Three-dimensional water/fat separation and Tmath image estimation based on whole-image optimization—Application in breathhold liver imaging at 1.5 T

Authors


Abstract

The chemical shift of water and fat resonances in proton MRI allows separation of water and fat signal from chemical shift encoded data. This work describes an automatic method that produces separate water and fat images as well as quantitative maps of fat signal fraction and Tmath image from complex multiecho gradient-recalled datasets. Accurate water and fat separation is challenging due to signal ambiguity at the voxel level. Whole-image optimization can resolve this ambiguity, but might be computationally demanding, especially for three-dimensional data. In this work, periodicity of the model fit residual as a function of the off-resonance was used to modify a previously proposed formulation of the problem. This gives a smaller solution space and allows rapid optimization. Feasibility and accurate separation of water and fat signal were demonstrated in breathhold three-dimensional liver imaging of 10 volunteer subjects, with both acquisition and reconstruction times below 20 s. Magn Reson Med, 2011. © 2011 Wiley Periodicals, Inc.

In magnetic resonance images of the human body, most of the signal originates from 1H nuclei in water and fat molecules. Because of chemical shift, the water resonance is faster than most fat resonances. This allows separation of the signal components through postprocessing of chemical shift-encoded data (1).

Chemical shift-based water/fat separation is of interest for several reasons. First, water-only images may reveal pathology that would have been obscured by the fat signal in conventional images. Second, fat-only images give information about the location of fat, which may be of clinical value (2, 3) and useful in studies of body composition (4). Third, the fat signal fraction of the total signal provides a quantitative measure of fat content (5).

In the original method by Dixon (1), images are acquired with water and fat in-phase and opposed-phase, respectively. Separate water/fat images can then be obtained by addition/subtraction of the in-phase and opposed-phase images. For accurate separation, the off-resonance due to B0 inhomogeneity must be known at each voxel. An off-resonance map can be estimated from the data and demodulated prior to water/fat separation (6).

In voxels with a small water or fat signal component, the off-resonance estimation is ambiguous (7). An alternative to estimate the off-resonance is to discard phase information before water/fat separation (1). This results in water/fat ambiguity, as well as increased noise in the estimates (8), particularly for certain choices of echo times (9).

The key to resolve the signal ambiguity is to use information from neighboring voxels under the assumption that the off-resonance varies smoothly across the image (6). One way to spread information between voxels is region growing (6, 10–13). A more elegant approach is optimization on a whole-image basis (14, 15), which allows separation of the formulation and the solution of the problem. A method by Hernando et al. (15) jointly resolves ambiguity and regularizes the off-resonance map. However, three-dimensional (3D) datasets were suggested to be computationally demanding. The meaning of “regularization” here is to prevent data overfitting by using the prior knowledge of spatial smoothness, with the goal to decrease noise in the estimates.

In this work, we use the problem formulation proposed by Hernando et al. (15), with a small modification that reduces the solution space by using periodicity of the residual as a function of the off-resonance. This requires uniform spacing of the echo times. The modified problem is solved in two distinct steps, corresponding to resolving ambiguity and regularizing the off-resonance map, respectively. The purpose is to simplify the problem and reduce the reconstruction time.

The proposed method is analyzed by application to in vivo breathhold imaging of the liver using a clincial 1.5 T scanner. The method is shown to produce accurate 3D water and fat images and maps of fat fraction (FF) and Tmath image covering most of the liver, with both acquisition and reconstruction times below 20 s.

THEORY

Signal Model

In a spoiled gradient-echo acquisition with echo time t, the complex signal in each voxel can be modeled as (16)

equation image(1)

where W and F are complex signal components originating from 1H nuclei in water and fat molecules, Rmath image = 1/Tmath image is the transversal signal decay, and αm are the relative amplitudes of the M fat resonances, normalized so that ∑math image αm = 1. Relative to the water resonance, each fat resonance m has a frequency offset ωm = γB0m − δW), where γ/2 π = 42.6 MHz/T, B0 is the static magnetic field, δm is the chemical shift of m, and δW is the chemical shift of water. In addition, all resonances are shifted with the off-resonance ω. The parameters ωm and αm are considered known a priori. Because of imperfections in the applied static field and tissue susceptibility, ω is unknown and needs to be estimated in each voxel along with W, F, and Rmath image.

