k-Space Water-Fat Decomposition With ${\text{T}}_2^{\ast }$ Estimation and Multifrequency Fat Spectrum Modeling for Ultrashort Echo Time Imaging

Abstract

Purpose

To demonstrate the feasibility of combining a chemical shift-based water-fat separation method (IDEAL) with a 2D ultrashort echo time (UTE) sequence for imaging and quantification of the short T₂ tissues with robust fat suppression.

Materials and Methods

A 2D multislice UTE data acquisition scheme was combined with IDEAL processing, including ${\text{T}}_2^{\ast }$ estimation, chemical shift artifacts correction, and multifrequency modeling of the fat spectrum to image short T₂ tissues such as the Achilles tendon and meniscus both in vitro and in vivo. The integration of an advanced field map estimation technique into this combined method, such as region growing (RG), is also investigated.

Results

The combination of IDEAL with UTE imaging is feasible and excellent water-fat separation can be achieved for the Achilles tendon and meniscus with simultaneous ${\text{T}}_2^{\ast }$ estimation and chemical shift artifact correction. Multifrequency modeling of the fat spectrum yields more complete water-fat separation with more accurate correction for chemical shift artifacts. The RG scheme helps to avoid water-fat swapping.

Conclusion

The combination of UTE data acquisition with IDEAL has potential applications in imaging and quantifying short T₂ tissues, eliminating the necessity for fat suppression pulses that may directly suppress the short T₂ signals.

Tissues with relatively long T₂ relaxation components in the human musculoskeletal (MSK) system can be visualized with conventional magnetic resonance imaging (MRI) techniques. However, there are a variety of tissues with short T₂ values that cannot be directly visualized with conventional techniques due to rapid T₂ or ${\text{T}}_2^{\ast }$ signal decay. Such tissues include the deep radial and calcified layers of articular cartilage, menisci, ligaments, tendons, and entheses (1–4). Novel MRI techniques have been developed to visualize these short T₂ tissues in the MSK system (1–5).

One method used to detect the short T₂ signal components is ultrashort TE (UTE) imaging (1–4,6). UTE methods employ a variety of approaches, including short half pulses for slice selection, rapid transmit/receive (T/R) switching, radial ramp sampling of k-space, and variable-rate selective excitation (VERSE) to acquire the signal immediately after radiofrequency (RF) excitation. A minimum echo time (TE) of 8 μs has been achieved (7). In conventional UTE imaging, fat saturation is typically required to improve the contrast for tissues with short T₂ times. However, conventional chemical shift-based fat saturation may be problematic when imaging short T₂ tissues because the broad spectral width of these species may overlap with that of fat. As a result, fat saturation pulse may inadvertently saturate the signal from short T₂ tissues, either directly or as a result of magnetization transfer.

One possible approach to separate fat signal from water in imaging of short T₂ tissues is to combine UTE with spectroscopic imaging (UTESI) (1,8,9). Similar to echo-planar spectroscopic imaging (EPSI) (10), spectroscopic images with high spatial resolution can be obtained by performing Fourier transform in the spectral dimension on a pixel-by-pixel basis, providing spectral information including ${\text{T}}_2^{\ast }$ spectrum, bulk susceptibility effect, water-fat separation, etc. (8,9).

While UTESI has been a successful technique to separate different chemical species (primarily water and fat), chemical shift artifacts from the non-Cartesian UTE technique and B₀ inhomogeneities have not been addressed in previous UTE applications. With conventional MRI, many chemical shift-based decomposition techniques, such as the multipoint Dixon approach (11), direct phase encoding (12), or IDEAL (iterative decomposition of water and fat with echo asymmetry and least-squares estimation) (13–17), have been increasingly used for long T₂ tissues, providing excellent water-fat separation with correction of some of these artifacts. Therefore, new methods, such as a UTE IDEAL approach which combines conventional water-fat separation methods with UTE data acquisition may overcome some of the aforementioned challenges and eliminate the necessity for fat saturation pulses. From both the data acquisition and reconstruction perspectives, the UTE IDEAL approach shares a lot of similarities with conventional IDEAL, and is expected to preserve signal from short T₂ tissue and provide high contrast water-only images as well as ${\text{T}}_2^{\ast }$ mapping.

