### Abstract

- Top of page
- Abstract
- THEORY
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- CONCLUSIONS
- Acknowledgements
- REFERENCES

In this work, a new two-point method for water–fat imaging is described and explored. It generalizes existing two-point methods by eliminating some of the restrictions that these methods impose on the choice of echo times. Thus, the new two-point method promises to provide more freedom in the selection of protocol parameters and to reach higher scan efficiency. Its performance was studied theoretically and was evaluated experimentally in abdominal imaging with a multigradient-echo sequence. While depending on the choice of echo times, it is generally found to be favorable compared to existing two-point methods. Notably, water images with higher spatial resolution and better signal-to-noise ratio were attained with it in single breathholds at 3.0 T and 1.5 T, respectively. The use of more accurate spectral models of fat is shown to substantially reduce observed variations in the extent of fat suppression. The acquisition of in- and opposed-phase images is demonstrated to be replaceable by a synthesis from water and fat images. The new two-point method is finally also applied to autocalibrate a multidimensional eddy current correction and to enhance the fat suppression achieved with three-point methods in this way, especially toward the edges of larger field of views. Magn Reson Med, 2010. © 2010 Wiley-Liss, Inc.

As hyperintense signal from fat may obscure underlying pathology, its partial or complete suppression is a basic requirement in various applications of magnetic resonance imaging. Its characteristics result from the comparatively short relaxation times and large chemical shifts of the dominant methylene protons and serve as the basis for its elimination.

Fat suppression is often an integral part of the acquisition. Popular methods include short-tau inversion recovery, which exploits the specific relaxation times, and selective saturation, which relies on the specific chemical shifts (1, 2). However, these methods all have individual drawbacks, such as longer scan times, lower signal-to-noise ratio (SNR), higher specific absorption rate, or less tolerance to field inhomogeneities. Postponing the separation of water and fat signals until the reconstruction allows avoiding most of these disadvantages. So-called Dixon methods perform for this purpose measurements at different echo times to encode the chemical shift (3). Besides fat suppression, they also permit efficient water–fat imaging, providing additional diagnostic information of relevance to selected applications.

Several Dixon methods have been proposed over the last two decades (4). Apart from different strategies for the separation, they are mainly characterized by the number of echoes, or points, that they sample, and by the constraints that they impose on the echo times. We focus in this work on two- and three-point methods, as multipoint methods are usually very similar to three-point methods, and one-point methods are generally too error-prone without a priori information (5).

Both two- and three-point methods originally required so-called in-phase and opposed-phase echo times at which the water and fat signals after demodulation to baseband are parallel and antiparallel in the complex plane, respectively (3, 6). The three-point methods were then gradually generalized to allow flexible echo times (7–9). Thus, they do not restrict the angle or phase between the water and fat signals at the echo times to certain values anymore. In this way, they provide more freedom in sequence design and enable in particular a trade-off between SNR gains from the acquisition and SNR losses in the separation. We will explicitly refer to Reeder's method (9) in this work, also as flexible three-point method. By contrast, the existing two-point methods still require at least one in-phase echo time. For instance, Ma's method (10), as the original Dixon method (3), assumes one in-phase and one opposed-phase echo time, and Xiang's method (11) assumes one in-phase and one partially opposed-phase echo time.

Sampling only two instead of three echoes is especially desirable in time-critical applications, such as abdominal imaging in single breathholds. However, constraints on the echo times may actually render dual-echo acquisitions slower than triple-echo acquisitions (12). We, therefore, aim at eliminating such constraints in dual-echo Dixon imaging in this work.

In the following sections, we first derive a flexible two-point method by generalizing Xiang's semiflexible two-point method. We then characterize its noise performance and investigate, representative for model errors, the effect of fat signal dephasing and decay and of their consideration in the separation by a more accurate spectral model of fat. We also propose the application of this method for an eddy current correction in triple-echo acquisitions. We present selected results of abdominal imaging in single breathholds at 1.5 T and 3.0 T, and we compare the new two-point method with existing two- and three-point methods.