If multiple echoes (n = 1,…,N; N ≥ 3) are acquired with echo times tn = t1 + Δ t(n − 1), the signal model can be written in matrix form:

equation image(2)

where y = [y1 y2 ··· yN]T, B = diag[1 ei ωΔt ··· eiωΔt(N − 1)], x = [W F]T, and A = [ai,j]N × 2, where an,1 = emath image and an,2 = ∑math image αmemath image.

It should be noted that the model in Eq. 2 differs from Eq. 1 as the factor emath image has been merged into W and F. This is not of practical importance, as neither the magnitudes nor the relative phase between W and F are affected. This also neither affects the calculation of FF nor affects the residual defined in Eq. 4.

Parameter Estimation

The signal model is separable with linear terms W and F (17). Provided the nonlinear terms Rmath image and ω, the least squares estimates for W and F are given by

equation image(3)

where A+ = (AHA)−1AH is the pseudoinverse and H denotes the conjugate transpose. The squared error residual associated with equation image is given by (18, 19):

equation image(4)

The nonlinear parameters Rmath image and ω are chosen so that Eq. 4 is minimized. W and F can then be estimated from Eq. 3. We propose to find Rmath image and ω using discrete optimization (15). Following Ref.20, we examine both “joint” and “decoupled” estimation of ω and Rmath image. In joint estimation, the residual minimum with respect to Rmath image is determined for each value of ω in each voxel and stored in memory. Then, ω is determined in each voxel using whole-image optimization as described below. In decoupled estimation, ω is determined first by whole-image optimization, assuming Rmath image = 0. Then, minimization with respect to Rmath image is performed in a second step. The joint estimation is slower and requires more memory, but avoids assumptions on Rmath image.

In general, the residual may have several local minima with respect to Rmath image, whereas in practice there is often only one minimum (15). We propose to find the minimum using the Fibonacci search technique (21).

It can be seen from the definition of B that J(Rmath image, ω) is periodic in ω with a period of Ω = 2 π/Δt (19). In each period, several local minima may exist (12, 18), reflecting the signal ambiguity. This is most easily understood assuming a single fat resonance model. In this case, it is impossible to differentiate a voxel of only fat from a voxel of only water, as the resonance offset ω is unknown. For multiple fat resonances, in contrast, the global minimum of Eq. 4 generally corresponds to the correct solution for any water/fat ratio. In practice, the ambiguity problem remains due to noise and model imperfections.

In Fig. 1, the residuals from several voxels at different anatomical sites are plotted as a function of the off-resonance.

Figure 1.

Plots of the squared error (residual) as a function of the off-resonance ω in seven voxels at different anatomical sites in one of the study subjects. The multiple local minima reflect the ambiguity of the signal. All voxels have a minimum close to ω = 0, which is the true minimum as the field inhomogeneity is small in this region. In voxels with significant amounts of both water and fat (adipose tissue, bone marrow, water/fat interface), the true minimum can be identified as having the smallest residual. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Whole-Image Optimization

If Q is the set of all voxels, equation image is the set of neighboring voxel pairs, and ω = {ωq}qQ is the off-resonance map, we seek the configuration of ω that minimizes:

equation image(5)

For joint estimation, J(ω) = minmath image J(Rmath image, ω), whereas J(ω) = J(Rmath image = 0, ω) for decoupled estimation. The residuals Jqq) assert data fidelity in each voxel q, whereas Vp, ωq) imposes the prior knowledge of spatial smoothness by penalizing discontinuities in the off-resonance map. The amount of regularization is controlled by the constant μ. The weights wp,q are defined as

equation image(6)

where d(p,q) is the Euclidean distance in mm between voxels p and q, and J′′min) is the second derivative with respect to ω at the smallest minimum ωmin. Using these weights, the arbitrary scale of the signal y is included in both terms of Eq. 5 through J(ω) and J′′(ω), affecting only the scale of the energy E(ω), but not the location of its minima. Thus, the impact of the regularization parameter μ does not depend on the scaling of y.

