Compensation for Spin-Lock Artifacts Using an Off-Resonance Rotary Echo in T₁ρ_off-Weighted Imaging

Abstract

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 T₁- and T₂-weighted images as well as the more recent steady-state free precession (SSFP) T₁/T₂-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 T₁ and T₂ relaxation contrast techniques, a small but increasing number of studies are devoted to examining T₁ρ relaxation contrast, in which tissue magnetization is “locked” by an on-resonance RF field. Redfield (1) showed that the T₁ρ relaxation time characterizes the spin-lattice relaxation in the rotating frame. In most cases, as the amplitude of the locking RF field approaches zero, T₁ρ- weighted imaging is functionally equivalent to T₂- weighted imaging, since both generate tissue contrast based on the disappearance of transverse magnetization.

In practice, T₁ρ-weighted imaging offers several advantages over T₁- and T₂-weighted imaging. For example, T₁ρ-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, T₁ρ-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 T₁ρ sequence is high, and both B₀ and B₁ imperfections are significant sources of artifacts.

To overcome three of these difficulties (coil damage, high SAR, and B₁ imperfections), we developed a T₁ρ_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 ω₁ by increasing the off-resonance component Δω₀ to achieve the same effective field strength, and thus reduces the SAR and ω₁ power demands on the RF coil. A rotary echo reduces artifacts due to B₁ inhomogeneity. T₁ρ_off is an intrinsically different relaxation time from T₁ρ, and instead combines both T₁ρ- and T₁-type contrast. In particular, when an off-resonance RF pulse is delivered far from resonance (Δω₀ ≫ ω₁), T₁ρ_off approaches T₁ and on-resonance T₁ρ_off becomes T₁ρ. Here we generalize the rotary echo to the off-resonance case using product operator theory and show that it reduces imperfect B₁ 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 T₁ρ_off in the human breast (15). Despite the decrease in sensitivity to tissue pathology that occurs with decreasing ω₁/ωΔ_RF (16), off-resonance spin locking may be useful for situations in which Δω₀ inhomogeneity artifacts prohibit an on-resonance spin lock, particularly at higher B₀ 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 = ω₁ and ϕ = 90°.

FIG. 1.

Off-resonance spin-locking geometry. The effective field ω_eff makes an angle φ = tan⁻¹(ω₁/Δω) with the z-axis. To perform off-resonance spin locking, the magnetization must be flipped parallel to the effective field such that the flip angle α = ϕ; however, B₁ imperfections cause deviations in the expected flip angle α = ϕ − Δϕ. In on-resonance spin-locking experiments, ω_eff = ω₁ and ϕ = 90°.

Solomon (17) first introduced the rotary echo to correct for imperfections in the RF field. On-resonance, spins accumulate a phase ω₁τ because of nutation due to the RF field. In particular, an inhomogeneous B₁ 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 B₁ pulse that is 180° out of phase with the first is applied for the same duration, the spins will accumulate the exact opposite phase ω₁τ 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 CFCl₃. 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 T₁ρ-weighted pulse sequences with artifact correction have been implemented in MRI (see Table 1). Charagundla et al. (21) eliminated artifacts from both B₁ imperfections and flip angles α ≠ 90° (usually also the result of an imperfect B₁) in T₁ρ-weighted imaging. By reducing these artifacts, it became possible to measure the spatial distribution of T₁ρ 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 B₁ imperfections and flip angles α ≠ ϕ, and can also complement an adiabatic excitation.

Table 1.

Sources of Artifacts in T1ρ_off-Weighted Imaging and Their Pulse Sequence Correction Schemes.

	Reference	B1 imperfections (on-resonance)	B1 imperfections (off-resonance)	Flip angle α ≠ φ(φ = 90° on-resonance)
Conventionala
Rotary echo	Wheaton et al. (8)	X		X
Adiabatic excitation	Santyr et al. (16)			X
Off-resonance rotary echo	This work	X	X	X

^aA conventional spin-lock is an on-resonance spin lock without a rotary echo and is sensitive to all three sources of artifacts in T1ρ-weighted imaging.

A traditional off-resonance, T₁ρ_off-weighted imaging sequence is shown in Fig. 2 (sequence 1). Magnetization is T₁ρ_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.

FIG. 2.

Two preparatory pulse clusters for T₁ρ_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. b: In sequence 2, the magnetization is refocused by a frequency- and phase-inverted spin-locking pulse after τ/2.

FIG. 3.

Compensation for B₁ inhomogeneity during on-resonance (Δω_RF = 0) and off-resonance (Δω_RF = 400 Hz) spin locking in 3% agarose phantoms (α = 65°, ω₁ = 400 Hz). Magnetization nutates about the effective field ω_eff, and without both phase and frequency inversion, banding artifacts are severe.

