Multiecho chemical shift–based water-fat separation methods have seen a recent increase in clinical use (1–6), particularly in challenging applications where inhomogeneous magnetic fields cause failure of conventional fat saturation methods. Dixon (1) first used in-phase (IP) and out-of-phase (OP) images to analytically calculate the water and fat images, in the so-called the 2-point Dixon method. Glover (2) and Glover and Schneider (3) then extended the idea to collect three echoes such that the water-fat separation can be performed with the correction for B0 field inhomogeneity. In the last decade, numerous variations have been proposed based on the 2-point and 3-point Dixon methods. These previous methods assumed a relatively simple signal representation that models both water and fat as a single resonant frequency. For most applications, this is a satisfactory model and excellent qualitative water-fat separation can be achieved.
Although water is well modeled by a single frequency, this is not true for fat. In general, it is assumed that fat resonates at a single frequency ∼3.5 ppm downfield from water (approximately 210 Hz at 1.5T, and 420 Hz at 3T). However, it is well known that fat has a number of spectral peaks (7–17). In particular, the spectral peak from olefinic proton (5.3 ppm) is close to the water resonant frequency, which will manifest as a baseline level of signal within adipose tissue on the separated water images (2, 14, 16). This effect is also commonly seen on images acquired with either conventional fat saturation (18) or spatial-spectral excitation (19). In general, this small signal within the fatty tissues is clinically acceptable for most qualitative applications. However, the incomplete suppression of fat in the water image may reduce the desired contrast between the water tissue and the surrounding fatty tissues. In addition, there has been growing interest in the use of chemical shift–based methods for quantification of fatty infiltration of organs, such as the liver (hepatic steatosis) (20), and vertebral bodies (16, 21). In these situations the multiple spectral peaks of fat must be considered. While the impact of the fat spectral peaks on 2-point or 3-point water-fat separation has been recognized and discussed in the past (2, 14), a modification of the algorithm to include the fat spectral peaks has not been demonstrated. In this work, we describe an improved method for 3-point water-fat separation by modeling a more accurate fat spectrum that consists of multiple spectral peaks. We combined this model with the iterative decomposition of water and fat with echo asymmetry and least-squares estimation (IDEAL) (6, 22) algorithm. The modified IDEAL processing that includes multipeak fat modeling is referred to as multipeak IDEAL (MP-IDEAL) in this work. With MP-IDEAL we demonstrate that greatly improved fat suppression can be achieved with improved contrast between nonfatty tissues and the adjacent adipose tissue.
The T decay is typically neglected in conventional 2-point or 3-point water-fat separation methods because T is much longer than the echo times (TEs) in most applications. However, for quantitative applications or imaging with substantially shortened T, it is important to consider the effects from both fat and T, as they may interfere with the estimation of each other. Therefore, a method to simultaneously estimate fat and T can be useful. Glover (2) and Ma et al. (23) showed that T can be estimated from the signals with echo shifts of 0 and 2π in multipoint fast spin-echo (FSE) acquisitions. However, these techniques require a specific sampling pattern and use only part of the collected signals for T estimation, and therefore they are not signal-to-noise ratio (SNR)-optimal. More recent approaches use multiecho gradient-echo sequences and perform nonlinear curve-fitting to an assumed signal model (17, 24–26). We have shown that the IDEAL method can be modified to estimate a common T of water and fat in addition to water and fat separation, using an algorithm called T-IDEAL (24, 26). Bydder et al. (17, 25) proposed a comprehensive signal model with different Ts for water and fat, which is solved by a sophisticated nonlinear curve-fitting algorithm and carefully chosen fitting parameters. These gradient-echo-based techniques permit flexible and rapid acquisitions, and have great potential as biomarkers of chronic liver diseases where fatty infiltration and iron overload may coexist (27–30). However, as pointed out by multiple studies (2, 10–12, 15, 17, 31), the presence of fat spectral peaks can effectively increase the linewidth of the main fat peak, causing an apparent shortened T decay. To correct for this error, Wehrli et al. (15) acquired reference fat signals in subcutaneous fat tissues. Bydder et al. (10) employed a spectroscopy-measured fat spectrum in the signal model. In another study from the same group, it is tentatively estimated that the fat spectral peaks cause an apparent R increase of 1/12 ms−1, which is then included in the model as the R difference of water and fat (17). In this work, we will study the impact of the fat spectral peaks on the T-IDEAL technique. We will show that the presence of the fat spectral peaks will result in an overestimated T decay within tissues containing fat for a 6-point T-IDEAL acquisition. Like the MP-IDEAL method, the T-IDEAL algorithm is modified to model a more accurate fat spectrum. In addition to utilizing a predetermined fat spectrum model, we also propose a novel method to estimate the relative amplitudes of fat spectral peaks from the 6-point data directly. We show that this enhancement results in substantially improved T estimation with no additional echoes needed.
