Balanced spin‐lock preparation for B1‐insensitive and B0‐insensitive quantification of the rotating frame relaxation time T1ρ

Accurate and artifact‐free T1ρ quantification is still a major challenge due to a susceptibility of the spin‐locking module to B0 and/or B1 field inhomogeneities. In this study, we present a novel spin‐lock preparation module (B‐SL) that enables an almost full compensation of both types of inhomogeneities.


| Basics of the SL preparation method
The general structure of a SL preparation module (standard spin lock 19 ) consists of a 90° excitation pulse, tilting the magnetization by convention to the x′-axis in the rotating frame. Subsequently, a continuous-wave SL pulse is applied on-resonant and in phase to the magnetization with the amplitude f SL , leading to a decay of the SL component with the relaxation time T 1ρ . 1 A second 90° flip-back pulse stores the prepared magnetization on the z′-axis. After crushing the residual transverse magnetization, imaging is performed, such as using a spin-echo acquisition. Different T 1ρ weightings are generated by variation of the spin-lock time t SL . For quantitative T 1ρ mapping, a series of images using different t SL is acquired. The calculation of a T 1ρ map is done by fitting the data pixel by pixel to a monoexponential model. 3

| Influence of B 0 and B 1 field imperfections
The main challenge of accurate and artifact-free T 1ρ mapping is the susceptibility against field imperfections. Inhomogeneities in the B 1 field result in an incorrect execution of excitation and refocusing pulses and change the effective SL field strength. An inhomogeneous B 0 field equals an off-resonant application of the SL pulse. The effect on the preparation process can be described in the rotating frame by tilting the effective SL field B e toward the z′-axis direction by the angle θ. 24 Both B 0 and B 1 imperfections result in an unsatisfactory SL condition, with the locked magnetization precessing around the direction of B e ( Figure 1A,B). This phenomenon is responsible for the emergence of banding artifacts in the acquired images and leads to a disturbed contrast of mixed T 1ρ and T 2ρ weighting. 20,23

| Approaches of field compensation
In the past decade, several methods have been proposed to compensate for field imperfections, thus ensuring a more robust SL preparation. In this context, the use of adiabatic excitation pulses and adiabatic spin locking is often discussed as an effective method. [25][26][27] In this work we focus on improving the on-resonant spin-lock technique, which concerns the spin-lock block of the pulse sequence itself. The first approach for optimizing the SL block was introduced by Charagundla et al 21 and is known as rotary echo (RE-SL) ( Figure 1C). Here, the SL pulse is separated into two pulses with equal duration and opposite phases. This method aims at compensation of B 1 inhomogeneities, but does not provide a suppression of artifacts caused by B 0 imperfections. For this purpose, Witschey et al introduced an extension known as composite SL (C-SL) ( Figure 1D). 23 In this case, a 180° refocusing pulse is applied between the SL pulses, resulting in an additional compensation for B 0 imperfections. A further extension of this method was presented by Mitrea et al, which again divides the two SL pulses. 28 This paired self-compensated module (PSC-SL) ( Figure 1E) consists of four SL pulses of alternating phases and one 180° refocusing pulse. As shown in Mitrea et al, 28 the PSC-SL module offers lower artifact susceptibility compared with the C-SL module in the particular case of small f SL and large inhomogeneities.

| Concept of balanced spin locking
The addition of a 180° refocusing pulse is designed specifically to compensate for B 0 inhomogeneities. However, a fatal error occurs if the refocusing pulse is misapplied due to B 1 imperfections. 20 In this case, even an increase in artifact formation is possible. For this reason, a second refocusing pulse has been included in our novel SL preparation module ( Figure 1F). Here we use an opposite phase (-x) according to the rotary-echo principle. Refocusing issues caused by B 1 imperfections during application of the refocusing pulses will therefore be balanced. The spin locking itself has been subdivided into three pulses also using alternating phases. Consequently, the B-SL module uses maximum symmetry, in which each pulse is compensated by a complementary pulse of opposite phase, ensuring an optimized B 0 and B 1 inhomogeneity compensation with minimal artifact formation.

| METHODS
To analyze the susceptibility of our new B-SL module to field inhomogeneities, various comparisons with the established SL modules described before have been performed. The comparisons were performed in three different ways. First, detailed analytical calculations of the SL trajectories were examined for pure B 1 and B 0 imperfections. Second, extensive numerical simulations were performed, which compare the T 1ρ quantification accuracy and artifact susceptibility. In the final step, the theoretical results were F I G U R E 1 A,B, Spin-lock (SL) process in rotating frame (x′ y′ z′) and tilted rotating frame (x″ y″ z″) coordinates. The graphs show 3D trajectories of magnetization during the SL process in the presence of a B 1 (A) and B 0 (B) field imperfection. The SL component decays according to an exponential function with T 1ρ . The envelope of the oscillating spintip components decays with T 2ρ . 24 The schematics (C-F) show pulse sequences of SL preparation modules that use different strategies for the compensation of field imperfections: rotary echo (RE-SL) (C), composite (C-SL) (D), and paired selfcompensated (PSC-SL) (E). F, Pulsesequence schematic of the presented balanced SL module, consisting of three SL pulses and two refocusing pulses using opposite phases for maximum symmetry validated on a 7T small animal imaging system in phantom experiments.

