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Keywords:

  • two-point Dixon;
  • water and fat separation;
  • chemical shift imaging;
  • fat suppression

Abstract

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. THEORY
  5. MATERIALS AND METHODS
  6. RESULTS
  7. DISCUSSION
  8. CONCLUSION
  9. 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

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. THEORY
  5. MATERIALS AND METHODS
  6. RESULTS
  7. DISCUSSION
  8. CONCLUSION
  9. 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 T1-weighted and T2-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 T1 rather than chemical shift. Consequently, all tissues with a T1 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 T1 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 Tmath image (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 Tmath image 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

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. THEORY
  5. MATERIALS AND METHODS
  6. RESULTS
  7. DISCUSSION
  8. CONCLUSION
  9. REFERENCES

Two complex images are acquired with echo times TE1 and TE2, 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) b0 representing the phase of the water signal at TE1, and the phasor b representing the phase accumulated during ΔTE = TE2TE1, mostly due to static field inhomogeneity. Since b0 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:

  • equation image(1)

The parameters a1 and a2 are defined as

  • equation image(2)

where γ is the hydrogen gyromagnetic ratio (42.6 MHz/T), B0 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 Tmath image of fat due to spectral broadening. The Tmath image of water is assumed to be much larger than ΔTE, 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 = |S1| + |S2|. A schematic overview of the method is given in Fig. 1.

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Figure 1. Graphical scheme giving an overview of the water/fat separation method, from the complex dual-echo source images (bottom left) to the reconstructed water-only and fat-only images (bottom right). The ambiguity associated with choosing between error phasors bA and bB is resolved by applying a message-passing algorithm (TRW-S) at low resolution (in this case at resolution level 3). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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Variable Estimation

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

  • equation image(3)

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

  • equation image(4)

where

  • equation image(5)

It follows from Eq. 1 that

  • equation image(6)

Through Eqs. 4 and 6 two alternative solutions of b can be obtained; bA and bB. If ν = 0, c1 = c2 so that QA = 1 − QB, 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, b0 is calculated as:

  • equation image(7)

where it was used that |b0| = 1 and W + F > 0 (reasonable for spoiled gradient echo and spin echo sequences). Weighted smoothing of the b0-map is performed in the same manner as for the b-map, described above.

The model (Eq. 1) can now be linearized by removing b0 and b. If split into real and imaginary parts, it can be described on matrix form as:

  • equation image(8)

where

  • equation image(9)

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

  • equation image(10)

where A = (ATA)−1AT 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 bA and bB in all voxels simultaneously. In particular, the b-map b is assumed to be spatially smooth. In b, each voxel s has a phasor bs ∈ {bA;s,bB;s}. The quality of a configuration b is measured by the energy

  • equation image(11)

where equation image 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

  • equation image(12)

where m is the voxel magnitude, and d2(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 Vst = Vts 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 bA and bB in the voxel sending the message. In each neighbor pair (s,t) ∈ equation image, messages will be sent from s to t and from t to s, denoted Ms [RIGHTWARDS ARROW] t and Mt [RIGHTWARDS ARROW] 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 + Nx(y + Nyz), where x, y, and z are the Cartesian coordinates of voxel s, and Nx and Ny 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:

  • 1
    Initialize all messages to zero.
  • 2
    Repeat (for a fixed number of iterations or until some stopping criterion):
    • a
      For all voxels s, in order of increasing i(s):
      • Calculate s = equation imageMt [RIGHTWARDS ARROW] s.

      • For each neighbor pair (s,t) ∈ equation image where i(s)< i(t) (so that t are the right, down, and below neighbors of s), update the message Ms [RIGHTWARDS ARROW] t for k ∈{A,B}:

        • equation image
      • Normalize Ms [RIGHTWARDS ARROW] t (subtract minimum element from both elements).

      • Repeat (a), but in order of decreasing i(s) and update the messages Ms [RIGHTWARDS ARROW] t where i(t) < i(s) (so that t are the left, up, and above neighbors of s).

      • In order of increasing i(s), choose the solution bs = bj;s, j ∈{A,B} that minimizes:

        • equation image

        Note that the left sum is taken over the left, up, and above neighbors of s and that the right sum is taken over the right, down, and below neighbors.