Figure 1 shows two representative fat spectra acquired in a phantom containing peanut oil and subcutaneous fat of a knee, both at 3T, showing very similar appearances. At least six discrete peaks are identified, whose chemical shift frequencies (at 3T) relative to the water resonant frequency are labeled. Although the majority of the signal lies in the main peak (peak 1, CH2) at 420 Hz, there is a substantial amount of signal contained within the other peaks. Therefore, a signal model that assumes fat resonates at a single frequency is inaccurate.
MP-IDEAL Signal Model and Reconstruction
From spectra such as those seen in Fig. 1, we can measure the relative chemical shift of the different peaks. In addition, if the relative signal contributions from each peak (relative amplitudes) are measured, we can also use this information to improve the accuracy of the signal model without increasing the number of unknowns in our reconstruction. We can write the signal from a voxel containing a mixture of water and fat as:
where fp is the resonant frequency of the pth fat peak (p = 1, …, P) relative to water, and αp is the relative amplitude of the pth peak, such that ∑αp = 1. For a group of N echoes measured at specific TEs, tn (n = 1, …, N), we can write Eq.  in a matrix format:
and diagonal matrix
Compared to the signal model used in conventional IDEAL (22), the second column of the A matrix is changed from a single exponential term (e) to a weighted sum of exponentials ((∑ αpe). Therefore, ρ can be estimated by following the IDEAL algorithm (22). First, the field map (ψ) is estimated using a modified version of the iterative field map estimation method (22), where e is replaced by ∑ αpe. The phase shifts introduced from the field map (D(ψ) matrix) are then demodulated, and final estimates of water and fat are obtained by performing the Penrose-Moore pseudoinverse (22), i.e.:
where the superscript “H” is the complex Hermitian transpose operator, and we have used the property of diagonal matrices and complex exponentials, i.e., D−1(ψ) = D(−ψ).
T-IDEAL and R Overestimation in Fatty Tissue
Various methods have been proposed to achieve T estimation in the presence of fat (2, 17, 23, 25, 26, 32), where nonlinear curve-fitting algorithms are often involved (17, 25, 26, 32). In the T-IDEAL algorithm, both the B0 field inhomogeneity and the T decay are modeled in a “complex field map” term, = ψ + jR2*/2π, enabling modification of the IDEAL algorithm (22) to estimate R in addition to water and fat (26). We have used a six-echo acquisition strategy to achieve a clinically acceptable scan time (20∼30-s breath-hold) (26), with an echo train typically extending to approximately 10 ms at 1.5T. However, as reported by other studies (2, 10, 11, 15, 17, 31) and also illustrated in Fig. 2a, the fat spectral peaks can have confounding effects on the estimation of R within fatty tissues. The signal curve of 16 echoes acquired in an oil phantom (dotted black line) does not follow a simple exponential decay. The high-frequency oscillatory pattern of the curve (dashed arrows) comes from the presence of peak 3 in Fig. 1. More importantly, the signal curve is modulated by a low-frequency oscillation, featured by a low point at the sixth echo near 10 ms. This effect arises from the signal at the peak 2, which completes a π phase shift from in-phase to out-of-phase with respect to the main peak near 10 ms. As a result, if only the first six echoes are used for R estimation, the effect from the peak 2 can simulate a fast T decay, resulting in an overestimated R (T = 8 ms, the solid blue curve). This shortened T in fat can be approximated as a consequence of effective spectral broadening caused by the fat spectral peaks (17). The effect will be clinically relevant when assessing liver iron level in fatty liver patients.
