### Abstract

- Top of page
- Abstract
- INTRODUCTION
- THEORY
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- CONCLUSION
- REFERENCES

The two-point Dixon method is a proton chemical shift imaging technique that produces separated water-only and fat-only images from a dual-echo acquisition. It is shown how this can be achieved without the usual constraints on the echo times. A signal model considering spectral broadening of the fat peak is proposed for improved water/fat separation. Phase errors, mostly due to static field inhomogeneity, must be removed prior to least-squares estimation of water and fat. To resolve ambiguity of the phase errors, a corresponding global optimization problem is formulated and solved using a message-passing algorithm. It is shown that the noise in the water and fat estimates matches the Cramér-Rao bounds, and feasibility is demonstrated for in vivo abdominal breath-hold imaging. The water-only images were found to offer superior fat suppression compared with conventional spectrally fat suppressed images. Magn Reson Med, 2010. © 2010 Wiley-Liss, Inc.

### INTRODUCTION

- Top of page
- Abstract
- INTRODUCTION
- THEORY
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- CONCLUSION
- REFERENCES

Relaxation times have traditionally been the main source of contrast in MRI. A fundamentally different source of image contrast is resonance frequency, the foundation of MR spectroscopy. Differences in resonance frequency for various chemical species is also known as chemical shift. This phenomenon is important in medical MRI, not least because of the distinct chemical shift between water and fat, which are the major contributors to the MR signal in proton imaging of the human body. The signal from fat is hyperintense in both *T*_{1}-weighted and *T*_{2}-weighted images, and can obscure the signal from contrast agent or pathology. Therefore, suppression of the fat signal is often desired. This can be done by short tau inversion recovery, STIR (1), where fat is suppressed with respect to *T*_{1} rather than chemical shift. Consequently, all tissues with a *T*_{1} similar to that of fat are suppressed, ruling out the possibility to use contrast agents. An alternative is spectral fat suppression (2), where the fat protons are excited by a selective radiofrequency pulse with respect to chemical shift. The excited fat magnetization is spoiled before the imaging sequence proceeds. Spectral fat suppression can be used with contrast agents as it does not alter *T*_{1} contrast. Unlike STIR, spectral fat suppression is sensitive to field inhomogeneity, which often causes local failure of the fat suppression. Rather than suppressing the fat signal, the idea of water excitation is to excite only the signal from water (3). Water excitation is also sensitive to inhomogeneity of the static field but less sensitive to excitation field inhomogeneity.

An alternative approach to fat suppression is that of the Dixon method (4), where multiple images are acquired with different echo times. Due to chemical shift, the contrast in the images will be altered, and the water and fat signals can be separated in a postprocessing step. The outcome is a water image (fat suppressed) and a fat image (water suppressed). This is a special case of chemical shift imaging (5) with proton spectrum modeling and extremely low spectral resolution. Depending on the number of acquired images, the Dixon methods can be classified as single-point (6), two-point (4, 7–11), three-point (12–17), or multi-point as in three or more (18–22).

An important feature of these methods is the possibility to account for phase errors resulting from field inhomogeneity, allowing uniform fat suppression over a large field of view. The phase errors are ambiguous, as they have the same effect as chemical shift on voxels that contain only water or only fat. However, this ambiguity can be resolved, since the phase errors are known to vary smoothly across the image. A common strategy to determine the phase error map is region growing algorithms (9, 14, 16, 17, 23). These are fast and tractable but greedy as the classification of a voxel is based on the incomplete information of previously grown voxels. A more sophisticated approach is to formulate the phase error estimation as a global optimization problem (21, 22).

Often, the three-point or multi-point Dixon methods are preferred over the two-point methods as they tend to be less vulnerable to reconstruction errors. However, the two-point methods are desirable in situations where the acquisition time is critical, such as dynamic imaging or breath-hold imaging. In the original two-point method (4), the echo times were constrained so that one image was required to have water and fat in-phase, and the other opposed-phase. Xiang showed that the opposed-phase constraint could be relaxed (10), and Eggers et al. (11) suggested that also the remaining constraint could be removed, allowing both echo times to be set as short as possible.