Note that, by defining Vp, ωq) = |ωp − ωq|2 and omitting the distance factor in Eq. 6, the energy function proposed in Ref.15 is obtained. Here, we propose the modified discontinuity cost:

equation image(7)

With this formulation, ω can be restricted to be within bounds 0 ≤ ω < Ω, giving a smaller solution space (19). This puts no restriction on W, F, or Rmath image, as both equation image (Eq. 3) and J (Eq. 4) are periodic in ω with period Ω. With this modification, the discontinuity cost is no longer convex. We propose to minimize the energy of Eq. 5 in two distinct steps.

First, the solution space of ω is further restricted to the two smallest J(ω) minima in each voxel. A solution to this binary subproblem can be found using quadratic pseudoboolean optimization (QPBO) (22), a graph cut-based technique that provides a partial solution. This means that all voxels will not receive a solution, but those that do are guaranteed to be part of an optimal solution of the binary subproblem. Note that a standard graph cut is not possible, as the modified discontinuity costs are not submodular.

Second, the problem considering the full ω period is solved using iterated conditional modes (ICM) (23) as described elsewhere (14), using the QPBO solution as the initial state. Alternatively, the global minimum of Eq. 5 could be found using continuous optimization (24).

As the fundamental ambiguity relates to the similarity of the water and fat components, we hypothesize that the ambiguity is well captured by the two smallest minima and that the solution provided by QPBO is close to the global optimum. The locally converging ICM can be regarded as an optional fine-tuning step corresponding to the actual regularization of the off-resonance map.

MATERIALS AND METHODS

Subjects

The feasibility and the properties of the proposed reconstruction were examined by imaging of 10 volunteer subjects, who gave written informed consent. Eight of the subjects were also included in a cross sectional study of obesity and metabolic disease in 50-year-olds. Two subjects (27 and 44 years) contributed to this methodology study only. The four female and six male subjects had a body mass index of 27.9 ± 3.7 kg/m2 (range 22.2–33.1).

Image Acquisition

Three-dimensional spoiled gradient-echo multiecho complex images were acquired from each of the subjects during a single breathhold using a 1.5 T clinical scanner (Achieva, Philips Healthcare, Best, The Netherlands). Standard phase correction procedures were turned off and the built-in quadrature body coil was used for signal transmit and receive.

A 384 × 288 × 150 mm3 field-of-view (sagittal × coronal × transverse) was placed to include most of the liver. The acquired voxel size was 3 × 3 × 10 mm3, and the pixel bandwidth was 2.7 kHz. Six echoes were acquired with monopolar readout (flyback) using t1t: 0.92/1.32 ms. The repetition time TR was 8.7 ms, and the flip angle was set to 5° to minimize T1 weighting (25). The total breathhold time was 15.9 s.

To evaluate the method performance in more challenging datasets, an additional six-echo acquisition was made in one of the subjects. A field-of-view of 384 × 40 × 512 mm3 was placed to cover the entire abdomen in the coronal plane. Imaging parameters were: acquired voxel size 2 × 4 × 2 mm3, pixel bandwidth 383 Hz, t1t: 1.98/3.83 ms, TR 23.4 ms, flip angle 10°, and breathhold time 41.5 s.

Image Reconstruction

The reconstructed multiecho images were exported from the imager and processed on a standard laptop computer. Figure 2 gives an overview of the procedure outlined in the theory section, which was followed to reconstruct water-only and fat-only images, as well as Rmath image- and off-resonance maps. Additionally, a FF map was calculated as |F/(F + W)| for fat-dominant voxels and 1 − |W/(F + W)| for water-dominant voxels. Such a calculation avoids noise bias in the FF estimates (25). For the water and fat images, W and F were recalculated using Rmath image = 0 to reduce noise. However, the Rmath image-corrected estimates were used in the calculation of FF.

Figure 2.

Information flow of the proposed method. The input is N source images acquired with different echo time shifts. The output is water- and fat-only images, as well as quantitative maps of Rmath image (1/Tmath image) and fat signal fraction (FF). The voxel-level signal ambiguity is resolved by formulating a whole-image optimization problem, which is solved in two distinct steps with two different algorithms: QPBO and ICM.

