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Magn Reson Med. Author manuscript; available in PMC 2010 May 27.

Published in final edited form as:

PMCID: PMC2877388

NIHMSID: NIHMS193660

Walter R.T. Witschey,^{1,}^{*} Arijitt Borthakur,^{1,}^{2} Mark A. Elliott,^{1,}^{2} Eric Mellon,^{1,}^{2} Sampreet Niyogi,^{2,}^{3} Chenyang Wang,^{2,}^{3} and Ravinder Reddy^{1,}^{2}

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

See other articles in PMC that cite the published article.

The origin of image artifacts in an off-resonance spin-locking experiment is shown to be imperfections in the excitation flip angle. A pulse sequence for off-resonance spin locking is implemented that compensates for imperfections in the excitation flip angle through an off-resonance rotary echo. The off-resonance rotary echo alternates the frequency offset and phase of the RF transmitter during two spin-locking pulses of equal duration. The underlying theory is detailed, and MR images demonstrate the effectiveness of the technique in agarose gel phantoms and in in vivo human brain at 3T.

The different magnetic relaxation properties of tissues make magnetic resonance imaging (MRI) one of the most clinically useful imaging modalities. Relaxation-dependent contrast is inherent to conventional *T*_{1}- and *T*_{2}-weighted images as well as the more recent steady-state free precession (SSFP) *T*_{1}/*T*_{2}-weighted images. Since relaxation times vary among healthy and diseased tissues, these techniques can be used to distinguish the diseased tissues in MR images. In addition to the widely employed *T*_{1} and *T*_{2} relaxation contrast techniques, a small but increasing number of studies are devoted to examining *T*_{1}ρ relaxation contrast, in which tissue magnetization is “locked” by an on-resonance RF field. Redfield (1) showed that the *T*_{1}ρ relaxation time characterizes the spin-lattice relaxation in the rotating frame. In most cases, as the amplitude of the locking RF field approaches zero, *T*_{1}ρ- weighted imaging is functionally equivalent to *T*_{2}- weighted imaging, since both generate tissue contrast based on the disappearance of transverse magnetization.

In practice, *T*_{1}ρ-weighted imaging offers several advantages over *T*_{1}- and *T*_{2}-weighted imaging. For example, *T*_{1}ρ-weighted imaging is sensitive to the slow motional processes in the 0.5–3 kHz range, such as proton exchange with amides and hydroxyl groups in proteins and residual dipole–dipole interactions (2,3). To mention a few applications, *T*_{1}ρ-weighted imaging has shown improved contrast between healthy brain or breast tissues and tumors in mice (4) and humans (5,6), tracked the degeneration of the patellar cartilage matrix (7,8), is sensitive to posttraumatic cartilage injury (9) and has shown sensitivity to degradation of the nucleus pulposus in studies of degenerative intervertebral disc disease (10,11) and enabled the indirect detection of metabolic ${\mathrm{H}}_{2}^{17}\mathrm{O}$ in vivo (12,13). This approach presents some difficulties compared to traditional relaxation contrast techniques, however. Long-duration RF pulses may cause coil damage, the specific absorption rate (SAR) delivered to tissues during a *T*_{1}ρ sequence is high, and both *B*_{0} and *B*_{1} imperfections are significant sources of artifacts.

To overcome three of these difficulties (coil damage, high SAR, and *B*_{1} imperfections), we developed a *T*_{1}ρ_{off} pulse sequence that implements an off-resonance “rotary echo” during spin locking. An off-resonance spin lock reduces the spin-lock RF field strength ω_{1} by increasing the off-resonance component Δω_{0} to achieve the same effective field strength, and thus reduces the SAR and ω_{1} power demands on the RF coil. A rotary echo reduces artifacts due to *B*_{1} inhomogeneity. *T*_{1}ρ_{off} is an intrinsically different relaxation time from *T*_{1}ρ, and instead combines both *T*_{1}ρ- and *T*_{1}-type contrast. In particular, when an off-resonance RF pulse is delivered far from resonance (Δω_{0} ω_{1}), *T*_{1}ρ_{off} approaches *T*_{1} and on-resonance *T*_{1}ρ_{off} becomes *T*_{1}ρ. Here we generalize the rotary echo to the off-resonance case using product operator theory and show that it reduces imperfect *B*_{1} MR image artifacts during off-resonance spin locking. In previous studies, off-resonance spin locking showed sensitivity to acute rat cerebral ischemia for Δω_{RF} < 2.5 kHz (14), and was used to obtain single-shot measurements of *T*_{1}ρ_{off} in the human breast (15). Despite the decrease in sensitivity to tissue pathology that occurs with decreasing ω_{1}/ωΔ_{RF} (16), off-resonance spin locking may be useful for situations in which Δω_{0} inhomogeneity artifacts prohibit an on-resonance spin lock, particularly at higher *B*_{0} field strengths.