The signal model typically employed by Dixon techniques includes single delta function spectral peaks for water and fat. In practice, the water and fat spectral peaks are broadened due to diffusion, intravoxel susceptibility, and multiple spectral components of fat (24). It has been proposed to expand the signal model to account for *T* (24–27) and multiple fat spectral peaks (20, 27). A more accurate signal model results in significantly reduced bias of the fat signal fraction, which can be used to quantify fat content (26–28). In the time domain, spectral broadening can be modeled as an apparent *T* decay, corresponding to a Lorentzian lineshape in the spectral domain.

In this article, a two-point Dixon method is proposed without the usual constraints on the echo times. This allows flexibility to shorten the acquisition time or increase the spatial resolution. The signal model accounts for broadening of the fat spectral peak for improved water-fat separation. The estimation of the phase error map is formulated as a global optimization problem, and the noise performance of the proposed method is compared with the Cramér-Rao bounds. Feasibility is demonstrated for in vivo abdominal breath-hold imaging.

### THEORY

- Top of page
- Abstract
- INTRODUCTION
- THEORY
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- CONCLUSION
- REFERENCES

Two complex images are acquired with echo times *TE*_{1} and *TE*_{2}, where *TE* = 0 is defined as the time of excitation for a gradient-recalled echo. %and as the centre of the echo for a spin echo. This gives each voxel four degrees of freedom to estimate the following parameters: *W* and *F*—the real-valued signals from water and fat at *TE* = 0, the unit magnitude phase factor (phasor) *b*_{0} representing the phase of the water signal at *TE*_{1}, and the phasor *b* representing the phase accumulated during Δ*TE* = *TE*_{2} − *TE*_{1}, mostly due to static field inhomogeneity. Since *b*_{0} and *b* are nuisance parameters in this context, they are referred to as error phasors. In each voxel, the following model is used to describe the signals at the two echo times:

- (1)

The parameters *a*_{1} and *a*_{2} are defined as

- (2)

where γ is the hydrogen gyromagnetic ratio (42.6 MHz/T), *B*_{0} is the magnetic flux density of the static field and δ is the chemical shift of fat (≈ 3.4 ppm relative to the water resonance frequency). The parameter ν denotes the inverse of the apparent *T* of fat due to spectral broadening. The *T* of water is assumed to be much larger than Δ*TE*, resulting in no significant spectral broadening. The spectral width ν can be assumed known a priori (26), or calibrated from the image data as described under the subheading “calibration of the fat spectral width.” A constant ν is used for the whole image.

Under “variable estimation” below, it is described how to estimate the parameters of interest in each voxel. Two alternative phasors *b* are obtained. In the section “resolving error phasor ambiguity,” it is described how to find the correct *b* in each voxel by applying a message-passing algorithm at low resolution. A “magnitude image” is used, where each voxel equals *m* = |*S*_{1}| + |*S*_{2}|. A schematic overview of the method is given in Fig. 1.

#### Variable Estimation

Taking the square magnitude of the signals in Eq. 1 gives

- (3)

By introducing the fat signal fraction *Q* = *F*/(*W* + *F*), Eq. 3 becomes quadratic in *Q* with the solutions

- (4)

where

- (5)

It follows from Eq. 1 that

- (6)

Through Eqs. 4 and 6 two alternative solutions of *b* can be obtained; *b*_{A} and *b*_{B}. If ν = 0, *c*_{1} = *c*_{2} so that *Q*_{A} = 1 − *Q*_{B}, i.e. the fat fraction becomes a water fraction. This explains the water and fat “swap-artifacts” associated with choosing the incorrect solution of *b*.

After resolving the error phasor ambiguity, noise can be efficiently removed from the *b*-map by smoothing. To give less weight to noisy background voxels, the *b*-map is multiplied with the magnitude image prior to smoothing (and normalized afterwards). Smoothing of complex images equals smoothing of the real and imaginary channels separately. An efficient smoothing algorithm is given by Xiang (10).