MATERIALS AND METHODS
The MP-IDEAL reconstruction described above assumes that the fat spectrum is known a priori. One option to provide the required information is to assume that the frequencies and amplitudes of the fat peaks can be measured once and considered to be constant for a defined set of conditions (range of body parts, pulse sequence type, and parameters). The fat spectrum can be measured by MR spectroscopy. The frequencies of the fat peaks (fp) can be accurately determined from this spectrum. The normalized areas of the peaks represent the relative amplitudes in Eq.  (αp), normally measured with spectroscopy analysis software (e.g., jMRUI (17)). However, in our studies, it was found that some spectral peaks are close to each other and possess broad linewidths, making it difficult to differentiate them.
Given the knowledge of the frequencies of the fat side peaks as determined by spectroscopy, another method to determine the relative amplitudes a priori is to use an oversampled multiecho acquisition and the multispecies IDEAL reconstruction. In practice, 16 echoes can be acquired using a multiecho sequence to collect the echoes as rapidly as possible in a single repetition. It has been demonstrated previously that the IDEAL algorithm can be extended to fit for multiple chemical species (22). By treating each of the fat peaks as an independent chemical species with known frequency locations (fp), a 16-point IDEAL reconstruction can fit for the six fat peaks plus the water peak independently, providing separate images for each fat peak as well as an image of the water peak. A region of interest (ROI) is then drawn in the fat area and the pixel intensities in the ROI are averaged to form the final, calibrated relative amplitude αp at each peak frequency. Figure 2b shows an example of using 16 echoes to fit for six fat frequencies independently. The resynthesized signals using the calibrated αp are plotted in solid green, which demonstrates an excellent fit to the acquired data. This spectrum precalibration needs to be performed only once and the calibrated spectrum can be used in all MP-IDEAL reconstructions that fit the predefined criteria for that precalibration.
Spectrum Self-Calibration for 6-point T-IDEAL Acquisitions
As described above, the spectrum precalibration approach uses a predetermined fat spectrum for all scans. The frequency locations of the fat peaks (fp) are well documented in various studies (7, 9, 10, 12, 13, 33). We found that the frequency locations of peaks 1–4 vary in a 10-Hz (at 3T) range measured by spectral data collected in peanut oil, subcutaneous fat, and bone marrow. Therefore, they are considered relatively constant between scans in this work. However, the relative amplitudes αp may, in principle, change between different subjects, as suggested by a previous study (17). In addition, the fat peaks may not possess identical relaxation parameters (11, 13, 15), so αp may appear to vary for sequences with different T1 and T2 weighting. To reduce the reconstruction's sensitivity to possible fat spectrum variation, we introduce algorithms below to calibrate the relative amplitudes of three primary fat peaks directly from the acquired data set. This “spectrum self-calibration” is based on the assumption that all fat-containing pixels in a data set can be characterized by the same fat spectrum, i.e., αp and fp are spatially invariant.
We first describe the spectrum self-calibration algorithm for the 6-point T-IDEAL acquisition. Unlike the 16-point acquisition used in the spectrum precalibration approach described above, where there are sufficient measurements to fit for six fat frequencies, it is challenging to accurately estimate the relative amplitudes of all frequencies with only six samples (6-pt) of data, and further simplification is needed. As can be seen from the spectra shown in Fig. 1 and the precalibrated spectrum in Table 1, peaks 1–4 have substantially more energy than peaks 5 and 6. Therefore, peaks 5 and 6 are neglected. Additionally, although peak 4 has appreciable signal, it is close to the main peak, and therefore its effect is small for the echo train duration (typically 10 ms) used in T-IDEAL. With these considerations, only peaks 1–3 are included in our 6-point spectrum self-calibration procedure. If more echoes are collected, more peaks can be included in the self-calibration algorithm. With this simplified three-frequency spectrum model, the following self-calibration procedure, which is similar to the spectrum precalibration algorithm, is performed as illustrated in Fig. 3.
