Dual Echo Dixon Imaging with a Constrained Phase Signal Model and Graph Cuts Reconstruction

Abstract

Purpose

The purpose of this work is to derive and demonstrate constrained-phase dual-echo Dixon imaging within a maximum likelihood framework solved with a regularized graph-cuts-guided optimization.

Theory and Methods

Dual-echo Dixon reconstruction is fundamentally underdetermined; however, adopting a constrained-phase signal model reduces the number of unknowns and the nonlinear problem can be solved under a maximum likelihood framework. Period shifts in the field map (manifesting as fat/water signal swaps) must also be corrected. Here, a regularized cost function promotes a smooth field map and is solved with a graph-cuts-guided greedy binary optimization. The reconstruction shown here is compared to two other prevalent Dixon reconstructions in experimental phantom and human studies.

Results

Reconstructed images of the water and fat signal are shown for a phantom study, and in vivo studies of foot/ankle, pelvis, and CE-MRA of the thighs. The method shown here compared favorably with the other two methods. Large field inhomogeneities on the order of 20 ppm were resolved, thereby avoiding the fat and water signal swaps present in images reconstructed with the other methods.

Conclusion

Constrained-phase dual-echo Dixon imaging solved with a regularized graph-cuts-guided optimization has been derived and demonstrated to successfully separate water and fat images in the presence of large magnetic field inhomogeneities.

Introduction

The Dixon technique can be broadly defined as a method in which the differential chemical-shift-induced phase accumulation of fat and water in images acquired at one or more echo times is exploited in separating the fat and water signals into separate images. This technique has found many applications throughout the body and can be used whenever fat or water obscures the signal of interest or when wishing to quantify the amount of fat such as in studies of obesity or fat fraction (1). The first description of this method used two images, one with an echo time with fat and water exactly in-phase and a second exactly out-of-phase (2). Since that first description, many variations have been developed using one (3–7), two (8–11), or three or more (12–15) echoes.

In the original Dixon method, the unknown signals to be estimated were the fat and water magnitudes, and the measurements were the signal magnitudes at the two echo times. Although this simple model of two real measurements and two real unknowns clearly showed the feasibility of the method, it was recognized that some allowance for B₀ inhomogeneity-induced phase was necessary. Failure to account for this is typically manifest as “swaps” in which fat signal is artifactually included in the water image at some locations and vice versa. Thus, subsequent methods treated the ΔB₀-induced phase as a third unknown (12, 13), leading to early techniques using images at three echo times. It is now well recognized that regardless of the technique used, effective Dixon imaging requires an estimation and correction of the phase induced by B₀ inhomogeneity.

The fat and water magnetizations are complex quantities, and together with ΔB₀-induced phase comprise five real-valued unknowns. These quantities can be determined from a minimum of three complex signals, such as those from a three-echo acquisition. While three echoes is the minimum to estimate fat, water, and ΔB₀, the use of additional echoes may provide improved SNR (13) or allow for accurate estimation of additional unknowns such as $T_2^{\ast }$ (16). With any number of echoes, however, the estimate of ΔB₀-induced phase must be unwrapped to ensure that the resulting fat and water images are free of signal swaps. The unwrapping process is typically guided by the “smoothness” of the ΔB₀ map.

When water and fat are assumed to be complex, and ΔB₀ is an additional unknown, Dixon reconstruction with only two echoes is fundamentally an underdetermined problem. However, a number of methods have been described to address this, typically through ad hoc restrictions on acquisition or reconstruction. These methods are typically deterministic in nature and mainly differ in how the field map is calculated and how the ΔB₀-induced phase unwrapping is performed. Whether from arbitrary echo time images or assumed in- and opposed-phase images, most dual-echo Dixon techniques solve for the field map deterministically under the assumption of a noise-free signal. Phase unwrapping is then typically performed with either a region growing algorithm (8) or one of a number of heuristic methods (9, 10); however, more advanced optimization-based procedures have also been applied to the unwrapping problem (11) while still using deterministic solutions for the field map. While these methods can be easy to implement and computationally fast, the deterministic solutions do not account for noise within the signal, and the heuristic phase unwrapping methods do not provably progress toward a solution.

As stated above, the dual-echo Dixon problem is underdetermined. However, the constrained-phase signal model (17) addresses this by making the assumption that fat and water have the same initial phase ϕ ₀ at the time of excitation, leading to a problem with four real-valued unknowns: the fat and water magnitudes, ϕ ₀, and ω, the phase induced by ΔB₀. Field map solutions using this signal model have been found deterministically (17).

