Dipolar couplings between distant spins have been shown experimentally to produce unexpected signals in a wide variety of MRI and solution NMR experiments (1, 2). The prototype sequence is the CRAZED sequence: 90x − {delay τ} − {gradient pulse, area GT} − Θx − {gradient pulse, area nGT} − {delay nτ + TE} (3). In the conventional picture of magnetic resonance, this sequence is expected to produce no signal for n ≠ ±1. Instead, it produces signals in two-dimensional NMR experiments with all the experimental properties of intermolecular multiple-quantum coherences (iMQCs). Recent applications include contrast enhancement in magnetic resonance imaging (3–5) and functional imaging (6, 7), suppression of inhomogeneous broadening (8). Most of these applications have used intermolecular zero-quantum coherences (n = 0, iZQCs) or double-quantum coherences (n = ±2, iDQCs), which provide the largest signals.

These signals can be understood in two different frameworks (9). The dipolar field can be reduced to an ensemble-averaged magnetic field correction at each spin, called the DDF (dipolar demagnetizing field (10) or distant-dipole field (11)). This mean-field treatment was first derived to explain multiple echoes in continuous gradients (10). In this framework the CRAZED signal arises from nonlinear spin evolution (1, 2, 10). Alternatively, the coupled-spin treatment retains the dipolar couplings explicitly. In this framework the signal is predicted to come from iMQCs (8, 11) involving spins with a macroscopic separation, called the correlation distanced_{c}, which is controlled by the area of the gradient pulses (d_{c} = π/γGT is typically in the range 10–1000 μm). At one time the relationship between these two frameworks was controversial. It has become clear that the two treatments make equivalent predictions for the CRAZED sequence (12), but the coupled-spin picture offers the most intuitive account of this experiment.

It is found that tuning of the correlation distance in a heterogeneous material can alter the image contrast in a manner that depends on the microstructure. Both theoretical predictions and experimental results are presented for water phantoms containing packed arrays of parallel hollow structures. This article is the first to present a quantitative verification of the expected changes in observable magnetization in a three-dimensional heterogeneous material, with good agreement between theory and experiments.

THEORY

The prototype imaging sequence is sketched in Fig. 1. Previously published spectroscopic (1), rat brain (3), and human MR experiments (13, 14) showed that such a sequence detects iMQCs. The signal (typically 5–20% of the conventional water magnetization) arises from traditionally neglected, multiple-spin operators in the equilibrium density operator (11, 15):

(1)

where ω is the resonance frequency, T_{s} is the sample temperature, and ℋ is the Hamiltonian for the Zeeman interaction. The first RF pulse transforms bilinear terms (II) in Formula 1 into two-spin operators II which contain both zero-quantum and double-quantum terms (16). The sums run over all of the spins in a sample, so these terms involve pairs of spins separated by various distances. These coherences evolve freely during the time period τ. iZQCs evolve at the difference in resonance frequencies between pairs of spins, whereas iDQCs evolve at the sum of resonance frequencies (16).

The first correlation gradient pulse winds the transverse magnetization into a helix along its axis. The second RF pulse partially transforms iZQC and iDQC terms into two-spin single quantum operators (such as II). The pulse flip angle Θ that maximizes the signal is π/4 or 3π/4 for iZQC imaging, 2π/3 for iDQC imaging with n = +2, and π/3 for iDQC imaging with n = −2 (opposed gradients). As a result of this pulse, the longitudinal component of the magnetization is also modulated along the axis of the gradient, giving rise to a spatially modulated resonance frequency. The second gradient pulse in iDQC imaging acts as a filter which passes observable single quantum coherences originated in intermolecular double quantum coherences but blocks all other coherences from proceeding any further. The area of this gradient is twice that of the first because a phase accumulated during the τ period will have evolved at the sum of resonance frequencies, i.e., approximately twice the Larmor frequency.

