Imaging of Materials
NMR John M. Listerud Pendergrass Diagnostic Research Laboratories Philadelphia, PA 19104-6086
Steven W. Sinton Lockheed Palo Alto 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 reso nance 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 sys tems are designed with the liquid state in mind, for use with large biological 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 (1) consisted of some arrangement, using either nonlin ear or time-varying gradients, that con fined the source of signal at a particular point on the spectrum to a single vol ume element. This sensitive volume element would be moved, possibly by moving the object itself. These meth ods, 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
INSTRUMENTATION that the widespread, highly visible suc cess of medical imaging has given ma terials researchers an inkling of the possibilities. NMR imaging depends on the use of linear magnetic field gradients to dis tinguish 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 be tween spatial and spectral positions. For protons, an isochromat (the signal from a small volume element of the sample) offset —1/2 cm in a gradient of 1 G/cm (=4.25 kHz/cm) will be shifted upfield by 2.12 kHz. Dimensional anal ysis highlights an important theme in NMR imaging: the mapping of time and frequency to wavenumber and spa tial dimension, as shown in the box. The early history of imaging saw a number of contending techniques en0003-2700/89/0361-023A/$01.50/0 © 1988 American Chemical Society
become the method of choice for medi cal imaging. Backprojection, the math ematical technique t h a t supports X-ray computed tomography, has also been popular in the NMR literature (2, 5) and will receive special consider ation 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.
s)
t
*G (Hz/cm)
FT
frequency t> (Hz)
The current effort is concentrated on basic effects that determine the inten sity and phase of signal, such as spinlattice and spin-spin relaxation, mo lecular diffusion, magnetic susceptibil ity, 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 gra dient, 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, and'spe cial attention will be given to backpro jection and spin warp (SW) imaging. We will also discuss so-called slice se lection techniques, in which the image of a particular slice through an object can be obtained. Factors limiting spa tial resolution in the images of liquids and solids will be considered, and in particular the effect of molecular diffu sion on image intensity and resolution will be described. Finally, a number of applications of NMR imaging to the study of materials will be briefly re viewed. Future prospects for NMR im aging in materials sciences will be dis cussed. NMR image reconstruction techniques As mentioned above, NMR imaging is based on the use of linear magnetic field gradients to frequency label spa tial position. Any chemist who has been forced t o contend with a poorly shimmed NMR magnet (i.e., a magnet with poor field homogeneity) can im-
• wavenumber k (cnr1)
*G-1 (cm/Hz)
1FT *
•position χ (cm)
ANALYTICAL CHEMISTRY, VOL. 61, NO. 1, JANUARY 1, 1989 · 23 A
INSTRUMENTATION ment. The basic equation describing the evolution of transverse magnetiza tion in the xy plane from a single isochromat after a 90° (or the π/2) pulse is: | M =
df
7
Û X M - M/T2
(1)
where M is a magnetization vector, Ω is a vector describing the effects of chem ical shift and resonance offset terms such as the gradient, y is the nuclear magnetogyric ratio, and T2 is the spinspin relaxation time. Equation 1 has the form of a classical torque equation, where as a result of the cross product between Ω and M, a precession of the magnetization occurs at the Larmor frequency. The second term on the right of Equation 1 is essentially a dissipative or damping term, and I/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 = Mx + iMy and by recognizing that in the absence of rf pulses, Ω is a scalar: -7- m = -y{H0a + R · G„)m - m/T2 at (2) where the magnetic field vector Ω has
been divided into two parts. The first part arises from chemical shift effects; accordingly, Ho is the static field and σ is the chemical shift. The second term arises from the interaction of the mag netic field gradient Gs and the spin isochromat whose location is designat ed by the vector R. The general solu tion to Equation 2 is m(t') = Μ0β~ί'/Τ2β-ίφ(ί')
(3)
where Mo is the amplitude of the mag netization at t = 0. The phase φ is the time integral of Ω:
0(f)-7 J ' u V + K-GiW (4) What is the meaning of this phase term φ, 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 Ω on the magnetization M, and within this phase is stored information about the interactions that the spins are experi encing. In high-resolution NMR, for example, the dominant interaction is the chemical shift σ, thus the ίίοσ term in the integrand of Equation 4 will dominate φ. Fourier transforming
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Equation 3 will yield the NMR absorp tion spectrum with the NMR transi tion frequencies marked by the chemi cal shift. But suppose that very large, linear magnetic field gradients occur across the sample volume. Then the R · Gj term in the integrand of Equation 4 dominates , and this term contains spatial information. Under such cir cumstances, Fourier transformation of Equation 3 will yield the type of spec trum schematized in Figure 1, where, because of the dominance of the gradi ent term in the phase φ, NMR frequen cies 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 φ and how to use this infor mation 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 evalu ated 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 0 of a spin isochromat located within a sam ple that has been subjected to the spinecho pulse sequence in Figure 2b. The phase φ just before the 180° pulse, also called a π pulse, is calculat ed from Equation 4 as φ(Ιτ) = γ(Β · Ge)t„ + yH0atr
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(5)
where t„ is the time just before the π pulse and te is the duration of the gradi ent pulse. The effect of the π pulse is to negate the phase of the magnetization m, making φ = — φ(£τ) just after the π pulse. After the π pulse, the readout gradient is turned on, and at any time tr after the readout gradient is turned on, the phase φ is given by φ(ϊ) = T R · G,(f -tryH0a(t' - 2t„)
t„) + (6)
Note that at a single unique time TE = tr + te, the phase shift proportional to position is identically zero. The signal at this time TE is called the gradient echo. At time t' = 2tT the phase shift due to the chemical shift (the term con taining σ in Equation 6) is also zero. In most imaging experiments, the gradi ent is the dominant dephasing mecha nism and the echo at TE can be thought of as modulated by the enve lope of the spin echo at 2t„ {10). In the spin-echo sequence shown in Figure 2b, the acquisition window is
INSTRUMENTATION centered on the gradient echo. By off setting time by ί = t' — TE to reflect this, we can rewrite Equation 6 as 0(t + TE) = γ[Κ + (H 0 a/G,)l] · G„t + - 2t„) (7) ΊΗ0σ(ΤΕ where 1 is a unit vector in the direction of Gfl. This formula exhibits two impor tant effects of chemical shift, or any constant offset to frequency. There is a spatial "misregistration" because of chemical shift evolution during the ac quisition that depends on the ratio H0a/Ge, and a constant phase shift be cause of the difference in echo times. Unless the gradient magnitude effec tively dominates the chemical shift, the image will contain distortions t h a t might be incorrectly interpreted as true spatial features. The phase term given in Equation 7 corresponds to a single spin isochromat located at a given spatial position in the sample. To obtain an expression for the FID signal S{t) that has been encoded with spatial information, we must inte grate over the spatial extent of the object. Neglecting chemical shift effects:
S(t)= J J
hB G t P(x,y)e- - ' dxdy
(8) where p(x, y) contains information on the spatial distribution of spin density and the effects of Ti 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 backpro
jection method, we introduce the con cept of the wavenumber by the defini tion k = (kx, ky) = (yGxt, yGyt), which yields Equation 8 in the form: S(K,ky)=
f
=
f
f I J
J
p(x,y)e-iK-ndxdy p(x,y)e~'k'xe~'kyydxdy (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 θ through the origin of the two-di mensional spectrum. The wavenumber origin coincides with the echo peak. Equation 9 also implies that ρ can be recovered by a two-dimensional in verse transformation of S(kx, ky) with respect to kx and ky:
S{kx,ky)eik'x^dkxdky
P(x,y)= f f
(10) When an imaging experiment is per formed using PR, the FID S(t) is ac quired as a function of the rotation an gle Θ. Thus the data set can be thought of as a two-dimensional data matrix with the horizontal dimension labeled by wavenumber and the vertical di mension labeled by Θ. We can take ad vantage of the actual form of the data (S{K, Θ)) by transforming Equation 10 into polar coordinates to give: p(x,y)=
/
/
\K\S(K,e)e~iKrdKde (11)