Once these solutions for the field map are found, the task is to choose the proper single solution for the field map. Various methods can be used for this step. Graph-representable Markov models have been successfully used with greedy binary optimization for dual-echo imaging (11) (tree-reweighted message passing), and with three or more echoes (15) (graph cuts), but these have both relied on the signal model that assumes that water and fat are complex-valued. In Reference (17) the constrained-phase field map solution was chosen through parameterization of the field as a 2D polynomial in space and solved with a Gauss-Newton algorithm.

Here, the constrained-phase signal model is used to find the maximum-likelihood estimates of the water and fat signals, main magnetic field inhomogeneity, and initial phase. In contrast to the deterministic methods described above, the maximum likelihood framework defines a statistical method that finds the most likely fit of the nonlinear model to the data assuming a statistical model of the measurement noise. The maximum likelihood framework results in a periodic cost function with respect to ΔB₀ with two minima per period. The ΔB₀ map is then found using a state-of-the-art graph-cuts-guided greedy binary optimization that provably progresses toward the solution of a regularized cost function that promotes smoothness in the ΔB₀ map. A similar method was described by Hernando et al. in Reference (18), however that method relies on an in-phase echo image to estimate the initial phase of the water and fat signals and reduce the number of unknowns in the constrained-phase formulation to three.

The purpose of this work is to derive a cost-based reconstruction for constrained-phase dual-echo, multi-coil Dixon imaging within a maximum likelihood framework and show its implementation with a graph-cuts-guided greedy binary optimization. This technique is demonstrated with phantom results and in vivo results in the pelvis and the extremities at 1.5T and 3.0T.

Theory

The mathematical conventions used in this work are as follows. (·)* refers to the conjugate (Hermitian) transpose, (·)^T refers to the non-conjugate transpose of a real vector or matrix, $\mathbb{E}[\cdot ]$ denotes the expected value, tr(·) denotes the trace, and Re(·) and Im(·) denote the real and imaginary parts, respectively, of vector or matrix.

Phase Constrained Signal Model

As stated above, by constraining the initial phase of the water and fat signals to be equal, there are only four real-valued unknowns (magnitude of water and fat, shared initial phase, and the field map) which can be found with only two complex-valued measurements (dual-echo Dixon imaging). The dual-echo signal from a multi-coil phased array can be modeled using the voxel-wise (denoted by the subscript v) constrained-phase signal model shown in Equation [1].

$$G_v=A_v(\omega ,{\varphi }_0)X_v{S}_v+N_v$$ [1]

where G_v is an 2 × N_C (echoes × coils) matrix whose columns represent the acquired dual-echo signal from each of the coils for voxel v, A_v(ω, ϕ₀) is a matrix functional of Dixon system parameters, and $X_v=\left[\genfrac{}{}{0ex}{}{W_v}{F_v}\right]$ is a 2 × 1 column vector containing the true water and fat signals (W_v and F_v, both presumed to be real). S_v is a 1 × N_C matrix of the complex-valued sensitivities of the N_C coil elements, and N_v is an 2 × N_C matrix of (zero-mean) proper complex Gaussian noise whose rows have covariance Ψ. The noise covariance of the dual-echo signal model is $\mathrm{\Lambda }=\mathbb{E}[N_v^{\ast }{N}_v]=2{\mathrm{\Psi }}^T$ (19).

The matrix functional of Dixon system parameters, $A_v(\omega ,{\varphi }_0)=e^{i{\varphi }_{0,v}}{B}_v(\omega )C$, contains terms for the phase shifts from the voxel-wise shared initial phase (ϕ_0,v) of the water and fat signals, voxel-wise B₀ inhomogeneities (B_v(ω)) and chemical shift (C) of M lipid species (16):

$$B_v(\omega )=\left[\begin{array}{cc}{e}^{i{\omega }_v{t}_1}& 0\\ 0& e^{i{\omega }_v{t}_2}\end{array}\right]$$ [2a]

$$C=\left[\begin{array}{cc}1& {\displaystyle {\sum }_{m=1}^M{\alpha }_m{e}^{i2\pi \mathrm{\Delta }{f}_m{t}_1}}\\ 1& {\displaystyle {\sum }_{m=1}^M{\alpha }_m{e}^{i2\pi \mathrm{\Delta }{f}_m{t}_2}}\end{array}\right]$$ [2b]

where ω_v = γΔB_0,v, γ is the gyromagnetic ratio, t_n is the echo time of the nth acquired image, α_m is the relative amplitude of the mth lipid species, and Δf_m is the chemical shift in Hz of the mth lipid species.

