A method for measuring B₀ field inhomogeneity using quantitative double-echo in steady-state

Abstract

Purpose:

To develop and validate a method for B₀ mapping for knee imaging using the quantitative Double-Echo in Steady-State (qDESS) exploiting the phase difference (Δθ) between the two echoes acquired. Contrary to a two-gradient-echo (2-GRE) method, Δθ depends only on the first echo time.

Methods:

Bloch simulations were applied to investigate robustness to noise of the proposed methodology and all imaging studies were validated with phantoms and in vivo simultaneous bilateral knee acquisitions. Two phantoms and 5 healthy subjects were scanned using qDESS, water saturation shift referencing (WASSR), and multi-GRE sequences. ΔB₀ maps were calculated with the qDESS and the 2-GRE methods and compared against those obtained with WASSR. The comparison was quantitatively assessed exploiting pixel-wise difference maps, Bland-Altman (BA) analysis, and Lin’s concordance coefficient (ρ_c). For in-vivo subjects, the comparison was assessed in cartilage using average values in 6 sub-regions.

Results:

The proposed method for measuring ΔB₀ inhomogeneities from a qDESS acquisition provided ΔB₀ maps that were in good agreement with those obtained using WASSR. ΔB₀ ρ_c values were ≥ 0.98 and 0.90 in phantoms and in vivo, respectively. The agreement between qDESS and WASSR was comparable to that of a 2-GRE method.

Conclusion:

The proposed method may allow B0 correction for qDESS T₂ mapping using an inherently co-registered ΔB₀ map without requiring an additional B0 measurement sequence. More generally, the method may help shorten knee imaging protocols that require an auxiliary ΔB₀ map by exploiting a qDESS acquisition that also provides T₂ measurements and high-quality morphological imaging.

Introduction

Quantitative Magnetic Resonance Imaging (MRI) techniques are valuable tools for assessing macromolecular changes in collagenous tissues and studying osteoarthritis (OA) progression^1,2. These parameters are measured using a variety of 2D and 3D sequences, all of which have varying sensitivities to pulse sequence parameters and hardware imperfections³. Magnetic field inhomogeneities can affect quantitative accuracy in many of these techniques, such as T₂ relaxation time mapping^4,5, chemical exchange saturation transfer of glycosaminoglycans⁶ (gagCEST). More generally, obtaining information about static magnetic field (B₀) inhomogeneities is beneficial for a variety of musculoskeletal (MSK) MRI methodologies. For example, correction of field-induced distortions has shown to improve localization of bone in diffusion-weighted imaging (DWI)⁷. Furthermore, B₀ mapping can be used to check if the right condition exists for effective fat saturation, chemical shift-based imaging methodologies^8,9, or simultaneous fat-water imaging¹⁰. B₀ inhomogeneity corrections usually require ancillary scans for B₀ mapping using the Water Saturation Shift Referencing¹¹ (WASSR) method or the two Gradient-Echo (2-GRE) method¹². These additional scans can lead to time-consuming MRI protocols¹³ that can limit the application of these methods, especially in studies with large cohorts.

The 3D double-echo in steady-state (DESS) pulse sequence^14,15,16 has been widely applied in musculoskeletal imaging due to its image contrast and high SNR efficiency. DESS enables high-resolution imaging of cartilage, including assessment of cartilage thickness and defects^17,18,19. Separation of the two echoes in the DESS sequence in a method known as quantitative DESS^20,21 (qDESS) can further provide accurate and reproducible T₂ measurements^22,23 by fitting their signals to an extended phase graph signal model^24,25,26. Additionally, the SNR efficiency and multiple contrasts provided by this approach have shown promise for bilateral clinical knee assessment in five minutes or less^27,28,29,30. However, as for other T₂ mapping methods, T₂ estimation accuracy is affected by field inhomogeneities.

Since qDESS acquires two echoes it could be possible to exploit the phase difference between them to obtain a ΔB₀ map as commonly done for the 2-GRE method. However, phase evolution in qDESS is more complicated than a GRE multi-echo sequence. In the latter, all echoes are gradient-recalled, and the phase accumulation between two echoes is proportional to the echo time difference. In qDESS, neglecting higher order pathways in the steady state²², the first echo is gradient-recalled, whereas the second echo records the rephasing of the gradient-recalled echo created by the radio-frequency (RF) pulse in the preceding cycle. Thus, the phase evolution is very different from a 2-GRE case and, as it will be explained in the Theory section, the phase difference between the two echoes is proportional to twice the first echo time, whereas it is independent on the choice of the second echo time. Recently, an analytical equation for performing B₀ mapping exploiting any two unbalanced Steady State Free Precession (ubSSFP) signals was proposed by Leupold³¹. Since the qDESS sequence belongs to the family of ubSSFP sequences, the work provided a theoretical setting to measure ΔB₀ maps exploiting the difference between the two echoes acquired with qDESS. However, the work by Leupold was purely theoretical and did not provide insights into the actual applicability of the method for in-vivo B₀ mapping in complex body regions such as the knee. For example, as will be further explained in the Theory section, the wrap-free bandwidth (the off-resonances for which the phase difference between the two echoes does not present wraps) of the qDESS method is much smaller than that of a standard 2-GRE method. Thus, highly wrapped phase images could be obtained, which can be hard to unwrap in complicated regions such as the knee even with current phase unwrapping algorithms. Inability to correctly perform phase unwrapping would hinder the practical applicability of the method despite theoretical feasibility.

