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J Magn Reson. Author manuscript; available in PMC 2010 July 6.

Published in final edited form as:

Published online 2007 August 29. doi: 10.1016/j.jmr.2007.08.015

PMCID: PMC2897241

NIHMSID: NIHMS208260

Vanderbilt University Medical Center, Institute of Imaging Science (VUIIS), AA-1105 MCN, Nashville, TN 37232-2310, USA

The publisher's final edited version of this article is available at J Magn Reson

The possibility of improving the signal-to-noise efficiency of NMR signal refocused by long-range dipolar interactions has been discussed recently [R. T. Branca, G. Galiana, W. S. Warren, Signal enhancement in CRAZED experiments, J. Magn. Reson. 187 (2007) 3843]. For systems where *T*_{1} *T*_{2}, by including an extra radio-frequency pulse in a standard CRAZED sequence, it is possible to increase the available signal by exploiting its sensitivity to *T*_{1} relaxation. Here, we use analytical calculations to investigate the source of this improved signal and determine the maximum enhancement provided by the method.

The NMR signal refocused in the presence of long-range dipolar interactions, also known as distant dipolar field (DDF) interactions, is related to the observation of inter-molecular multiple-quantum coherences (iMQC). This signal exhibits very unusual properties and has found applications in high resolution spectroscopy and imaging [1, 2]. A more general utilization of DDF-related effects is prevented by the poor signal-to-noise efficiency inherent to current methodologies. Several methods have been suggested recently for improving the signal-to-noise ratio in DDF-based sequences [3–5]. One of these methods [5] exploits signal sensitivity to *T*_{1} relaxation by applying an extra pulse in a standard CRAZED sequence (see Fig. 1). Apparently when the *T*_{1} spin–lattice relaxation occurs in a time-scale much longer than the *T*_{2} spin–spin relaxation, which is the case in several biologically important systems, the *T*_{2}-dependent transverse magnetization can be partially replaced by some of the *T*_{1}-dependent longitudinal magnetization in a “stimulated-echo-like” approach [6] thereby providing some enhancement in the observed signal. Simulation results and experiments [5] corroborate this idea but the detailed mechanism of this enhancement and its maximum value are yet to be determined.

Modified CRAZED pulse sequence where a third pulse θ was included in order to enhance DDF refocused signal. (The time-intervals are not in scale.)

Here we investigate this effect analytically by solving the modified Bloch–Torrey equations where the DDF contribution is included via a mean-field theory approximation. Assuming *T*_{2} relaxation as the principal attenuation mechanism, the effect of diffusion in the transverse magnetization as well as contributions from higher order terms in a Bessel function expansion can be neglected. With this assumptions we are able to obtain an approximate solution that accounts for the enhanced signal effect and allows for evaluating the maximum enhancement available from this method.

The detection of iMQCs in liquids is based on the idea that the existence of a modulation helix in the magnetization can break the angular-symmetry of the dipolar interaction and recover the long-range part of the motionally averaged dipolar field between spins in separate molecules [7]. This long-range dipolar field is responsible for converting the iMQCs into observable signal. The theoretical framework for evaluating the detected signal follows two conceptually different views: the quantum [8] and the classical approach [9], which have proven to give the same results at least for non-confining geometries [10]. The classical approach, where the effect of the dipolar field interactions are considered, via a mean-field theory, as a non-linear additional term in the Bloch–Torrey equations is commonly adopted since it offers a straightforward approach to quantitative results. In a frame rotating at the Larmor frequency of a single species of spin, with gyromagnetic ratio γ, and assuming that radio-frequency (RF) inhomogeneities and background gradients can be neglected

$$\begin{array}{l}\frac{\partial {M}^{+}}{\partial t}=-\mathrm{i}\gamma [{\displaystyle \mathbf{G}\cdot}{\displaystyle \mathbf{r}+{B}_{\mathit{\text{dz}}}(r,t)]{M}^{+}-\frac{{M}^{+}}{{T}_{2}}+D{\mathrm{\nabla}}^{2}{M}^{+},}\\ \frac{\partial {M}_{z}}{\partial t}=-\frac{{M}_{z}-{M}_{0}}{{T}_{1}}+D{\mathrm{\nabla}}^{2}{M}_{z},\end{array}$$