Table 1. Values of the Precalibrated and Self-Calibrated Relative Amplitudes for the Data Shown in Figs. 4–9*
α1, 420 Hz
α2, 318 Hz
α3, −94 Hz
α4, 472 Hz
α5, 234 Hz
α6, 46 Hz
Chemical shift frequencies of the peaks relative to water are labeled for 3T.
1By treating the water (ρw) and three fat peaks (ρf · α1, ρf · α2, and ρf · α3) as four independent species at each pixel, a 6-point T-IDEAL reconstruction is performed, leading to images of the four “species” (Fig. 3b).
2Meanwhile, a conventional IDEAL reconstruction is performed. By comparing the signal intensity on the resulting water and fat images (Fig. 3c), a fat mask is automatically created to select only fat-rich pixels for spectrum calibration (Fig. 3d). A threshold-based algorithm to create such a mask is described in the next section for 3-point self-calibration. However, for 6-point spectrum self-calibration, the choice of the threshold is flexible because the algorithm does not require the pixels in the mask to be fat only.
3The signal intensities at the three fat peaks are normalized at each pixel such that ∑αp = 1,, leading to three αp maps (Fig. 3e).
4The αp values in each map are averaged over all pixels in the fat mask to form the final estimates of αp (Fig. 3f).
Spectrum Self-Calibration for 3-point IDEAL
The spectrum self-calibration algorithm for 6-point acquisitions cannot be directly applied in a 3-point IDEAL reconstruction because there are fewer measurements. The 3-point self-calibration algorithm is performed as follows:
1As in Fig. 3c, we first perform a conventional IDEAL reconstruction (6, 22, 34) on the same data. A water image, a fat image, and a field map are obtained.
2By comparing the fat and water images, a number of pure fat pixels can be found. The water content is neglected in these pixels. The identification of fat pixels can be achieved in several ways. For example, the maximum intensity in the fat image is determined. The fat intensity of all pixels is then compared with this maximum fat intensity. Those pixels whose intensity is larger than a fraction (e.g., 70%) of the maximum fat intensity are included in the spectrum self-calibration. For fat pixels whose signal exceeds the intensity threshold, Eq.  can be simplified to exclude the water signal, i.e.,
where F is a matrix described by the fat peak frequencies and TEs:
Note that with the matrix F, the matrix A in Eq.  can be described as: A = [1|F · α], where 1 is an N × 1 vector with 1 as the value of all elements, denoting the contribution from water, and F · α describes the fat signal formation at the TEs given a multipeak fat spectrum.
3At these fat pixels, we demodulate the signals with the field map obtained from conventional IDEAL in the first step. is estimated from a fat model with a single resonant frequency. However, as we show in the Appendix, because the same fat spectrum model is used in both the fat spectrum self-calibration step and the water-fat decomposition step, the impact of the field-map error on water-fat decomposition is minimal.
4Like the 6-point self-calibration algorithm, the relative amplitudes can be estimated at these fat pixels using the linear least-squares inverse:
Again, utilizing the fact that ∑αp = 1, the relative amplitudes are estimated at each pixel, which are averaged to form the final estimated αp for this data set.
Phantom and In Vivo Experiments
Phantom, volunteer, and patient scans were performed on GE 1.5T TwinSpeed and 3.0T VH/i (HDx; GE Healthcare, Waukesha, WI, USA) MRI systems. Informed consent and permission from our institutional review board (IRB) were obtained for all human scanning. Images were collected using a 2D FSE sequence and a 3D spoiled gradient-echo (SPGR) sequence modified for use with the IDEAL method (6, 35). The TEs that maximize the SNR performance for conventional 3-point water-fat separation were used for 3-point IDEAL scans (6, 35, 36). In addition, a multiecho 3D-SPGR pulse sequence was used to collect 6-point images for simultaneous separation of water, fat, and T using the T-IDEAL reconstruction (26).