This work aims i) to provide an empirical verification and validation of the theoretical framework proposed by Leupold for B₀ mapping using qDESS with phantom and in-vivo experiments again standard methods, ii) to set up and describe an optimal processing pipeline, and iii) to analyze strengths and limitations for a practical application to simultaneous bilateral knee imaging. These contributions may allow implementing a B₀ correction method for qDESS T₂ mapping without acquiring an additional scan with the additional benefit of inherent co-registration of the T₂ and ΔB₀ maps. More generally, the method may help shorten knee imaging protocols that require an auxiliary ΔB₀ map, such as gagCEST imaging, by exploiting a qDESS acquisition that also provides T₂ measurements and high-quality morphological imaging. Preliminary results for this work were presented at the 29th ISMRM annual meeting & exhibition³².

Theory

The qDESS is a gradient-spoiled steady-state sequence that samples two echoes per repetition (TR) that are separated by a spoiler gradient. The two signals have different contrast and are functions of the extrinsic scan parameters (TR, TE, flip angle α, gradient amplitude G, and gradient duration τ) as well as the intrinsic tissue parameters (T₁, T₂, and diffusivity). Figure 1(A) shows the pulse diagram of the qDESS sequence along with a phase graph representation. Multiple echo pathways contribute to the origin of the first and second echoes, S₁ and S₂, respectively. Neglecting the contribution from high dephasing orders pathways²², S1 can be thought of as a Free Induction Decay (FID) signal, and thus it has $T_2^{*}/T_1$ weighting. The second echo (S₂), conversely, is a combination of a spin-echo pathway, formed by pulse refocusing the first echo S₁ of the previous TR, and a gradient echo pathway and it has a higher T₂ and diffusion weighting. The pulse diagram of a multi-echo GRE with a phase graph representation is reported in Fig. 1(B) to highlight the difference between qDESS and a multi-echo GRE sequence.

Figure 1:

(A) qDESS pulse sequence diagram with phase graph representation. FGRE pulse sequence diagram with phase graph representation. (C) The schematization of the processing pipeline that was used to compute ΔB₀ map in Femoral Cartilage using qDESS. The coil-combined phase difference between the second and first echo is computed using the HiP. Femoral cartilage is segmented using a deep learning model with the DSOMA framework. The 3D wrapped phase difference is unwrapped and then ΔB₀ is computed according to eq. 2. The 3D ΔB₀ map in FC is projected onto a 2D space for visualization using DOSMA.

ΔB₀ field mapping with qDESS

Recently, Leupold proposed a theoretical equation to estimate the off-resonance, ΔB₀, based on any two unbalanced Steady State Free Precession (ubSSFP) signals³¹. This relationship is reported in equation 1.

$$\Delta B_0=\frac{{\Delta }_{k_1,k_2}\theta +\pi \cdot \left({\chi }_{k_1<0}-{\chi }_{k_2<0}\right)}{2\pi \cdot \left[T{E}_2-T{E}_1+\left(k_2-k_1\right)TR\right]}$$ [1]

Here ${\Delta }_{k_1,k_2}\theta$ is the phase difference between the second and first ubSSFP signals acquired at echo times TE₂ and TE₁, respectively, and k_i describes the dephasing order of a ubSSFP signal using the EPG framework. The indicator function ${\chi }_{k<0}$ is such that ${\chi }_{k<0}=0$ if k ≥ 0 and ${\chi }_{k<0}=1$ if k < 0.

For a qDESS sequence, the first echo (S₁) has dephasing order k₁ = 0, whereas the second echo (S₂) is has dephasing order k₂ = −1. Note that since TE₂ = 2TR − TE₁ in qDESS, eq 1 can then be expressed as eq. 2:

$$\Delta B_0=\frac{\angle \left(S_2\cdot S_1^{*}\right)}{2\pi \cdot 2T{E}_1}$$ [2]

where the numerator is the phase difference, Δθ, between S₂ and S₁, obtained applying the angle function (∠) to the Hermitian inner Product³³ (HiP), $S_2\cdot S_1^{*}$, of the two complex qDESS echo signals. It is worth noting that according to eq. 1, a factor of π is added to the actual phase difference so that in eq. 2 one should subtract π from the phase difference. In our qDESS implementation, the RF pulses are given by alternating the phase of the pulses by a factor of π. As such, no π factor is added to the actual phase difference, and thus it does not need to be subtracted from Δθ in eq. 2.

Contrary, with the 2-GRE method, both echoes have dephasing order k₁ = 0 and eq. 1 becomes the well known relationship:

$$\Delta B_0=\frac{\angle \left(S_2\cdot S_1^{*}\right)}{2\pi \cdot \left(T{E}_2-T{E}_1\right)}$$ [3]

Note that in the qDESS sequence, the second echo has a pulse-refocused component, so the phase accumulation due to off-resonances is also refocused. Thus, once the first echo time is set, varying TE₂ does not change the phase accumulation due to off-resonances. The ΔB₀ region for which Δθ does not present phase wraps, the wrap-free bandwidth, is then given by Eq. 4

$$\Delta \nu =\left(-\frac{1}{4T{E}_1},\frac{1}{4T{E}_1}\right)$$ [4]

Thus, it is possible to observe that the wrap-free bandwidth is much smaller than that of a 2-GRE method run using an echo spacing similar to qDESS TE₁. The minimum TE₁ values for qDESS are in the order of a few units of ms, implying that off-resonances in the order of a few tens of Hz give rise to phase wraps in qDESS difference maps. In MSK imaging, it is not uncommon to observe ΔB₀ in the order of a few hundreds of Hz. In such cases, phase unwrapping algorithms are necessary to restore the correct phase difference maps. However, for complex phase topographies such as those encountered in MSK imaging, phase unwrapping is a difficult task that can potentially introduce severe errors in phase estimation and thus in ΔB₀ estimation.