### THEORY

- Top of page
- Abstract
- THEORY
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- CONCLUSIONS
- Acknowledgements
- REFERENCES

The complex composite signal in image space *S*, sampled at echo times *t*_{n}, with *n* = 1,…,*N*_{E}, is modeled by

- (1)

*W* and *F* denote the real or complex water and fat signal in image space, respectively. The dephasing angle θ is given by

- (2)

where Δ*f* commonly represents the resonance frequency offset of the dominant spectral peak of fat with respect to water. φ denotes a phase error, and ϕ = *e*^{iφ} a corresponding phasor, that is usually attributed to field inhomogeneity, in which case it is proportional to both the field strength offset Δ*B*_{0} and the echo time TE. Unlike three-point methods, two-point methods do not rely on a linear relation between ϕ and TE, however.

The decomposition of *S* into *W* and *F* is carried out in three steps:

- 1
Computation of potential values of the phasor Δϕ = *e*^{i}^{Δφ}, with Δϕ = ϕ_{2} − ϕ_{1}, from the samples of the composite signal *S*,

- 2
Selection of one value for the phasor Δϕ,

- 3
Estimation of the water and fat signal *W* and *F* given the samples of the composite signal *S* and the phasor Δϕ.

Of these three steps, which are described in more detail in the following, the first and the last may be performed independently for each voxel.

#### Computation of Potential Values of the Phasor

*W* and *F* are considered as real in this step. As the phase error is still unknown, two-point methods may initially only rely on the amplitude of the composite signal, while the phase reflects inseparable contributions from chemical shift, field inhomogeneity, and eddy currents. Estimates of |*S*_{1}| and |*S*_{2}| permit the separation of the composite signal into two signal components, without resolving their correspondence to *W* and *F* (11). Applying the law of cosines to the triangles formed by the water, the fat, and the composite signal in the complex plane at the two echo times generally yields two potential values of *W*

- (3)

and two corresponding values of *F*, given by *F*_{1} = *W*_{2} and *F*_{2} = W_{1}, under certain restrictions. The first restriction is that cos θ_{1} ≠ cos θ_{2}. If (θ_{1} − θ_{2}) mod 2π = 0, with θ_{1} and θ_{2} in radians, *S*_{1} = *S*_{2} in the noiseless case, which makes a separation basically impossible. If (θ_{1} + θ_{2}) mod 2π = 0, *S*_{1} ≠ *S*_{2}, but |*S*_{1}| = |*S*_{2}|, which renders a separation completely unreliable, as an infinite number of solutions exist, covering a broad range of water–fat ratios. The second restriction is that *W*_{1} ≥ 0 and *W*_{2} ≥ 0, implying that both are real. The latter, for example, is violated when the amplitude of the measured composite signal is smaller at the more in-phase echo time, i.e., at the echo time for which |(θ mod 2π) − π| is larger. Different approaches to handling such exceptions exist, the discussion of which is beyond the scope of this work.

Given *W*_{1}, *W*_{2}, *F*_{1}, and *F*_{2}, two potential values of the phasor may generally be derived. We suggest multiplying the signal equations for *S*_{1}* and *S*_{2} for this purpose, leading to

- (4)

At least for the following step, Δϕ should be normalized by setting |Δϕ| = 1.

#### Selection of One Value for the Phasor

Any of various existing strategies for this selection may be applied in this step (10, 11, 13). We used the procedure outlined in Ref.11 in this work. It basically performs a spatial smoothing of Δϕ and a subsequent nonlinear mapping onto the closer of the potential values Δϕ_{1} and Δϕ_{2}. These operations are repeated in an iteration until convergence is reached, followed once more by a spatial smoothing. We usually set Δϕ initially to unity, relying on a reasonable shimming, i.e., small field strength offsets for the majority of voxels, and only rarely resorted to more complicated alternatives (11). Voxels containing pure noise were masked very conservatively, typically excluding only those close to receive coil elements, where the expected SNR is particularly high. Moreover, we extended the spatial smoothing to three spatial dimensions, if applicable, to improve consistency and robustness.

#### Estimation of the Water and Fat Signal

Given Δϕ, *W,* and *F* are re-estimated, for which we propose two approaches. In the first approach, they are considered as real. As only Δϕ is known, *e* and *e* are eliminated by using the squared magnitude and the conjugate complex product of the signal equations for *S*_{1} and *S*_{2}, leading to the nonlinear system of four real equations

- (5)

with Δθ = θ_{2} − θ_{1}.The sum of the squared or absolute error between the left and right hand sides of these equations is minimized numerically, for instance with a Newton method with backtracking.