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 bA and bB, 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, bA;t and bB;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 24 phasor combinations (28 for the 3D case). The combination with the largest magnitude of the sum is considered most phase coherent, and the phasor bA,T is assigned the same phase as this combination. The phasor bB,T is assigned the phase of the “complementary” combination. For instance, if the combination [bB;t1,bA;t2,bB;t3,bB;t4] has the largest magnitude of the weighted sum, then bA,T = exp(i ∠(mt1bB;t1 + mt2bA;t2 + mt3bB;t3 + mt4bB;t4)) and bB,T = exp(i ∠(mt1bA;t1 + mt2bB;t2 + mt3bA;t3 + mt4bA;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 a1, a2 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 b0 through Eqs. 6 and 7. For each ν, the sum of |WLS| 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 T2 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.

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Figure 2. Estimated fat fraction as a function of true fat fraction for three sampling schemes, with and without calibration of the fat spectral width. As substitutes for the echo-times, a = ∠a1 and Δa = ∠(a2/a1) are indicated. a: a = Δa = 90°, b: a = Δa = 180°, c: a = Δa = 270°.

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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):

  • equation image(13)
  • equation image(14)

where σmath image is the noise variance in the real and imaginary parts of the source images and σmath image, σmath image 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, b0, 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.

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Figure 3. In each plot, the Cramér-Rao bounds on the effective number of signal averages (NSA) are shown in false color, as a function of a = ∠a1 and Δa = ∠(a2/a1). The columns correspond to different fat fractions Q. The first row shows NSA of the water estimate with no spectral broadening of the fat peak. The corresponding NSA of the fat estimate is not shown, as it is symmetric with respect to fat fraction (i.e, NSA of fat for fat fraction Q equals the NSA of water for fat fraction 1 − Q). The second and third rows include broadening of the fat peak in the signal model, so that the NSA of water (second row) and the NSA of fat (third row) are no longer symmetric.

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MATERIALS AND METHODS

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. THEORY
  5. MATERIALS AND METHODS
  6. RESULTS
  7. DISCUSSION
  8. CONCLUSION
  9. REFERENCES

Noise Simulation

To experimentally examine the performance of the proposed estimation method, a Monte Carlo simulation study was performed. Using Eq. 1, signal samples were calculated for 1.5T with four fat fractions (0%, 40%, 60%, and 100%) with and without broadening of the fat peak (ν = 0 and ν−1 = 10 msec), varying α = ∠a1 and Δα = ∠(a2/a1) (as substitutes for TE1 and ΔTE) from 0 to 360° in 100 discrete steps. Note that the Cramér-Rao bound on the NSA for all these cases are shown in Fig. 3. For each combination of fat fraction, fat spectral width, α, and Δα, 1000 signal samples were created and gaussian noise was added to the real and imaginary parts of the signal samples giving a signal-to-noise ratio of 100. Estimates of W and F were obtained using the proposed algorithm (the b candidate closest to the correct solution was chosen). The smoothing step was omitted, since smoothing introduces bias in the estimates. The fat fraction error and NSA for the water and fat estimates were calculated.

Water/Fat Imaging

As a proof of concept, the method was tested for in vivo imaging of the abdomen. The proposed method might be of interest in abdominal imaging, since imaging time is often restricted by breath-holding. Imaging was performed at four occasions (A, B, C, and D) of totally three volunteer subjects (the same subject was scanned at occasions B and D). One 3T and two 1.5T MR scanners were used (Philips Healthcare, Best, The Netherlands). Scan details are given in Table 1. Dual-echo datasets (Adual, Bdual, Cdual, Ddual) were acquired with different combinations of echo times. Datasets Csingle and Dsingle were obtained using single-echo protocols with spectral fat suppression employed clinically at our institution for dynamic liver scans. These were modified by removing the fat suppression and adding a second echo to create the protocols used in Cdual and Ddual. The second echo gives a longer minimum repetition time (TR), but time is saved by omitting the fat suppression. The resulting breath-hold times were 17.2 sec and 18.7 sec for datasets Csingle and Cdual, and 18.5 sec and 17.5 sec for datasets Dsingle and Ddual. All datasets except Bdual were 3D scans. Gradient-recalled echo and phased array coils with parallel imaging (35) were used for all datasets. The study was performed according to the guidelines of the local ethics committee, and informed consent was given from all subjects.

Table 1. Scan Details
 AdualBdualCsingleCdualDsingleDdual
Scanner3T Achieva1.5T ACS3T Achieva1.5T Achieva
FOV [mm3]480 × 378 × 174375 × 297 × 175430 ×305 ×214430 × 305 × 162
Voxelsize [mm3]1.5 × 1.5 × 6.02.0 × 2.0 × 8.02.9 × 2.9 × 2.52.0 × 2.0 × 4.0
No. slices292017080
Flip angle80°10°10°
TR (msec)4.11032.84.12.84.2
TE1 (msec) (α)1.2 (180°)1.4 (110°)1.3 (200°)1.3 (200°)1.4 (110°)1.4 (110°)
TE2 (msec) (Δα)2.9 (270°)2.8 (110°)2.6 (200°)2.7 (110°)
Fat suppressionYesYes

Water/fat separation was performed offline for the dual-echo datasets using the proposed algorithm. The algorithm was run in 3D except for Bdual which was processed in 2D. The fat spectral width, ν, was calibrated from the data. The field map was determined at resolution level 3 using TRW-S with 100 iterations, and the smoothing kernel size was set to 10 mm. The execution time for the water/fat separation algorithm was measured.