Effect of ΔB₀ off-resonances in T₂ estimation with qDESS

Although not investigated in the current work, it is worth providing insights on the effect of ΔB₀ off-resonances in qDESS T₂ mapping estimation accuracy. Sveinsson et al.²² proposed a simple analytic model that can be used for T₂ estimation from the ratio between the second and first echoes of qDESS. The model is expressed in eq. 5:

$$\frac{S_2}{S_1}={\text{e}}^{-\frac{2(TR-TE)}{T_2}-\left(TR-\frac{\tau }{3}\right)\Delta k^{2}D}{\sin }^2\left(\frac{\alpha }{2}\right)\left(\frac{1+{\text{e}}^{-\frac{TR}{T_1}-TR\Delta k^{2}D}}{1-\cos \alpha {\text{e}}^{-\frac{TR}{T_1}-TR\Delta k^{2}D}}\right)$$ [5]

Here, TR and TE represent the repetition and the first echo time, respectively, α is the flip angle, D is the diffusivity, and the dephasing per unit length induced by the unbalanced gradient is denoted by Δk = γGτ, where G and τ are the spoiler amplitude and duration, respectively, and γ is the gyromagnetic ratio. This simple model, which allows estimating T₂ from qDESS, also allows one to appreciate the mechanism for which ΔB₀ off-resonances can introduce errors in qDESS T₂ mapping. The flip angle must be known to correctly estiamte T2 from eq. 5. It is clear how B₁ inhomogeneities across the imaged volume directly affect the actual flip angle experienced by the spins within voxels. When selective pulses are used, which is the standard for fat-suppressed qDESS acquisition, ΔB₀ inhomogeneities also affect the actual flip angle experienced by the spins within the voxels accordingly to the frequency profile of the selective pulse used.

Figure 2 reports Bloch simulations run to show the dependence of the T₂ estimation error as a function of the off-resonance experienced by the spin ensemble when considering the frequency profile of the selective pulse used in the qDESS sequence.

Figure 2: T2 error as a function of the off-resonance ΔB₀.

(A) qDESS RF pulse shape with slice-selective gradient (top) and corresponding spectral (bottom left) and slice (bottom right) profiles. (B) T₂ error as a function of the off-resonance ΔB₀ obtained running Bloch simulations considering the qDESS RF slice profile for a voxel composed of cartilage (T2 = 45 ms).

Methods

Simulation experiment

Bloch simulations of the qDESS pulse sequence were performed using an in-house developed MATLAB (MathWorks) library³⁴. The simulations considered the spectral profile of the selective pulse used for fat suppression in real qDESS acquisitions. The pulse and its corresponding frequency profile are reported in Fig. 2.

The accuracy and robustness to noise of the qDESS B₀ mapping method (eq. 2) were investigated by simulating the qDESS signal (#TR = 400) for different combinations of TE₁ and TE₂ considering a spin ensemble with T₁ = 1400 ms and T₂ = 45 ms (relaxation parameters of articular cartilage). Three first echo times were selected: TE₁ = 5, 10 and 15ms. For each TE₁, the simulations were run three times, progressively increasing TE₂. For each pair of (TE₁, TE₂), the signal was simulated for ΔB₀ values ranging from −90Hz to 90Hz with a step size of 0.25Hz resulting in an array of dimensions n x m, where n is the number of ΔB₀ values (n = 721) and m is the number of pulse repetition (m = 400). For each ΔB₀ value, white Gaussian noise was added progressively increasing the variance of the noise such that the signal-to-noise ratio (SNR) for the qDESS signal with TE₁ = 5ms and TE₂ = 25ms ranged from 250 to 5. The SNR was defined according to eq. 6, where P_signal and P_noise represent the average power of the qDESS signal and the noise, respectively, and E(signal) indicates the expectation value of the qDESS signal.

$$SNR=\frac{P_{signal}}{P_{noise}}=\frac{E(signal)}{{\sigma }_{noise}^2}$$ [6]

The noise addition procedure was repeated 721 times for each ΔB₀ and variance value. The first and second noisy echoes of the last time repetition (the steady state) were retained. For each echo, the data were then reshaped into a n x n matrix, and the phase difference between the second and first echo was computed pixel-wise, applying the angle function to the HiP of the two echoes (see eq. 2. Since phase unwrapping was always necessary given the values of TE₁ and the range of ΔB₀ considered, 2D unwrapping algorithms are more robust to noise than 1D unwrapping algorithms. Thus, a 2D phase unwrapping algorithm based on the transport of intensity equation³⁵ was used to unwrap the phase maps. The need for effective phase unwrapping was the rationale for reshaping the data in a 2D image, which was also considered closer to a real scenario.

Mean Error (ME) and Root Mean Squared Error (RMSE) between estimated ΔB₀ values and ground truth values were estimated for each noise variance and averaged across ΔB₀ values.

MRI acquisitions

All MRI acquisitions were performed on a GE 3T SIGNA Premier scanner (GE Healthcare, Milwaukee, WI, USA) using 16-channel flexible phased-array receive-only coils (NeoCoil, Pewaukee, WI, USA). Informed consent was obtained from all subjects in accordance with the Institutional Review Board protocol at our institution. Simultaneous bilateral scans used two coils wrapped around the knees.

The MR imaging was performed using 3D qDESS, 3D Magnetization Prepared SPGR (WASSR) and 2D fast multi-echo GRE (FGRE) sequences. 3D qDESS used a spectrally-selective excitation with nominal flip angle α = 20° and a spoiler gradient, which had a spoiler moment of 3.13mT/mms with duration τ = 1.6ms. WASSR magnetization preparation utilized a 200ms saturation pulse train with a root mean squared B₁ of 0.3μT, from −1ppm to 1ppm with a step size of 0.1ppm. Other imaging parameters are specified for each acquisition in the following sections.