The geometry of the system during an off-resonance spin lock in the rotating frame is shown in Fig. 1. Magnetization is flipped parallel to the effective field, such that the flip angle α = ϕ, where the effective field ω_{eff} makes an angle with the *z*-axis:

$$\mathrm{\phi}={\text{tan}}^{-1}\left(\frac{{\mathrm{\omega}}_{1}}{\mathrm{\Delta}\mathrm{\omega}}\right).$$

[1]

In the on-resonance case, ω_{eff} = ω_{1} and ϕ = 90°.

Off-resonance spin-locking geometry. The effective field ω_{eff} makes an angle = tan^{−1}(ω_{1}/Δω) with the *z*-axis. To perform off-resonance spin locking, the magnetization must be flipped parallel to the effective **...**

Solomon (17) first introduced the rotary echo to correct for imperfections in the RF field. On-resonance, spins accumulate a phase ω_{1}τ because of nutation due to the RF field. In particular, an inhomogeneous *B*_{1} field will cause the spin phase to vary throughout the sample and cause a decay of the net magnetization throughout the sample. Solomon (17) realized that if a second *B*_{1} pulse that is 180° out of phase with the first is applied for the same duration, the spins will accumulate the exact opposite phase ω_{1}τ and an echo will be formed at 2τ. Sears (18,19) extended the rotary echo to off-resonance spins to remove the dipolar broadening in solid CFCl_{3}. Further, Rhim et al. (20) used an off-resonance rotary echo to compensate for inhomogeneity in the RF field in time reversal experiments on dipolar coupled spins. Notably, the current implementation is used not only to compensate for the loss of coherence during the RF pulse, but also to correct for nutations about the effective field as a result of an imperfect excitation. Indeed, as we will show in Eq. [13], the origin of spin-lock artifacts is nutation about the effective field rather than the loss of phase coherence, which may shorten only the apparent relaxation time during the applied RF pulse.

More recently, several *T*_{1}ρ-weighted pulse sequences with artifact correction have been implemented in MRI (see Table 1). Charagundla et al. (21) eliminated artifacts from both *B*_{1} imperfections and flip angles α ≠ 90° (usually also the result of an imperfect *B*_{1}) in *T*_{1}ρ-weighted imaging. By reducing these artifacts, it became possible to measure the spatial distribution of *T*_{1}ρ relaxation times accurately (22). A complementary technique uses adiabatic excitation to correct for an imperfect flip angle (16). We suggest that an off-resonance rotary echo is useful for spin-locking off-resonance, corrects for both *B*_{1} imperfections and flip angles α ≠ ϕ, and can also complement an adiabatic excitation.

A traditional off-resonance, *T*_{1}ρ_{off}-weighted imaging sequence is shown in Fig. 2 (sequence 1). Magnetization is *T*_{1}ρ_{off}-weighted during the spin-locking cluster and spatially encoded in two or three dimensions by a spin-echo imaging sequence. If the flip angle α ≠ ϕ, images will have banding artifacts, as shown in Fig. 3.