Dual-Echo Multi-Coil Phase-Constrained Field Estimation

The crux of estimating the water and fat signals in dual-echo Dixon imaging is to properly estimate the shared initial phase ϕ_0,v and the ΔB₀-induced off resonance ω_v for each voxel. This is performed with a dual-echo multi-coil constrained-phase field estimation. In the presence of zero-mean Gaussian noise, the maximum likelihood estimates (20) are found from the measurements as the parameter values that minimize the four-dimensional least-squares cost function in Equation [3]:

$$\left[{\widehat{X}}_v,{\widehat{\omega }}_v,{\widehat{\varphi }}_{0,v}\right]=\arg \underset{\underset{[{\omega }_v,{\varphi }_{0,v}]\in \mathrm{ℝ}}{X_v\in {\mathrm{ℝ}}^2}}{\min }[\mathrm{Re}\{\text{tr}((A_v(\omega ,{\varphi }_0)X_v{S}_v-G_v){\mathrm{\Lambda }}^{-1}{(A_v(\omega ,{\varphi }_0)X_v{S}_v-G_v)}^{\ast })\}]$$ [3]

where Λ has been defined above as the noise covariance matrix of the dual-echo signal (19). Note that X_v, ω_v, and ϕ_0,v are minimized under the assumption that they are real-valued, in accordance with the constrained-phase signal model.

If ω_v and ϕ_0,v (within A_v(ω, ϕ ₀)) are assumed known, the maximum likelihood estimate of X_v is derived in Appendix A and equal to

$${\widehat{X}}_v(\omega ,{\varphi }_0)=Q^{-1}\mathrm{Re}\{A_v^{\ast }(\omega ,{\varphi }_0)F_v\}{Z}_v^{-1}$$ [4]

where Q = Re{C*C}, $Z_v=(S_v{\mathrm{\Lambda }}^{-1}{S}_v^{\ast })$, and $F_v=G_v{\mathrm{\Lambda }}^{-1}{S}_v^{\ast }$and is size 2 × 1. Note that F_v is a coil-combined quantity, with size 1 in the coil dimension.

Substitution of ${\widehat{X}}_v(\omega ,{\varphi }_0)$ into Equation [3] reduces the degrees of freedom in the cost function from four to two, namely ω_v and ϕ_0,v. Substituting the analytical solution for one or more variables into the separable nonlinear least squares cost function to reduce the dimensionality is called Variable Projection (VARPRO) (21). After the simplification shown in Appendix B, the reduced cost function becomes

$$\left[{\widehat{\omega }}_v,{\widehat{\varphi }}_{0,v}\right]=\arg \underset{[{\omega }_v,{\varphi }_{0,v}]\in \mathrm{ℝ}}{\min }[-D_v{(\omega ,{\varphi }_0)}^T{Q}^{-1}{D}_v(\omega ,{\varphi }_0)Z_v^{-1}]$$ [5]

where $D_v(\omega ,{\varphi }_0)=\mathrm{Re}\{A_v^{\ast }(\omega ,{\varphi }_0)F_v\}$.

The minima of Equation [5] with respect to ϕ_0,v can be found by setting the derivative equal to zero and solving for ϕ_0,v. As shown in Appendix C, this results in

$${\widehat{\varphi }}_{0,v}=\frac{1}{2}\angle [H_v{(\omega )}^T{Q}^{-1}{H}_v(\omega )]$$ [6]

where $H_v(\omega )=C^{\ast }{B}_v^{\ast }(\omega )F_v$. Equation 6 has a single coil analog in Equation [A5] in Reference (17). Performing VARPRO a second time and substituting the estimate of ϕ_0,v (which is now a function of only ω_v) into Equation [5] provides a one-dimensional cost function.

$${\widehat{\omega }}_v\equiv \arg \underset{{\omega }_v\in \mathrm{ℝ}}{\min }[J_D({\omega }_v)]\equiv \arg \underset{{\omega }_v\in \mathrm{ℝ}}{\min }[-P_v{(\omega )}^T{Q}^{-1}{P}_v(\omega )]$$ [7]

where $P_v(\omega )=\mathrm{Re}\{C^{\ast }{B}_v^{\ast }(\omega )\exp \{-i(\frac{1}{2}\angle [H_v{(\omega )}^T{Q}^{-1}{H}_v(\omega )])\left\}{F}_v\right\}$. Note that J_D(ω_v) has been implicitly defined in Equation [7].