Phantoms scans

To validate eq. 2 and study accuracy for different combinations of TE₁ and TE₂, an agar gel phantom was scanned multiple times using qDESS with three different first echo times: TE₁ = 5, 10 and 15ms. For each TE₁, the sequence was run three times, progressively increasing the TE₂. Using multiple first and second echo times also allowed us to evaluate the effect of SNR on qDESS ΔB₀ estimation accuracy. The voxel resolution was 0.7 × 0.7 × 4.0mm³, and all other qDESS imaging parameters were kept the same among acquisitions. A reference ΔB₀ map was acquired using WASSR with readout parameters of: TR = 3ms, TE = 1.2ms, α = 5°, voxel size 0.7 × 0.7 × 4.0mm³. The experiment was repeated by modifying the shimming gradient along the x-axis, from −12 mT/m (default prescan value) to 28mT/m, to obtain a greater B₀ heterogeneity. The ΔB₀ maps obtained using qDESS were compared to the WASSR method obtained reference ΔB₀ using pixel-wise difference and Bland-Altman (BA) analysis.

A second phantom, made of 6 vials containing doped water, was scanned using qDESS, WASSR and the Fast GRadient-Echo (FGRE) sequences. qDESS scan parameters were TE₁ = 5ms, TR = 25ms (i.e. TE₂ = 45ms), voxel resolution = 1 × 1 × 2mm³. WASSR scan parameters were TR = 3.5ms, TE = 2.5ms and voxel size = 1 × 1 × 2mm³. The FGRE scan parameters were TR = 30ms, TE = 2.5, 5, 7.5 and 10ms, voxel size = 1 × 1 × 2mm³. The first two echoes were used to compute the ΔB₀ map, i.e. 2-GRE method. The duration of the acquisitions (expressed in minutes:seconds) were 1:42, 3:28, and 3:58 for qDESS, WASSR, and FGRE, respectively. The ΔB₀ maps obtained with the qDESS and the 2-GRE methods were compared to the reference map obtained using the WASSR method. The 2-GRE method was added to the comparison to observe the agreement between another standard ΔB₀ mapping method with the WASSR reference method and also to observe the agreement between the qDESS and 2-GRE method.

The rationale for considering WASSR the reference method for ΔB₀ mapping was because this method has shown sub-Hz accuracy¹¹. This has made it the benchmark for CEST asymmetry ΔB₀ correction, which is particularly sensitive to these inhomogeneities. The 2-GRE method also provides accurate ΔB₀ mapping, but its absolute accuracy has shown dependence on the ΔTE used³⁶. Thus, the WASSR method was considered a more appropriate reference for evaluating the accuracy of the proposed qDESS ΔB₀ mapping method.

In vivo simultaneous bilateral knee scans

To validate the qDESS B₀ mapping method in vivo, five healthy subjects were scanned in the sagittal plane with the MRI protocol described above using a simultaneous bilateral knee acquisition²⁷. qDESS scan parameters were TE₁ = 5ms, TR = 15ms (i.e. TE₂ = 25ms), voxel size = 0.3 × 0.3 × 1.4mm³. WASSR scan parameter were TR = 3.5ms, TE = 2.5ms and voxel size = 0.6 × 0.6 × 4mm³. The FGRE scan parameters were TR = 30ms, TE = 1.4, 3.5 and 5.7ms, voxel size 0.6 × 0.6 × 4mm³. As for the phantoms, the first two echoes were used to compute the ΔB₀ map with the 2-GRE method. The voxel size was kept the same among acquisitions to minimize differences in the ΔB₀ maps that could arise from imaging parameters instead of reflecting a method-specific difference. Acquisition times (expressed in minutes:seconds) were 2:40, 11:27, and 3:29 for qDESS, WASSR, and 2-GRE, respectively.

Data processing and analysis

Multi-channel data: phase difference reconstruction

For qDESS and FGRE, phase difference images between two echoes were obtained to compute ΔB₀ maps. The phase difference images were obtained by applying the angle function to the coil-combined complex difference between the two complex echo signals using the HiP according to eq. 7³⁷.

$$\Delta \theta =\angle \left[\frac{1}{N}\sum _{m=1}^N{\lambda }_m{S}_m^{*}\left(T{E}_1\right)S_m\left(T{E}_2\right)\right]$$ [7]

Here, $S_m^{*}\left(T{E}_1\right)$ is the complex conjugate of the complex signal from the m-th channel at the first echo time TE₁. S_m(TE₂) is the complex signal from the m-th channel at the second echo time TE₂. N is the number of total channels, which was equal to 16. λ_m is a weight factor that accounts for varying noise in different coil channels³⁸ and was computed according to eq. 8.

$${\lambda }_m=\frac{\frac{1}{N}\sum _{k=1}^N{\sigma }_k^2}{{\sigma }_m^2}$$ [8]

In the equation above, σ_m is the noise amplitude in the corresponding m-th channel. For each channel, σ_m is calculated by computing the root mean squared error of pixel intensities from a 10×10 pixel area in the background region of the magnitude images³⁸.

Once obtained the Δθ maps, ΔB₀ maps were obtained using eq. 2, for qDESS, and using the 2-GRE method (eq. 3) for FGRE data.

ΔB₀ map analysis in femoral cartilage

The ΔB₀ maps were evaluated in the femoral cartilage (FC). The pipeline used to obtain the ΔB₀ maps in the FC with qDESS is summarized in Fig. 1 (C). For both knees, after coil-combination and phase difference computation, the femoral cartilage was segmented using the Deep Open-Source Medical Image Analysis (DOSMA) framework^39,40(https://github.com/ad12/DOSMA). When the 3D phase maps presented wraps, 3D phase unwrapping was used to perform 3D phase unwrapping. Since multiple phase unwrapping algorithms were tested, details are given in the following subsection. After phase unwrapping, the qDESS ΔB₀ maps in FC were computed using Eq. 2 and projected onto a 2D space for visualization⁴¹. The FC was automatically subdivided into 6 sub-regions: anterior/central/posterior compartments for the medial and lateral condyles.