Evolution during the final delay generates observable signal from dipolar couplings. The dipolar Hamiltonian contains terms such as D_{ij}II; the dipolar coupling D_{ij} equals (3 cos^{2}θ_{ij} − 1)/r, where r_{ij} is the separation between the two spins and θ_{ij} is the angle the internuclear vector makes with the Zeeman field. The time evolution of the density operator (dρ/dt = (i/ℏ) [ρ, H]) then includes terms such as i[II, II] = −I/4, thus it generates an observable signal. The echo forms naturally after a delay nτ (17). To enhance the signal, a delay time (TE) is applied to allow for evolution II → I under the dipolar Hamiltonian, which is very slow (D_{ij} is very small in frequency units). In this limit, the I term grows linearly while the transverse magnetization decays exponentially. The optimal signal is observed at TE ≈ T_{2} (3). In Fig. 1 the delay nτ can be negative (case n = −2), i.e., the delay TE/2 between acquisition of the center of k-space and the third pulse is longer than the delay between the second and third pulses. For n = +2 the reverse holds.

The dipolar couplings can be positive or negative and average to zero over a spherical surface with uniform magnetization. Thus, spherical symmetry must be broken for the couplings to produce a net signal. Furthermore, this symmetry breaking cannot be random: it has to alter the angular dependence (3 cos^{2}θ − 1) to make a nonzero spatial average. This is done by applying a gradient filter. The combination of gradient pulses modulates the magnetization to give a correlation distance d_{c} = π/γGT, thus letting the sum of the dipolar interactions produce a net nonzero effect over that distance. For some proton of interest (index j), the signal intensity at its location for short evolution times can be shown to be proportional to the dipolar field strength (15) Σ_{i}r (3 cos^{2}θ_{ij} − 1) cos(γGTs_{ij}) where r_{ij} is the Euclidean distance between the jth and ith protons, s_{ij} is the component of r_{ij} along the applied gradient direction ŝ (s = · ŝ), θ_{ij} is the angle between r_{ij} and the Zeeman field's direction, ẑ. The sum runs over all spins in the sample, so that the magnetization density is implicitly contained in the summation index. For uniform magnetization density, Warren et al. (15) have shown that the bulk of the signal comes from spins located at the correlation distance d_{c}. For the imaging signal in a voxel, the same formula applies but the ensemble average 〈I_{x}〉 = tr{I_{x}ρ} must be calculated over a voxel volume, where ρ is the reduced density operator, I_{x} = Σ_{i}I and the index i runs over all the spins belonging to the voxel. If the imaging signal is proportional to the dipolar field, this amounts to averaging all dipolar sums that originate from the voxel. At best, a qualitative picture can be obtained because the nonlinear evolution is completely ignored. Quantitative predictions of the imaging signal can be obtained by direct integration of the Bloch equations (18, 19).

As an illustrative example, let us consider a unit 3-cell I^{3} = [0, 1] × [0, 1] × [0, 1] partitioned into 64 × 64 × 64 cuboids of equal volume and let the magnetization be modulated sinusoidally along the cylinder axis (z). To each cuboid we assign a magnetization density of either 0 (cylinder wall) or 1 (pure water). Figure 2a shows a binary map of magnetization density obtained by thresholding a proton-density weighted MR image. This map was replicated to create a third dimension (slice thickness), and the resulting 3D matrix was used to simulate an array of parallel cylinders. Figure 2b–d are maps of the magnitude of the transverse magnetization sampled at the center of the spin echo and summed along z, the slice thickness, for an iDQC pulse sequence (τ = 15 ms, TE = 500 ms, 170 MHz proton Larmor frequency, T_{1} = 4 sec, T_{2} = 2 sec, over a 1-cm^{3} region of 48 × 48 × 48 cuboids located inside a 64 × 64 × 64 matrix, thus allowing 8 pixels of zero-padding on each side of the cubic sample) with different choices of correlation gradient strengths: (Fig. 2b) 6 cycles per sample side length, (Fig. 2c) 3 cycles and (Fig. 2d) 2 cycles. It is apparent that, with a helix pitch of 6 cycles (Fig. 2b), the signal loss occurs primarily in the smaller gaps between the cylinders. Important signal losses are also observed very near the cylinder walls, giving the illusion of an apparent thickening of the walls. At the longest correlation distance (Fig. 2d), the signal loss between the cylinders is no longer as important. What is more noticeable is the signal loss at the edges of the sample, due to the absence of magnetization in the nearby regions. Figure 2c is shown to illustrate that this transition is gradual. In Fig. 2f is a 128 × 128 × 64 (x,y,z) simulation for the geometry shown in Fig. 2e. The FOV is twice as small as Fig. 2a and the helix pitch is 8 cycles across the side length of the image plane shown. This produces a correlation distance d_{c}, which is 2.7 times shorter than that of image 2b. It is readily seen that, at this shorter correlation distance, the signal in the gaps is much stronger than in 2b. (This helix pitch is now smaller than the short axis of the quasi-triangular threefold intersections between the cylinders.) The signal losses occur primarily at the edge of the cylinder walls. We conclude from these simulations that the expected image contrast depends strongly on the choice of correlation distance in relation to the sample structure.