where r = χ cos θ + y sin θ is the dis placement along the gradient direction from the gradient origin of the point (x,y). The factor of \K\ arises from the transformation from Cartesian to polar coordinates and the change in sign of the argument of the exponential. Equa tion 11 means that the image can be reconstructed from the FID by per forming a one-dimensional Fourier transform of \K\S with respect to Κ fol lowed by an integration over θ from 0 to χ. In practice, the first step is done by Fourier transformation of each row of the two-dimensional data set followed by a convolution with a filter function that approximates the Fourier trans form of \K\ (11). The final step is han dled by placing each transformed and convoluted row at an angle θ to the χ axis of a Cartesian coordinate system, which we call the image buffer (Figure 3, left). Intensities in the image buffer along all rays perpendicular to this ro tated 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 θ » 0, is backprojected onto the image buffer, resulting in an intense band running vertically through the buffer. The upper righthand frame corresponds to backprojec-
Flgure 3. The reconstruction of images by backprojection of NMR data. An NMR data set is accumulated as a function of the sample rotation angle Θ. (left) The backprojection step is handled by laying the NMR spectrum corresponding to a rotation angle θ onto the image buffer at an angle Β to the χ axis. The intensity at a point (r, Θ) contributes to a point (x, y) if (x, y) lies on a ray that is perpen dicular to the spectral line, (right) The reconstruction of a circular phantom by backprojection. The four frames show formation of the image as NMR data are backprojected through 0°, 45°, 90°, and 180°.
ANALYTICAL CHEMISTRY, VOL. 61, NO. 1, JANUARY 1, 1989 · 29 A
INSTRUMENTATION tion of data from 0° through 45°, and the lower left-hand frame corresponds to backprojection through 90°. The lower right-hand frame shows the com plete reconstruction of the image fol lowing backprojection through 180°. Spin warp imaging The second principal NMR imaging technique is known as spin warp imag ing. 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 medi cine. A version of this experiment is shown in Figure 4. The effects of the rf pulses and the Gx gradient are the same as those found above in the discussion of PR imaging. Instead of rotating the gradient or the sample through differ ent angles, a second orthogonal gradi ent Gy is applied during the first half of the pulse sequence. The phase shift in duced in a single isochromat at y from this gradient is given by (neglecting chemical shift effects):
S(yGttz,
yGyty) = fp(x,y)e-^G't'e-nyG^dxdy (13a)
/
or, using the wavenumber notation as introduced in the last section, S(kx, k)
- / /
(12) The amplitude of this gradient is incre mented systematically for each row of data collected. Thus the two-dimen sional data matrix is indexed by yGyty for each row and yGxtx for each col umn. The overall echo signal is:
7Γ/2
"1
7Γ
Echo signal
1
Figure 4. Spin warp pulse sequence using spin-echo refocusing. The rf and Gx pulse timing are the same as in the projection-reconstruction spin-echo sequence. An additional gradient Gy is applied between the rf pulses to phase encode the echo signal with spa tial information in the y direction. The amplitude of this gradient is incremented for each new row collected.
^dxdy
In Equations 13a and 13b, tx is the ac quisition time and ty is the duration of the y 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 ex periment is in a form that will yield the image ρ in Cartesian coordinates di rectly by two-dimensional Fourier transformation. The phase encoding proceeds by holding ty constant and in crementing the amplitude of Gy, and the increment in Gy and the duration ty are chosen to give the same field of view (FOV) in the χ and y directions: y
= yyGyty
l
(13b)
FOV v = y(ty) = yft'yGydt
p(x, y)e ' **e
2ττ
2
l yàGYtv
yGAh
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 at the Larmor frequency about the magnetic field di rection). At the same time, the reso nance frequencies of the spins along the gradient direction G2 are spread out by the gradient. This is dia grammed in the upper half of Figure 5. The only spins that will be excited are those that resonate within the band width Δω of the rf pulse; thus, only those spins that lie within the slice de fined by Δζ are affected by the rf. From Figure 5 it is obvious that the adjust able parameters Δω and Gz control the width of the selected slice, and we therefore require either a narrow-band rf irradiation or a strong gradient Gz to obtain a thin slice. The location of the slice is controlled by the offset frequen cy of the rf transmitter. A number of schemes have been de vised to produce narrow-band rf
FOV, (14)
Gy is known as the phase-encoding gradient to reflect the fact that this gradient encodes the echo with spatial information through its phase.