The water-only and fat-only images were examined by an experienced radiologist. The presence and localization of water/fat swap-artifacts was determined. The conventionally fat suppressed Csingle and Dsingle were compared with the water images reconstructed from Cdual and Ddual, respectively. Any difference in the quality of fat suppression was subjectively identified by the radiologist, blinded to the method used.

RESULTS

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. THEORY
  5. MATERIALS AND METHODS
  6. RESULTS
  7. DISCUSSION
  8. CONCLUSION
  9. REFERENCES

Noise Simulation

The bias in terms of fat fraction error in the Monte Carlo simulation was almost zero, except for at echo times with extremely low upper bounds on the NSA. For echo times were the Cramér-Rao bound on the NSA was >0.1 for both water and fat, the fat fraction error was 0.00 ± 0.10% without spectral broadening and 0.03 ± 0.14% with spectral broadening (mean ± standard deviation).

The NSA of the water and fat estimates determined from the simulation were visually compared to the Cramér-Rao bounds (Fig. 3) by creating corresponding plots (not shown). Excellent agreement with the Cramér-Rao bounds was found. The difference between the Cramér-Rao bound on the NSA and the simulated NSA was 0.01 ± 0.06 for water and 0.01 ± 0.08 for fat without spectral broadening, and 0.00 ± 0.07 for water and 0.00 ± 0.06 for fat with spectral broadening.

Water/Fat Imaging

Examples of the water and fat separated images are shown in Figs. 4–7. Execution times and water/fat swaps found by the radiologist are reported in Table 2. Figure 4 shows the reconstructed water-only and fat-only images from dataset Adual, demonstrating consistent water/fat separation. Figure 5 shows the water-only and fat-only images reconstructed from dataset Bdual, both with the fat spectral width ν set to 0 (corresponding to the standard signal model), and with ν calibrated from the data. For this combination of echo times, the standard model water-only image contains residual fat signal, while the broad fat peak model achieves stronger fat suppression. In Fig. 6, the spectrally fat-suppressed dataset Csingle is compared with the reconstructed water-only image from dataset Cdual. The spectral fat suppression results in more residual fat signal, and fails in regions of large field inhomogeneity. Figure 7 compares the spectrally fat-suppressed dataset Dsingle with the water-only image reconstructed from dataset Ddual. The spectral fat suppression causes undesired suppression of the water signal in regions with large static field inhomogeneity. The water-only image offers better visualization of organs compassed by adipose tissue, such as the pancreas and the adrenal glands. Both water-only images reconstructed from Cdual and Ddual were found to achieve fat suppression with superior quality compared with the spectrally fat suppressed Csingle and Dsingle.

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Figure 4. One slice of reconstructed water-only (a) and fat-only (b) images acquired at 3T. Consistent separation of water and fat signal is demonstrated, even in the arms near the periphery of the field of view.

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Figure 5. One slice of reconstructed water-only (a, c) and fat-only (b, d) images acquired at 1.5T. water/fat separation was performed both using the standard signal model (a, b) and the broad fat peak model (c, d). In the standard model water-only image (a), residual fat signal results in poor contrast between water-dominant and fat-dominant tissue. The broad fat peak model gives superior fat suppression (c). The spectral broadening also affects the water and fat estimates in water-dominant tissue. In this case, the fat signal in the liver is somewhat higher for the broad fat peak model (d) compared with the standard signal model (b).

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Figure 6. Two slices from datasets covering the entire liver, acquired at 3T. A single echo with spectral fat suppression (a, c) is compared with the water-only image reconstructed from a dual-echo acquisition (b, d). Superior fat suppression of the reconstructed water-only image can be seen in the subcutaneous fat. Additionally, the spectral fat suppression causes undesired suppression of the water signal in regions far from the magnet isocentre, due to inhomogeneity of the static field (c).