The simulated cubic region can be thought of as an imaging voxel containing cylindrical structures. In this case, the measured signal is an integral over this voxel. In relative units, with 1.00 being the largest value, the cubic samples of Fig. 2b–d give a total signal (magnitude of the total transverse component) of 1.00, 0.85, and 0.75, respectively. This trend, however, is mainly due to the important signal losses at the edges of the cubic sample. If we restrict the summation far away from the edges, say, to the middle cube of side length 1/3 smaller than the cubic sample, we get: 0.68, 1.00, and 0.89, respectively. Thus the tuning of the correlation distance modulates the signal intensity for this two-dimensional geometry. Other geometries can be expected to exhibit a quite different dependence on correlation distance. The experimental results presented later are spatially resolved and, therefore, do not represent the integrated transverse magnetization over cylinders.

These 3D simulations were carried out on a 750 MHz dual processor Hewlett-Packard workstation (HP-UX 9000/785/J6700 with 10 GB RAM, Hewlett-Packard Co., Palo Alto, CA) in less than an hour each, using in-house software described elsewhere (18, 19). The time evolution of the Bloch equations is calculated using a Runge-Kutta algorithm while the DDF is computed directly in k-space by contraction of the local magnetization vector field M(k) with the dyadic tensor (μ_{0}/6)(3(k̂ · ẑ)^{2} − 1)(−k̂_{x}k̂_{x} − k̂_{y}k̂_{y} + 2k̂_{z}k̂_{z}) (10). If M(r) varies only along a single direction, say, ŝ, the DDF in real space is local: (μ_{0}/2)(3(ŝ · ẑ)^{2} − 1) (M_{z}(s)ẑ − M(s)/3). This expression, although easier to calculate, completely neglects those Fourier components along directions other than ŝ. It is imperative to include these components if the material contains heterogeneities on a length scale comparable to the helix pitch. This is certainly the case for the experiments presented herein, and virtually any other experiment, where the dependence on microstructure and correlation distance is investigated in a highly structured sample. The importance of 3D simulations over the 1D case for structured samples has been stressed by Enss et al. (18).

In theory, it is possible to predict the signal intensities as a function of microstructure for a given choice of correlation distance. Information about the microstructure contained in a volume can be obtained by tuning the correlation distance, even if the volume or voxel is larger than the heterogeneities. For example, bulk signal measurements have been shown (Capuani et al. (5)) to be sensitive to trabecular bone structure. It is likely that localized iMQC spectroscopy may be useful in characterizing materials and tissue structure in vivo.

In real applications, the structure of the sample will be observable even in the limit of short evolution times, provided the magnetization density varies sufficiently over the correlation distance. Variations in the evolution frequency or relaxation rates will alter the magnetization distribution, but only at longer evolution times. Finally, transport phenomena such as flow or diffusion are expected to spoil the periodic magnetization pattern and to alter the signal. These effects including nonlinear evolution can all be accounted for by directly solving the full Bloch equations.

METHODS

All experiments were done on a 4 T whole-body magnet (GE Signa Echospeed 5.8, GE Medical Systems, Waukesha, WI) at the University of Pennsylvania Medical Center with a birdcage quadrature head coil. The phantoms were positioned as close as possible to the center of the coil. The pulse sequence in Fig. 1 was used to generate iDQC images by applying a correlation gradient along the z axis, parallel to the cylinder axes. The preparation period is followed by a phase and frequency encoded single-line acquisition of a spin echo. Although this sequence may be used with multiple refocusing pulses to improve SNR, a single echo was used in this study. Two-step acquisitions with phase reversal of the first RF pulse provided suppression of single quantum contamination (3). A four-steps RF phase cycling scheme provides an even better selection of multiple-quantum coherence pathways (16).