(1)

where the transverse magnetization density *M*^{+} *M _{x}* + i

Analytical expressions for *B _{dz}*(

Here we will be interested in the regime where the *T*_{2} spin–spin relaxation dominates the attenuation process. Under this regime it is safe to neglect any effect of diffusion on *M*^{+}. If diffusion attenuation is considered only in the *z*-magnetization component, the modified Bloch–Torrey equations can be solved analytically. After the application of the first (π/2)_{x} pulse and the gradient pulse of area *G*δ along a direction *ŝ* as described in Fig. 1, the magnetization density becomes *M*^{+}(*r*,δ) = i*M*_{0}exp(−i*k _{m}s*) and

$$\begin{array}{ll}{M}^{+}(r,{t}_{1}+\delta )=\hfill & \mathrm{i}\frac{{M}_{0}}{2}{\mathrm{e}}^{-({t}_{1}+\delta )/{T}_{2}}\times [(\text{cos}\beta +1){\mathrm{e}}^{-\mathrm{i}(n+1){k}_{m}s}+(\text{cos}\beta -1){\mathrm{e}}^{-\mathrm{i}(n-1){k}_{m}s}],\hfill \\ {M}_{z}(r,{t}_{1}+\delta )=\hfill & -{M}_{0}{\mathrm{e}}^{-({t}_{1}+\delta )/{T}_{2}}\text{sin}\beta \text{cos}{k}_{m}s.\hfill \end{array}$$

(2)

The magnetization along the *z*-direction generates the distant dipolar field that will refocus signal. The transverse magnetization that relaxes back to the *z*-direction during *t*_{1} is negligible if *T*_{1} *t*_{1} and will be discarded hereafter. During the interval Δ the signal evolves in the presence of the DDF as given below:

$$\begin{array}{ll}{M}^{+}(r,t1+\Delta )\hfill & ={M}^{+}(r,{t}_{1}){\mathrm{e}}^{-\Delta /{T}_{2}}\text{exp}(-\mathrm{i}\gamma {\displaystyle {\int}_{{t}_{1}}^{\Delta}}{B}_{dz}(r,t)\mathrm{d}t)\hfill \\ \hfill & ={M}^{+}(r,{t}_{1}){\mathrm{e}}^{-\Delta /{T}_{2}}\text{exp}(\mathrm{i}{\xi}_{1}(\Delta )\text{cos}{k}_{m}s)\hfill \\ \hfill & ={M}^{+}({t}_{1}){\mathrm{e}}^{-\Delta /{T}_{2}}{\displaystyle \sum {\mathrm{i}}^{l}{J}_{l}({\xi}_{1}(\Delta )){\mathrm{e}}^{\mathrm{i}l{k}_{m}s}}\hfill \\ \hfill & =\mathrm{i}\frac{{M}_{0}}{2}{\mathrm{e}}^{-({t}_{1}+\Delta )/{T}_{2}}\{{J}_{0}({\xi}_{1}(\Delta ))[(\text{cos}\beta +1){\mathrm{e}}^{-\mathrm{i}(n+1){k}_{m}s}+(\text{cos}\beta -1){\mathrm{e}}^{-\mathrm{i}(n-1){k}_{m}s}]+{\mathrm{i}}^{n+1}{J}_{n+1}({\xi}_{1}(\Delta ))(\text{cos}\beta +1)+{\mathrm{i}}^{n-1}{J}_{n-1}({\xi}_{1}(\Delta ))(\text{cos}\beta -1)+\cdots \}\hfill \end{array}$$

(3)