WASSR and FGRE images were registered to qDESS using Elastix^42,43. The FC segmentation mask computed from qDESS was then applied to the registered WASSR and FGRE images. The 3D ΔB₀ maps of FC were then obtained and processed with DOSMA as described above.

Phase unwrapping

Due to the relatively narrow wrap-free bandwidth of qDESS phase difference (see 4), phase unwrapping is likely to be necessary for restoring the correct phase difference and thus obtaining accurate ΔB₀ maps.

To investigate the most reliable phase unwrapping algorithm, we performed phase unwrapping of in vivo knee qDESS phase difference maps with 3 different algorithms: Phase Region Expanding Labeller for Unwrapping Discrete Estimates⁴⁴ (PRELUDE), Speedy rEgion-Growing algorithm for Unwrapping Estimated phase⁴⁵ (SEGUE) and the Rapid Opensource Minimum spanning treE algOrithm⁴⁶ (ROMEO). PRELUDE belongs to the region-growing spatial unwrapping approach class. Although it is considered the gold standard, it is computationally demanding and masking of the image is normally required to speed up computation. SEGUE is based on PRELUDE, but it can reduce computation time by simultaneously unwrapping and merging multiple regions. In contrast, ROMEO is a recently proposed phase unwrapping method belonging to the class of path-following approaches, which usually provide solutions within seconds, even for highly wrapped images with large matrix sizes. Thus, masking is not necessary.

For each algorithm, the phase unwrapping procedure was tested with and without applying a cartilage mask to the 3D volume. However, a mask removing the background and low-intensity regions (such as the bones) was applied to reduce computation time for PRELUDE and SEGUE. The background mask was automatically obtained by applying intensity thresholding to the 3D volume of the first echo of qDESS. The cartilage mask was added to the background mask to ensure that no pixels belonging to cartilage was wrongly removed by the binary operation.

Successful phase unwrapping was assessed by manually comparing the unwrapped phase maps with the reference WASSR B₀ maps. Since eq. 2 describes the relationship between Δϕ and ΔB₀ for qDESS it is possible to verify if phase unwrapping was successful.

Statistical analysis

The comparison between ΔB₀ maps was quantitatively assessed by exploiting pixel-wise difference maps, the mean deviation (MD), root-mean-squared-deviation (RMSD) and Bland-Altman (BA) analysis. The Lin’s concordance coefficient⁴⁷ (ρ_c) was also evaluated as a global metric for assessing the agreement between ΔB₀ maps. For the in vivo acquisitions, the BA analysis was performed considering the average ΔB₀ values measured in the 6 FC sub-regions. All statistical analysis was performed using in-house scripts written in Python 3

Results

Bloch simulations

Figure 3 reports the RMSE and ME between qDESS estimated ΔB₀ values and ground truth values as a function of the noise variance for different combinations of TE₁ and TE₂. The ME was always below 0.5 regardless of the level of noise that was injected into the data. This implies that there’s no bias between qDESS ΔB₀ values and ground truth values, and the method was very robust to noise in terms of bias. Regarding the RMSE, it rapidly fell below 5 and 2Hz when the SNR of the first echo was above 30 and 70, respectively. Once fixed the first echo time, increasing the value of the second echo time lowered the SNR, which increased the RMSE. Using a TE₁ equal to 15ms yielded the best SNR for the first echo, which resulted in the best RMSE values despite the SNR for the second echo being the worst among the other values of TE₁ used.

Figure 3: Robustness to noise investigated with Bloch simulations.

RMSE (blue line) and ME (orange line) between qDESS estimated ΔB₀ values and ground truth values as a function of the noise variance for different combinations of TE₁ and TE₂. The coordinate axis is expressed as the reciprocal of the noise variance. The top coordinate axis reports the corresponding SNRs for the first and second echos.

Validation in phantoms

Agar phantom

The qDESS ΔB₀ maps obtained using different combinations of TE₁s and TE₂s are reported in Fig. 4 along with their difference with the reference ΔB₀ (left column). Due to the small range of ΔB₀ values (−15 to 15Hz), phase unwrapping was not necessary. Furthermore, isolated dark red spots in the difference maps are due to the presence of air bubbles in the agar gel phantom (See Figure S1 of supplementary material). All the computed ΔB₀ maps were similar across different (TE₁, TE₂) combinations, and had a small positive MD with the reference values (bias) ranging from 2.3 to 4.6Hz and a RMSE ranging from 2.5 yo 4.8Hz. The highest deviation and RMSE were observed for the pair TE₁ = 5ms and TE₁ = 85ms, which was consistent with what was found in the simulation experiment. Overall, there was good agreement between the qDESS and WASSR ΔB₀ maps as supported by the Lin’s coefficients, which were all greater or equal to 0.72, with the only exception of the map computed from TE₁ = 5ms and TE₁ = 85ms (ρ_c = 0.65) for which SNR was the smallest.

Figure 4: Results of the experiment with the agar phantom.

(left panel) Reference ΔB₀ map obtained with WASSR. (right panel) qDESS ΔB₀ maps for different combinations of TE₁ and TE₂ in the agar phantom. The estimated off-resonance ΔB₀ maps have good agreement with the reference map regardless of the combination of TE₁ and TE₂. The worst agreement is when TE₁ = 5ms and TE₂ = 85ms, which also shows the lowest SNR. Isolated dark red spots in the difference maps are due to the presence of air bubbles in the agar gel phantom (See Fig. S1 of supplementary material).

