, the ratio of ID to IS is given by:
'.=
e - ~ ~ ~ . ~ y ~ ) *(isa) . ~ / i ~
IS
where ux = is the standard deviation of position and AX is the spatial resolution. Equations 1%and 18b indicate that severe attenuations of echo amplitude may occur a t high gradient strengths if the diffusion coefficient is large enough. The results of Equation 18a can be appreciated by again referring to Figure 7, where several attenuation curves have been superimposed on the log-log plot. For eaample, assuming a diffusion coefficient D equal to 2.5 X cm2 s-l (a value of D typical for water a t room temperature), the graph in Figure 7 indicates that at an A T of 10 ns an imaging experiment carried out using a gradient strength of 10Glcm would yield a spatial resolution of 0.024 mmlpt but would also result in an attenuation raS4A
r-I,
in each frame) and water (lower left in each frame). Acquired at a constam field of vlew. the lour frames mnespond to gradients 01 5.25 Glcm (upper lell frame). 8.15 Glcm (upper rlpm frame), 18.90 Glcm (lower len Irame). and 33.25 G/cm (lower right
Irsme).
ANALYTICAL CHEMISTRY, VOL. 61. NO. 1, JANUARY 1, 1989
more than 20 G/cm would be required. To resolve spins 1mm apart, gradients of 100 G/cm or more would be required, and to achieve higher spatial resolution, even larger gradients would be required. The situation is even more challenging than illustrated by our example, because solid-state NMR line widths can easily exceed 30 kHz. These practical limitations to spatial resolution in liquids and solids are not entirely set in stone, however. For example, the attenuation of image intensity by large field gradients in liquids is only a problem if long echo times (large values of TE) are used. If high-resolution image reconstruction requires the use of very large magnetic field gradients, TE must be minimized or eliminated altogether. TE can be minimized through the use of a current pulsergradient coil system with a short rise time to peak current. On the other band, some PR pulse sequences (e.g., Figure 2a) do not require the use of echos a t all, allowing in principle the detection of an NMR signal unattenuated by diffusion effects. In the case of solids, if line widths are dominated by large anisotropic interactions, incorporation of techniques that reduce or eliminate anisotropic interactions into imaging experiments may be advantageous. Such techniques include composite pulse decoupling to remove heteronuclear dipolar couplings, multiple-pulse techniques to remove homonuclear dipolar coupling, and high-speed magic angle spinning techniques to remove broadening that results from the anisotropy of the chemical shift (14,15). Recent appllcallons In this section we will describe current efforts designed to apply NMR imaging to the study of solid materials. We will consider a straightforward structural question involving a polymer matrix filled with solid particles. There are a number of applications of such composite materials in the aerospace field, and the ways in which these materials are formed are of great importance to their end use and mechanical properties. Often, the solid particles are introduced by mixing with a viscous polymer melt, which is later cured to a solid matrix. At issue is the uniformity of the solids concentration throughout a composite, a question that ideally could be addressed by an NMR imaging experiment in which T2 or proton density is used as a contrast agent to distinguish between the polymer matrix and the solid particles. The primary aim of this discussion is to compare and contrast the results obtained by PR and by SW imaging studies.
We begin with a study by P R metbods of a cross-linked rubber impregnat-
ed with aluminum particles about 3 pm in diameter and sodium sulfate crystals about 10 pm in diameter. The weight percent composition of the material was aluminum, 5%and sodium sulfate, 65% the remainder was composed primarily of the rubber. PR images were obtained with a home-built NMR probe designed to perform both P R and SW experiments. The probe was designed to be inserted into a superconducting magnet with an 89-mm diameter bore, and all P R experiments were performed on a CXP-200 NMR spectrometer (Bruker Instruments Inc.) a t a proton resonance frequency of 200 MHz. Figure 9 is a view of the imaging probe head, looking down the axis of a solenoidal rf coil (10-mm id., 12-mm length). Typical proton 90° pulse times were less than 4 ps. PR experiments were performed by mechanically rotating the sample using a worm geadscrew arrangement in which the screw was driven by a stepping motor that in turn was controlled by a trigger from the pulse programmer of the CXP-200. In a typical experimental run, an imaging pulse sequence was performed for 180 rotation angles with a rotation angle increment of lo.Two gradient coils were mounted on the probe head. A Helmholtz coil generated a G, gradient with an efficiency of 7.01 G/cm/amp, and a quadrupolar coil (seen as the flat array of conductors to the immediate right and left of the rf coil) generated an orthogonal gradient
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Figure 9. A view of a home-built probe head designed to perform both proJection-reconstruction and spin warp imaging experiments, looking down the axis of the rf coil. The probe 16 equipped wRh two wadiem coils; the Hsimholb coil is esen 89 the two law windings 01 wire above and below the solenold, and a quadruwlar mll wlm a rectilinear aperture is smn BS me fiat may of wirdingr lo the right and ien 01 the solenold.