${\xi}_{1}(\Delta )=\frac{{\Delta}_{s}}{{\tau}_{d}}\frac{\text{sin}\beta}{D{k}_{m}^{2}+1/{T}_{1}}{\mathrm{e}}^{-{t}_{1}/{T}_{2}}(1-{\mathrm{e}}^{-(D{k}_{m}^{2}+1/{T}_{1})\Delta})$ and we have ignored terms in *B _{dz}*(

$$\begin{array}{l}{M}_{n\pm 2}^{+}(r,{t}_{1}+\Delta +{t}_{2}^{\prime})\\ \text{\hspace{1em}}=\mathrm{i}\frac{{M}_{0}}{2}{\mathrm{e}}^{-({t}_{1}+\Delta )/{T}_{2}}\{{J}_{0}({\xi}_{1}(\Delta ))(\text{cos}\beta \mp 1)\phantom{\rule{thinmathspace}{0ex}}\times \phantom{\rule{thinmathspace}{0ex}}[\text{cos}\theta \text{cos}{k}_{m}s-\text{sin}{k}_{m}s]+\mathrm{i}{J}_{1}({\xi}_{1}(\Delta ))(\text{cos}\beta \mp 1)\}-\mathrm{i}{M}_{0}{\mathrm{e}}^{-{t}_{1}/{T}_{2}}{\mathrm{e}}^{-{(D{k}_{m}^{2}+1/{T}_{1})}^{\Delta}}\text{sin}\beta \text{sin}\theta \text{cos}{k}_{m}s,\end{array}$$

(4)

$$\begin{array}{l}{M}_{z(n\pm 2)}(r,{t}_{1}+\Delta +{t}_{2}^{\prime})\\ \text{\hspace{1em}}=-{M}_{0}\left[{\mathrm{e}}^{-{t}_{1}/{T}_{2}}{\mathrm{e}}^{-(D{k}_{m}^{2}+1/{T}_{1})\Delta}\text{sin}\beta \text{cos}\theta +\frac{1}{2}{\mathrm{e}}^{-({t}_{1}+\Delta )/{T}_{2}}{J}_{0}({\xi}_{1}(\Delta ))(\text{cos}\beta \mp 1)\text{sin}\theta \right]\text{cos}{k}_{m}s,\end{array}$$

(5)

where ${t}_{2}^{\prime}$ is the instant immediately after the θ pulse. For the case *n* = 0:

$$\begin{array}{l}{M}_{n=0}^{+}(r,{t}_{1}+\Delta +{t}_{2}^{\prime})\\ \text{}=\frac{{M}_{0}}{2}{\mathrm{e}}^{-({t}_{1}+\Delta )/{T}_{2}}\{\mathrm{i}{J}_{0}({\xi}_{1}(\Delta ))[\text{cos}\beta \text{cos}\theta \text{cos}{k}_{m}s+\text{sin}{k}_{m}s]-{J}_{1}({\xi}_{1}(\Delta ))\text{cos}\beta \}-\mathrm{i}{M}_{0}{\mathrm{e}}^{-{t}_{1}/{T}_{2}}{\mathrm{e}}^{-(D{k}_{m}^{2}+1/{T}_{1})\Delta}\text{sin}\beta \text{sin}\theta \text{cos}{k}_{m}s,\end{array}$$

(6)

$$\begin{array}{l}{M}_{z(n=0)}(r,{t}_{1}+\Delta +{t}_{2}^{\prime})\\ \text{\hspace{1em}}=-{M}_{0}[{\mathrm{e}}^{-{t}_{1}/{T}_{2}}{\mathrm{e}}^{-(D{k}_{m}^{2}+1/{T}_{1})\Delta}\text{sin}\beta \text{cos}\theta +{\mathrm{e}}^{-({t}_{1}+\Delta )/{T}_{2}}{J}_{0}({\xi}_{1}(\Delta ))\text{cos}\beta \text{sin}\theta ]\text{cos}{k}_{m}s.\end{array}$$

(7)