| Analytical comparison
The analytical comparison of the preparation modules was based on an approach for calculating spin-lock trajectories using matrix propagators. 23 The formalism is described in detail in Supporting Information S1. Dependencies of banding and relaxation behavior can be derived from the solution functions in the case of only B 0 and only B 1 inhomogeneities.

| Numerical Bloch simulations
The numerical simulations take the simultaneous occurrence of B 0 and B 1 field inhomogeneities into account and were performed in MATLAB (R2017a; The MathWorks, Natick, MA). The simulations reflect the effect on the T 1ρ quantification accuracy in the experimental setup. For the B 0 imperfections, the range of ±600 Hz (2 ppm at 7 T) and for the B 1 imperfections the range of ±50% was observed. Here, 100 × 100 cases were distinguished for each module. Using the matrix propagator formalism, the prepared magnetization M z was calculated for different SL times using N = 2000 linearly spaced sampling points (Δt SL = 0.1 ms) in the range 0 … 2T 1ρ . These nearly continuously sampled trajectories were used to analyze the banding susceptibility by calculating the residual sum of squares (RSS) in relation to a monoexponential fit. In contrast, the quantification accuracy was not determined from the global fit (2000 sample points), as this does not correspond to the realistic conditions of a T 1ρ quantification experiment. Instead, as in most in vivo experiments, a small sample (8 points) of the SL trajectory was taken randomly and T 1ρ was fitted on this basis. This procedure was repeated 100 times for each trajectory; thus, the mean quantification error ΔQ was determined as a measure of quantification accuracy.
Here, Δq indicates the quantification error of a single experiment, which was calculated from the fitted value T 1 ,fit and the value T 1 ,true = 100 ms, which was actually defined in the simulation. For the selection of random sampling points, the range 0 … 200 ms was divided into eight equally sized areas from which uniformly distributed samples were drawn. This ensures a sufficient sampling of the exponential decay. The comparison of the different modules was always based on identical random numbers. The simulation was repeated and evaluated for various f SL (100 Hz … 4 kHz) and various T 1ρ :T 2ρ ratios (1:5 … 5:1) to finally compare and rate the performance of the preparation modules.

| Experimental validation
For experimental validation, the preparation modules (RE-SL, C-SL, PSC-SL, and B-SL) were implemented on a 7T small animal imaging system (Bruker BioSpec 70/30, Bruker BioSpin MRI, Ettlingen, Germany) based on a turbo spin-echo acquisition. In contrast to the numerical simulations, the experiment cannot be based on the quantification accuracy. This is because an equivalent procedure cannot be carried out in any realistic measurement time, and the true T 1ρ relaxation time of a phantom is not known due to the lack of a gold-standard method for a definitely determination. Therefore, the simulation results were validated by a comparative study analyzing the banding susceptibility and the quality of the mono-exponential fit (RSS) in a large number of phantom measurements. The phantom in use consisted of a homogeneous cylindrical sample tube filled with an aqueous solution of agar (2%). The T 1ρ -weighted images were taken in a transverse slice (FOV = 32 × 32 mm 2 , matrix = 96 × 96, thickness = 2.5 mm). Further imaging parameters were TE = 7.3 ms, TR = 5000 ms, and turbo factor = 4.
To compare the preparation modules in different cases of B 0 /B 1 field inhomogeneities, targeted disorders of the SL process were carried out in addition to the natural inhomogeneities of the selected slice. The natural inhomogeneities were obtained by B 0 /B 1 mapping and determined to be ΔB 1,max ≈ 10.6% and ΔB 0,max ≈ 0.11 ppm. An additional disorder of the B 1 field was caused using incorrect flip angles (up to −25%) of the excitation and refocusing pulses within the preparation modules. Furthermore, B 0 imperfections were increased using specific off-resonances (up to +1 ppm) for the SL pulses. Both cases lead to a violation of the SL condition and cause distinct banding artifacts in the T 1ρ -weighted images. Measurements were performed for all modules at different t SL (4,12,20,28,36,44,52, and 60 ms), f SL (500, 1000, 1500, and 2000 Hz), B 1 (0, −5, −10, −15, −20, and −25%), and B 0 (0, 0.2, 0.4, 0.6, 0.8, and 1.0 ppm) disorders. With 1536 measurements in total, a high coverage of an experimentally relevant parameter scope was achieved. As a measure of artifact susceptibility, the SD σ of the signal intensities has been determined within a circular region of interest. The quality of mono-exponential fitting was specified by the mean observed RSS. To compare the overall performance of the different modules, the SDs σ and the RSS values of identical experiments were each normalized on a scale of 0 … 1. With this procedure it was possible to compare a large number of individually different experiments on an equal basis and to statistically evaluate the overall result.