The signal model comprised nine fat resonances (M = 9). Values of the chemical shifts δW, δm, and of the relative amounts αm were taken from a recent liver spectroscopy study of 121 human subjects (26, Table 3). The off-resonance 0 ≤ ω < Ω was discretized into 100 values and 0 ≤ Rmath image ≤ 233 s−1 into 234 values. A six-neighborhood was used to define equation image, and μ was set to 1. The second-order central finite differences were used to obtain J′′min) before calculating the weights wp,q. After running QPBO, any voxels that did not receive a solution were set to the smallest minimum. The solution was then iteratively updated by ICM for 10 iterations. Decoupled estimation of ω and Rmath image was performed before solving Eq. 3. Further implementational details are given in Appendix.

Analysis of the Reconstruction Method

The water-only and fat-only images from the datasets reconstructed with μ = 1 were inspected by one of the authors (J.K.) and an experienced radiologist to locate any fat/water swaps. The location of each swap was noted and the volume was measured by manual segmentation. Consensus was reached based on initial individual analysis.

For all subjects, the execution times for different parts of the reconstruction algorithm (residual calculation, QPBO, ICM, Rmath image estimation, and total) were noted. The number of background and nonbackground voxels that received a solution from QPBO was counted.

The effect of regularization was evaluated by reconstructing the datasets with different values of the regularization parameter μ (0.001, 0.01, 0.1, 1, 10, 100, and 1000). In addition, a nonregularized reconstruction was performed by setting μ = 0.1 and omitting the regularization step (ICM). In each subject, a region of interest was drawn in a homogeneous area of the liver (as judged from the nonregularized FF and Rmath image maps). The mean values and standard deviations (SD) of FF and Rmath image were calculated inside each region of interest for each value of μ.

To test the hypothesis that the fundamental signal ambiguity is captured by the two smallest minima in each voxel, the number of local minima and their residuals were measured in each nonbackground voxel in one of the datasets. The same dataset was used to compare Rmath image-maps obtained using both the proposed Fibonacci search and “brute-force search” for minimizing J(Rmath image, ω) with respect to Rmath image. Additionally, the Rmath image and FF maps were compared using both joint and decoupled estimation. The coronally oriented dataset was also reconstructed using both joint and decoupled estimation of ω and Rmath image.

Background voxels were defined using Otsu's thresholding method (27) on a magnitude image defined by ∑math image|yn|.

RESULTS

Data acquisition and image reconstruction were successful in all subjects. Reconstructed water-only and fat-only images, as well as maps of Rmath image and FF from one of the subjects are presented in Fig. 3.

Figure 3.

Resulting images in the same slice from one of the 10 subjects. (a) fat-only image; (b) water-only image; (c) fat fraction map; and (d) Rmath image map.

The visual inspection revealed that lung blood vessels sometimes appear in the fat-only image. Such water/fat swaps were found in all but one dataset, with a total volume of 6.7 ± 4.6 mL (range: 3.1–18.4 mL). One water/fat swap with a volume of 62.5 mL was found in one of the datasets in an area subject to aliasing in the phase encoding direction. In the coronally oriented dataset, a swap with a volume of 2.2 mL was found in the left lung muscle wall. The cause was believed to be phase errors due to an aortic flow ghost artifact. No other swaps were found. Examples of swap artifacts are shown in Fig. 4.

Figure 4.

Examples of water/fat signal swap artifacts are pointed out (arrows) in fat-only images (a, d), water-only images (b, e), and off-resonance maps (c, f) from two different subjects. Small artifacts in the lung blood vessels were found in nine of the ten datasets. An example slice is given in (a–c). Beyond this, only one additional artifact was found, shown in (d–f). The cause was considered to be signal corruption due to aliasing in the phase encoding direction.

The reconstruction times were: residual calculation, 7.9 ± 0.3 s; QPBO, 0.6 ± 0.5 s; ICM, 2.4 ± 0.5 s; Rmath image estimation, 1.3 ± 0.5 s. The total reconstruction time was 13.0 ± 0.9 s. QPBO provided a solution to all nonbackground voxels and to 80.9 ± 6.5% of the background voxels. For the coronally oriented dataset, the reconstruction times were 34 s (decoupled estimation) and 270 s (joint estimation).