During the *t*_{2} interval for either *n* = 0 or *n* = ±2 the DDF field, given, respectively, by Eq. (5) and Eq. (7), will refocus the magnetization described by Eq. (4) and Eq. (6). The process is similar to the one described in Eq. (3) and yields

$$\begin{array}{l}\overline{{M}_{n=0}^{+}({t}_{1}+\Delta \text{+}{t}_{2})}\\ \text{\hspace{1em}}=-{M}_{0}{\mathrm{e}}^{-({t}_{1}+{t}_{2})/{T}_{2}}\{{\mathrm{e}}^{-\Delta /{T}_{2}}[\text{cos}\beta \text{cos}\theta {J}_{0}({\xi}_{1}(\Delta )){J}_{1}({\xi}_{2}^{0}({t}_{2}))+\text{cos}\beta {J}_{1}({\xi}_{1}(\Delta )){J}_{0}({\xi}_{2}^{0}({t}_{2}))]+{\mathrm{e}}^{-(D{k}_{m}^{2}+1/{T}_{1})\Delta}\text{sin}\beta \text{sin}\theta {J}_{1}({\xi}_{2}^{0}({t}_{2}))\},\end{array}$$

(8)

$$\begin{array}{l}\overline{{M}_{n=+2}^{+}({t}_{1}+\Delta +{t}_{2})}\\ \text{}=-\frac{{M}_{0}}{2}{\mathrm{e}}^{-({t}_{1}+{t}_{2})/{T}_{2}}\{{\mathrm{e}}^{-\Delta /{T}_{2}}(\text{cos}\beta -1)[\text{cos}\theta {J}_{0}({\xi}_{1}(\Delta )){J}_{1}({\xi}_{2}^{+2}({t}_{2}))+{J}_{1}({\xi}_{1}(\Delta )){J}_{0}({\xi}_{2}^{+2}({t}_{2}))]+2{\mathrm{e}}^{-(D{k}_{m}^{2}+1/{T}_{1})\Delta}\text{sin}\beta \text{sin}\theta {J}_{1}({\xi}_{2}^{+2}({t}_{2}))\},\end{array}$$

(9)

$$\begin{array}{l}\overline{{M}_{n=-2}^{+}({t}_{1}+\Delta +{t}_{2})}\\ \text{}=-\frac{{M}_{0}}{2}{\mathrm{e}}^{-({t}_{1}+{t}_{2})/{T}_{2}}\{{\mathrm{e}}^{-\Delta /{T}_{2}}(\text{cos}\beta +1)[\text{cos}\theta {J}_{0}({\xi}_{1}(\Delta )){J}_{1}({\xi}_{2}^{-2}({t}_{2}))+{J}_{1}({\xi}_{1}(\Delta )){J}_{0}({\xi}_{2}^{-2}({t}_{2}))]+2{\mathrm{e}}^{-(D{k}_{m}^{2}+1/{T}_{1})\Delta}\text{sin}\beta \text{sin}\theta {J}_{1}({\xi}_{2}^{-2}({t}_{2}))\},\end{array}$$

(10)

where the overline means spatial average over the sample and the DDF contributions for *n* = 0 and *n* = ±2 are, respectively,

$$\begin{array}{ll}{\xi}_{2}^{0}({t}_{2})=\hfill & \frac{{\Delta}_{s}}{{\tau}_{d}}[{\mathrm{e}}^{-{t}_{1}/{T}_{2}}{\mathrm{e}}^{-(D{k}_{m}^{2}+1/{T}_{1})\Delta}\text{sin}\beta \text{cos}\theta \phantom{\rule{thinmathspace}{0ex}}+\text{}{\mathrm{e}}^{-({t}_{1}+\Delta )/{T}_{2}}{J}_{0}({\xi}_{1}(\Delta ))\text{cos}\beta \text{sin}\theta ]\phantom{\rule{thinmathspace}{0ex}}\times \phantom{\rule{thinmathspace}{0ex}}\frac{1}{D{k}_{m}^{2}+1/{T}_{1}}(1-{\mathrm{e}}^{-(D{k}_{m}^{2}+1/{T}_{1}){t}_{2}}),\hfill \end{array}$$