In this work we present a chemical shift separation method, based on the IDEAL water-fat separation algorithms and UTE data acquisition, to image short T₂ tissues with water-fat separation and ${\text{T}}_2^{\ast }$ estimation. This technique employs multifrequency modeling of fat to improve water-fat separation. The combination of k-space formulation of IDEAL with ${\text{T}}_2^{\ast }$ estimation is also proposed to efficiently correct chemical shift artifacts. This technique was employed to image and quantify the Achilles tendon and meniscus both in vitro and in vivo using a clinical 3T scanner.

Theory

Drawbacks of Fat Saturation for Short T₂ Tissue

Chemical shift-based fat saturation pulses are usually spectrally selective. Because the width of the water peak of short T₂ tissues may be very wide (due to the short T₂) and may overlap with that of the fat peak, a fraction of the short T₂ signal (for example, up to 32% for the Achilles tendon; unpubl. results) may be suppressed by the fat saturation pulse. Another drawback of the saturation pulse is that fat has multiple peaks, some of which are very close to the water peak. Therefore, a spectral selective pulse cannot effectively suppress the signal from all of the fat protons (17), resulting in reduced image contrast for short T₂ tissues.

Brief Review of IDEAL and Its Variations

In 2004 and 2005, Reeder et al (13,14) described the IDEAL technique for water-fat separation. In this method, the B₀ inhomogeneity map is iteratively solved by minimizing the residue of the signal model and acquired images. It is well known that water-fat separation may suffer from water-fat ambiguity, and in 2005 Yu et al (18) presented a region-growing (RG) scheme to obtain a more accurate field map. Since then, a number of different approaches have been introduced to more robustly estimate the field map (19,20). In 2007 Yu et al (15) presented a variation of this method, termed ${\text{T}}_2^{\ast }$-IDEAL, which considers the effective ${\text{T}}_2^{\ast }$ decay between the acquisition of different echoes. ${\text{T}}_2^{\ast }$-IDEAL introduces the concept of the complex field map, whose real part is the conventional B₀ inhomogeneity and whose imaginary part corresponds to the ${\text{R}}_2^{\ast }(=1/{\text{T}}_2^{\ast })$ map. In 2008, a k-space formulation of the IDEAL algorithm, named k-space IDEAL, was described by Brodsky et al (16). The key principle of the k-space approach is that it introduces a timestamp, τ_k⃗,n, to account for the relative time between the center of k-space and the sampled point and then corrects the chemical shift artifacts experienced with non-Cartesian trajectories. In the same year, Yu et al (17) described the use of multifrequency spectral modeling in combination with the ${\text{T}}_2^{\ast }$-IDEAL approach to account for the spectral complexity of fat. This spectral modeling was included in Brodsky et al's work (16) as well. Meanwhile, Bydder et al (21) also described the use of multifrequency spectral modeling for quantification of fat. For the purposes of this work, we will refer to the use of multifrequency fat spectral modeling as “multipeak” (MP) IDEAL.

Combining ${\text{T}}_2^{\ast }$-IDEAL and k-space IDEAL

The reported image space ${\text{T}}_2^{\ast }$-IDEAL can be combined with the k-space IDEAL formulation by implementing the two algorithms sequentially. Specifically, a basic chemical shift decomposition signal model can be mathematically described by:

$$s_n(\overrightarrow{r})=\left(\sum _{m=1}^M{\mathrm{\rho }}_m(\overrightarrow{r})e^{i2\mathrm{\pi }\mathrm{\Delta }{f}_m{t}_n}\right)e^{i2\mathrm{\pi }\mathrm{\psi }(\overrightarrow{r})t_n}$$ [1]