Figure 5 shows off-resonance, FF, and Rmath image maps from another subject for different amounts of regularization. The smoothness of the off-resonance map is clearly affected by the regularization amount as expected, but the effect on Rmath image and FF is more subtle. Setting μ = 0.001 resulted in several large fat/water swaps in all of the datasets. Such swaps were not seen using any of the other examined values of μ. Therefore, μ = 0.001 was excluded from the analysis. In each region of interest, the mean Rmath image difference compared with the nonregularized case was less than 1.5 s−1 for all μ, and less than 0.2 s−1 for μ ≤ 1. The maximum relative decrease in Rmath image SD was 0.2%. Setting μ ≥ 10 gave increased FF SD in six of the 10 cases (nine of the cases for μ ≥ 100) and a mean FF difference up to 3% compared with the nonregularized case. The largest decrease in FF SD was given by μ = 1 in five cases, by μ = 0.1 in four cases, and by μ = 0.01 in one case. For μ = 1, the difference in mean FF was <0.5% for all cases, and the relative decrease in FF SD was 1.3% on average. Taken together, we concluded that μ = 1 achieves a reasonable tradeoff between bias and noise.

Figure 5.

Effect of regularization of the off-resonance map. Different amounts of regularization are represented by the regularization parameter μ. (a) Off-resonance maps; (b) fat fraction maps; and (c) Rmath image maps. The effect of different amounts of regularization is seen clearly in the off-resonance maps, but is subtle in the fat fraction and Rmath image maps.

The detailed analysis carried out in one of the datasets revealed that 100% of the nonbackground pixels had three or four local minima in a period of ω. The residual for the second smallest minimum was on average nine times greater than the smallest, whereas the residual for the third smallest minimum was on average 141 times greater than the smallest. The reconstruction time for this dataset was 13 s using the proposed method. Replacing the Fibonacci search with “brute force” minimization of Rmath image added 18 s to the reconstruction time, but resulted in the same Rmath image for each nonbackground voxel. However, 2% of the background voxels were assigned a different Rmath image, demonstrating the possibility of several local residual minima with respect to Rmath image. Joint estimation of Rmath image and ω added 87 s to the reconstruction time compared with the proposed decoupled estimation and gave different estimates in 1.7% of the nonbackground voxels. The average absolute difference in the affected voxels was 2.5 ± 2.4 s−1 (Rmath image) and 2.8 ± 2.2% (FF).

The reconstructed images from the coronally oriented dataset are shown in Fig. 6. Excellent separation of water and fat signal was achieved despite field inhomogeneity across the large field-of-view. Because of the relatively long interecho spacing, the spectral bandwidth Ω encompasses only ±2 ppm. This gives “wrapped” off-resonance values in the superior part of the liver, where the off-resonance exceeds +2 ppm. The correct separation of water and fat is not affected by such wraps. Near the edges of the field-of-view, the large field inhomogeneity gradient causes intravoxel dephasing, reflected by the high Rmath image values. In these regions, the signal is only visible in the first echo, not supporting enough information for reliable water/fat separation. Importantly, errors do not propagate into regions of more benign field inhomogeneity.

Figure 6.

Reconstructed images from a coronally oriented dataset with a 512 mm field-of-view. (a) Fat-only image; (b) water-only image; (c) Rmath image map; (d) off-resonance map. Correct separation of water and fat signal is achieved despite “wraps” in the off-resonance map. Near the edges of the field-of-view, extreme intravoxel dephasing makes water/fat separation unreliable.

DISCUSSION

An automatic method for water/fat separation has been described in detail. The method shares several features with previously described methods, including: resolving signal ambiguity (6); allowing flexible number of echo times ≥3 (18); allowing flexible choice of time shifts t1 and Δt (7); exploiting periodicity of the residual function (18, 19); allowing simultaneous estimation of Tmath image (28); allowing modeling of multiple fat resonances (16); propagating information in 3D (13); allowing regularization of the off-resonance (14); quantifying FF avoiding bias due to T1 and noise (25).