Once the estimate of ω_v is found via Equation [7], that value can be substituted into Equation [6] to find ${\widehat{\varphi }}_{0,v}$. And further, both ${\widehat{\omega }}_v$ and ${\widehat{\varphi }}_{0,v}$can be used in Equation [4] to find estimates of W_v and F_v. Repeating this process for every voxel produces a voxel-wise estimate of all quantities.

From Voxel to Whole Volume Estimation

The one-dimensional cost function in Equation [7] is a voxel-wise measure of the fit of the maximum-likelihood estimate of the constrained-phase signal model to the data. To ensure consistency between neighboring voxel estimates, this measure is extended to a whole volume estimation, leveraging a priori knowledge of the spatially smooth nature of the whole image ω map to regularize the solution.

A whole volume cost function that promotes smoothness is shown in Equation [8]

$$J(\omega )=\arg \underset{\omega \in \mathrm{ℝ}}{\min }\left[{\displaystyle \sum _{v\in \mathrm{\Omega }}{J}_D({\omega }_v)}+\mu {\displaystyle \sum _{v\in \mathrm{\Omega }}{\displaystyle \sum _{k\in \eta }{J}_{SM}({\omega }_v,{\omega }_{v+k})}}\right]$$ [8]

where Ω is the set of all voxels within the image volume and η is the local neighborhood around a specific voxel v. J_D (Equation [7]) promotes data fidelity and J_SM promotes smoothness. J_SM is typically a measure of the distance between values of neighboring voxels such as the weighted squared finite difference

$$J_{SM}({\omega }_v,{\omega }_{v+k})={\mathrm{\Gamma }}_k|{\omega }_{v+k}-{\omega }_v{|}^2$$ [9]

where Γ_k is used to weight neighbors in comparison to the other neighbors. For example, the weight given to neighbors in the slice direction may be smaller than that given to in-plane neighbors if the slice thickness is larger than the in-plane resolution (15).

The other component of the whole volume cost function is the regularization parameter μ, which controls the balance between J_D(ω) and J_SM(ω_v, ω_v,k). Choosing an appropriate value of μ ensures that the smoothing term (J_SM) has enough of an effect to promote smoothness upon the map of ω, but not so much of an effect that the field map is over-smoothed.

Cost Function Optimization

Given that a whole volume cost function has been defined, the task is to find a solution that minimizes it. A brute force search for a global minima is intractable. Additionally, in non-convex problems such as this, descent methods can get stranded in local minima. Heuristic methods perform well in some situations, but the current state-of-the-art combines a sequential greedy binary algorithm with methods that are guaranteed to progress toward the solution with each iteration. Examples of these methods include graph cuts (15) and quadratic pseudoboolean optimization (22). In essence, at each iteration a binary choice is made in each voxel between two solutions to optimize the cost function on that iteration. As this binary process is repeated sequentially and with a sufficient search coverage, the solution is guaranteed to be within close proximity to the globally optimal solution (23). Here, graph cuts is used to make the binary choice between the two solutions in each voxel at each iteration.

Graph cuts recasts the binary decision problem as a max-flow/min-cut graph problem. That is, each voxel of the image is considered a node on a graph with edges connecting it to its neighbors and the source and sink. For each iteration of the graph cuts algorithm, four steps are performed:

1. A binary update function (ω_new) of the field map is computed.

2. A graph is constructed from ω_new and the previous estimate (ω_old) of ω such that the edges between neighboring nodes are proportional to J_SM computed from the combinations of ω_new and ω_old, and the edges adjoining nodes with the source and sink are dependent on values of J_D and J_SM computed from the combinations of ω_new and ω_old. Details on graph construction can be found in References (15, 23, 24).

3. A min-cut is made through the graph and leaves some nodes connected to the source and the others connected to the sink.

4. The estimate of the whole image field map, ω, is updated. Nodes connected to the source keep ω_old, and nodes connected to the sink take the value of ω_new.

This process, shown in Figure 1, continues until the solution converges.

Figure 1.

Flowchart of the procedure used in the graph cuts reconstruction.