(11)

$$\begin{array}{ll}{\xi}_{2}^{\pm 2}({t}_{2})=\hfill & \frac{{\Delta}_{s}}{{\tau}_{d}}[{\mathrm{e}}^{-{t}_{1}/{T}_{2}}{\mathrm{e}}^{-(D{k}_{m}^{2}+1/{T}_{1})\Delta}\text{sin}\beta \text{cos}\theta \phantom{\rule{thinmathspace}{0ex}}+\frac{1}{2}\text{}{\mathrm{e}}^{-({t}_{1}+\Delta )/{T}_{2}}{J}_{0}({\xi}_{1}(\Delta ))(\mathrm{cos}\beta \mp 1)\mathrm{sin}\theta ]\text{}\times \text{}\frac{1}{D{k}_{m}^{2}+1/{T}_{1}}(1-{\mathrm{e}}^{-(D{k}_{m}^{2}+1/{T}_{1}){t}_{2}}).\hfill \end{array}$$

(12)

The parameters τ_{d} = 240 ms, *T*_{2} = 50 ms and *T*_{1} = 1 s were utilized in all calculations and the gradients were considered parallel (*ŝ*||) with the *B*_{0} field (Δ_{s} = 1) in order to provide a comparison with previous simulated results from the literature [5].

Fig. 2 shows curves where the absolute value of the analytical solution of the modified Bloch-equations for the cases *n* = 0 (see Eq. (8)) and *n* = +2 (see Eq. (9)) are plotted as a function of the interval *t*_{2} (see Fig. 1). Curves were plotted for values of the Δ interval varying from 10 to 50 ms for the case of no diffusion. The maximum signal is obtained with Δ = 40 ms using the parameters given in the caption. In both cases the maximum signal-peak obtained is higher than that obtained by the maximum signal available by conventional CRAZED sequence represented by the dashed lines in each panel [11]. The maximum signal for the cases *n* = 0 and *n* = +2, when compared to the corresponding standard CRAZED maximum signal, show an enhancement of 15% and 10%, respectively. Furthermore, in a conventional CRAZED sequence, in the regime where *T*_{2} relaxation is the dominant attenuation process, the maximum refocused signal occurs at Δ + *t*_{2} = *T*_{2} which is 50 ms in the case described. As such, one would expect the maximum signal for the modified sequence to be given by a similar relation. In fact, the results indicate that the maximum signal does not necessarily occurs at that point. For instance, for the case of *T*_{2} = 50 ms, the maximum signal for the modified sequence was obtained at Δ + *t*_{2} ≈ 60 ms.

In Fig. 3 we consider the effects of diffusion in the longitudinal magnetization and the case with $1/D{k}_{m}^{2}=464\text{ms}$ is presented. Again, the dashed lines indicate the result from the conventional CRAZED sequence. The effect of diffusion attenuation as seen in the analytical results can be understood as an effective *T*_{1} relaxation. So, if $(1/{T}_{1}+D{k}_{m}^{2})$ increases approaching 1/*T*_{2} the enhancement is reduced which indicates that the magnetization that evolved in the *z*-direction during Δ, given by the last term in Eq. (7)–Eq. (9), is responsible for the improved signal.

Fig. 4 shows maps of the absolute signal-amplitude for the case *n* = 0 as a function of the pulse-angles β and θ for different values of the Δ interval. The maximum signal can then be calculated and, for Δ = 30 ms, is up to 15% higher than the original CRAZED-sequence signal. Apart from the overall change in the maps appearance as a function of Δ, the areas where the signal reaches a maximum are rather stable.

Fig. 5 shows maps of the absolute signal-amplitude for the case *n* = +2 in the same way as described for Fig. 4. The maximum signal is up to 10% higher than the original CRAZED-sequence signal for Δ = 40 ms. Again, besides the overall change in the maps appearance as a function of Δ, the areas where the signal reaches a maximum are rather stable. The case for *n* = −2 is not shown but overall produces enhancement similar to the *n* = +2 except for the differences in the combinations of β and θ angles.

Analytical calculations for the distant dipolar field refo-cused signal in a modified CRAZED sequence have been presented. For systems where *T*_{2} relaxation is the main source of signal attenuation, diffusion effects can be neglected in the transverse magnetization. This allows for a closed-form analytical solution based on the Bloch–Torrey equations modified to include the distant dipolar field contribution and considering the role of diffusion on the longitudinal magnetization. This is a good approximation if $1/D{k}_{m}^{2}\gg {T}_{2}$. For more general regimes analytic solutions including diffusion turn out to be complicated [12]. The solution presented here, valid if (*t*_{2} + Δ) ≈ *T*_{2} τ_{d}, neglects terms coming from combinations of Bessel function contributions of added order higher than one.