Imaging was performed with a variety of applications to demonstrate the improved water-fat decomposition using MP-IDEAL, including knee, ankle, breast, spine, brachial plexus, pelvis, and abdominal scans. For each data set, three online reconstructions were performed using the same source data: conventional IDEAL reconstruction with no multipeak correction, MP-IDEAL using a precalibrated spectrum (precalibrated MP-IDEAL), and MP-IDEAL using the self-calibrated spectrum (self-calibrated MP-IDEAL). T-IDEAL and its multipeak reconstruction was used in the case of 6-point acquisitions. Images from the three reconstructions were compared and the residual fat signal in the water images was examined as a measure of fat suppression. Contrast-to-noise ratios (CNRs) were obtained in one knee scan to quantify the contrast improvement. Residue maps were also created to visualize and compare the goodness of fit for different methods. In addition, a region-growing algorithm was applied in all reconstructions to avoid water-fat swapping (34).
Representative results from phantom and human scans are shown in Figs. 4–9. The calibrated relative amplitudes are listed in Table 1. Figure 4 is a 1.5T FSE scan of a water-oil phantom with echo shifts from the spin echo of [−0.4 ms, 1.2 ms, 2.8 ms] (6). The conventional 3-point IDEAL reconstruction results in “gray fat” in the water image due to the multipeak fat effect, a common appearance shared by many fat-suppression methods. The fat-signal fraction image, calculated as fat/(water+fat), suggests an averaged 83.0% fat-signal fraction in the fat area by the conventional IDEAL. A complex sum of water and fat has been used in fat-signal fraction calculations to reduce noise bias (37). In contrast, the MP-IDEAL reconstructions with both the precalibrated and self-calibrated spectra resulted in dark fat in the water images. Fat-signal fractions measured by the precalibrated and self-calibrated MP-IDEAL methods are 98.6% and 99.97% in the fat area, respectively. The residue images also demonstrate considerably improved agreement between the acquired signals and the self-calibrated MP-IDEAL signal model.
Figures 5 and 6 present results from three FSE scans in knee, ankle, and spine with IDEAL optimized echo shifts (6). Superior fat suppression was achieved in the MP-IDEAL water images, which is typical for all FSE scans. Precalibrated MP-IDEAL and self-calibrated MP-IDEAL are very similar in terms of quality of fat suppression. The fat-saturation image in the knee scan (Fig. 5d and h) also suffered from the same “gray fat” effect due to the presence of the olefinic proton (peak 3), in addition to the fat-suppression failure from the B0 field inhomogeneity, resulting in poor contrast of the cartilage (arrow in Fig. 5h). The CNR between the knee cartilage and the nearby bone is measured to be 17, 24, and 24 for the conventional IDEAL, precalibrated MP-IDEAL, and self-calibrated MP-IDEAL, respectively. Superior depiction of the cartilage with the MP-IDEAL is demonstrated.
The MP-IDEAL technique was also evaluated with 3-point 3D-SPGR data, and representative results are shown in Fig. 7. For both ankle and breast scans, among the water images produced from the three reconstructions, the self-calibrated MP-IDEAL achieves the best fat suppression.
Figures 8 and 9 demonstrate the improved R estimation in tissues containing fat with the multipeak T-IDEAL reconstruction. Figure 8 is a 6-point abdominal 3D-SPGR scan. The self-calibration algorithm described above for 6-point acquisitions was used. The averaged T value measured in the subcutaneous fat area (arrows) is 10 ms for the conventional T-IDEAL reconstruction, demonstrating the R overestimation effect described in Fig. 2. In contrast, the multipeak corrected T value is 26 ms in the same subcutaneous fat area. The improved R estimation is also confirmed by the residue maps of the two reconstructions, showing markedly better fit with the multipeak model in the fat area. As expected, the T values in liver are the same measured from the two reconstructions.
Figure 9 illustrates the clinical impact of the R overestimation effect. Results from two patient scans are shown. The conventional T-IDEAL reconstruction results in a moderately reduced T of approximately 15 ms in the liver of both patients, suggesting the presence of mild iron concentration (38). However, with the multipeak corrected T-IDEAL, the estimated T is within normal limits (22 ms) for patient 1 (38). The change of the T estimates is caused by the presence of substantial amounts of fat, which biased the estimation of T to a shorter value. In addition, with the multipeak correction, the fat-signal fraction measured in patient 1 increased from 35% to 41%, likely due to the appropriate inclusion of the signal from the fat spectral peaks. On the other hand, the water-fat decomposition of patient 2 shows no fat in the liver, and as a result the T value measured in the liver remains the same (T =15 ms) even with the multipeak correction. The presence of mild iron concentration for patient 2 was also confirmed by biopsy. These examples demonstrate the importance of correcting for the multipeak fat effect when attempting to estimate R in liver patients.