In step 1 of the process, the new estimate of the field map is computed based on a “jump schedule” for the graph cuts method. The jump schedule defines how much the estimate of ω changes or “jumps” from one update to the next. A schematic representation of three common jumps is shown in Figure 2. These jumps consist of 1) “period jumps” which shift ω one period, 2) “minima jumps” which shift ω to another minimum within the same period, and 3) “refinement jumps” which shift ω by a smaller amount in an attempt to refine the field map even further. These “refinement” jumps increase the cost of the data fidelity term by moving out of the J_D minimum, but reduce the overall cost due to the reduction of J_SM.

Figure 2.

Schematic representation of the jumps allowed in a graph cuts optimization scheme.

Methods

Reconstruction Implementation

To evaluate the efficacy of a graph-cuts-based field map estimation method for a constrained-phase signal model, three dual-echo Dixon reconstruction algorithms were compared: (a) the constrained-phase with graph cuts algorithm described in this work, (b) a Flex reconstruction (8) included in Orchestra (version 1.4-722), a reconstruction framework distributed by GE Healthcare (Waukesha, WI), and (c) an algorithm inspired by Reference (10), but with the addition of a multi-resolution process for field map estimation. The constrained-phase with graph cuts algorithm, (a), was implemented in C++ using the GridCut library (25), while algorithms (b) and (c) were implemented in MATLAB (MathWorks, Natick Massachusetts, USA).

The descriptions of the comparative reconstructions (b) and (c) can be found in References (8) and (10), respectively. In this work, the minima of J_D(ω) (Equation [7]) were found via line search in each voxel. In practice, it is computationally beneficial to calculate the minima to J_D(ω) once at the beginning and reach convergence by performing period jumps and minima jumps without additional cost calculation before moving on to the refinement jumps. Once the refinement process starts, J_D(ω) needs to be calculated on each iteration. J_SM needs to be calculated on each iteration in all cases. In this work, Γ_k was equal for all 6 neighbors in the 3-dimensional neighborhood for the calculation of J_SM.

The graph cuts algorithm used in this work utilized the following process that takes advantage of the computational benefits of upfront computation of the minima of J_D(ω):

Algorithm 1.

Unwrapping and refinement of the field map with graph cuts.

Initialize field map to min(JD(ωv)) in each voxel
while not converged do	▹ Unwrapping
+ Minima Jump
+ Period Jump
− Minima Jump
− Period Jump
end while
while desired precision is not yet attained (e.g. 1 Hz resolution) do	▹ Refining
+ Refinement Jump
− Refinement Jump
if converged then reduce size of refinement jump
end if
end while

Note that overall convergence is reached by first unwrapping the field by iterating within the precomputed minima of J_D and then iterating to refine the field until the field is smooth within the desired precision.

In this work, convergence was considered to be reached if one of the following occurred over two loops of either unwrapping or refining within Algorithm 1:

1. No change to the field

2. The average fractional change (over two loops) of the total cost (J(ω)) was less than 1e-6, where the fractional change of the total cost is defined as [J_c(ω) − J_c−1(ω)]/J_c(ω) at iteration c.

3. The total cost (J(ω)) was found to oscillate with respect to iterations

When the field map estimation is complete, ${\widehat{\varphi }}_{0,v}$ is calculated from Equation [6] and unwrapped using graph cuts with period jumps of ±π. The graph cuts optimization only includes period jumps as it is used solely for unwrapping (ϕ₀ has period π). When ${\widehat{\varphi }}_{0,v}$ has converged, estimates of W_v and F_v are calculated from Equation [4].

The effect of choosing μ to be too high or too low can be seen in Figure 3, where a range of regularization parameter values is used to reconstruct a dataset from the ISMRM Fat-Water Toolbox (26) (Knee_8ch from the “MultiChannel 3 Echo” directory of the data provided by the University of Southern California (USC)). When μ is chosen much too low (A–D), swaps of water and fat signal occur, but when chosen only a little too low (E–H), the effect is not as great. Likewise, when μ is chosen too high (M–P and Q–T), fat signal leaks into the water image, and vice-versa. The appropriate selection of μ (I–L) results in a clear separation of the water and fat signals. In this work, the regularization parameter was selected by visual inspection.

Figure 3.

An illustration on the importance of selecting the appropriate regularization parameter, μ, for the regularized cost function optimization with graph cuts. A dual-echo Dixon reconstruction is performed on a knee dataset from the ISMRM Fat-Water Toolbox (26). When the parameter is chosen much too low, fat water swaps occur. Likewise, when the parameter is chosen too high, signal leakage can be seen (arrows). When μ is chosen appropriately, the field map and initial phase images are smooth and the fat water separation is successful.