It is solved analytically

- (7)

and yields directly the magnitude of the water and fat signal, as |*W*′| = |*W*| and |*F*′| = |*F*|.

#### Propagation of Noise

In first approximation, the influence of noise on the value of the phasor may be neglected due to the spatial smoothing applied at the end of the selection procedure (11). Limiting us to the analytically tractable second approach to estimating *W* and *F*, we define

- (8)

and obtain the amplification of the noise variance in the separation by evaluating the diagonal elements of the matrix XX^{H}, assuming that the noise in the composite signal at the two echo times after phase correction is uncorrelated. The reciprocal value

- (9)

for *m* = 1, 2, corresponds to the so-called effective number of signal averages (NSA) (6, 9). A contour plot of it is provided in Fig. 1a. A comparison with the result for the special case of one in-phase echo time in Ref.11

- (10)

shows that the first approach to estimating *W* and *F*, which considers them as real, may provide the higher effective NSA, as 3 + cos Δθ ≥ 2 for all Δθ.

In second approximation, the uncertainty about the value of the phasor has to be considered (14). We performed Monte-Carlo simulations of the worst case, in which no spatial smoothing at all is applied at the end of the selection procedure. An SNR of 50 was assumed, and a set of 1000 samples was drawn per point. In Fig. 1b, a selected result is reproduced, which was obtained with the second approach to estimating *W* and *F* to facilitate a comparison with Fig. 1a. Besides the dependency on Δθ already seen in Fig. 1a, the contour plot most notably reflects the additional restriction (θ_{1} + θ_{2}) mod 2π ≠ 0, with θ_{1} and θ_{2} in radians, as discussed above. It leads to high SNR losses close to the two diagonals 2θ_{1} + Δθ = 2π and 2θ_{1} + Δθ = 4π.

#### Fat Signal Dephasing and Decay

Various effects render the signal equation used so far simplistic. We limit us in this work to an analysis of fat signal dephasing and decay. For this purpose, Eq. 1 is generalized to (15)

- (11)

with

- (12)

The weights *w* add up to 1.0. The index *m* distinguishes multiple spectral peaks of fat with individual resonance frequency offsets Δ*f*_{m} and optionally individual transverse relaxation rates *R*_{m}. θ is then given by

- (13)

We relied on a spectral model of fat with seven peaks based on Ref.16 throughout this work. To simulate the influence of fat signal dephasing and decay, we evaluated Eq. 11 and separated the resulting signal based on Eq. 1. The discrepancy between both models most noticeably leads to leakage of fat signal into water images. Its extent is quantified for a range of θ_{1} and Δθ values in Fig. 2. As to be expected, it increases with growing echo spacing. Remarkably, it is higher for a more in-phase than for a more opposed-phase first echo time. As the separation tends to polarize the water and fat signals, i.e., to overestimate the major signal component, when the signal at the more in-phase echo time is relatively reduced, the fat suppression apparently improves under this condition.

The generalized signal equation may also be incorporated into the separation. Starting from the squared magnitude of the signal equations at the two echo times

- (14)

where the subscripts R and I refer to the real and imaginary components, a biquadratic equation is obtained for *W* or *F*. Solving for F, for instance, yields

- (15)

and thus

- (16)

and

- (17)

where

- (18)

In general, *F*_{1} ≠ *W*_{2} and *F*_{2} ≠ *W*_{1} now. For a single-peak spectral model of fat without relaxation, Eq. 17 may be simplified by radical denesting to Eq. 3. Again, two pairs of values of *W* and *F* are obtained, to which two potential values of Δϕ

- (19)

correspond. The selection of one of them remains unchanged. Re-estimating *W* and *F* can be performed analogous to Eqs. 5 and 7, using Eq. 11 instead of Eq. 1 as underlying signal equation.