ANALYTICAL CHEMISTRY, VOL. 61, NO. 1. JANUARY 1, 1989
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with an efficiency of 9.37 Glcmfamp (16). An attendant current pulser was capable of delivering more than 10 amps to the probe coils, with a rise time to peak gradient of just over 100 ps. Figure 10 is a PR image of a 1-mm thick, 7-mm diameter disk of aluminum-filled cross-linked rubher recon-
structed from a set of 180 FIDs. The spin-echo pulse sequence shown in Figure 2b was used, with the duration of the dephasing gradient set to 300 ps and the total echo time TE less than 1 ms. The gradient amplitude was 28 Gfcm, and the total experimental run time was about 2 h. Surprisingly, the image intensity was not uniform across the sample; it displayed a number of regions of low signal intensity distributed throughout the material. Althoughthe probe can perform both PR and SW experiments, it is equipped with only two coils and thus cannot perform slice selection experiments (slice seleetion can be accomplished in a PR experiment with a gradient parallel to the goniometer axis). To further characterize the extent of these features, we used a commercial SW imager equipped for slice selection. Figure 11 shows four spin warp images of the same aluminum-filled rubber that was imaged hy PR methods and displayed in Figure 10. These SW images were obtained with a commercial NMR spectrometer (MSL 200, Bruker Instruments Inc.) fitted with a microscope imaging attachment. Each frame is the image of a slice through a square sample with a diagonal dimen-
Figure 10. An image of aluminurnfilled rubber obtained by proJection-reconstruction methods. The diameter of the -le is about 7 mm with a thlckneu of 1 mm. A gradiem of 28 Glcm was used with medsnlcai rotation of the 88mDie. as described in Uw texi
b
I
1 P Figure 11. Spin warp images . of aluminum-filled rubber. Each 61108 is 1 mm thick. and the four siiws were taken from adjcent locations. The diagonal dimension of the square sample is 10 mm.
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ANALYTICAL CHEMISTRY, VOL. 61, NO. 1, JANUARY 1. 1989
sion of 10 mm. Slice selection was accomplished using a sinc excitation pulse in the spin-acho pulse sequence shown in Figure 3. Because of the short TZ of this material, the selective rf pulse was limited to just 1 ms,and to retain the narrow slice width, the sinc function covered only one cycle. This means that the excitation profile is not the ideal shape shown in Figure 5, but is instead nearly triangular with the apex located in the midplane of the slice. With the z gradient used (4.7 GI cm), the slice thickness was 1mm. The echo time TE for each image was 4.3 ms, which means that the duration of the pulse sequence was actually much longer than the apparent Tz (0.3 ms) that was estimated from the line width. This indicates the existence of an inhomogeneous broadening that causes the apparent Tz to be shorter than the actual TZ(16 ms). The likely sources of this broadening are chemical shift dispersion of the protons in the rubber or susceptibility broadening caused by the presence of aluminum particles. Although the 180° pulse refocused decay that resulted from susceptibility effects and chemical shift dispersion, substantial loss in signal intensity still occurred. This loss was a result of T2 decay during TE and made it necessary to signal average much longer than in the PR experiment, where a much shorter TE was used. In all, 512 echoes were accumulatedforeachrowofthe128x128pixel images in Figure 11;the overall image acquisition time was 14 h per slice. This same sample was examined by X-ray imaging in the region where NMR images show regions of low intensity. No such features were observed by the X-ray imager, which had been calibrated to be sensitive to as little as 1%variation in the X-ray absorption. Our interpretation of the features shown in the NMR image is that the low-intensity regions correspond to low rubber concentration in areas where the aluminum powder was not depleted. These regions were formed before curing of the cross-linked rubber. Figure 12 shows another example of SW imaging. In this case the sample is an unplasticized, solid-filled phenolic resin held in place by four wooden sticks. The sticks are 2 mm in diameter, and the entire sample was wrapped with Teflon tape. Spin-spin relaxation of protons in this sample is so fast (see below) that slice selection with shaped rf pulses was not possible. Instead, the nonselective sequence of Figure 3 was used with an echo time TE of 3 ms. The image is therefore the projection of the entire 2-cm length of the sample along the I axis. Protons from the tape and
from water adsorbed on the surface of the sticks show up in the image. The proton line shape of the resin has two components: a broad component with a full width at half maximum (fwhm) of 25 kHz from hydrocarbon protons and a narrow component (--1 kHz fwhm) from a small amount (