Given some specific sample parameters, a maximum signal enhancement up to 15% of the signal available from a conventional CRAZED sequence was obtained for the sequence depicted in Fig. 1 with either *n* =0 or *n* = ±2. Similar enhancement was found (data not shown) for other parameter data-sets in the regime *T*_{1} τ_{d} > *T*_{2}. The results generally agree with the numerical data given in reference [5] with a few minor discrepancies [13]. We have not considered effects of the background gradient generated by the polarizing *B*_{0} field. Thus, to reproduce experimentally the results calculated here, 180° refocusing pulses have to be symmetrically inserted within each time-interval of the sequence in Fig. 1. The imaging case has to be taken carefully since encoding gradients during the *t*_{1} interval affect the DDF. Thus it is wise to perform the imaging encoding during the time immediately before acquiring the signal where the DDF is pointing along the polarizing field and will not be affected by these gradients.

Phase cycling strategies for cancelling out potential spurious signal were not investigated. Along these lines, there might be some concern that the signal generated with the modified CRAZED sequence could originate from single-quantum coherence pathways [14]. However, if the effect of the DDF is neglected there is no observable signal which indicates that the signal originates from dipolar interactions. Nevertheless, it should be mentioned that similar modifications in standard CRAZED sequences have proved recently to be more robust to pulse angle imperfections [15,16].

This modified CRAZED sequence is intended to benefit from *T*_{1}-sensitive signal contribution for systems where *T*_{1} *T*_{2} with negligible effect of diffusion. The results show that diffusion effects may minimize the enhancement available even if *T*_{1} *T*_{2}. The small increase in the signal, up to 15% in the best scenario, contrasts with the much higher signal enhancement available in standard single-quantum methods. For instance, in a Hahn-stimulated-echo experiment, the signal is attenuated exclusively by *T*_{1} relaxation. This is possible because the longitudinal signal is immediately available for observation when flipped into the transverse plane which mitigates the effect of *T*_{2} relaxation. On the other hand, for the DDF signal, although the longitudinal part of the magnetization relaxes under *T*_{1} effect during the Δ interval, after the θ pulse (see Fig. 1) this dephased magnetization needs some time (in the order of τ_{d}) in the transverse plane to build up in the presence of the DDF, thereby suffering from *T*_{2} attenuation during this interval. One might think that going to higher field, where τ_{d} gets shorter, could alleviate this issue; however, at higher fields the signal-peak will appear in a time-scale independent of any relaxation time which makes the use of relaxation-dependent magnetization to improve signal meaningless.

Finally, the goal of this contribution was, via analytical calculations, to investigate the origin of the enhanced signal when an additional pulse is included in a standard CRAZED sequence and evaluate the maximum enhancement possible for this type of sequence implementation. Unfortunately, the way the NMR signal builds up in the presence of the DDF prevents an appreciable signal enhancement based on the more favorable *T*_{1} sensitivity. Nevertheless, it should be pointed out that these conclusions do not rule out the possibility of using this method to obtain contrast enhancement in some properly designed experiment.

W. Barros thanks Dr. R.T. Branca for helpful clarifications about their work. This work was supported by NIH Grants No. R21 EB04040 and RO1 EB000214.

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13.
For the case *n* = ±2 there was some discrepancy in the maximum enhancement obtained here and that presented in reference [5]. Examining the conventional CRAZED maximum signal for the case (*n* = +2, β = 120°, θ = 0°) and (*n* = 0, β = 45°, θ = 0°) represented, respectively, by the dotted lines in Fig. 8 and Fig. 6 of reference [5], it was noticed that standard theoretical predictions for the ratio between these maxima was not respected which made a reasonable comparison with the analytical solution difficult. Furthermore, in the analytic results presented here, no signal enhancement was detected for *n* = ±2 if the second gradient was inserted after the θ pulse.

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