#### Eddy Current Correction

In acquisitions that use a bipolar readout gradient to increase scan efficiency and decrease echo spacing, the phase error φ may also comprise significant contributions from eddy currents. These primarily alternate with the polarity of the readout gradient and thus violate the linear relation between φ and TE assumed by three-point methods. We propose to use the flexible two-point method for an autocalibration of a 3D correction of eddy currents in bipolar acquisitions to enhance in particular the fat suppression achieved with three-point methods.

For this purpose, the composite signal is first, separately for each echo time, decimated in space by some factor *N*. The flexible two-point method is then applied to the resulting data for pairs of echo times that were sampled with opposite polarity of the readout gradient. In the case of three echoes, to which we limit us in the following, the pairs *S*_{1}, *S*_{2} and *S*_{2}, *S*_{3} are selected. The resulting phasors Δϕ_{12} and Δϕ_{23} may be described by

- (20)

The original phasor is split up into a direct contribution Δϕ and an alternating contribution Δϕ_{EC}. The latter is attributed to eddy currents and derived from the conjugate complex product of the two equations

- (21)

Depending on the range of the corresponding phase error 2 Δφ_{EC}, an unwrapping may have to be applied in this process, for which we used the algorithm proposed in Ref.17. Finally, Δϕ_{EC} is upsampled by the factor *N*, and its complex conjugate is multiplied with *S*_{2}. A standard three-point method is then applied to the original *S*_{1} and *S*_{3}, and to the corrected *S*_{2}.

### MATERIALS AND METHODS

- Top of page
- Abstract
- THEORY
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- CONCLUSIONS
- Acknowledgements
- REFERENCES

Experiments were performed on 1.5 T and 3.0 T scanners (Philips Healthcare, Best, The Netherlands), equipped with gradient systems providing a maximum field strength of 23 mT/m and 33 mT/m and a maximum slew rate of 180 T/m/s and 200 T/m/s, respectively. Volunteers were scanned after obtaining informed consent in accordance with the guidelines of our institutions.

Data were acquired with a 3D T_{1}-weighted multi-gradient-echo sequence using receive coil arrays with 16 and 32 elements. Slope and partial echo sampling were disabled. Typical parameters of the employed abdominal imaging protocol included a field of view of 370 × 260 × 240 mm^{3}, a voxel size of 1.5 × 1.5 × 3.0 mm^{3}, and a flip angle of 10°. Scan times were decreased by parallel imaging and optionally Half Fourier imaging, which were applied in both phase encoding directions (18). Acceleration factors ranged between 2.0 and 5.0 for parallel imaging and between 1.0 and 1.5 for Half Fourier imaging. Scans were completed in single breathholds of about 20 sec and included a short calibration of a 1D eddy current correction (19).

The commonly assumed first in-phase echo times in gradient-echo imaging of 4.6 msec at 1.5 T and 2.3 msec at 3.0 T, which approximately correspond to the resonance frequency offset of the dominant spectral peak of fat, served as reference for the calculation of the nominal values of the dephasing angle at the echo times. These were used in the separation in conjunction with the standard single-peak spectral model of fat.

The flexible two-point method was evaluated by varying primarily two parameters of the protocol, the echo times and the spatial resolution. Its performance was compared to that of existing three- and two-point methods (9–11), both at 1.5 T and 3.0 T, with respect to scan time and with respect to SNR and fat suppression in the resulting water images.

The relative optimal SNR for each combination of dual- or triple-echo acquisition and two- or three-point method was predicted based on the pixel bandwidth of the acquisition and the effective NSA of the separation. The latter was determined using Eq. 9 for the flexible two-point, Eq. 10 for the existing two-point, and Eq. 9 from Ref.9 for the flexible three-point method(s). The absolute actual SNR in the resulting water images was measured in a homogeneous area of the liver parenchyma. The fat suppression was visually assessed, as was the benefit of using a more accurate spectral model of fat in the flexible two-point method and a 3D eddy current correction in the flexible three-point method on selected examples.

In- and opposed-phase images were synthesized by addition and subtraction, respectively, of the real or complex water and fat images produced with the flexible two-point method, and they were qualitatively compared to corresponding images reconstructed from dual-echo acquisitions with one in-phase and one opposed-phase echo time.