Two preparatory pulse clusters for *T*_{1}ρ_{off}-weighted imaging. **a**: In sequence 1, if the excitation flip angle α is not the same as that of the effective field ϕ, the magnetization nutates about ω_{eff} and produces imaging artifacts. **...**

The origin of the banding artifacts can be shown with the use of product operator theory. Although the on-resonance spin-lock theory of the rotary echo is well described by quantum mechanical (20) or Bloch (21) treatments, the off-resonance case has not been examined. Following the notation of Levitt (23), the density matrix at equilibrium is given by

$$\mathrm{\rho}({0}^{-})={\mathrm{I}}_{\mathrm{Z}},$$

[2]

where we omit the unity operator 1 and the Boltzmann factor for compactness. Ideally, a uniform pulse of flip angle α = ϕ nutates the magnetization parallel to the effective magnetic field ω_{eff} so that immediately after the pulse

$$\mathrm{\rho}({0}^{+})={\mathrm{R}}_{\mathrm{X}}(-\mathrm{\phi}){\mathrm{I}}_{\mathrm{Z}}{\mathrm{R}}_{\mathrm{X}}(\mathrm{\phi}),$$

[3]

where the pulse nutation is assumed to be instantaneous, and thus R_{X}(−) = e^{−iIX}. The density matrix evolves under the influence of the effective field ω_{eff}, so

$$\mathrm{\rho}({\mathrm{\tau}}^{-})={\mathrm{R}}_{\mathrm{Z}\prime}({\mathrm{\omega}}_{\text{eff}}\mathrm{\tau}){\mathrm{R}}_{\mathrm{X}}(-\mathrm{\phi}){\mathrm{I}}_{\mathrm{Z}}{\mathrm{R}}_{\mathrm{X}}(\mathrm{\phi}){\mathrm{R}}_{\mathrm{Z}\prime}\xb7(-{\mathrm{\omega}}_{\text{eff}}\mathrm{\tau}),$$

[4]

where the effective field in the rotating frame is ${\mathrm{\omega}}_{\text{eff}}=\sqrt{{\mathrm{\omega}}_{1}^{2}+\mathrm{\Delta}{\mathrm{\omega}}^{2}}$, the evolution propagator is R_{Z′}(ω_{eff}τ) = e^{−iωeff}I_{Z′}τ and *z′* denotes the axis of the effective field. After *T*_{1}ρ_{off}-weighting, the magnetization is stored along the *z*-axis with a −ϕ pulse:

$$\mathrm{\rho}({\mathrm{\tau}}^{+})={\mathrm{R}}_{\mathrm{X}}(\mathrm{\phi}){\mathrm{R}}_{\mathrm{Z}\prime}({\mathrm{\omega}}_{\text{eff}}\mathrm{\tau}){\mathrm{R}}_{\mathrm{X}}(-\mathrm{\phi}){\mathrm{I}}_{\mathrm{Z}}{\mathrm{R}}_{\mathrm{X}}(\mathrm{\phi}){\mathrm{R}}_{\mathrm{Z}\prime}(-{\mathrm{\omega}}_{\text{eff}}\mathrm{\tau}){\mathrm{R}}_{\mathrm{X}}(-\mathrm{\phi}).$$

[5]

To examine the density matrix evolution during the spin-locking pulse, we transform the density matrix to the tilted rotating frame using

$${\mathrm{R}}_{\mathrm{Z}\prime}({\mathrm{\omega}}_{\text{eff}}\mathrm{\tau})={\mathrm{R}}_{\mathrm{X}}(-\mathrm{\phi}){\mathrm{R}}_{\mathrm{Z}}({\mathrm{\omega}}_{\text{eff}}\mathrm{\tau}){\mathrm{R}}_{\mathrm{X}}(\mathrm{\phi}).$$

[6]

Combining Eqs. [5] and [6] yields

$$\mathrm{\rho}({\mathrm{\tau}}^{+})={\mathrm{R}}_{\mathrm{X}}(\mathrm{\phi}){\mathrm{R}}_{\mathrm{X}}(-\mathrm{\phi}){\mathrm{R}}_{\mathrm{Z}}({\mathrm{\omega}}_{\text{eff}}\mathrm{\tau}){\mathrm{R}}_{\mathrm{X}}(\mathrm{\phi}){\mathrm{R}}_{\mathrm{X}}(-\mathrm{\phi}){\mathrm{I}}_{\mathrm{Z}}\mathrm{R}(\text{inv}),$$

[7]

where R(inv) denotes the inverse operators applied to the right of the density matrix at ρ(0^{−}). Now, R_{X}()R_{X}() = 1. Since at τ = 0^{−} the density matrix is proportional to I_{Z}, the commutator [R_{Z}(ω_{eff}τ),I_{Z}] = 0. Consequently, the density matrix in Eq. [7] is reduced to

$$\mathrm{\rho}({\mathrm{\tau}}^{+})=\mathrm{\rho}({0}^{-}),$$

[8]

in addition to the usual *T*_{1}ρ_{off} relaxation.