Table 1 lists the values of the precalibrated αp as well as the self-calibrated αp from the data shown in Figs. 4–9. The precalibration procedure utilizes 16 echoes, thus allowing fitting of all six visible peaks. The self-calibration algorithms use a simplified three-frequency spectrum model. As shown in Eq. [A4] in the Appendix, αp estimated from the 3-point self-calibration algorithm is not a direct representation of the relative amplitude at each peak, but is also modulated by the terms contributed from the error in the field map and the spectrum model. Therefore, for 3-point self-calibration, α2 and α3 are in general complex relative to the main peak α1. As also shown in the Appendix (Eq. [A7]), during the multipeak-IDEAL decomposition of a fat-only pixel, the modulation terms in αp caused by the field map and fat spectrum errors can automatically demodulate them out provided that the same three-frequency fat spectrum model is used in the calibration and the multipeak-IDEAL reconstruction. Therefore, it is important to keep the phase information of αp. In practice, αp are normalized such that ∑|αp| = 1. The phase in αp is highly consistent from different data sets with the same echo shifts; for example, the phase values of α2 and α3 from all FSE data (Figs. 4–6) are in close agreement. For 6-point acquisitions, the additional measurements allow fitting for more unknowns, including the field map. Therefore, the estimated α2 and α3 are in-phase with α1 for all three 6-point data.
In this work we have modified the signal model for the IDEAL/T-IDEAL chemical shift-based methods in order to create a more accurate representation of the fat signal. Fat has a complex spectrum with a considerable fraction of its signal resulting from peaks other than the largest peak. In fact, one of the larger peaks (olefinic proton, peak 3 in Fig. 1) is located near the water frequency, which results in some fat signal being incorrectly included in the water image. With more accurate multipeak modeling of fat, improved separation of water and fat is achieved with decreased fat signal in the water images. In addition, improved fitting of the signals within fatty tissues permits more accurate R mapping and T correction of the water-fat separation. However, further validation work is required to quantify the improvement of fat-fraction and R measurements and establish correlation to gold standard techniques. We chose IDEAL/T-IDEAL as the baseline algorithms; however, other nonlinear curve-fitting algorithms, such as AMARES (17, 39), may be used to solve Eq. , thus achieving the same purpose of multipeak correction. One source of error in the T-IDEAL technique is the assumption that the T values of water and fat are the same if they coexist within a voxel (26). In practice, the Ts of water and fat may be different due to their difference in T2 (8). The effective T shortening effect of fat described due to multiple adjacent fat peaks may substantially magnify this difference, leading to increased error from T-IDEAL. Therefore it is very important to correct for the fat spectral peaks in T-IDEAL. For methods that model different T values for water and fat (17, 25, 32), it may not be necessary to correct for the T shortening effect of fat if only the T of water is of interest. When the R difference of water and fat is considered known and constant (e.g., 1/12 ms−1 as suggested by Ref.17), it can be included in the model without increasing the number of unknowns. In this work, the J-coupling effect (peak splitting) between the fat molecule groups (40) is not considered, as the effective signal modulation for the short echo train length (<10 ms) is small compared to the typical T decay. If the frequencies of the peak splitting are known or measured by high-resolution spectroscopy scans, they can be included in the model.