The results obtained modifying the shimming gradient along the x-axis are summarized in Fig.5. Figure 5 (A) reports the ΔB₀ map obtained using the WASSR method, and the Δθ maps obtained from qDESS for TE₁s equal to 5ms and 15ms. The ΔB₀ map spanned the range −40Hz to 40Hz, with a linear dependence with the x-position. The qDESS phase difference Δθ obtained using TE₁ = 15ms presented phase wraps.

Figure 5: Relationship between Δθ and ΔB₀ evaluated in the agar phantom.

(A) ΔB₀ map obtained using the WASSR method, and the Δθ maps obtained from qDESS for TE₁s equal to 5ms and 15ms. (B) and (C) show the scatter plots between the qDESS phase difference Δθ as a function of the reference ΔB₀ evaluated along a horizontal line across the phantom (black dashed line in (A)). For the case in which TE₁ was set equal to 15ms, the scatter plot is also presented after phase unwrapping was applied to the Δθ map (bottom of (C)). A linear model was fit to each scatter plot and the experimental slope of the model is reported along with the theoretical slope obtained by computing the denominator of eq. 2. The theoretical ΔB₀ wrap-free bandwidth (Δν), for which the qDESS Δθ does not present wraps, is also reported for both the selected TE₁s.

Figures 5 (B) and (C) show the scatter plots between the qDESS Δθ as a function of the reference ΔB₀ evaluated along a horizontal line across the phantom (black dashed line in Fig. 5 (A)). For the case in which TE₁ was set equal to 15ms, the scatter plot is also presented after phase unwrapping was applied to the Δθ map (bottom of Fig. 5 (C)). A linear model was fit to each scatter plot and the experimental slope was compared to the theoretical slope (the denominator of eq. 2). The theoretical ΔB₀ wrap-free bandwidth, for which the qDESS Δθ does not present wraps, is also reported for both the selected TE₁s. For both TE₁ values, the theoretical slope was comparable to the experimental slope: 0.063rads vs 0.060rads, and 0.188rads vs 0.185rads for the case in which TE₁ was set to 5ms and 15ms, respectively. When TE₁ was set to 5ms, the phase difference did not present phase wraps, and the theoretical wrap-free bandwidth was from −50Hz to 50Hz. When TE₁ was set to 15ms, the phase difference presented phase wraps in accordance with the theoretical wrap-free bandwidth, which was from −15Hz to 15Hz. Finally, in both cases, when ΔB₀ was equal to 0 Hz, Δθ was not equal to 0rad but had a small positive value.

Doped water phantom

Figure 6 shows the comparison between the ΔB₀ maps obtained with WASSR, qDESS and the 2-GRE method for the phantoms made of 6 vials of doped water. The ΔB₀ maps are reported along with the difference between the reference WASSR map, and the BA plots. For completeness, the difference between the ΔB₀ map obtained with the qDESS and the 2-GRE methods is also reported. The ΔB₀ values spanned the range (−60Hz, 60Hz). Overall, the qDESS and 2-GRE methods produced ΔB₀ maps that were in good agreement with those obtained with WASSR as visually highlighted by the quantitative maps, the difference maps, and statistically confirmed by the BA plots and corresponding statistics (Fig. 6 (B)). ρ_c was equal to 0.98 and 0.91 with a bias of −1Hz and 9.4Hz for qDESS and 2-GRE, respectively. The LOAs were in the order of ±8Hz for both the qDESS and 2-GRE methods. The qDESS ΔB₀ map was also in good agreement with the 2-GRE map. The Lin’s coefficient was equal to 0.90 with a negative bias of −10.5Hz and LOAs of ±5.2Hz.

Figure 6: Results in Phantom.

[A] B0 maps obtained using the WASSR, qDESS, and 2-GRE methods (first row) and the difference maps with WASSR (second row). The third row shows the difference between B0 maps obtained with qDESS and the 2-GRE methods. [B] Bland-Altman plots for assessing the agreement of qDESS (first row) and 2-GRE (second row) with WASSR. The third row reports the BA plot between qDESS and 2-GRE. The solid line indicates the mean of the differences, while the dashed lines represent the upper and lower limits of agreement (LoA). Lin’s coefficients are displayed in each plot.

Validation in vivo

Phase unwrapping algorithms

Figure 7 summarizes the results and processing times of phase unwrapping with PRELUDE, SEGUE, and ROMEO for cases where at least one was unsuccessful. A sagittal representative slice is reported for the reference ΔB₀ map, wrapped qDESS Δθ map and unwrapped qDESS Δθ maps obtained with the tested algorithms. In 3 out of 10 total knees, at least one phase unwrapping algorithm was unsuccessful.

Figure 7: In vivo results: phase unwrapping algorithms.

Representative sagittal slices for the knees in which at least one algorithm did not correctly perform phase unwrapping. The first column reports the reference ΔB₀ map obtained with the WASSR method. The second column reports the wrapped qDESS phase difference along with the unwrapped maps obtained with: PRELUDE applying a cartilage mask (third column), SEGUE applying a cartilage mask (fourth column), SEGUE without applying a cartilage mask (firth column), ROMEO applying the cartilage mask (sixth column) and ROMEO applying without applying the cartilage mask. For each algorithm, the time required to process a knee is also reported.

• PRELUDE. It was not possible to assess PRELUDE unwrapping accuracy without using the cartilage mask. No results were obtained in 5+ hours when not using the cartilage mask. When the cartilage mask was used, 1 over 10 knees were wrongly unwrapped (third row, third column in Fig. 7).

• SEGUE. Using the cartilage mask provided the same result as PRELUDE. However, without applying the cartilage mask, SEGUE successfully unwrapped all the knees.

• ROMEO. In 2 over 10 cases, ROMEO wrongly estimated the unwrapped phase when not applying the cartilage mask. Interestingly, the two knees belonged to the same subject (first and second row in Fig. 7), for which all the other methods correctly unwrapped the phase maps. On the other hand, with the cartilage mask, the phase was correctly unwrapped in all knees.