As Soloman (17) showed empirically, Eq. [8] is not correct in the presence of an imperfect *B*_{1} field, but can be corrected with a rotary echo. In addition, as the result of an imperfect spin flip where α ≠ ϕ, the magnetization now makes a small angle Δ to ω_{eff}:

$$\mathrm{\rho}({0}^{+})={\mathrm{R}}_{\mathrm{X}}(-\mathrm{\phi}+\mathrm{\Delta}\mathrm{\phi}){\mathrm{I}}_{\mathrm{Z}}{\mathrm{R}}_{\mathrm{X}}(\mathrm{\phi}-\mathrm{\Delta}\mathrm{\phi}).$$

[9]

Since the magnetization is no longer parallel to ω_{eff}, Eq. [7] no longer simply reproduces ρ(τ^{+}) = ρ(0^{−}) in Eq. [8]. As a result, ω_{eff} modulates the density matrix by R_{Z}(ω_{eff}τ), and signal oscillations proportional to both ω_{eff} and τ are observed. Figure 3 shows the artifacts produced because of this oscillation.

Suppose now that ω_{1} is broken into two pulses of equal duration, as shown in Fig. 2 (sequence 2). During the first period τ/2 (Fig. 2a), ω_{eff} modulates the density operator as in Eq. [6]. During the second period τ/2 (Fig. 2b), the phase of ω_{eff} is rotated 180°. The transformation to the tilted rotating frame during this period may be written as

$${\mathrm{R}}_{\mathrm{Z}\prime}({\mathrm{\omega}}_{\text{eff}}\mathrm{\tau}/2)={\mathrm{R}}_{\mathrm{X}}(\mathrm{\pi}){\mathrm{R}}_{\mathrm{X}}(-\mathrm{\phi}){\mathrm{R}}_{\mathrm{Z}}({\mathrm{\omega}}_{\text{eff}}\mathrm{\tau}/2){\mathrm{R}}_{\mathrm{X}}(\mathrm{\phi}){\mathrm{R}}_{\mathrm{X}}(-\mathrm{\pi}).$$

[10]

Since [R_{X}(π), R_{X}(−)] = 0, the π rotation inverts the direction of nutation:

$${\mathrm{R}}_{\mathrm{Z}\prime}({\mathrm{\omega}}_{\text{eff}}\mathrm{\tau}/2)={\mathrm{R}}_{\mathrm{X}}(-\mathrm{\phi}){\mathrm{R}}_{\mathrm{Z}}(-{\mathrm{\omega}}_{\text{eff}}\mathrm{\tau}/2){\mathrm{R}}_{\mathrm{X}}(\mathrm{\phi}).$$

[11]

As before, the density matrix at time τ/2 is

$$\mathrm{\rho}(\mathrm{\tau}/2)={\mathrm{R}}_{\mathrm{Z}\prime}({\mathrm{\omega}}_{\text{eff}}\mathrm{\tau}/2){\mathrm{R}}_{\mathrm{X}}(-\mathrm{\phi}+\mathrm{\Delta}\mathrm{\phi}){\mathrm{I}}_{\mathrm{Z}}{\mathrm{R}}_{\mathrm{X}}(\mathrm{\phi}-\mathrm{\Delta}\mathrm{\phi}){\mathrm{R}}_{\mathrm{Z}\prime}(-{\mathrm{\omega}}_{\text{eff}}\mathrm{\tau}/2).$$

[12]

The density matrix now evolves according to Eq. [11], so

$$\mathrm{\rho}({\mathrm{\tau}}^{-})={\mathrm{R}}_{\mathrm{X}}(-\mathrm{\phi}){\mathrm{R}}_{\mathrm{Z}}(-{\mathrm{\omega}}_{\text{eff}}\mathrm{\tau}/2){\mathrm{R}}_{\mathrm{X}}(\mathrm{\phi}){\mathrm{R}}_{\mathrm{X}}(-\mathrm{\phi})\phantom{\rule{thinmathspace}{0ex}}\times \phantom{\rule{thinmathspace}{0ex}}{R}_{Z}({\mathrm{\omega}}_{\text{eff}}\mathrm{\tau}/2){R}_{X}(\mathrm{\phi}){R}_{X}(-\mathrm{\phi}+\mathrm{\Delta}\mathrm{\phi}){\mathrm{I}}_{\mathrm{Z}}\mathrm{R}(\text{inv}).$$