THEORY

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(−φ) = e^−iφI_X. The density matrix evolves under the influence of the effective field ω_eff, so

$$\mathrm{\rho }({\mathrm{\tau }}^{-})={\mathrm{R}}_{\mathrm{Z}\prime }({\mathrm{\omega }}_{\text{eff}}\mathrm{\tau }){\mathrm{R}}_{\mathrm{X}}(-\mathrm{\phi }){\mathrm{I}}_{\mathrm{Z}}{\mathrm{R}}_{\mathrm{X}}(\mathrm{\phi }){\mathrm{R}}_{\mathrm{Z}\prime }·(-{\mathrm{\omega }}_{\text{eff}}\mathrm{\tau }),$$ [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_effτ) = e^−iω_effI_Z′τ and z′ denotes the axis of the effective field. After T₁ρ_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(φ)R_X(φ) = 1. Since at τ = 0⁻ the density matrix is proportional to I_Z, the commutator [R_Z(ω_effτ),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 T₁ρ_off relaxation.

As Soloman (17) showed empirically, Eq. [8] is not correct in the presence of an imperfect B₁ 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 Δφ 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_effτ), 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 ω₁ 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(−φ)] = 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

$$\mathrm{\rho }({\mathrm{\tau }}^{-})={\mathrm{R}}_{\mathrm{X}}(-\mathrm{\phi }){\mathrm{R}}_{\mathrm{Z}}(-{\mathrm{\omega }}_{\text{eff}}\mathrm{\tau }/2){\mathrm{R}}_{\mathrm{X}}(\mathrm{\phi }){\mathrm{R}}_{\mathrm{X}}(-\mathrm{\phi })\times R_Z({\mathrm{\omega }}_{\text{eff}}\mathrm{\tau }/2)R_X(\mathrm{\phi })R_X(-\mathrm{\phi }+\mathrm{\Delta }\mathrm{\phi }){\mathrm{I}}_{\mathrm{Z}}\mathrm{R}(\text{inv}).$$ [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_effτ) 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 = ω₁) or off-resonance (ω_eff ≠ ω₁). We examine both of these cases below.

RESULTS

Phase inversion of ω₁ (±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 ω₁ is parallel to the y-axis during the first spin-locking period, then ω₁ 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 (ω₁ → −ω₁) and the magnetization returns to its orientation prior to spin locking.

Combined inversion of ω₁ 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, ω₁ → −ω₁ 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(−φ), 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).

DISCUSSION

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 Δω₁ and Δω₂ in the rotating frame. Frequency inversion of the first spin (Δω₁ → −Δω₁) 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 B₀ 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 ω₁ ≫ Δω₀. 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 ω₁ = Δω_RF = 400 Hz. For this reason, surface coil T₁ρ_off-weighted imaging in particular benefits from the compensated rotary echo.

FIG. 4.

T₁ρ_off-weighted brain images obtained in a healthy volunteer at 3T (α = 65°, ω₁ = 400 Hz, Δω_RF = 400 Hz). Phase and frequency alternation of the spin-lock pulse compensates for imperfect flip artifacts; however, there is signal loss in regions of Δω₀ inhomogeneity. The flip angle α = 65° deviates from the nominal tan⁻¹(ω₁/Δω) = 45° flip necessary for an off-resonance spin lock, and was chosen to amplify the image artifacts.

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

$$\mathit{\text{SAR}}(\mathrm{\alpha },\mathrm{\tau })=f{\left(\frac{3\mathit{\text{ms}}}{\mathrm{\tau }}\right)}^2{\left(\frac{\mathrm{\alpha }}{90°}\right)}^2{\mathit{\text{SAR}}}_{90,3\mathit{\text{ms}}},$$ [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 T₁ρ-weighted 3D gradient-echo (GRE) or T₁ρ-weighted balanced-SSFP sequence, the average SAR can often surpass FDA limitations. For example, the average SAR delivered during a T₁ρ-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 T₁ρ_off sequence with ω_eff = 500 Hz, ω₁ = 325 Hz and Δω₀ = 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 T₁ρ-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_eff. During a rotary echo, M_eff is expected to change halfway through the spin lock and consequently change the resultant image contrast. There are no obvious differences in contrast for τ < 100 ms, although they certainly may exist. Δω_RF inversion may even null tissue magnetization if it is timed appropriately.

CONCLUSIONS

We have introduced a spin-locking sequence that reduces B₁ artifacts by means of a frequency- and phase-inverted spin-locking pulse cluster. In agreement with theory, we demonstrated that B₁ nutation can be reversed by a 180° inversion of the effective field on- or off-resonance, and showed that B₁ 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 T₁ρ_off decay of the signal while it reduces the required spin-locking RF amplitude.