| Analytical comparison
The analytical calculations of SL trajectories provide information about the susceptibility of the SL pulses to artifact formation and the influence of the excitation and refocusing pulses as well as off-resonance effects. The results show that perfect compensation of field inhomogeneities is not possible with any module (Supporting Information Table S1). However, clear differences can be identified. In contrast to RE-SL and PSC-SL, the B-SL module shows neither banding terms for B 0 nor B 1 imperfections. Compared with PSC-SL and C-SL, B-SL is not dependent on the quality of the refocusing pulse.

| Numerical Bloch simulations
The concept for evaluating the simulation results is shown in Figure 2. Comparing C-SL and B-SL for a special B 0 /B 1 scenario, it appears that B-SL exhibits a lower oscillation level (lower RSS) and leads to fewer quantification errors (lower ΔQ), yielding a higher accuracy in the synthetic experiments ( Figure 2C). The quantitative indicators ∆Q ( Figure 3A) and RSS (Supporting Information Figure S2A) were calculated on a grid of 100 × 100 different field constellations and are visualized as heat maps. The RE-SL module appears to offer an effective compensation mechanism for B 1 inhomogeneities, as expected. Both indicators rise rapidly with increasing B 0 imperfections. In the case of C-SL, PSC-SL and B-SL, the compensation mechanism for B 0 is clearly visible for both indicators. The B-SL module provides the best B 1 compensation. The results of the f SL analysis are depicted in Figure 3B and Supporting Information Figure S2B. Here, a lower susceptibility to bandings and a better quantification accuracy in the case of high amplitudes can be identified for all modules. Figure 3C and Supporting Information Figure S2C show the results of the detailed analysis of the relaxation-time ratio influence. The best behavior is reached for the case of T 1ρ ∶T 2ρ ≈1. Comparing the overall performance of the modules, B-SL achieves the best result for both indicators, various amplitudes, as well as various relaxation-time ratios. The average performance is outmatched by a factor of 3.58 (RE-SL), 2.04 (C-SL), and 2.87 (PSC-SL).

| Experimental validation
The measurements with no additional B 0 or B 1 imperfections show the same T 1ρ quantification for all preparation modules. The B-SL module achieves the values T 1 ,500 Hz = 48.05 ± 0.66 ms, T 1 ,1000 Hz = 49.69 ± 0.67 ms, T 1 ,1500 Hz = 50.57 ± 0.66 ms, and T 1 ,2000 Hz = 51.69 ± 0.69 ms with the four spin-lock amplitudes used. The modules indicate the same dispersion behavior, with the highest deviation between the modules being 0.77%. Figure 4 shows exemplary T 1ρ -weighted images, T 1ρ maps, and RSS maps of the four different preparation modules. In Figure 4A, in which a B 1 disorder for the excitation and refocusing pulses has been used, C-SL and PSC-SL exhibit slight artifacts. The RE-SL and B-SL modules provide the most reliable T 1ρ -weighted images, as proven by the corresponding T 1ρ and RSS maps. Figure 4B shows images with off-resonant SL pulses. Here, the occurrence of banding artifacts is clearly visible for all modules. The RE-SL module, without any B 0 compensation, exhibits the highest banding intensity. Comparing the remaining three modules, B-SL achieves a substantial improvement.
The statistical analysis of the 1536 individual experiments using the normalized indicators σ and RSS are depicted in Figure 5. As expected from theory, RE-SL yields good banding and RSS performance in the case of B 1 imperfections, but achieves the worst overall results in the case of B 0 imperfections. The C-SL module shows the opposite behavior to RE-SL. We found poor banding and RSS performance in the case of B 1 disorders and good performance in the case of B 0 disorders. The PSC-SL module shows an anomaly for B 1 disorders, with a high banding susceptibility and good RSS performance at the same time. This effect occurs when the T 1ρ -weighted images show distinct artifacts, which decay mono-exponentially. In the case of B 0 disorders, a medium banding susceptibility and a good RSS performance could be determined. The results of B-SL show good banding and RSS performance for both B 1 and B 0 imperfections. This is also evident in the combined statistics of both scenarios. Compared with the established modules, the banding performance could be increased by 84% (RE-SL), 79% (C-SL), and 86% (PSC-SL). The RSS performance was increased by 70% (RE-SL), 58% (C-SL), and 30% (PSC-SL).