Once *b* is determined, *b*_{0} is calculated as:

- (7)

where it was used that |*b*_{0}| = 1 and *W* + *F* > 0 (reasonable for spoiled gradient echo and spin echo sequences). Weighted smoothing of the *b*_{0}-map is performed in the same manner as for the *b*-map, described above.

The model (Eq. 1) can now be linearized by removing *b*_{0} and *b*. If split into real and imaginary parts, it can be described on matrix form as:

- (8)

where

- (9)

The least-squares solution for *W* and *F* is given by

- (10)

where **A**^{†} = (**A**^{T}**A**)^{−1}**A**^{T} is the Moore-Penrose pseudoinverse (which needs to be calculated only once for the entire image).

#### Resolving Error Phasor Ambiguity

This section describes how to use global information to choose between *b*_{A} and *b*_{B} in all voxels simultaneously. In particular, the *b*-map **b** is assumed to be spatially smooth. In **b**, each voxel *s* has a phasor *b*_{s} ∈ {*b*_{A;s},*b*_{B;s}}. The quality of a configuration **b** is measured by the energy

- (11)

where is the set of all neighbor pairs using a 4-neighborhood for 2D images and a six-neighborhood for 3D images. The discontinuity cost *V* for a neighbor pair (*s*,*t*) is defined as

- (12)

where *m* is the voxel magnitude, and *d*^{2}(*s*,*t*) is the squared euclidean distance between voxels *s* and *t*. Typically, there will be two different values of this distance: one for in-plane neighbors and one for through-plane neighbors. This discontinuity cost gives a small energy for phase coherent neighbors and vice versa, thus imposing smoothness on **b**. Moreover, phase incoherence is more costly for voxels with large magnitudes and for neighbors that are spatially close. Note that *V*_{st} = *V*_{ts} and that one pair of neighbors has four possible configurations with four associated discontinuity costs.

Finding the lowest energy for Eq. 11 corresponds to finding the smoothest configuration of **b**. This problem is NP-hard since the discontinuity costs are nonsubmodular, but an approximation can be found using the *sequential tree-reweighted message-passing* (TRW-S) algorithm (29, 30).

TRW-S is an iterative algorithm, related to belief propagation (31), where information is sent between neighbor voxels in the form of messages. A message *M* is a vector of size 2, with elements denoted *M*(*A*) and *M*(*B*) corresponding to the minimum energies currently associated with the phasors *b*_{A} and *b*_{B} in the voxel sending the message. In each neighbor pair (*s*,*t*) ∈ , messages will be sent from *s* to *t* and from *t* to *s*, denoted *M*_{s t} and *M*_{t s}.

Since the messages in opposite directions will be updated in a forward and a backward pass, respectively, they can be stored in the same memory location. Thus, one message needs to be stored for each neighbor pair. All messages are precisely defined by the initialization and the message update rule described below.

TRW-S uses a trivial ordering of the voxels according to *i*(*s*) = *x* + *N*_{x}(*y* + *N*_{y}*z*), where *x*, *y*, and *z* are the Cartesian coordinates of voxel *s*, and *N*_{x} and *N*_{y} are the number of voxels along the *x*- and *y*-axis, respectively. The reweighting constant λ = 1/3 for 3D images and λ = 1/2 for 2D images. A description of the algorithm follows:

More details of TRW-S can be found elsewhere (29, 30).

##### Low Resolution Solution

Since **b** is spatially smooth, the TRW-S algorithm can be run at a lower resolution and still find an accurate solution. According to our experience, a correct solution is even more likely to be found at lower resolutions. This is believed to depend on the larger physical neighborhood and reduced noise. An additional advantage is that the algorithm runs faster for the smaller grid associated with a lower resolution.

To downsample the alternative phasors *b*_{A} and *b*_{B}, an image pyramid structure is used, where each “parent voxel” on a coarse level *L* corresponds to four “child voxels” on a finer level *L* − 1. If 3D images are used, each parent voxel corresponds to eight child voxels. Alternatively, in the case of nonisotropic 3D images, the downsampling can be used to make the voxels more isotropic (for example, if the voxel size on level 1 is 2 × 2 × 5, level 2 can be downsampled to 4 × 4 × 5, and level 3 to 8 × 8 × 10. Thus, each voxel at level 3 has eight child voxels at level 2, which each have four child voxels at level 1).