[13]

Finally, Eq. [13] reduces to

$$\mathrm{\rho}({\mathrm{\tau}}^{-})={\mathrm{R}}_{\mathrm{X}}(-\mathrm{\phi}+\mathrm{\Delta}\mathrm{\phi}){\mathrm{I}}_{\mathrm{Z}}{\mathrm{R}}_{\mathrm{X}}(\mathrm{\phi}-\mathrm{\Delta}\mathrm{\phi}).$$

[14]

The effects of the nutation R_{Z}(ω_{eff}τ) in Eq. [13] are completely eliminated in Eq. [14] by R_{X}(π). Of course, the form of R_{X}(π) in Eq. [10] depends on whether spin locking is performed on-resonance (ω_{eff} = ω_{1}) or off-resonance (ω_{eff} ≠ ω_{1}). We examine both of these cases below.

Phase inversion of ω_{1} (±180°) meets the condition of R_{X}(π) in Eq. [10] on-resonance, and is shown in Fig. 3 to reduce artifacts in 3% agarose phantoms. For example, if ω_{1} is parallel to the *y*-axis during the first spin-locking period, then ω_{1} must be parallel to the −*y*-axis during the second spin-locking period. Physically, when the flip angle α ≠ ϕ, the magnetization makes an angle Δϕ with the *xy*-plane. During the first spin-locking period, magnetization nutates about the spin-locking axis *y* for a time τ/2. In the second spin-locking period, the sense of precession is reversed (ω_{1} → −ω_{1}) and the magnetization returns to its orientation prior to spin locking.

Combined inversion of ω_{1} and inversion of Δω, the transmitter offset frequency, satisfies R_{X}(π) in Eq. [10] and is shown experimentally in Fig. 3. Once again, if α ≠ ϕ, the magnetization makes an angle Δϕ with ω_{eff}. During the first spin-locking period, magnetization nutates about ω_{eff}. Without inversion of the ω_{eff}, the signal oscillates. If during the second spin-locking period, ω_{1} → −ω_{1} and Δω → −Δω, the axis of ω_{eff} will be inverted and the sense of precession about ω_{eff} will be reversed. This condition is called phase and frequency inversion.

The pulse sequence used to reduce off-resonance spin-locking artifacts is shown in Fig. 2, sequence 2, and is contrasted with the conventional off-resonance spin lock shown in Fig. 2, sequence 1. Magnetization is excited along the direction of the effective field with a hard pulse R_{X}(−), where it is spin-locked by two phase and frequency symmetric spin-locking pulses (+*y*, +Δω and −*y*, −Δω). Consequently, the magnetization is stored along the *z*-axis with another hard pulse. During spin locking, the magnetization nutates first around the effective field (Fig. 2a) and is refocused during the second τ/2 period by nutation in the opposite direction (Fig. 2b).

While it may be useful for single-spin systems, the off-resonance rotary echo has limited applicability to NMR spectroscopy because it cannot simultaneously rephase spins precessing at different frequencies. For example, suppose two spins are spin-locked off-resonance, where the frequencies of the two precessing spins are Δω_{1} and Δω_{2} in the rotating frame. Frequency inversion of the first spin (Δω_{1} → −Δω_{1}) cannot simultaneously bring the second spin back into phase with the first at the end of the spin-locking sequence (*t* = τ). Despite this limitation, the off-resonance rotary echo pulse sequence in Fig. 2, sequence 2, is still useful for MRI because most imaging systems have a single resonant frequency. An additional complication is the presence of any appreciable *B*_{0} inhomogeneity, which may cause local spin density to precess off-resonance; however, the condition for an effective off-resonance spin lock ${\mathrm{\omega}}_{\mathit{\text{eff}}}=\sqrt{{\mathrm{\omega}}_{1}^{2}+\mathrm{\Delta}{\mathrm{\omega}}_{\mathit{\text{RF}}}^{2}}\gg \mathrm{\Delta}{\mathrm{\omega}}_{0}$ is in fact an easier requirement to fulfill compared to the on-resonance condition ω_{1} Δω_{0}. We demonstrate the utility of the off-resonance spin lock for imaging the human brain at 3T in Fig. 4 and call attention to the severe deviation of the initial excitation flip angle (α = 65°) from the nominal flip angle (α = 45°) for ω_{1} = Δω_{RF} = 400 Hz. For this reason, surface coil *T*_{1}ρ_{off}-weighted imaging in particular benefits from the compensated rotary echo.