### RESULTS

- Top of page
- Abstract
- THEORY
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- CONCLUSIONS
- Acknowledgements
- REFERENCES

Selected water and fat images obtained with the flexible two-point method at 3.0 T are shown in Fig. 3. Essential parameters of the used protocols, as well as of corresponding protocols with at least one fixed echo time, are summarized in Table 1. The comparatively high SNR at 3.0 T permitted increasing the in-plane resolution from 1.5 mm to 1.0 mm. The benefit of a flexible choice of echo times becomes more and more significant toward higher resolution. Most notably, the use of one in-phase and one opposed-phase echo time leads to impracticable scan times at the highest resolution. Additionally, scan times vary discretely with the resolution, among others, employing existing two-point methods.

Table 1. Selected Protocol Parameters of the Dual-Echo Acquisitions with Flexible Echo Times at 3.0 T Used for Fig. 3 and of Corresponding Acquisitions with One In-Phase (IP) and Optionally One Opposed-Phase (OP) Echo Time (Gray-Shaded Rows). Listed Are the In-Plane Resolution (RES), the Scan Time (*T*), the Repetition Time (TR), the Two Echo Times (TE), and the Acceleration Factor for Half Fourier Imaging (HF) |

Water images reconstructed with different three- and two-point methods are compiled in Fig. 4. Protocol parameters of the underlying acquisitions at 1.5 T are listed in Table 2. First, starting from the minimum repetition time (TR) of 6.0 msec for an in-phase echo time, four protocols with identical scan times were designed: one flexible triple-echo acquisition, one flexible dual-echo acquisition, and two dual-echo acquisitions with at least one fixed echo time. The pixel bandwidth was individually minimized given the TR. The SNR is visibly higher for the flexible acquisitions shown in the first row. The fat suppression is slightly better for the flexible dual-echo acquisition and the dual-echo acquisition with one in-phase and one opposed-phase echo time shown in the right column, notably in the right and anterior subcutaneous fat layer. Then, the two flexible acquisitions were accelerated. At the minimum TR of 4.8 msec for the triple-echo acquisition, the SNR stays for both as good as for the two slower dual-echo acquisitions with at least one fixed echo time. Finally, at the minimum TR of 3.7 msec for the dual-echo acquisition, the SNR becomes limiting for the intended application at the employed fixed acceleration factor for parallel imaging. Moreover, the fat suppression is degraded. Relative optimal and actual SNR values for all protocols are provided in Table 2. Although they match reasonably, the actual SNR values are consistently smaller than the optimal SNR values for the flexible dual-echo acquisitions.

Table 2. Selected Protocol Parameters of the Flexible Dual- and Triple-Echo Acquisitions and the Dual-Echo Acquisitions with One In-Phase (IP) and Optionally One Opposed-Phase (OP) Echo Time (Gray-Shaded Rows) at 1.5 T Used for Fig. 4. Listed are the Number of Echoes (*N*_{E}), the Scan Time (*T*), the Repetition Time (TR), the Echo Times (TE), the Shift of Fat (FS), and the Theoretically Predicted Optimal and Experimentally Measured Actual Signal-to-Noise Ratio (SNR opt/act) Relative to the First Protocol |

Differences between the proposed two approaches to estimating the water and fat signal in the flexible two-point method are highlighted in Fig. 5. The water images were obtained at 3.0 T with a TE_{1}/TE_{2} of 1.0 msec/2.3 msec. Primarily interfaces are affected by a direct calculation of real water and fat signals. They appear enhanced and even pixelated in severe cases. By contrast, a higher SNR was not observed in the water images. The images in all other figures were, therefore, produced with the approach that considers the water and fat signals as complex.

The influence of the choice of echo times on the fat suppression is illustrated in Fig. 6. Using a fixed TR of 4.8 msec, TE_{1}/TE_{2} were collectively shifted. In the shown four examples, they correspond to a θ_{1} between 110° and 141° and a Δθ of 102°. TE_{1} is closer to an in-phase echo time than TE_{2} in the first two. As predicted, the fat suppression is clearly inferior under this condition. Using the referenced seven-peak instead of the standard single-peak spectral model of fat in the separation, a considerable improvement is achieved. This is demonstrated for the worst case, for which a further enhancement is attained by including transverse relaxation. Missing in the series of the shown four examples is the result for a TE_{1}/TE_{2} of 1.6/2.9 msec. The separation grossly failed in this case due to almost perfect symmetry to an opposed-phase echo time. Remarkably, a perturbation of TE_{1}/TE_{2} by only ±0.1 msec apparently suffices to prevent such instability.