On level 1, each voxel *t* has two phasors, *b*_{A;t} and *b*_{B;t}, calculated through Eqs. 4 and 6. To obtain corresponding phasors for a “parent voxel” *T* on a coarser level, the phasor sum, weighted by the magnitude *m*, is taken for all 2^{4} phasor combinations (2^{8} for the 3D case). The combination with the largest magnitude of the sum is considered most phase coherent, and the phasor *b*_{A,T} is assigned the same phase as this combination. The phasor *b*_{B,T} is assigned the phase of the “complementary” combination. For instance, if the combination [*b*_{B;t1},*b*_{A;t2},*b*_{B;t3},*b*_{B;t4}] has the largest magnitude of the weighted sum, then *b*_{A,T} = exp(*i* ∠(*m*_{t1}*b*_{B;t1} + *m*_{t2}*b*_{A;t2} + *m*_{t3}*b*_{B;t3} + *m*_{t4}*b*_{B;t4})) and *b*_{B,T} = exp(*i* ∠(*m*_{t1}*b*_{A;t1} + *m*_{t2}*b*_{B;t2} + *m*_{t3}*b*_{A;t3} + *m*_{t4}*b*_{A;t4})). In this manner, two phasor candidates are obtained for each voxel on every level.

When TRW-S has found the best configuration of **b** on level *L*, the solution is chosen in each voxel at level 1 that is most coherent with the phase of its parent.

#### Calibration of the Fat Spectral Width

The model parameter ν may be calibrated from the image data, similar to the approach described by Yu et al. (20). This requires an initial water/fat separation where ν = 0, corresponding to the standard signal model.

A binary fat mask is composed of all voxels with a fat signal fraction *F*/(*W* +*F*) > 75% and with a magnitude larger than the median magnitude of all nonbackground voxels (the background threshold was obtained using Otsu's method (32)). We want to calibrate ν so that the signal from water becomes as small as possible inside this mask. This is done by varying ν and updating *a*_{1}, *a*_{2} and the **A** matrix following Eqs. 2 and 9. Since fat is dominant inside the fat mask, the largest fat fraction obtained through Eq. 4 is chosen before obtaining *b* and *b*_{0} through Eqs. 6 and 7. For each ν, the sum of |*W*_{LS}| given by Eq. 10 is taken over all voxels inside the fat mask. The value of ν is chosen that minimizes this sum, and the water/fat separation algorithm is performed a second time, now including ν in the model.

In Fig. 2, the effect of including a fat spectral broadening parameter is demonstrated. Using a seven-peak model of the fat spectrum including *T*_{2} decay, with parameters taken from the literature (33), signals at 1.5 T were calculated for fat fractions in the range 0–100%. Estimated fat fraction is given by Eqs. 4,6,7, and 10. The fat spectral width ν was either calibrated from the 100% fat fraction signal, or assumed to be zero.

#### Cramér-Rao bounds

The Cramér-Rao bounds provide a lower limit on the variance of the water and fat estimates for any unbiased estimator. As this variance will depend on the echo times, the Cramér-Rao bounds give guidance in choosing echo times, regardless of estimation method. Associated with a lower bound on the variance is an upper bound on the effective number of signal averages (NSA). The NSA serves as a normalization of the variance in the estimates, and is defined as (24):

- (13)

- (14)

where σ is the noise variance in the real and imaginary parts of the source images and σ, σ are the variances of the estimates. For a two-point acquisition and an unbiased estimator, this gives a maximum NSA of 2, irrespective of noise variance. Given the signal model (Eq. 1) and the parameters to be estimated (*W*, *F*, *b*_{0}, and *b*), the Cramér-Rao bounds can be calculated for any combination of echo times, water and fat magnitudes, and fat spectral width. The details of this procedure are described elsewhere (34).

In Fig. 3, the upper bounds on the NSA of the water and fat estimates are shown for different echo times and fat fractions, with and without broadening of the fat peak.