Reconstruction parameters, such as the regularization parameter, μ, are shown in Tables 1 and 2.

Table 1.

Imaging and reconstruction parameters used in this work. Field of view, sampling matrix, and resolution are reported as S/I × L/R × A/P. All imaging was performed using an interleaved TE, multi-echo 3DFT acquisition.

	Phantom (3.0T) Fig. 4	CE-MRA (3.0T) Fig. 5	Foot/Ankle (1.5T) Fig. 7	Pelvis (3.0T) Fig. 8
Imaging:
TE (msec)	2.30/3.50	2.30/3.50	2.30/4.60	2.30/3.50
Field of View (cm3)	22.0 × 22.0 × 16.0	42.0 × 42.0 × 22.4	28.0 × 14.4 × 28.0	15.4 × 28.0 × 28.0
Sampling Matrix	224 × 224 × 160	260 × 260 × 140	280 × 144 × 280	128 × 224 × 224
Resolution (mm3)	0.98 × 0.98 × 1.00	1.62 × 1.62 × 1.60	1.00 × 1.00 × 1.00	1.20 × 1.25 × 1.25
TR (msec)/FA/BW (kHz)	12.5/12°/±62.5	5.7/18°/±62.5	7.2/20°/±62.5	6.8/12°/±62.5
Number and arrangement of receive coil elements	1 birdcage	12 circumferential	1 birdcage	12 circumferential
Reconstruction:
Regularization Parameter, μ	1e12	6e12	1e13	1e13
Refinement reduction factor	$\sqrt{10}$	$\sqrt{10}$	$\sqrt{10}$	$\sqrt{10}$

Table 2.

Imaging and reconstruction parameters used in for the datasets from the ISMRM Fat-Water Toolbox (26). Sampling Matrix is reported as S/I × L/R × A/P.

	Knee (3.0T) Fig. 3	Abdomen (3.0T) Fig. 9	Ankle (3.0T) Fig. 10
Imaging:
TE (msec)	2.18/3.77	2.05/3.54	2.18/3.77
Sampling Matrix	256 × 4 × 256	8 × 192 × 192	256 × 4 × 256
Number of receive coil elements	8	8	8
Reconstruction:
Regularization Parameter, μ	6e-6	5e1	1e-7
Refinement reduction factor	$\sqrt{10}$	$\sqrt{10}$	$\sqrt{10}$

Imaging Studies

Phantom imaging was performed with an anthropomorphic fat-water phantom consisting of a bovine gelatin center surrounded by solid vegetable shortening to simulate a water-like abdomen surrounded with subcutaneous fat. A cylindrical void in the center of the bovine gelatin allowed vials of doped bovine gelatin to be inserted to simulate an enhancing abdominal aorta. The phantom measured 18×18×12.5 cm³ (length×width×height), and the cylindrical vials were 1.6 cm in diameter and 6 cm long. In this study, a single vial doped with approximately 0.5 mM of gadopentetate dimeglumine (Magnevist; Bayer HealthCare Pharmaceuticals, Wayne, NJ, USA) was used. All 3D phantom imaging was performed at 3.0T (Signa; GE Healthcare, Waukesha, WI), and additional scan parameters are shown in Table 1.

All human studies were IRB approved and informed consent was obtained from each subject. Three dimensional in vivo datasets depicting the peripheral vasculature, foot/ankle, and the pelvis were used to determine the utility of the dual-echo graph cuts reconstruction. The peripheral vasculature was imaged as part of a contrast-enhanced MR angiography (CE-MRA) study, while the pelvis was imaged to monitor a perianal fistula. The interleaved, multi-TE acquisition utilized corner cutting in k_Y-k_Z-space (27), but no additional undersampling. The use of an interleaved, multi-TE acquisition avoids flow-induced artifacts that can occur with bipolar gradients (28). Re-construction was performed using a multi-peak fat spectrum (29). All studies were performed at either 1.5T or 3.0T (GE Healthcare, Waukesha, WI) with the scan parameters shown in Table 1.

Additional studies were performed using data from the ISMRM Fat-Water Toolbox (26). Specifically, the datasets AbdomenAxial2 and Ankle were used from the “MultiChannel 3 Echo” directory of the data provided by the University of Southern California (USC). As no coil sensitivity profiles were available for the datasets from the ISMRM Fat-Water Toolbox, ESPIRiT was used to estimate sensitivity profiles (30). The available imaging parameters for these studies are summarized in Table 2.