The calculation of in- and opposed-phase images from complex water and fat images is evaluated in Fig. 7. The dual-echo acquisition with one in-phase and one opposed-phase echo time and the flexible dual-echo acquisition with a TR of 6.0 msec underlying Fig. 4 were reused for this purpose. Overall, the acquired and synthesized in- and opposed-phase images agree satisfactorily. Primary differences between acquired and synthesized images are attributable to a misregistration between both acquisitions, which were performed in separate breathholds. Secondary differences are seen at interfaces, but no clear preference for their depiction in the acquired or the synthesized images was ascertainable. Besides, the calculated images show a higher SNR, resulting from the higher SNR in the underlying images reconstructed from the flexible dual-echo acquisition.

The benefit of exploiting the flexible two-point method for a 3D eddy current correction of triple-echo acquisitions is exemplified in Fig. 8. The displayed transverse and reformatted coronal slices indicate that a 1D correction compensates phase errors only in the center of the field of view satisfactorily. Toward the edges, substantial leakage of fat signal into the water images arises from remaining phase inconsistencies between odd and even echoes. Applying a 3D correction instead, which was autocalibrated with the fourfold subsampled composite signal in this case, restores a homogeneous fat suppression across the entire field of view.

### DISCUSSION

- Top of page
- Abstract
- THEORY
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- CONCLUSIONS
- Acknowledgements
- REFERENCES

A new two-point method for water–fat separation was introduced that assumes neither of the used two echo times to be in- or opposed-phase. It imposes as a remaining, fundamental restriction that the sampled two echoes must not provide redundant information. Most notably, it thus excludes echo times that are exactly symmetric to in- or opposed-phase echo times. A similar generalization of Xiang's method, which originally demanded one in-phase echo time, was briefly discussed, but not explored, in Ref.11. It was suggested that the phasors *e* and *e* may be determined and eliminated separately, and it was argued that this procedure may be error-prone, as the smoothness of the phasors deteriorates with growing echo times and not growing echo spacing. In this work, we pursued a direct calculation of Δϕ = *e*^{i}^{Δφ} instead, which considerably improved robustness of the separation at the expense of a somewhat more complicated estimation of the water and fat signals. It still permits an identification of water and fat signals, very similar to Xiang's method, provided that Δθ is not equal to, or for the sake of robustness not close to, 0° or a multiple of 180°.

For most of the cited two- and three-point methods, a dedicated strategy for determining Δφ or Δϕ was developed. Reeder's three-point method was combined with a region growing approach that tries to recover the field map Δφ (13). Ma's two-point method was linked to a region growing approach that attempts to determine the phasor map Δϕ (10). Xiang's two-point method was coupled to an iterative filtering and mapping procedure for estimating Δϕ (11). Ma's and Xiang's methods both calculate first several alternative solutions for the field or phasor map for each voxel and only then select one solution by considering a local neighborhood. As Reeder's method is in principle compatible with this approach, as recently demonstrated in Ref.20, all of these strategies may be cast into the proposed framework for the separation, and any procedure for choosing one solution for Δφ or Δϕ based on the assumption that Δ*B*_{0} varies smoothly may be used with the described flexible two-point method. For this work, we chose the iterative filtering and mapping procedure, because it seemed conceptually attractive. It is immediately amenable to parallel processing, completely independent of the otherwise critical selection of a starting point, and entirely satisfied with recovering the phasor map, which is a substantial simplification compared with the field map. It proved in practice to allow a robust selection of one value for the phasor in a few seconds for typical data sets of a few million voxels on state-of-the-art workstations, equipped with up to two processors with eight cores each.