where ρ_m is the intensity of the m^th species and is, in general, a complex term with its own magnitude and phase. Δf_m is the chemical shift (in Hz) for the m^th species which is modeled in Eq. [1] as having a single spectral peak, ψ(r⃗) is the local magnetic resonance offset (in Hz), and n refers to the echo index. If the effect of ${\text{T}}_2^{\ast }$ decay between the acquired echo images is significant, such as in UTE applications where the tissues of interest have very short T₂ relaxation times, an exponential decay term can be added into Eq. [1], as in the ${\text{T}}_2^{\ast }$-IDEAL model:

$$\begin{array}{ll}{s}_n(\overrightarrow{r})& =\left(\sum _{m=1}^M{\mathrm{\rho }}_m(\overrightarrow{r})e^{i2\mathrm{\pi }\mathrm{\Delta }{f}_m{t}_n}\right)e^{i2\mathrm{\pi }\mathrm{\psi }(\overrightarrow{r})t_n}{e}^{-R_2^{\ast }(\overrightarrow{r})t_n}\\ & =\left(\sum _{m=1}^M{\mathrm{\rho }}_m(\overrightarrow{r})e^{i2\mathrm{\pi }\mathrm{\Delta }{f}_m{t}_n}\right)e^{i2\mathrm{\pi }\stackrel{\sim }{\mathrm{\psi }}(\overrightarrow{r})t_n}\end{array}$$ [2]

In the above equation, $R_2^{\ast }(\overrightarrow{r})=1/{\text{T}}_2^{\ast }(\overrightarrow{r})$ and $\stackrel{\sim }{\mathrm{\psi }}(\overrightarrow{r})=\mathrm{\psi }(\overrightarrow{r})+i\frac{R_2^{\ast }(\overrightarrow{r})}{2\mathrm{\pi }}$ which is the complex field map. Here $R_2^{\ast }(\overrightarrow{r})$ or $T_2^{\ast }(\overrightarrow{r})$ is a spatially variant but species-independent parameter; therefore, this parameter assumes that all species share the same ${\text{T}}_2^{\ast }$ value if they coexist in a single voxel (15). This complex field map can be determined iteratively and then demodulated from the source signals, which can be written as:

$${\widehat{s}}_n(\overrightarrow{r})=s_n(\overrightarrow{r})e^{-i2\mathrm{\pi }\stackrel{\sim }{\mathrm{\psi }}(\overrightarrow{r})t_n}=\sum _{m=1}^M{\mathrm{\rho }}_m(\overrightarrow{r})e^{i2\mathrm{\pi }\mathrm{\Delta }{f}_m{t}_n}$$ [3]

Performing a Fourier transform on the image ŝ_n (r⃗) yields the k-space representation of the demodulated signal, ŝ_n (τ_k⃗,nk⃗):

$${\widehat{s}}_n({\tau }_{\overrightarrow{k},n},\overrightarrow{k})=\sum _{m=1}^M{\mathrm{\rho }}_m(\overrightarrow{k})e^{i2\mathrm{\pi }\mathrm{\Delta }{f}_m(t_n+{\tau }_{\overrightarrow{k},n})}$$ [4]

where the new term τ_k⃗,n refers to the relative time between the acquisition of sample point k⃗ and the center of k-space (22-24). Unlike the image space model in Eq. [3] where it is assumed that all signals of an echo are collected at a single timepoint (t_n), the k-space formulation allows the characterization of the sample time difference for each k-space point during the readout. As a result, this method corrects for the phase accumulated during the actual readout time due to the chemical shifts of different species (ie, water and fat). It will be shown later that the k-space formulation can significantly improve the quality of the fat image by removing blurring caused by chemical shift.