Heterogeneous phantoms consisted of a plastic bottle filled with plastic straws of various sizes (all measurements ±0.5mm): round straws with 3.6 mm (phantom A) and 2.7 mm (phantom B) diameters, and coffee stirrers (phantom C) with cross sectional area shaped in a figure eight: 1.1 mm (1.7 mm, resp.) inner diam. for the small axis (large axis, resp.); 2.0 mm (3.0 mm, resp.) outer diam. for the small axis (large axis, resp.). The 2.7 mm straws were put in a 15 cm long, 7 cm diam. cylindrical neoprene bottle (phantom B), while the phantoms A and C were made using 12 cm long, 2.7 cm diam. cylindrical plastic bottles. Water was added to fill all the spaces inside and outside the straws and care was taken in keeping air bubbles to a minimum. Ten-millimeter thick axial slices of the phantoms were acquired with the axis of the bottle parallel to the physical z-axis of the magnet.

Prior to each experiment, we verified that all cylinders were parallel to the z-axis by inspection of high-resolution images over the entire volume of the phantom. For all phantom images we used: TR = 6 sec, TE = 500 ms, τ = 15 ms, first correlation gradient pulse = 8 ms × G (G was varied over the range 0.14–2.2 G/cm), FOV = 8 cm for phantom B and 4 cm for phantoms A and C, 10 mm thick, matrix = 256 × 256, two acquisitions co-added with phase reversal of the first RF pulse. All images of the same run (with different values of the correlation gradient strength) were acquired using the same RF transmitter, receiver, and shim settings. Independent control experiments on a uniform water phantom over a similar period yielded a stable behavior when varying the correlation distance. The experiments were repeated over a 6 month period with consistent results.

Proton spectra of phantom B were acquired using a modified gradient echo sequence with all imaging gradients removed except for slice selection. The sampling rate was 4 kHz, number of readout points = 512, number of averages = 32, axial slice thickness = 10 mm. Diffusion weighted spin echo (control) images of the phantom were acquired using a modified spin echo sequence with two identical trapezoid z-gradient pulses, placed symmetrically around the π pulse, separated by an interval Δ = 480 ms (center of the first gradient pulse to the second).

RESULTS AND DISCUSSION

Single quantum coherence (SQC) control experiments were done on phantom B to assess the effects of long evolution times (T) on image contrast. In Fig. 3, a comparison of gradient-recalled echo images at short (10 ms) and long (200 ms) evolution times shows no difference, indicating that resonance frequency is essentially uniform everywhere. This is corroborated by 1D spectra (Fig. 4) for the same phantom at the same axial slice, with and without cylinders. The proton linewidths are nearly identical at about 20 Hz. Estimated four points saturation-recovery T_{1} and four-echo T_{2} maps (not shown) did not show any unusual heterogeneity over the water-filled regions, apart from slight partial volume effects near the cylinder walls.

Conventional spin echo images with and without diffusion weighting gradients (2.2 G/cm × 11 ms, Δ = 480 ms) are shown in Fig. 3. Apart from a near total loss of signal at strong diffusion weightings, the image contrast of the spin-warp acquisition is unaltered by diffusion gradients and remains uniform over the water-filled regions. Flow weighting is not likely to be a contributor to the image contrast, for had flow been a problem, this image would also have been affected because only the zeroth gradient moment is nulled.

Phantoms A and B were imaged at 4 cm and 8 cm fields of view, respectively. Close-up regions of these images are shown in Fig. 5. In (Fig. 5a,d) are conventional SQC spin echo images for phantoms A and B, respectively, while 5b,c and 5e,f are from iDQC z-gradient images for the corresponding regions. In Fig. 5b (d_{c} = 67 nm), the contrast is very similar to SQC (5a) except for a slight apparent thickening of the cylinder walls. The effect is more pronounced at d_{c} = 180 μm (5c), where not only the walls appear thicker but there is additional signal loss in the gaps between the cylinders. This signal loss occurs primarily in the smaller gaps and few changes are seen in the larger gaps. It is seen that in some gaps the signal loss easily reaches 20%. Some signal losses are also seen inside the cylinders, near the walls.