The signal model used in MP-IDEAL requires accurate knowledge of the fat spectrum, including the frequency shifts and the relative amplitudes of the peaks. The fat spectrum can be measured by a separate acquisition, and the precalibrated spectrum is then assumed to be constant and incorporated into the signal model for nonlinear curve fitting, as demonstrated by other studies (10, 17). However, it is possible that the fat spectrum may change from subject to subject and sequence to sequence with different T1 and T2 weightings. Therefore, we also introduced spectrum self-calibration methods, where the relative amplitudes of the three primary fat peaks are estimated directly from the data. Figure 2b suggests an excellent fit of signals using the three-frequency model at the first six echoes. However, in applications where more echoes are collected, it is possible and may be important to include more peak frequencies (e.g., peak 4) in the self-calibration procedure. Like the precalibration methods, the self-calibration methods assume that all fat pixels in the data set possess the same spectrum. Although it was found in a previous study that subcutaneous fat and bone marrow have almost identical spectral components (15), one limitation of this assumption is that it does not allow for the possibility of intradata spectrum variation in different disease states or fatty tissues. However, the modified multipeak model is certainly more representative of the true fat spectrum than the previous single peak model. For all of our self-calibrated MP-IDEAL results, fat appears uniformly dark in all slices in the water images, supporting the assumption that the fat pixels within a data set can be characterized by the same spectrum. While precalibrated MP-IDEAL, in general, is sufficient to allow improved water-fat separation compared to conventional IDEAL, the self-calibrated MP-IDEAL method provided a more consistent and superior performance.
Compared with conventional IDEAL and T-IDEAL, the additional computation cost of the multipeak methods comes primarily from the spectrum self-calibration procedure. In our implementation, the spectrum calibration is performed using the center slice data, and the resultant fat spectrum is used for all slices. After the fat spectrum is determined, the matrix A (Eq. ) and its pseudo-inverse, which rely only on the fat spectrum and TEs, needs to be calculated once and is then used for all voxels. Therefore, the processing time at each voxel is the same for IDEAL and MP-IDEAL provided that the proper precalculated A matrix is used. With these considerations, the reconstruction penalty for self-calibrated MP-IDEAL is equivalent to the time needed to reconstruct one additional slice using conventional IDEAL. For precalibrated approaches in which both the fat spectral frequencies and relative amplitudes are presumed to be known a priori, there is little to no additional computational time penalty.
The modification of the signal model will have an impact on the noise performance of the water-fat decomposition. Glover (2) first defined the effective number of signal averages (NSA) to characterize the noise property of a multipoint water-fat separation method. The optimum echo shifts of a 3-point water-fat decomposition technique were found by Pineda et al. (36) using the Cramer-Rao bounds (CRB) theory, however, based on the model that fat has a single peak. For a decomposition method that models multiple fat spectral peaks, such as MP-IDEAL, it is likely that the optimum sampling strategy may be slightly different from those for the conventional single-peak model. Despite the complication in noise performance, the MP-IDEAL potentially allows a superior CNR in anatomy and lesions due to the improved fat suppression. A complete analysis of the noise performance and CNR for the modified signal model will be addressed in future work.
It is well known that multiecho water-fat separation methods may suffer from water-fat ambiguity, which often causes water-fat swapping, and as a result, region-growing methods that are similar to phase-unwrapping algorithms are often needed (4, 5, 34). In this work, the region-growing algorithm designed for IDEAL (34) is applied without any modification. The region-growing algorithm estimates the field map following a square-spiral trajectory. At each pixel, it aims to obtain a better initial guess for the iteration by averaging the field-map values from the neighboring pixels. Therefore, the algorithm is independent of the model used in the IDEAL iteration. The intrinsic ambiguity arises from the fact that water and fat are both modeled as single peaks separated by 210 Hz at 1.5T. As a result, a fat pixel with −210 Hz off-resonance behaves identically as an on-resonance water pixel. With the multipeak fat modeling, this ambiguity problem should, in theory, be eliminated because now a water pixel with a single resonant frequency should be distinguished from a fat pixel with multiple resonant frequencies regardless of the B0 off-resonance value. In practice, however, 3-point and even 6-point signals may not have sufficient temporal samples to reveal the difference between the two signal evolutions. In particular, local minima still exist; however, they are associated with larger fitting residues. Therefore, although region-growing algorithms are necessary, it is possible to better distinguish local minima and true solutions, making the region-growing algorithm more robust. This is an advantage of applying the region-growing methods specifically developed for IDEAL rather than using general phase-unwrapping algorithms.
Finally, the presence of multiple peaks in the fat spectrum complicates the ability to correct for spatial chemical-shift artifacts. For a multipoint water-fat decomposition method with the single peak model, water-fat recombined images free from the chemical-shift artifact can be obtained by realigning the separated water and fat images in the readout direction. However, when we consider multiple peaks in the spectrum model, each peak will have a different spatial shift in the readout direction. As a consequence, in the fat images there may be blurring due to the different chemical-shift artifact of the different spectra peaks of fat. Further work will address this challenge and will require k-space-based water-fat separation approaches (41, 42).