Multipeak Signal Model

Although it is often approximated by a single peak, the fat spectrum has multiple spectral peaks. As demonstrated in recent studies (16,17,25), the presence of multiple fat peaks can effectively increase the line width of the main fat peak (3.5 ppm downfield from water), simulating a faster ${\text{T}}_2^{\ast }$ decay. For qualitative water-fat separation, the single peak model often results in “gray fat” in the water image, resulting from incomplete water-fat separation due to fat peaks near the water peak. Assuming that the relative amplitudes of these peaks are measured and known as prior knowledge, the fat spectral peaks can be modeled (“multipeak” model) accordingly. Therefore, the image space formulation, Eq. [2], becomes:

$$s_n(\overrightarrow{r})=\left({\mathrm{\rho }}_w(\overrightarrow{r})+{\mathrm{\rho }}_f(\overrightarrow{r})\sum _{p=1}^P{\alpha }_p\cdot e^{i2\mathrm{\pi }\mathrm{\Delta }{f}_p{t}_n}\right)e^{i2\mathrm{\pi }\stackrel{\sim }{\mathrm{\psi }}(\overrightarrow{r})t_n}$$ [5]

and the k-space formulation Eq. [4] becomes:

$${\widehat{s}}_n({\tau }_{\overrightarrow{k},n},\overrightarrow{k})={\mathrm{\rho }}_w(\overrightarrow{k})+{\mathrm{\rho }}_f(\overrightarrow{k})\sum _{p=1}^P{\alpha }_p\cdot e^{i2\mathrm{\pi }\mathrm{\Delta }{f}_p(t_n+{\mathrm{\tau }}_{\overrightarrow{k},n})}$$ [6]

where Δf_p is the resonant frequency of the p^th fat peak (p = 1, 2, …, P) relative to water, and α_p is its relative amplitude, such that ${\sum }_{p=1}^P{\alpha }_p=1$. In Eqs. [5] and [6] we have already simplified our model by assuming that water and fat are the only two species present in the image, ie, m = 1 means water (w), m = 2 means fat (f), and M = 2.

Materials and Methods

The proposed chemical shift decomposition algorithm, described by Eq. [6], is a k-space formulation of IDEAL with ${\text{T}}_2^{\ast }$ estimation and multifrequency fat spectrum modeling, and can be summarized schematically in Fig. 1. The implementation of this method includes the following steps as denoted in the figure:

Figure 1.

Schematic diagram of the proposed decomposition method; (1–5) correspond to the five steps described in Materials and Methods.

1. Perform the image space IDEAL algorithm with ${\text{T}}_2^{\ast }$ estimation and multipeak fat spectral model (Eq. [5]) on the reconstructed echo images to generate a B₀ inhomogeneity map (ψ(r⃗)) and an ${\text{R}}_2^{\ast }$ map. Advanced field map estimation methods, such as an RG scheme, can be included in this step.

2. Use the B₀ inhomogeneity map and the ${\text{R}}_2^{\ast }$ map to process the complex echo images to obtain demodulated echo sources, ie, ŝ_n(r).

3. Perform a Fourier transform (if Cartesian acquisition) or non-Cartesian sampling operator on the demodulated echo images, generating a k-space representation of these images, ŝ_n(k).

4. Decompose the water and fat signal in k-space to obtain ρ_w(k) and ρ_f(k) using a least-square approach and the actual readout time τ_k⃗,n, which was determined using known pulse sequence timing (Eq. [6]).

5. Perform an inverse Fourier transform or non-Cartesian reconstruction method to generate the final water and fat images.

Institutional Review Board approval and informed consent were obtained prior to all scanning. The proposed method was applied to the Achilles tendon and meniscus using a 2D multislice radial UTE technique in cadaveric ankle specimens (Experiments 1 and 4) and healthy volunteers (Experiments 2 and 3) using a 3 T Signa TwinSpeed scanner (GE Healthcare Technologies, Milwaukee, WI) with a maximum gradient performance of 40 mT/m and 150 mT/m/msec.