The proposed method was based on the problem formulation by Hernando et al. (15), with a few modifications. First, a distance term was included in Eq. 6 to handle 3D data with nonisotropic voxels. Second, a first order neighborhood was used rather than second order, giving six neighbors in 3D rather than 26. Third and most important, the discontinuity cost (Eq. 7) was formulated to consider the periodicity of the residual function. With this formulation, it suffices to determine the off-resonance in a single period. The modified formulation also enables the off-resonance to be further restricted to the two smallest minima of the residual function within a period, enabling the use of a single QPBO graph cut rather than multiple standard graph cuts. Ultimately, this allows practical reconstruction times even for 3D datasets.

Interpretation of Results

This study demonstrates that the proposed method is feasible and correctly separates water and fat signal. Fractional reconstruction errors may occur in the lung blood vessels, although we speculate that this behavior is not unique to the proposed method. The fact that the reconstruction time (13.0 s) was smaller than the acquisition time (15.9 s) suggests that online implementation should be practical. Given that only 20% of the background voxels and none of the foreground voxels were unsolved by QPBO, we conclude that QPBO is an appropriate algorithm for this application.

Reconstruction of the coronally oriented dataset demonstrated robustness to field inhomogeneity. Accurate water/fat separation was achieved with the use of relatively long Δt. This gives a smaller spectral bandwidth Ω and is potentially more challenging due to spectral aliasing.

A broad range of values were examined for the regularization parameter μ. As demonstrated in Fig. 5, the resulting images are quite insensitive to μ. We concluded that μ = 1 offers a reasonable tradeoff between bias and noise, although this choice is not critical. Rather, the question arises whether regularization is motivated at all, in light of the apparently modest noise reduction in the estimates. Omitting the regularization is straightforward with the proposed method by leaving out the ICM step. Regularization might be more important with fewer echoes or other echo times than those examined in this work.

Our hypothesis that the two smallest residual minima capture the fundamental signal ambiguity seems reasonable when studying the residual functions in Fig. 1. The hypothesis is supported by the accurate separation of water and fat signal with the proposed method, as well as by the much larger residual of the third smallest minimum compared with the two smallest minima.

The results show that there may be several residual minima with respect to Rmath image. In practice, only one minimum is found in the relevant range, enabling the use of search techniques such as the Fibonacci search.

Decoupled estimation of ω and Rmath image reduces computational complexity and memory requirements compared with joint estimation, while giving identical estimates in the vast majority of voxels. Voxels with nonidentical estimates are still in close agreement, and the impact of the difference is probably negligible. Our results are in agreement with a previous analysis of this matter (20).

Limitations

In this study, only two different acquisition protocols were examined. In particular, varying the number of echoes was not evaluated. All results may not apply for the general case. However, this study demonstrates feasibility using both short and long echo time spacings. In addition, our initial experience of using the method with three-point datasets with other echo time combinations shows promise.

The reconstruction was validated by visual inspection only, indicating qualitative but not necessarily quantitative accuracy. Possible references for quantitative validation include phantom studies, MR spectroscopy, and biopsy. Fat quantification using the applied signal model (Eq. 1) has previously been validated both in phantom (29, 30) and in vivo as a biomarker for fatty infiltration in the liver, using MR spectroscopy as a reference (30, 31). Good reproducibility has also been demonstrated (32). In addition, quantification of liver fat in mice using this signal model correlates well with both qualitative and quantitative histology analysis, and lipid extraction (33).

Only moderate liver Rmath image values were encountered in the study subjects. It is thus unclear if the proposed method gives accurate estimates in patients with hepatic iron overload, which may shorten the Tmath image of the liver to a few milliseconds only (34). The use of short echo times is reasonable for estimating short Tmath image values. Decoupled estimation of ω and Rmath image initially assumes that Rmath image = 0. This assumption is more defective for shorter Tmath image values. Hence, it is of interest to compare the joint and decoupled estimations in subjects with iron overload. An alternative approach is to perform joint estimation with only a few discrete values of Rmath image and then refining the Rmath image estimate in a second step.

A possible source of error that was not addressed in this work is eddy currents. It has been reported that eddy currents may cause phase errors, particularly in the first echo (35). If such errors are found to be significant for a given imaging sequence, the impact on the estimates can be reduced by using only magnitude data from the first echo (36).