ζ
Slope =
y G
z /
I Δζ
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 perpen dicular to the image plane. The image in this case is a projection of spin densi ty along this third orthogonal direc tion. In the case of SW imaging, resolu tion in this third dimension could be 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 unnec essary as well as time consuming. An alternative to a full three-dimensional experiment has been devised that re stricts 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 limita tions. Before discussing these limita tions we present the basic concepts be hind slice-selective imaging. Figure 5 shows how slice selection works. A narrow-band rf field is ap
30 A · ANALYTICAL CHEMISTRY, VOL. 6 1 , NO. 1, JANUARY 1, 1989
7GZZ
S,
•«-Δω - * · Frequency = (ω - ω0)
Figure 5. The basic slice selection con cept in NMR imaging. The bottom diagram shows the amplitude of the rf field Si as a function of frequency in the rotating frame. The rf field is zero everywhere outside the region Δω. The top diagram illustrates the con nection between frequency offsets produced by a linear gradient in the ζ direction and the corre sponding z-axis coordinate. The dotted lines indi cate how the rf profile will affect only those isochromats whose coordinates lie within the slice defined by Δζ.
INSTRUMENTATION pulses. Figure 6 shows one commonly used method. The amplitude of the rf pulse is modulated while a slice-selec tion gradient is applied. The rf modula tion is chosen so that the frequency profile of the pulse is as square as possi ble, thus affording as selective an irra diation as possible. A sin(ir/i')/iri' (or sine) function is typically used, al though other shapes have been imple mented (12, 13). The duration of the pulse t„ defines the bandwidth Δω: Δω = 2irn/tp, where η depends on the num ber of cycles in the sine function. The amplitude of the rf pulse is adjusted to give either τ/2 rotation of mz in the case of selective excitation or τ rotation of transverse magnetization for selec tive spin-echo refocusing. For selective excitation (Figure 6a) the slice selection gradient is reversed following the rf pulse. 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 π pulse in the spin-echo pulse se quences can be replaced by the se quence shown in Figure 6b. In this case, the slice selection gradient does not need to be reversed because the π rota tion of the magnetization already ne
gates the phase of each of the isochro mats 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 be collected while ζ magnetization from the other slices is restored through spin-lattice relaxation. Factors limiting spatial resolution We have discussed the two principal methods whereby images may be re constructed from NMR data: projec tion reconstruction and spin warp. We have also explained how, with the con trolled application of one or more lin ear magnetic field gradients, the FID can be encoded with information on the spatial distribution of spin density, and how NMR spectral information can be 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 DW, and the spectral width SW: DW-SW=1
(15)
If Ν 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 (Hz/pt), N- W-DW=
W-AT=l
(16)
where AT is the total acquisition time. Suppose G is the gradient strength in units of Gauss c m - 1 and ΔΧ is the spa tial resolution in units of cm/pt. Then yG • AX • ΑΤΙ2π = 1
(17)
Equation 17 means that the product of the gradient strength yG, the spatial resolution Δ.Χ, and the acquisition time AT equals a constant. The rela tionship 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 horizon tal axis. Diagonal lines are lines of equal spatial resolution. From Figure 7 we see that to attain a spatial resolu tion of 0.24 mm/pt we might use a gra dient strength of 10 G/cm and an ac quisition time of 1 ms. If the acquisi tion time is held constant at 1 ms and
42.5
c
to Ο
4.25
0.425
1
100 ms 15.8 μΓη
Figure 6. Selective sequences consist ing of amplitude-modulated rf pulses and slice gradients. (a) Selective excitation scheme used to convert mz to — mt within a slice and (b) selective refo cusing pulse that converts (m„ my) to (m„, — my) within a slice.