In Experiments 1 and 2, single-echo acquisition was used since the Achilles tendon has a short ${\text{T}}_2^{\ast }$ of around 2 msec. For each echo image, 512 half projections were acquired and each half projection contained 284 points sampled at a rate of 8 μs per points. Six echo images were acquired with the center of k-space sampled at TE = 8 μs, 400 μs, 800 μs, 1200 μs, 2000 μs, 3000 μs. Common imaging parameters include: FOV = 10 cm, readout = 512, slice thickness = 3.0 mm, bandwidth (BW) = ±62.5 kHz, NEX = 2, flip angle = 45°, TR =100 msec, and total scan time = 10.6 minutes. For Experiment 3 a multiecho UTE sequence was used for meniscus imaging with improved scanning efficiency. The meniscus has a longer ${\text{T}}_2^{\ast }$ allowing multiecho acquisition. In this experiment, four echoes were acquired with TE = 8 μs, 3.3 msec, 6.6 msec, and 9.9 msec, TR = 500 msec, FOV =15 cm, slice thickness = 3 mm, 22 slices, FA = 45°, BW = ±125 kHz, resulting in a total scan time of 8.5 minutes. Each image contained 512 half projection with 262 sampling points. We also investigated the feasibility of multiecho UTE acquisition for the Achilles tendon in Experiment 4, where a total number of six echo images were acquired at three separate TRs: TE = 12 μs/3.4 msec acquired in the first TR, TE = 0.8 msec/4.2 msec in the second and TE = 1.6 msec/5 msec in the third (9). Taking the shorter ${\text{T}}_2^{\ast }$ of the Achilles tendon into account, we increased the sampling bandwidth to ±125 kHz, reduced the spatial resolution by increasing the FOV to 15 cm, and reducing readout to 384 (262 sampling points) to reduce the echo spacing down to 3.4 msec. Other imaging parameters included: TR = 200 msec, FA = 45°, NEX = 2, 16 slices with slice thickness = 2 mm, 512 half projections, scan time = 10.5 minutes. In this experiment, a fat saturation image obtained with a fat-selective saturation pulse followed by a spoiling gradient was also included for comparison. Slice thickness = 3 mm, TR = 300 msec, FOV =12 cm with 512 readout.

In the reconstruction, half projections that were 180° apart were combined into a full projection prior to standard filtered back-projection (FBP) reconstruction to generate echo source images. In the third step of the proposed method, the demodulated echo images, ŝ_n(r), were reprojected (ie, Radon transformed) and Fourier transformed to produce the radial k-space representation, ŝ_n(k).

Images reconstructed from Experiments 1 and 2 using IDEAL with ${\text{T}}_2^{\ast }$ estimation and multipeak fat model, with and without the k-space formulation, were compared in order to evaluate the importance of the different reconstruction steps. In addition, the significance of the accurate spectral modeling of fat was studied. Subsequently, the role of the RG field map estimation was investigated using data from Experiment 3. Finally, data from Experiment 4 were processed to show the feasibility of achieving scan time reduction using multipass multiecho sequences for UTE IDEAL tendon imaging.

Results

Figure 2 shows the results from Experiments 1 and 2. The image space (Eq. [5]) and k-space IDEAL (Eq. [6]), both with ${\text{T}}_2^{\ast }$ estimation and multipeak modeling, were compared. In both experiments the readout duration was 8 μs/point × 284 points = 2.3 msec, which corresponds to an approximate 2π phase for fat at 3 T As a result, severe blurring artifacts, which manifest as ringing artifacts in radial imaging of fat, are observed in Fig. 2b,c and h,i (arrows). The blurring artifacts are removed using the k-space decomposition approach, as shown in Fig. 2e,f and k,l, which corrects for accumulated phase during readout.

Figure 2.

Comparison of the results from Experiments 1 and 2 using IDEAL with and without k-space formulation. ${\text{T}}_2^{\ast }$ compensation and multipeak model were included for both reconstruction methods. The chemical shift-induced blurring, indicated by the arrows, was significantly reduced by the k-space formulation.

Figure 3 shows the comparison between the single peak (Eq. [4]) and multipeak model (Eq. [6]) using data from Experiment 2. The frequencies and amplitudes of the multiple fat resonant peaks, shown in Table 1, were chosen based on previously reported work (26). The multipeak approach has two effects. First, it helps to suppress the remaining residual fat signal (“gray fat”) in the water image, as indicated by the arrows in the water image. Second, it corrects the chemical shift-induced artifacts more accurately than the single peak model (arrows in fat images).