Based on these results, we concluded that SEGUE, run without applying the cartilage mask, and ROMEO, run applying the cartilage mask, were the most reliable methods. SEGUE was utilized to obtain the qDESS ΔB₀ maps. Representative ΔB₀ maps of sagittal slice for WASSR, qDESS, and 2-GRE methods are reported in Figure 8.

Figure 8: In vivo results: representative sagittal slices for a subject.

Left (A) and right (B) representative ΔB₀ maps of a saggital slice obtained using the WASSR, qDESS and 2-GRE methods for a simultaneous bilateral knee acquisition. The WASSR and 2-GRE images were registered to the qDESS images. The maps are shown after the application of a mask that removed the background and regions of low signal intensity.

ΔB₀ map comparison in articular cartilage

Figure 9 shows the comparison between the ΔB₀ maps obtained with WASSR, qDESS, and the 2-GRE method for the left and right knees of a healthy subject. For each knee, the 2D projections of the ΔB₀ maps are reported along with their difference with the reference WASSR map and the BA plots. Overall, the qDESS and 2-GRE methods produced ΔB₀ maps in good agreement with those obtained with WASSR as visually highlighted by the 2D projected maps, and statistically confirmed by the BA plots and corresponding statistics (Fig. 9 A2 and B2). For the left knee, ρ_c was equal to 0.98 and 0.92 with a bias of −2.1Hz and 6Hz for qDESS and 2-GRE, respectively. The LOAs were on the order of ±6 Hz for both the qDESS and 2-GRE methods. For the right knee, ρ_c was equal to 0.74 and 0.85 with a bias of −8.1Hz and 5.3Hz for qDESS and 2-GRE, respectively. The LOAs were on the order of ±6Hz for both the qDESS and 2-GRE methods. Overall, although comparable in magnitude, the qDESS method had a negative bias with WASSR, whereas the 2-GRE method had a positive bias with WASSR.

Figure 9: In vivo results: example of ΔB₀ in FC.

Left (A1) and right (B1) unrolled 2D ΔB₀ projection maps in FC obtained using the WASSR, qDESS and 2-GRE methods for a simultaneous bilateral knee acquisition. A pixel-wise unrolled 2D difference map with WASSR is also displayed. BA plots for the left (A2) and right (B2) knees to evaluate the agreement between the WASSR method and the qDESS and 2-GRE methods, respectively. The BA plots were made using the average ΔB₀ values measured in the 6 sub-regions of FC.

Figure 10 shows the BA plots for the comparison of the reference average ΔB₀ values in the 6 cartilage regions for 5 subjects, obtained with the WASSR method, with the corresponding average ΔB₀ values obtained with the qDESS (Fig. 10 (A)) and 2-GRE (Fig. 10 (B)) methods, respectively, produced by pooling together the data for all 5 subjects. Left and right knees were considered separately. For completeness, the comparison between ΔB₀ values obtained with the qDESS and 2-GRE methods is also reported (Fig. 10 (A)). Regarding the comparison between qDESS and WASSR, for both the left and right knees, a negative bias of about −10Hz was observed with LOA in the order of ±10Hz. ρ_c was equal to 0.95 and 0.90 for the left and right knees, respectively. Based on Lin’s concordance coefficients, an overall very good agreement was found between the qDESS and the WASSR methods. The agreement was comparable to that observed between the 2-GRE and the WASSR methods in terms of LOA (in the order of ±10Hz) and ρ_c (0.90 and 0.98 for the left and right knee, respectively). Conversely to the qDESS method, a positive bias with the WASSR method was observed for the 2-GRE method, which was 11.7Hz and 2.8Hz for the left and right knees, respectively. The agreement between the qDESS and the 2-GRE methods is less strong than when each method is compared to the reference WASSR method. The Lin’s coefficient is 0.77 and 0.87 for the left and right knee, respectively.

Figure 10: In vivo results: pooled Bland-Altman plots.

BA plots for the comparison of the reference average ΔB₀ values in the 6 FC regions, obtained with the WASSR method, with the corresponding average ΔB₀ values obtained with the qDESS (A) and 2-GRE (B) methods, respectively. (C) BA plot between ΔB₀ values obtained with the qDESS and 2-GRE methods. The BA plots are made by pooling together the average ΔB₀ values of all the 5 subjects measured in the 6 sub-regions of FC, considering left and right knees separately.

Discussion

The Bloch simulations and the experiments run on the agar phantom validated eq. 2 and investigated the impact of SNR on ΔB₀ accuracy. The slope of the linear relationship between ΔB₀ and the phase accumulated between the first and second qDESS echos, Δθ, was proportional to 2TE₁. Thus, in accordance with the theory, for the qDESS B₀ mapping method, the choice of TE₂ does not affect the relationship between ΔB₀ and Δθ. In the Bloch simulations, the qDESS method showed good noise robustness for different combinations of TE₁ and TE₂. We run the simulations for a spin ensemble characterized by relaxation parameters similar to those of articular cartilage, which was the target of interest in this study. Thus, since SNR will depend on the actual relaxation parameters, the performance of the method might be different for the same combination of TE₁ and TE₂. However, the results reported in Fig. 5 showed good robustness to noise for the same combination of TE₁ and TE₂ tested with Bloch simulations, even for a compound with different relaxation properties than cartilage.