It is possible to estimate the reduction in SAR during the off-resonance spin lock using a model for SAR deposited in the human head by a quadrature birdcage coil developed by Collins et al. (24). It was shown that

$$\mathit{\text{SAR}}(\mathrm{\alpha},\mathrm{\tau})=f{\left(\frac{3\mathit{\text{ms}}}{\mathrm{\tau}}\right)}^{2}{\left(\frac{\mathrm{\alpha}}{90\xb0}\right)}^{2}\phantom{\rule{thinmathspace}{0ex}}{\mathit{\text{SAR}}}_{90,3\mathit{\text{ms}}},$$

[15]

where the SAR is a function of the shape factor *f*(= 1 for a rectangular pulse), flip angle α, pulse duration τ, and coefficient SAR_{90°,3 ms} = 1.46 W/kg for a 3-ms, 90° rectangular pulse at 1.5T. Considering a single 500 Hz spin-locking pulse delivered for 100 ms, the average SAR delivered with TR = 3 s is approximately 2 W/kg, which is well under the 8 W/kg FDA mandated restriction in the head. For pulse sequences with shorter TR, such as a *T*_{1}ρ-weighted 3D gradient-echo (GRE) or *T*_{1}ρ-weighted balanced-SSFP sequence, the average SAR can often surpass FDA limitations. For example, the average SAR delivered during a *T*_{1}ρ-weighted 3D GRE sequence with TR = 300 ms is nearly 18 W/kg, but can be reduced to less than 8 W/kg by implementing a *T*_{1}ρ_{off} sequence with ω_{eff} = 500 Hz, ω_{1} = 325 Hz and Δω_{0} = 380 Hz. Even if assumptions about the filling factor or SAR_{90,3ms} are incorrect, Eq. [15] predicts that the reduction in SAR will be (*B*_{1,off}/*B*_{1,on})^{2} and the SAR will be reduced to a fraction of that obtained in an on-resonance *T*_{1}ρ-weighted experiment. In addition, one can reduce the total scan time by lowering the minimum TR necessary to maintain FDA guidelines.

One unusual consequence may emerge from samples with an asymmetric *z*-spectrum, where the magnetization transfer (MT) effect may vary between the two spin-locking pulses. For example, suppose an off-resonance spin-locking or saturation experiment is performed in the presence of an asymmetric *z*-spectrum. The signal intensity might be expected to vary with the spin-locking length and the asymmetry in the *z*-spectrum. Also, in the conventional off-resonance spin lock, as τ → ∞ the magnetization approaches a steady state *M _{eff}* along the direction of the effective field ω

We have introduced a spin-locking sequence that reduces *B*_{1} artifacts by means of a frequency- and phase-inverted spin-locking pulse cluster. In agreement with theory, we demonstrated that *B*_{1} nutation can be reversed by a 180° inversion of the effective field on- or off-resonance, and showed that *B*_{1} artifacts were reduced in experiments on agarose phantoms. We expect that this off-resonance rotary echo will be useful for off-resonance spin locking, since the technique removes complicating image artifacts and preserves exponential *T*_{1}ρ_{off} decay of the signal while it reduces the required spin-locking RF amplitude.

The authors thank Luke Bloy, Jeremy Wellen, and Susanta Sarkar for stimulating discussions and technical expertise. This study was performed at the Metabolic Magnetic Resonance Research and Computing Center, an NIH-supported resource center (NIH RR02305).

Grant sponsor: National Institutes of Health (NIH); Grant numbers: R01AR045404; R01AR051041.

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