Figure 3.

Multifrequency spectral modeling of fat spectrum (right) helped eliminating the “gray fat” signal in the water image with single peak fat spectrum model (left), as shown by the arrows in the water images. Multipeak model also corrected the chemical shift artifacts more accurately than single peak model, as shown by the increased sharpness (arrows in the fat images).

Table 1. Resonant Peaks of Fat at 3.0 T (26).

Peak	#1	#2	#3	#4	#5	#6
fp (Hz)	434	332	486	−76	52	250
αp	0.69	0.13	0.09	0.05	0.04	< 0.01

Figure 4 shows images from one slice in Experiment 3. Using a simple pixel independent approach (as defined in Ref. 18), the algorithm yields an incorrect field map and therefore results in water-fat swap. After applying the RG method the field map is correctly estimated throughout the image.

Figure 4.

Water, fat, and field map images from the central slice in Experiment 3. A simple pixel independent approach yielded water-fat swapping (arrows). The RG method provided a more accurate field map and avoided the phase swapping.

Figure 5 shows the ${\text{R}}_2^{\ast }$ map obtained from the first step in Fig. 1 for the first three experiments. A region of interest (ROI) containing 10 × 10 pixels was placed in the central portion of the Achilles tendon and meniscus and their ${\text{T}}_2^{\ast }$ were measured to be ≈2.2 ± 0.24 msec, 1.5 ± 0.20 msec, and 9.0 ± 1.6 msec for Experiments 1, 2, and 3, respectively. The ${\text{R}}_2^{\ast }$ map for Experiment 3 is noisier because only four echoes were acquired (please see Discussion, below).

Figure 5.

The ${\text{R}}_2^{\ast }$ maps from the first three experiments. The tendon (arrows) was found to have a ${\text{T}}_2^{\ast }$ of ≈2.2 ± 0.24 msec for Experiment 1 and 1.5 ± 0.20 msec for Experiment 2. For Experiment 3 the ${\text{T}}_2^{\ast }$ of meniscus, pointed by the arrow, was about 9.0 ± 1.6 msec.

Figure 6 shows comparison between the fat saturation and UTE IDEAL techniques using data from Experiment 4. The water images of all slices from the UTE IDEAL method provides superior fat suppression than the chemical shift-based fat saturation technique, and the fine structures surrounding the tendon are much better depicted.

Figure 6.

Results from Experiment 4, a 2D multislice multiecho and multipass UTE IDEAL scan of an ankle specimen. Images from multiple slices are shown. A fat saturated image is also provided for comparison. UTE IDEAL provides superior fat suppression and better depiction of the tendon and its surrounding structures than the conventional fat saturated UTE acquisition technique.

Discussion

In this work we present a k-space water-fat decomposition method for UTE imaging of short T₂ tissues. The proposed method integrates the advantages of ${\text{T}}_2^{\ast }$-IDEAL, k-space IDEAL, and MP-IDEAL to compensate for ${\text{T}}_2^{\ast }$ decay between echo source images, to correct for chemical shift-induced ringing artifacts in UTE imaging, and to account for the spectral complexity of fat.

In this work we used a single echo acquisition with variable TE delays for tendon imaging in Experiments 1 and 2 to demonstrate the importance of each reconstruction step in the proposed water-fat separation method, such as k-space formulation and multifrequency modeling of a fat spectrum. The strategy to select the TEs is as follows. First, it is clear that collecting more echoes will improve the accuracy of the fitting of all parameters, especially ${\text{R}}_2^{\ast }$. However, due to scan time limitations, we found that six-echo acquisitions in 8–10 minutes achieved a good balance between total scan time and accuracy in ${\text{T}}_2^{\ast }$ fitting. Second, since the imaging object contains tissue with very short ${\text{T}}_2^{\ast }$ (≈2 msec), we chose the first echo time, TE₁, to be the shortest echo time achievable using our UTE technique (8 μs). Third, as suggested in Ref. (15), when acquiring more than three echoes the noise performance, measured by the effective number of signal averages (NSA), is less sensitive to the change of echo spacing, unless echo spacing is close to 0 or 2π. According to these considerations, we chose TE₁ = 8 μs and ΔTE = 0.4 msec for the first three echoes in order to acquire images with better signal-to-noise ratio (SNR), and ΔTE = 0.8 msec spacing for later echoes in order to allow sufficient phase difference between fat and water and enough decay for better ${\text{T}}_2^{\ast }$ estimation. While single echo acquisition is very robust in terms of avoiding water-fat ambiguity and acquiring more echoes at earlier TEs with higher signal, the TR is limited by scan time and precludes sufficient slice coverage.