In Fig. 5e,f we observe a similar effect for phantom B. The signal loss in the gaps for d_{c} = 1077 μm (5f) reaches as much as 30% compared to the SQC (5d) and short d_{c} (5e) cases. Again, the short correlation distance (d_{c} = 67 μm) image (5e) has contrast similar to the SQC image (5d). All images were normalized to their maximum intensity to show the relative changes.

In Fig. 6 we compare theoretical and experimental measurements for phantom C. Image 6a is a conventional SQC spin echo image and 6b,c are iDQC z-gradient images with d_{c} = 67 μm and d_{c} = 1077 μm, respectively. Again, an important loss of signal in and around the straws is seen at the longer correlation distance (6c). The image contrast in (6c) is clearly different from the SQC contrast (6a) and the short d_{c} case (6b). A proton density weighted image was thresholded to produce a binary map of the water distributions (6d). Sixty-four replicas of this 2D map were stacked vertically to provide a third dimension (slice thickness), and the resulting 64 × 64 × 64 matrix with zeroed edges (8 pixels) was used to simulate the expected iDQC signal with the same parameters (TE = 500 ms, τ = 15 ms, 170 MHz proton Larmor frequency). The proton relaxation times were taken to be uniform over the water region: T_{1} = 4 sec, T_{2} = 2 sec. The resulting projection of the transverse magnetization along the slice thickness (Fig. 6f) is in excellent agreement with the experimental image (6c) for the same choice of correlation distance. In Fig. 6e, a SQC sequence produces a map that is similar to the proton density distribution, as expected for conventional SQC images because their evolution depends on the distant dipolar field only weakly.

It is worth noting for all these measurements that, even though the cylinder walls may have a proton density comparable to the surrounding water, it is the magnetization density that matters. For T_{2} (plastic) ≪ T_{2} (water), as is the case for most plastics, the magnetization density available for transfer along the z axis by the second pulse is effectively zero after almost any selective excitation pulse.

An advantage of iMQCs in microscopy studies arises when the conventional imaging resolution does not allow resolution of heterogeneities smaller than the voxel size. Selection of the correlation distance represents an additional degree of freedom that may help detect smaller structures. The direction of the correlation gradient may also be used to probe anisotropy along various directions. Complications may arise, however, if imaging gradients are used to spatially resolve the signal. In small field of view studies, where the imaging gradients are relatively strong, a cancellation of the correlation gradient can occur if the latter is too weak. This causes a gradient-recall of the SQC echoes, and the effect is most easily seen in k-space. Figure 7a shows an iDQC image with correlation gradient applied along the phase-encode direction. A strong SQC echo is refocused by the phase-encode gradient (7a), resulting in a spatial modulation along the phase encode direction in the reconstructed image (7b). Cycling the phase of the first RF pulse between 0 and 180° cancels most of the contamination (7c) but not completely. The reconstructed image (7d) contains residual modulations.

Correction of the image artifact by zeroing out the erratic region directly in k-space is not advisable since it is hard to assess the full extent of the contaminated region. Precise cancellation of the correlation gradient by the imaging gradient is not absolutely necessary for contamination to occur, as residual signal can leak due to the possibility of having a noninteger number of helix cycles across the imaging region. The implication is that, for highly structured samples, a correlation gradient along the phase-encode or frequency-encode directions should only be applied with great care.

The spurious signal can be removed by RF phase cycling (for example, double-quantum filtering by phase cycling the first pulse) but only if the magnetization is sufficiently small that other nonlinear effects, such as radiation damping (20), do not perturb the time evolution from the refocused magnetization. It should also be possible to circumvent the problem by storing the peak of the echo along the z direction, applying crusher gradients and a long delay to attenuate modulated magnetization components, and detecting with a FLASH readout.

CONCLUSION

The present article aims at demonstrating the unique feature of iMQC contrast based on distance selectivity in a structured sample. It was verified theoretically and experimentally that the image contrast in a material containing differences in magnetization density depends on the choice of dipolar correlation distance. In particular, the tuning of the correlation distance to the longer values relative to the gap size between the cylinders in the phantom produces image contrast that is not seen in conventional SQC images. This contrast allows microscopic structures to be detected; it is thus unique and there exists no analog in conventional single quantum imaging. It is anticipated that this technique can serve as a tool for probing structure in a variety of heterogeneous materials, with applications to biomedical and materials imaging (4, 19, 21).