AT σχ
TE
Figure 7. A log-log plot of G vs. Α Γ with the diagonal lines defining spatial resolu tion. The relationship between gradient strength G, acquisition time AT, and spatial resolution AXis given by the equation yG· ΑΧ· ΑΤ/2π = 1. The red curves show lines of attenuation lD/ls corresponding to a diffusion constant of 2.5 Χ 10 - 5 cm 2 s" 1 .
32 A · ANALYTICAL CHEMISTRY, VOL. 6 1 , NO. 1, JANUARY 1, 1989
INSTRUMENTATION the gradient strength is increased to 100 G/cm, the spatial resolution be comes 0.024 mm/pt. It seems too good to be true. Equa tion 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 factors, 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 noth ing 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 case, because our discussion of im age reconstruction has thus far omitted any mention of molecular motion. If we assume that the molecules in a sample are undergoing Brownian mo tions with diffusion coefficient D, what is the effect of this motion on the NMR signal that we detect during the acqui sition 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 = 2tx, where 2tx is the duration of the readout gradient. The ratio of the spinecho intensity in the presence of diffu sion ID to the intensity of the spin echo in the hypothetical case of no diffusion Is is given by: b. = e-ATHyGf.Dll2
( l g a )
tio of ID/IS - 0.86. If, however, the gra
dient is increased to somewhat over 80 G/cm, theoretically yielding a spatial resolution of better than 1 jim/pt, ID/IS decreases to about 0.009. This means that higher spatial resolution can be realized by using ever-increasing gradi ent strengths, but only at a severe cost in signal-to-noise ratio. These expectations are borne out by experimentation. Figure 8 shows a se ries of four image frames consisting of two juxtaposed glass bulbs (diameter of each bulb is 4 mm). In each frame 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 dis cussion 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 G/cm, the sec ond (upper right) with a gradient of 8.75 G/cm, the third (lower left) with a gradient of 18.90 G/cm, and the fourth (lower right) with a gradient of 33.25 G/ cm. The imaging experiments were car ried out at 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 im ages 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 attenuat ed hardly at 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 fore most 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 be cause of the averaging effect of rapid and isotropic molecular tumbling. For example, the NMR spectrum of pro tons 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 1 G/cm gradient to a solid sample, two spin isochromats sep arated 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 iso chromats 5 mm apart, gradients of
h If, on the other hand, TE greatly ex ceeds 2tx(AT = 2tx), then considerable diffusion takes place between the gra dient pulses. In this case, the ratio of Io to Is is given by: IR _ e-ATa-(yG)2-Dm
(lga)
where σχ = -J2D · TE is the standard deviation of position and AX is the spa tial resolution. Equations 18a and 18b indicate that severe attenuations of echo amplitude may occur at high gra dient strengths if the diffusion coeffi cient is large enough. The results of Equation 18a can be appreciated by again referring to Figure 7, where sev eral attenuation curves have been superimposed on the log-log plot. For example, assuming a diffusion coeffi cient D equal to 2.5 X 10~5 cm2 s _ 1 (a value of D typical for water at room temperature), the graph in Figure 7 in dicates that at an AT of 10 ms an imag ing experiment carried out using a gra dient strength of 10 G/cm would yield a spatial resolution of 0.024 mm/pt but would also result in an attenuation ra
Figure 8. A series of four frames showing juxtaposed images of glycerol (upper right in each frame) and water (lower left in each frame). Acquired at a constant field of view, the four frames correspond to gradients of 5.25 G/cm (upper left frame), 8.75 G/cm (upper right frame), 18.90 G/cm (lower left frame), and 33.25 G/cm (lower right frame).
34 A · ANALYTICAL CHEMISTRY, VOL. 61, NO. 1, JANUARY 1, 1989
more than 20 G/cm would be required. To resolve spins 1 mm 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 hand, some PR pulse sequences (e.g., Figure 2a) do not require the use of echos at 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).