| DISCUSSION
In the present work, a new spin-lock preparation module has been presented. The new method has been validated and compared in analytical, numerical, as well as experimental studies, showing a superior behavior to other previously published SL preparation techniques. The results of the three investigation methods are consistent and demonstrate an improved robustness as well as a much less artifact formation for combined B 0 /B 1 inhomogeneities. This confirms that the concept of a second complementary refocusing pulse within the B-SL module provides further stabilization of the spinlock process.
When interpreting the simulation results, it must be considered that spin relaxation and spin dynamics under the influence of the SL pulse are highly complex processes. It is known from relaxation theory that T 1ρ and T 2ρ depend on both amplitude and off-resonance of the locking field. 24 The simulation examines the influence of spin dynamics on the accuracy of a T 1ρ mapping experiment. Therefore, we studied banding artifacts leading to quantification errors. The influence of T 1ρ dispersion, which is described in the theory of spin relaxation, 24 was not considered. Furthermore, matrix propagators were used for the Bloch simulation, which represent efficient modeling of a preparation module. Using this approach, however, the influence of the pulse shape and pulse amplitude of the excitation and refocusing pulses is not taken into account. As a result, the B 0 susceptibility of these pulses is not considered in the simulation results. Low amplitude pulses are expected to increase artifact formation. Furthermore, the simulation is based on the single pool model. Influences of the excitation and refocusing pulses on F I G U R E 4 Examples of experimental results based on phantom measurements with agar. For each preparation module, four T 1ρ -weighted images (t SL = 12, 28, 44, and 60 ms) and the corresponding T 1ρ and RSS maps are listed for f SL = 1500 Hz. A, We set a specific B 1 deviation of −15%. B, An offresonance of the SL pulses of 0.4 ppm was used. The banding artifacts (SD of intensity) of the T 1ρ -weighted images and the RSS values of the mono-exponential fits are used as quantitative indicators for statistical analysis in Figure 5 the multipool situation were therefore not examined. The results of the simulation using matrix propagators are nevertheless clear and indicate a significantly improved quantification accuracy for B-SL in the context of strong B 0 /B 1 field inhomogeneities. Compared with the C-SL module, which has been used in numerous studies, an increase in absolute performance by a factor of 2.04 could be observed.
The simulation results were validated using a large series of measurements, in which the different modules were compared directly for identical experiments. The dispersion behavior of T 1ρ could be observed equally for all modules in the case of no additional field imperfections. Using the simultaneous analysis of the banding intensity and the RSS value, both the image quality and the goodness of the monoexponential T 1ρ mapping were validated in the statistical evaluation. This prevents a result showing good RSS but high spatial artifact formation (and vice versa) from being incorrectly rated as a high performance. It could not be proven that the B-SL module delivers the best result in every single combination of B 0 /B 1 inhomogeneity. However, B-SL shows a significantly increased performance in the evaluation of the entire series of measurements. Above all, an increased robustness against banding artifacts could be determined, which leads to a significant improvement in image quality and quantification accuracy. Compared with C-SL, the banding performance was increased by 79% in the combined B 0 /B 1 evaluation. Under the same conditions, the goodness of mono-exponential T 1ρ fitting, which was determined through the RSS performance, was improved by 58%.
The results of the phantom experiments show a lower increase in performance than predicted by the simulation. However, the absolute performance values are not directly comparable and depend on the choice of simulation and measurement parameters. In the simulation, very large parameter spaces were used, whereas in the experimental setup parameters were selected that are most relevant for T 1ρ -based imaging.
The presented work only considers on-resonant spin-lock techniques. The use of adiabatic excitation pulses enables the spin-lock preparation to be further stabilized, as previous studies have already shown. 25 Furthermore, the new balanced spin-lock technique was not compared with fully adiabatic methods. This or a combination of B-SL with a constant amplitude technique 26,27 is interesting for future studies.
Due to its high robustness against field imperfections, B-SL is particularly suitable for the use at high field strengths or for cardiac applications. 13 A drawback of our new method is the increased specific absorption rate, due to the additional refocusing pulse. In the worst case, the rise in specific F I G U R E 5 Statistical evaluation of the normalized indicators σ (banding susceptibility) and RSS for only B 1 and only B 0 disorders and the combination of both. The indicators normalized to [0 … 1] were presented as histograms for each preparation module. The classes were divided into the three groups: "low" [0 … 1/3], "mid" [1/3 … 2/3], and "high" [2/3 … 1] σ or RSS. The performance of a module was determined from how often indicators were observed in the low or high class. In nearly all cases, B-SL provides the most σ and RSS indicators in the low class, and the fewest indicators in the high class