In conclusion, we have described a new multipeak signal model for water-fat separation and R estimation in the presence of fat. We demonstrated that residual fat signal in the water images can be removed with the 3-point multipeak-IDEAL, leading to improved contrast between the water tissue and the surrounding fatty tissues. Furthermore, R is more accurately estimated in the presence of fat with the multipeak corrected T-IDEAL. The multipeak fat spectrum, characterized by frequency shifts and relative amplitudes, can be measured by a separate scan and can be considered to be known in the modified decomposition algorithms (10, 17). Novel spectrum self-calibration algorithms were also introduced to reduce the technique's sensitivity to potential spectrum variation. Compared to existing water-fat separation methods, the MP-IDEAL and MP T-IDEAL methods offer improved fat suppression and improved R estimation in the presence of fat for applications such as quantification of hepatic steatosis and hepatic iron overload.
In this section we evaluate the impact of using the field map from conventional 3-point IDEAL reconstruction for 3-point spectrum self-calibration, described in Eqs. –. We first assume the real fat spectrum to have P (P ≥ 3) discrete peaks at frequencies fr1, fr2, … frP. The corresponding relative amplitudes are: αr1, αr2, … αrP, with ∑αrp = 1. From Eqs.  and , the acquired signals can be described as:
During the 3-point spectrum self-calibration process, we assume that the conventional 3-point IDEAL results in a field map estimate of . In addition, a simplified fat spectrum that consists of three frequencies is used for calibration: F̂, where:
Therefore, the 3-point spectrum self-calibration using pure fat pixels leads to an estimated set of relative amplitudes that can be described by:
where ρfsc is a scaling factor such that the sum of elements is 1. We have inserted the expression for s in Eq. [A2] and utilized the fact that D(ψ − ) = D(ψ) · D(− ).
As can be seen, in general is not equal to the true relative amplitudes αr. However, we will show in the following that the error in the calibrated spectrum will compensate for itself in the calibration step and the water-fat decomposition step because the same spectrum model (F̂) is used in the two steps.
First we show that a fat pixel can still be correctly identified as a fat pixel. For a pure fat pixel, the acquired signals can be described with Eq. [A2]. The model, however, uses a simplified spectrum F̂ and the associated self-calibrated relative amplitudes . During the water-fat decomposition step, the acquired signals are used to fit the following model:
where and represent the unknown values that are to be determined in the decomposition step, with the criterion of minimizing the least-squares error:
where it should be noted that D( ) · D(ψ − ) = D(ψ). It can be seen that the solutions that will make the least-square error 0 are:
Therefore, a fat pixel can be correctly identified as a fat-only pixel. Note that the solution of the field map is the same as the field map estimated from 3-point conventional IDEAL .
With a mixture of water and fat, Eq. [A7] no longer has an exact set of solutions. We used simulations to study the impact on fat-fraction measurements. A precalibrated spectrum that consists of six discrete frequencies is used as the real spectrum Fr · αr. The synthesized three-echo source signals are produced using this real fat spectrum and water-fat contents with increasing fat fraction. The spectrum self-calibration then is performed with a pure fat pixel. Both the conventional 3-point IDEAL and MP-IDEAL reconstructions are performed with the source signals at each fat fraction. The calculated fat fraction and field map were plotted against the true fat fraction. This procedure was repeated with different TEs, including FSE IDEAL TEs ([−0.4 ms, 1.2 ms, 2.8 ms]) (6, 36) and SPGR IDEAL TEs ([1.9 ms, 3.6 ms, 5.2 ms]) (35). Results are shown in Fig. 10. The conventional IDEAL results in biased fat-fraction measurements as the fat content increases. In contrast, MP-IDEAL-calculated fat-fraction values are in close agreement with the true fat fraction. As predicted from Eq. [A8], in the case of a pure fat pixel, the field-map values from the two reconstructions are identical and the fat fraction measured from MP-IDEAL is one.