Since meniscus has a longer T₂, single-pass, multiecho acquisition is possible to acquire all echoes needed within one TR, as demonstrated in Experiment 3. This allows sufficient slice coverage within acceptable scan time. However, the minimal echo spacing, ie, ΔTE, is limited by the hardware (maximum gradient, maximum slew rate, bandwidth, etc.). For the protocol used in Experiment 3, ΔTE corresponds to a water-fat phase difference larger than 2π. The major impact of this is the increased number of local minima in the residue curve for field map estimations (18). That is why advanced field map estimation methods, such as an RG scheme, are necessary for single-pass multiecho UTE IDEAL acquisition.

In order to increase the TR and slice coverage for tendon imaging, we implemented a multipass multiecho acquisition, as demonstrated in Experiment 4. Since tendon has short ${\text{T}}_2^{\ast }$(≈2 msec), increasing echo train length is not helpful due to low signal at later echoes. For a more robust field map estimation, a three-pass protocol with two echoes per pass was chosen in this work. Since the first three echoes were equally spaced within a 2π water-fat phase difference in Experiment 4, a very basic MP-IDEAL algorithm can be first performed on the earliest three echoes to generate an initial guess of the field map, subsequently incorporating this initial guess in the proposed water-fat separation method (Eq. [6]) to process all the echo images acquired. Other protocols such as two passes with two echoes per pass are also possible. However, because of the large ΔTE and only four echo images acquired, more advanced field map estimation methods are needed and ${\text{T}}_2^{\ast }$ estimation is expected be less accurate.

The major limitation of this work is that it uses two assumptions associated with the signal model described by Eq. [6] in this article. First, it assumes that any species coexisting in a voxel will have the same ${\text{T}}_2^{\ast }$ and this may be problematic for some human MSK tissues or at interfaces between two species. However, the ROIs in this work, ie, tendon and meniscus, are mostly comprised of water. Therefore, species-independent ${\text{T}}_2^{\ast }$ estimation is a valid assumption for tendon or meniscus imaging. Recently, Chebrolu et al (27) provided an approach to estimate ${\text{T}}_2^{\ast }$ for water and fat independently, and can in principle be included in the UTE IDEAL application. Another assumption in this work is that water is a single T₂ compartment in the ROI, which may not be true for tendon or meniscus imaging. Even though we can model water in multiple compartments as multiple independent species in Eq. [6], more echo images are needed due to the increase of unknowns in the equation. One possible approach may include incorporating prior knowledge of the multiple water compartments in the signal model, similar to the current multifrequency modeling of fat spectrum. These two assumptions imply that the ${\text{T}}_2^{\ast }$ obtained by the proposed model is an average value of all the species and all compartments within a voxel.

In conclusion, we have presented a water-fat decomposition method for UTE applications. The proposed method combines a chemical shift-based water-fat separation method that corrects for the effects of ${\text{T}}_2^{\ast }$ decay ( ${\text{T}}_2^{\ast }$-IDEAL) with a k-space formulation (k-space IDEAL) and includes the effects of accurate spectral modeling (MP-IDEAL). This combined UTE IDEAL method provides high contrast imaging of the short T₂ tissues with robust fat water separation, as well as automatic ${\text{T}}_2^{\ast }$ mapping.