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thumbnail image

Figure 7. Two slices from datasets covering the entire liver, acquired at 1.5T. A single echo with spectral fat suppression (a, c) is compared with the water-only image, reconstructed from a dual-echo acquisition (b, d). The reconstructed water-only image shows superior fat suppression, enabling better visualization of water-dominant tissue surrounded by fat, such as the pancreas and the adrenal glands (a, b, zoomed in dotted box). The spectral fat suppression is prone to cause unwanted suppression of the water signal in regions with large static field inhomogeneity (c).

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Table 2. Results of the Water/Fat Separationa
 Time (sec)Water/fat swaps
  • a

    Execution time and localization of water/fat swap artifacts are given for the dual-echo datasets.

Adual92None
Bdual34None
Cdual270Half the forearm in both arms, below the elbow
Ddual102None

DISCUSSION

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. THEORY
  5. MATERIALS AND METHODS
  6. RESULTS
  7. DISCUSSION
  8. CONCLUSION
  9. REFERENCES

This article provides tools to perform water-fat separation from dual-echo datasets with flexible echo times. The following topics deserve more in-depth discussion.

Signal Model

The proposed signal model and variable estimation does not make any assumptions on the echo times. In contrast, previous two-point Dixon methods constrain the echo times so that the water and fat signal components are sampled in-phase (IP), and opposed-phase (OP) (4, 7–9), or partially opposed-phase (POP) (10). Even if the echo times are totally flexible in theory, they are in practice limited by noise performance and capacity of resolving error phasor ambiguity, as discussed below.

The relevance of Tmath image decay in water/fat imaging has been long recognized (36). Glover (24) argued that diffusion, susceptibility dephasing, and multiple spectral components of fat independently contribute to signal decay. For the multi-point Dixon method, signal models have been proposed that account for multiple fat spectral peaks (20), a common Tmath image for water and fat (20, 25), and independent Tmath image for water and fat (27). The relative magnitudes and chemical shifts of the fat spectral peaks are assumed known a priori (or calibrated from the data) and constant over the image, while Tmath image values are determined in each voxel.

In this work, the inclusion of a global fat spectral width ν in the signal model is regarded as a crude means to account for the multiple spectral components of fat (26), manifested as an apparently shortened Tmath image. It is understood that this apparent Tmath image depends on the combination of echo times, due to the complex interference of the fat spectral peaks. The broadening of the fat peak due to multiple spectral components is believed to be more relevant for water/fat separation than the actual Tmath image decays of water and fat, which are assumed to be large compared with ΔTE. However, this assumption may be violated if Tmath image is shortened, for instance due to hepatic iron overload (37). Employing a multiple fat spectral peak model in a phantom experiment, Chebrolu et al. (27) showed that Tmath image correction does not improve the fat signal fraction estimates, unless iron contamination is present.

The capacity of improved water/fat separation using the broad fat peak model is demonstrated in Fig. 2 for three different sampling schemes, assuming a true seven-peak model of the fat spectrum. Overall better estimates are seen with calibrated ν. The fat spectral width calibration algorithm is designed to minimize water signal in adipose tissue to improve the visual quality of the fat suppression (Fig. 5). Although, the fat fraction accuracy remains a subject for future investigation.

Note that the proposed estimation procedure does not make any assumptions on a1 and a2. It is thus straightforward to expand the model to account for multiple peaks with a priori known chemical shifts, relative magnitudes and Tmath image relaxation.

Resolving Error Phasor Ambiguity

The problem of error phasor ambiguity, associated with the ambiguity of identifying the separated water and fat signal components, was formulated as a global energy minimization problem. A desirable feature of this approach is that the formulation of the energy function and its minimization can be addressed as separate problems. In particular, the performance of different minimizers can be compared in terms of the minimum energy found. The energy minimization approach has previously been taken in the context of the multi-point Dixon method (21, 22). In those works, the energy was minimized using iterated conditional modes (21, 38) and graph cuts (22, 39). Since iterated conditional modes is obsolete and the graph cut technique employed in (22) cannot be used for the proposed nonsubmodular energy function (40), TRW-S (29, 30) was proposed. This algorithm was recently shown to compare favorably with other optimization methods for a set of benchmark vision problems (41).

Xiang proposed a regional iterative phasor extraction (RIPE) procedure to resolve the error phasor ambiguity (10). RIPE could be used instead of the energy minimization proposed in this paper. However, a comparison of the performance of RIPE and different energy minimization algorithms is beyond the scope of this work.