We begin with a study by PR methods of a cross-linked rubber impregnated with aluminum particles about 3 jum in diameter and sodium sulfate crystals about 10 μτη in diameter. The weight percent composition of the material was aluminum, 5% and sodium sulfate, 65%; the remainder was composed pri marily of the rubber. PR images were obtained with a home-built NMR probe designed to perform both PR and SW experiments. The probe was designed to be inserted into a super conducting magnet with an 89-mm di ameter bore, and all PR experiments were performed on a CXP-200 NMR spectrometer (Bruker Instruments Inc.) at 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 i.d., 12-mm length). Typical proton 90° pulse times were less than 4 μβ. PR experiments were performed by me chanically rotating the sample using a worm gear/screw arrangement in which the screw was driven by a stepping mo tor 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 1°. Two gradient coils were mounted on the probe head. A Helmholtz coil generat ed a Gz 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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Recent applications 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 T 2 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.
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IMO In awmd, V&tkmmwx, Figure 9. A view of a home-built probe head designed to perform both projec tion-reconstruction and spin warp imag ing experiments, looking down the axis of the rf coil. The probe is equipped with two gradient coils; the Helmholtz coil is seen as the two large windings of wire above and below the solenoid, and a quadrupolar coil with a rectilinear aperture is seen as the flat array of windings to the right and left of the solenoid.
ANALYTICAL CHEMISTRY, VOL. 61, NO. 1, JANUARY 1, 1989 · 35 A
INSTRUMENTATION with an efficiency of 9.37 G/cm/amp (16). An attendant current puiser was capable of delivering more than 10 amps to the probe coils, with a rise time to peak gradient of just over 100 μ&. Figure 10 is a PR image of a 1-mm thick, 7-mm diameter disk of alumi num-filled cross-linked rubber recon-
Figure 10. An image of aluminum-filled rubber obtained by projection-recon struction methods. The diameter of the sample Is about 7 mm with a thickness of 1 mm. A gradient of 28 G/cm was used with mechanical rotation of the sample, as described in the text.
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 /ts and the total echo time TE less than 1 ms. The gradient amplitude was 28 G/cm, 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 distrib uted throughout the material. Al though the probe can perform both PR and SW experiments, it is equipped with only two coils and thus cannot perform slice selection experiments (slice selection can be accomplished in a PR experiment with a gradient paral lel to the goniometer axis). To further characterize the extent of these fea tures, we used a commercial SW imager equipped for slice selection. Figure 11 shows four spin warp im ages of the same aluminum-filled rub ber that was imaged by PR methods and displayed in Figure 10. These SW images were obtained with a commer cial 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 11. Spin warp images of aluminum-filled rubber. Each slice is 1 mm thick, and the four slices were taken from adjacent locations. The diagonal dimension of the square sample is 10 mm.
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sion of 10 mm. Slice selection was ac complished using a sine excitation pulse in the spin-echo pulse sequence shown in Figure 3. Because of the short T2 of this material, the selective rf pulse was limited to just 1 ms, and to retain the narrow slice width, the sine 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 ζ gradient used (4.7 G/ cm), the slice thickness was 1 mm. 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 T2 (0.3 ms) that was estimated from the line width. This indicates the existence of an inhomogeneous broadening that causes the apparent T2 to be shorter than the ac tual T2 (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 chemi cal shift dispersion, substantial loss in signal intensity still occurred. This loss was a result of T% decay during TE and made it necessary to signal average much longer than in the PR experi ment, where a much shorter TE was used. In all, 512 echoes were accumu lated for each row of the 128 X128 pixel 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 im ages show regions of low intensity. No such features were observed by the X-ray imager, which had been calibrat ed to be sensitive to as little as 1% vari ation in the X-ray absorption. Our in terpretation of the features shown in the NMR image is that the low-intensi ty regions correspond to low rubber concentration in areas where the alu minum powder was not depleted. These regions were formed before cur ing 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 ζ axis. Protons from the tape and
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are not excessively large (