In addition, in the agar gel phantom, when ΔB₀ was equal to 0Hz, Δθ was not equal to 0rad, which is the theoretical Δθ value for ΔB₀ = 0Hz according to eq. 2. This is because the reference ΔB₀ was evaluated with WASSR, for which a bias (order of a few Hz) with the qDESS method was observed both in phantom and in vivo acquisitions. Although the origin of the bias was not directly investigated, biases among B₀ mapping methods have been previously observed in the literature⁴⁸. The actual phase accumulation in qDESS can be altered by hardware imperfections such as eddy currents. Furthermore, in the agar phantom, we observed that using long TE₁ and TE₂ produced a stronger agreement with WASSR than using shorter echo times. This might be because, with a longer time for phase accumulation, phase biases introduced by hardware imperfections might be mitigated more. At the same time, utilizing longer echo times increases scan time and exacerbates phase unwrapping. Nonetheless, ΔB₀ maps evaluated using the proposed qDESS method were in good agreement with those obtained using the WASSR method, both in phantom and in-vivo. The observed bias with the reference WASSR method is considered acceptable with the aim of implementing a B₀ correction for qDESS T₂ mapping. The Bloch simulations investigating the effect of ΔB₀ off-resonances in qDESS T₂ estimation, reported in Fig. 2 (B), showed that an error in ΔB₀ estimation of about ±10Hz underestimates or overestimates T₂ by less than 1ms, which is below the reproducibility of T₂ estimation with qDESS. Furthermore, the agreement between qDESS and WASSR was comparable to that of a standard 2-GRE method and WASSR.

The reported results demonstrated that we could obtain a reliable ΔB₀ map using qDESS, a widely applied sequence in knee MRI that allows combining high-resolution morphological information with quantitative T₂ mapping. The ability to obtain ΔB₀ map can further benefit the impact of qDESS for qMRI knee imaging protocols for multiple reasons. First, the Bloch simulations shown in Fig.2(B) showed that when ΔB₀ was above the ±60Hz range, the error in T₂, measured with qDESS, was higher than 1.5ms. This error may be relevant as studies investigating the longitudinal variation of T₂ in cartilage in knee OA have found variations in the order of 1.5 – 2ms^49,50. Similarly, longitudinal studies of T₂ in knee cartilage after ACL reconstruction for over a period of 24 months have observed average differences of about 5 ms between T₂ at 6 months after surgery and at 24 months⁵¹. As we observed ΔB₀ values above the ±60Hz range in each subject, these preliminary considerations suggest that taking into account ΔB₀ in qDESS T₂ mapping might be beneficial for accurate quantification of longitudinal changes. The proposed ΔB₀ mapping method will allow such a correction without the need for additional scan time, with the benefit of having inherently co-registered images.

Future studies could systematically investigate the impact of a B₀ correction on qDESS T₂ mapping, which may be particularly relevant for simultaneous bilateral knee imaging, where B₀ is normally less homogeneous compared to unilateral acquisitions. Furthermore, our method can be used to shorten qMRI protocols that require an ancillary scan for B₀ mapping to correct the quantitative analysis, such as gagCEST imaging, DWI and to check if the right condition exists for effective fat saturation, chemical shift-based imaging methodologies or simultaneous fat-water imaging. Since a qDESS sequence is normally already acquired in a knee MRI protocol, the saved scan time may allow for cheaper MRI exams, or it can be exploited to enrich the MRI protocol with sequences that will generate information that are biologically meaningful instead of merely correct for hardware imperfections.

The fact that the dynamic range of off-resonances for which qDESS Δϕ does not present wraps is considerably smaller than a standard 2-GRE method can potentially be a limitation of the proposed methodology. In our study, since TE₁ was set equal to 5 ms, the field dynamic range was ± 50 Hz. For most of the knees, off-resonance values reached values in the order of 60–80 Hz in the trochlea region of FC, while in other regions, the values were within the field dynamic range for which phase wraps did not occur. By comparing the performance of three state-of-the-art phase unwrapping algorithms it was interesting to observe that only ROMEO and SEGUE, run with and without and applying the cartilage mask, respectively, correctly unwrapped all the Δϕ maps. However, since SEGUE is based on PRELUDE, it is reasonable to think that the same result would have been achieved by PRELUDE without applying the cartilage mask to the phase difference map. Nonetheless, even without considering the background and areas of low intensity (such as bones), we were unable to perform phase unwrapping in a reasonable time without applying the cartilage mask with PRELUDE. Thus, our experiments showed that phase wraps could be a potential limitation for effective qDESS ΔB₀ mapping in MSK, but with the proper phase unwrapping algorithm, we were able to restore the correct phase difference maps for all the knees under study. However, the application of the proposed method to body regions where a higher B₀ heterogeneity is expected, such as the hip, might be further challenged by phase wraps that might not be easily recovered. One solution would be selecting a shorter TE₁, which would increase the field dynamic range and mitigate phase wraps. However, there are limitations to the shortest TE₁ that can be used, and, depending on the application, such a short TE₁ might not be ideal for SNR optimization and T₂ mapping accuracy. Future studies should investigate the application of the proposed method in other regions of the body to better characterize the limitations of the current method and the feasibility of strategies to overcome such limitations.

Conclusions

A method for accurately measuring B₀ field inhomogeneity with qDESS was proposed and validated in phantom and in-vivo against the gold standard WASSR B₀ mapping method. The proposed qDESS method provided ΔB₀ maps that were in good agreement with those obtained using the reference WASSR method both in phantom and in-vivo. The proposed methodology may allow for implementing a B₀ correction method for qDESS T₂ mapping using an inherently co-registered ΔB₀ map without additional scan time. More generally, the method may help shorten knee imaging protocols requaring an auxiliary ΔB₀ map, such as gagCEST imaging, by exploiting a qDESS acquisition that also provides T₂ measurements and high-quality morphological imaging.

Supplementary Material

SUPINFO

Figure S1 qDESS first (left) and second (right) echoes of a representative slice of the agar gel phantom. Spots of lower signal intensity within the phantom are due to air bubbles.

Data Availability Statement

The MRI data that support the findings of this study are openly available in Zeonodo at http://doi.org/10.5281/zenodo.7034725.