The analysis of the noise amplification in the separation revealed a complicated dependence on the choice of echo times. To minimize SNR losses, the dephasing angles at the echo times should be distributed evenly around the unit circle (9). This condition is met for two-point methods by Δθ = 180°. If the uncertainty about the value of the phasor is considered, however, only dephasing angles of 0° and 180°, for which cos θ_{1} differs the most from cos θ_{2}, lead to no SNR losses, while dephasing angles of 90° and 270°, for which cos θ_{1} equals cos θ_{2}, result in complete SNR losses. Obviously, besides the echo spacing, which determines the increment in the dephasing angle, the first echo time also has to be chosen judiciously.

A comparison of the SNR analyses in Table 2 and Fig. 1 shows that actual SNR values are far closer to the best case than to the worst case. For the three studied dual-echo acquisitions with flexible echo times, θ_{1}/Δθ are 141°/171°, 133°/117°, and 102°/86°, for which an effective NSA of 1.4, 0.1, and 0.5 is read from Fig. 1, respectively. The uncertainty about the value of the phasor is consequently small in practice. Moreover, the particularly critical case of a water–fat ratio close to one is rarely encountered, usually only at interfaces. Any local smoothing of the field or phasor map suppresses excessive SNR losses in these few voxels very effectively. The images in Fig. 6 suggest that minor adaptations to the echo times are sufficient not only to comply with the remaining restriction cos θ_{1} ≠ cos θ_{2} but also to enable a robust separation with acceptable noise amplification.

Two advantages of flexible echo times were demonstrated in this work, a reduction of scan times by shortening echo times, and an increase in SNR in the resulting images by lowering the pixel bandwidth. To benefit from the latter, a joint optimization of SNR gains from the acquisition and SNR losses in the separation is required. While a flexible placement of the echo times in a given repetition time permits minimizing the pixel bandwidth and thus maximizing the SNR for the composite signal, the subsequent separation entails a noise amplification that grows with the deviation from in- and opposed-phase echo times. As shown, the gains may more than offset the losses, leading to an overall better SNR in water and fat images.

The use of more accurate spectral models of fat improved the extent and consistency of the fat suppression. Considering the existence of multiple spectral peaks of fat, the concept of in- and opposed-phase echo times may be questioned fundamentally, at least in gradient-echo imaging. The described generalization to an independent complex weighting of the fat signal at the two echo times reduces the systematic error introduced by such an idealization. In theory, it should also enhance the contrast in synthesized opposed-phase images, which has not unmistakably been observed yet, but warrants further studies.

Two-point methods, in contrast to most three-point methods, do not assume a linear evolution of the phase error over echo time. Therefore, they are inherently less susceptible to inconsistencies in the phase error, as they occur in bipolar acquisitions between the data from odd and even echoes. In first approximation, the spatial variations of the phase error are limited to the frequency encoding direction. These proved to be removable prior to the separation based on a short calibration measurement. Although a 1D correction of eddy currents is conceptually not required for two-point methods, it increases the robustness of the separation (21). For larger volumes, the accuracy of this first approximation turned out to be insufficient for three- and multi-point methods. Improving it by mapping the phase error in 3D would require substantial scan time. By contrast, the proposed autocalibrating correction relies on the available imaging data only. As it is applied on a low resolution level, the loss in SNR is negligible, and the occasionally required phase unwrapping remains simple and robust. Performing a 1D correction before the 3D correction further reduces the need for or the complexity of the phase unwrapping. The concept can easily be extended to multipoint methods. It may be an attractive alternative to fitting the phase error as a nuisance parameter in this case (22).

Several other potential sources of artifacts have not explicitly been addressed in this work. One source is the shift of fat. We neglected it, because the strong readout gradients used to reach short scan times usually reduce it to a fraction of a pixel in the frequency encoding direction. However, in cases where the pixel bandwidth is deliberately lowered to enhance the SNR, it may become significant and may cause normally benign artifacts. For bipolar acquisitions, it may be considered by moving the separation from image space to *k*-space or by coupling the separation for adjacent voxels (23). Another source is flow. It is of particular concern in Dixon imaging, as the phase errors arising from it may mislead the separation, resulting in additional artifacts. In some of the figures, ghosting from pulsatile flow in the aorta is noticeable. It may be eliminated for the most part by a simple addition of saturation pulses to the sequence. By contrast, a flow compensation by gradient moment nulling complicates multi-echo sequences considerably and increases echo times substantially. The described flexible two-point method may contribute to rendering it practicable, however.