This work describes water/fat separation for gradient-recalled acquisitions only, but the method can also be used with spin-echo sequences. Chemical shift encoding can then be obtained by varying the echo time, or by keeping a constant echo time and shifting the timing of the refocusing pulse (7). The first strategy enables estimation of Rmath image as in the gradient-echo case, whereas the second strategy enables estimation of the reversible decay Rmath image = Rmath image − 1/T2 (37).

In this study, we used the same fat profile (i.e., values of αm) for all subjects, not accounting for individual variations in fatty acid composition. However, it has been shown that diagnostic accuracy of fatty liver is not sensitive to expected individual variations in the fat profile (38).

An evident limitation of the proposed method is the requirement of a constant interecho spacing Δt. This is necessary to obtain a periodic residual function. The acquisition of multiecho gradient-echo data typically involves a constant echo spacing to achieve the shortest possible echo times, and hence TR to minimize acquisition times. Therefore, constant echo spacing is common in practice. To obtain even shorter interecho spacing, two interleaved echo trains are sometimes used. The combination of these echo trains does not necessarily have a constant interecho spacing. For three-point datasets, the echo time combination shown to give optimal noise performance has a constant echo spacing (39).

As the off-resonance is restricted to a single period, a wrapped map is obtained. If an unwrapped off-resonance map is desired, phase unwrapping can favorably be performed in a separate step. This is necessary if the off-resonance map is to be used for correction of spatial distortion.

It has been proposed to perform the signal separation in k-space after estimating the off-resonance map in image space to eliminate spatial misregistration caused by chemical shift (40). Such an approach is compatible with the proposed method, but spatial misregistration was considered insignificant due to the high readout bandwidth used in this work.

Future Work

The described method may be valuable in quantitative applications, such as assessing the amount of fat and iron overload in the liver and other organs, as well as in qualitative water/fat separation, were the water image may reveal structures underlying bright fat signal and the fat image provides additional information about fat localization. The focus of this work was image reconstruction, and the clinical value of the described reconstruction method and imaging sequences remains to be evaluated.

A possible extension of the proposed method is to estimate the off-resonance map at a lower resolution than the acquired voxelsize. This may be of interest for high-resolution images and is motivated by the slow-varying nature of the off-resonance. The low-resolution residual functions can then be set as the sum of the residual functions of all voxels belonging to each low-resolution voxel, reducing the impact of noise and increasing the amount of voxels with signal contributions from both water and fat.

It is also of interest to evaluate the method for three-point water/fat separation (without Tmath image estimation). If regularization is omitted, two candidates for the off-resonance can be obtained analytically rather than evaluating the complete residual function (41, 42). The duality can then be resolved by QPBO. The proposed method may also be modified for use in two-point water/fat separation (43, 44).

CONCLUSIONS

The proposed method is feasible for 3D datasets and enables accurate separation of water and fat signal jointly with Tmath image estimation. The reported reconstruction times promise practical online implementation. For the described imaging protocol, the FF and Tmath image maps are not sensitive to the regularization amount. The noise reduction in the estimates due to regularization appears to be small. The voxel-level signal ambiguity is well captured by the two smallest minima of the residual function. Fibonacci search for Tmath image optimization and decoupled estimation of Tmath image and off-resonance can be used for fast algorithm performance.

Acknowledgements

The authors thank Prof. Håkan Ahlström for inspection of the reconstructed images.

APPENDIX: DETAILS OF IMPLEMENTATION

The reconstruction algorithm was implemented in C++ within our in-house image analysis environment. We used the QPBO implementation made publicly available along with Ref.45, including the fast graph cut technique described in Ref.46. All cost terms were truncated to integer values. A+ was precalculated for each value of Rmath image, and (IAA+)BH was precalculated for each combination of Rmath image and ω to allow fast calculation of the residuals J. A small speedup was gained by precalculating Vp, ωq) = V(|ωp − ωq|) for each possible |ωp − ωq|. In ICM, the ω update candidates were checked in order of proximity to the previous solution. The addition of terms to the cost of the current candidate was aborted if the cost exceeded the current minimum cost. The ω update was restricted to be within Ω/10 from the previous iteration for speedup and to stabilize the solution.

Ancillary