In the determination of the error phasor map, it is assumed that the correct error phasor configuration is the most phase coherent. However, for the IP/OP sampling scheme, bA = −bB for all voxels. As a consequence, the energy of the correct error phasor map given by Eq. 11 is equal to the energy of its complement. Thus, the IP/OP sampling scheme rules out the possibility of identifying the separated water and fat signals based on phase information. Instead, the identification can be based on manual seed point selection (8) or intensity distribution (9). Under this “global ambiguity,” the energy minimization tends to fall into local minima. Therefore, for the IP/OP sampling scheme, other approaches are recommended, such as region growing (9). Alternatively, a priori knowledge of intensity distribution or phase error may be incorporated into the energy function.

Noise Properties

The noise properties of the two-point Dixon method have been examined in terms of the NSA as a function of α = ∠a1 and Δα = ∠a2/a1, directly corresponding to TE1 and ΔTE through Eq. 2. Since too noisy water and fat estimates are of limited interest, the choice of echo times is in practice restricted by the NSA.

First, the Cramér-Rao bounds were calculated, providing an upper bound on the NSA for any unbiased estimation procedure. Then, the NSA for the proposed method were measured in a simulation experiment, showing that the proposed estimator is unbiased and matches the Cramér-Rao bounds. In other words, the proposed estimation method was shown to be efficient for the examined cases.

Echo times giving low NSA can be explained in terms of sampling redundancy. For instance, Δα = 0°,360°, gives the same relative water and fat phase for both echoes and results in low NSA. Moreover, low NSA is seen (Fig. 3) for echo times corresponding to symmetrical sampling around 180° or 360°. This results in Re(a1) ≈ Re(a2) and |S1| ≈ |S2| so that the denominator of Eq. 4 approaches zero. For voxels with one dominant species, high NSA is achieved for symmetrical sampling around 90° or 270°. For voxels with mixed water and fat content, the NSA is best when α or α + Δα is close to 180°. For the IP/OP sampling scheme, NSA is high for any fat fraction. The asymmetry between water and fat in the broad fat peak model is reflected in its NSA properties. Specifically, the NSA of fat decreases with longer echo times due to signal decay.

The intention of the smoothing step is to remove noise from the b0 and b-maps. In practice, this results in higher NSA than predicted by the Cramér-Rao bounds. However, smoothing also introduces bias in regions were the smoothness assumption is violated. For this reason, the smoothing kernel should not be too large.

It can be noted that complex-valued water and fat signals (defined as Wc = Wb0 and Fc = Fb0) may also be used. Given b, the complex water and fat signals can then be found from Eq. 1. This approach is simple, since no b0-map or least-squares estimation is needed. However, the somewhat lower NSA of Wc and Fc after smoothing (not shown) motivates the use of the real-valued W and F as proposed in this work.

In Vivo Water/Fat Imaging

The resulting water-only and fat-only images demonstrate the feasibility of the method. Consistent separation of water and fat signal is provided, with exception of dataset Cdual, where some water/fat swaps were found in the image periphery. The echo times used for this dataset are close to those of the IP/OP sampling scheme. Therefore, the error phasor determination may suffer from the global ambiguity problem discussed above, being more prone to give swap artifacts. The execution time of the algorithm, given in Table 2, increases with datasize but remains within reasonable limits.

The in vivo experiments also demonstrate the ability of the proposed method to acquire water and fat images faster than with the IP/POP or IP/OP sampling schemes, as shorter echo times allow shorter minimum TR (11). As an example, keeping all other parameters fixed, the minimum TR for the Cdual protocol for the flexible, IP/POP and IP/OP sampling schemes are 4.1, 5.1, and 7.2 msec, respectively. The corresponding minimum TR for the Ddual protocol are 4.2, 5.7, and 5.7 msec. Moreover, the flexible echo times are associated with flexibility in the resolution and readout bandwidth.

Compared with the protocols with spectral fat suppression employed clinically at our institution for dynamic liver imaging, the water-only images produced by the proposed method were found to yield fat suppression with superior quality. This promises better visualization of the contrast uptake in dynamic liver imaging. This may also have an impact on delineating adrenal and pancreatic tumors and assessing the extent of tumor growth into the adipose tissue surrounding the glands.

Future studies of interest include evaluation of the method performance and application in oncology and dynamic imaging of the liver.

CONCLUSION

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. THEORY
  5. MATERIALS AND METHODS
  6. RESULTS
  7. DISCUSSION
  8. CONCLUSION
  9. REFERENCES

A two-point Dixon method has been proposed without the usual constraints on the echo times. The noise in the water and fat estimates has been shown to achieve the Cramér-Rao lower bound, and feasibility has been demonstrated in breath-hold abdominal imaging.

REFERENCES

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. THEORY
  5. MATERIALS AND METHODS
  6. RESULTS
  7. DISCUSSION
  8. CONCLUSION
  9. REFERENCES