Echo times for the two images were chosen to provide near optimal SNR in the water and fat images (19) while adhering to the acquisition restrictions presented in Reference (10). Dual-echo images from the ISMRM Fat-Water Toolbox were chosen from multi-channel three-echo datasets to provide the greatest SNR in the water and fat images from the available images.

To compare the ΔB₀ fields from the different techniques, a common unit (radians) was chosen for display. For the constrained-phase with graph cuts method of this work, the ΔB₀ map in radians per second was multiplied by ΔTE to obtain a map in radians. For the other techniques, which provide the ΔB₀ map as a phasor, the phase component (in radians) was unwrapped (31) for comparison.

Results

Images from the phantom experiment are shown in Figure 4. The reconstruction executed successfully and the unwrapping stage converged in 40 iterations while refinement converged in 137 iterations for a total reconstruction time of 40:04 (m:s). The constrained-phase with graph cuts method results in distinct separation of the water (A) and fat (B) signals without any signal swaps. The ΔB₀ map (C) shows the smoothness expected from the regularized reconstruction, even in the presence of large susceptibility-induced field inhomogeneities near the air cavity (indicated by the arrowheads). Likewise, the initial phase (D) retains the smoothness it derives from the ΔB₀ map. Note that the straight edge of the air cavity remains sharp without any signal leakage between the water and fat images.

Figure 4.

Dual-echo Dixon reconstructions of a phantom dataset with the technique described in this work. The phantom is constructed of water composed gelatin (g), fat-mimicking shortening (s), a contrast-enhanced vial (v), and surrounding air (a). From the first echo (A) and the second echo (B) (real channel shown) the (C) water image, (D) fat image, (E) field map, and (F) initial phase are all produced as expected from the construction of the phantom and a priori knowledge of the field map and initial phase. Imaging parameters can be found in Table 1.

Coronal slices from the in vivo CE-MRA study are shown in Figure 5. All three reconstructions executed successfully and produced reasonable images of fat signal, water signal, and ΔB₀. The reconstruction presented here converged in 44 iterations for the unwrapping stage and 144 iterations for refinement for a total reconstruction time of 1:51:26 (h:m:s). The constrained-phase reconstruction described in this work also produces an initial phase image, shown in Figure 5D. A fat/water signal swap can be seen near the superior region (Figure 5, red arrows) of the images produced with Flex (E–G) and the method inspired by Reference (10) (H–J). In the images produced with constrained-phase with graph cuts (A–D), however, the swap is resolved. A line profile through each of the ΔB₀ maps in the area of signal swap (K) shows that the ΔB₀ in the region of the signal swap is on the order of 6π or ≈20 parts per million, and estimated by the graph-cuts-guided optimization scheme, while the region growing method of Flex and the heuristic method inspired by Reference (10) are unable to resolve the field.

Figure 5.

Dual-echo Dixon reconstructions of a CE-MRA dataset. From left to right: the technique described in this work (A–D), Flex (E–G), and the technique inspired by Reference (10) (H–J). Note that the technique described in this work resolves the field map in the right superior region, while the other techniques leave residual fat/water signal swap. Line profiles through the ΔB₀ maps (K) show that the technique described in this work is able to resolve the very large inhomogeneity while the other techniques are not.

Enlarged sections of the water and fat images in Figure 5 (yellow boxes) are shown in Figure 6. Note the improved conspicuity of the contrast-enhanced vessel (red arrowheads) and the reduced signal “leakage” in the fat image in the femoral arteries and veins (yellow outline). Additionally, at the tissue interface near the contrast-enhanced vessel the image reconstructed with Flex has a “jagged” appearance due to small fat/water swaps in the low signal region.

Figure 6.

Enlarged sections of the dual-echo Dixon reconstructions of a CE-MRA dataset shown in Figure 5. From left to right: the technique described in this work (A–B), Flex (C–D), and the technique inspired by Reference (10) (E–F). Note the improved depiction of the vasculature (red arrowheads) with the method presented in this work, and the partial signal leakage in the fat image reconstructed with the technique inspired by Reference (10) (yellow outline).

Results from the foot/ankle study are shown in Figure 7, depicting an axial slice of the foot. The reconstruction of this work was completed in 1:22:53 (h:m:s) and used 25 iterations for unwrapping and 145 iterations for refinement. The constrained-phase with graph cuts method (A–D) shows definite signal separation throughout the images. Signal swaps are present in the images reconstructed with Flex (E–G) and the method inspired by Reference (10) (H–J) in the toes and in the heel (red arrowheads). Errors in the unwrapping of the field map (white arrowheads in G and J, compared to C) are seen in the regions of signal swap.

Figure 7.

A single axial slice of a foot/ankle dataset reconstructed with the method presented in this work (A–D), Flex (E–G), and the technique inspired by Reference (10) (H–J). Signal swap is apparent in the Flex images and those produced with the technique inspired by Reference (10) (red arrowheads). This is due to choosing the wrong solution for the field map (white arrows).

Figure 8 shows results from a pelvis study performed on a patient with a perianal fistula. The constrained-phase with graph cuts reconstruction unwrapped ΔB₀ with 38 iterations and used 166 more for refinement for a total reconstruction time of 36:23 (m:s). The perianal fistula (white arrowheads in A, C, and E) is depicted well with all three techniques, and no fat/water signal swaps are visible. Figure 8f, however, does show some partial signal swap in the contrast-enhanced perianal fistula anatomy (enlarged in B3, D3, and F3) and within the iliac arteries (enlarged in B1–2, D1–2, and F1–2) when the method inspired by Reference (10) is used. Note also that a bulk swap is present in the Flex images where the thighs wrapped through the field-of-view.

Figure 8.

Dual-echo Dixon reconstructions of a pelvis dataset at 3T. Note that there is a bulk fat-water swap in the Flex images where the thighs have wrapped in the field-of-view (arrow, C and D). Additionally, some partial signal swap is visible within the lumen of the iliac vessels (enlargements 1 and 2) and within the contrast-enhanced perianal fistula anatomy (enlargement 3) in the fat image produced with the technique inspired by Reference (10).

Figure 9 shows results using dual-echo abdominal data from the ISMRM Fat-Water Toolbox reconstructed with the constrained-phase with graph cuts method. The water (A) and fat (B) images show a clear separation, while the field map (C) and initial phase (D) are smooth as expected. Reconstruction of this volume was completed in 1:38 (m:s) – faster than the previous examples due to the reduced number of slices.

Figure 9.

Dual-echo Dixon reconstruction of an abdomen dataset from the ISMRM Fat-Water Toolbox (26). The water (A) and fat (B) signals are clearly separated throughout the abdomen, showing that the field map (C) and initial phase (D) are well estimated.

A foot/ankle study from the ISMRM Fat-Water Toolbox is shown in Figure 10. Reconstruction was completed in 0:42 (m:s). The water (A) and fat (B) images depict the different signal components, even in regions of relatively large susceptiblility induced field inhomogeneities in the skin folds of the ankle and sole of the foot. Despite these large field map inhomogeneities (C), when the appropriate regularization parameter is chosen, the separation is successful.

Figure 10.

Dual-echo Dixon reconstruction of an ankle dataset from the ISMRM Fat-Water Toolbox (26). The water (A) and fat (B) signals are correctly assigned, even in areas of large susceptibility-induced field inhomogeneity near the skin folds on the sole of the foot and ankle as seen in the field map (C) and initial phase (D) images.

Discussion

We have described, derived, and demonstrated a statistical method for dual-echo Dixon image reconstruction using a constrained-phase signal model. Through the use of the maximum likelihood estimator and state-of-the-art graph-cuts-guided greedy binary optimization, the method presented here is able to avoid fat/water signal swaps in areas of large field inhomogeneity and at tissue interfaces. These advantages have been demonstrated in a phantom study and in 1.5T and 3.0T human studies depicting contrast-enhanced peripheral vasculature, the foot/ankle, and the pelvis.

The method presented here provides improved image quality compared to Flex and the method inspired by Reference (10), particularly near the edge of the field of view and at tissue interfaces. The reason for this is three-fold. First, because the graph-cuts-guided optimization allows for a shift of any number of periods to find the appropriate ΔB₀ value, it has the capacity to unwrap very large field inhomogeneities. Second, because this method is objective-based, and makes provable progress toward a solution, with the choice of an appropriate regularization parameter it can balance data fidelity and smoothness to avoid local minima. Third, because the method presented here is a statistical method that accounts for measurement noise, the resulting solutions represent a better fit to the noisy data than other, deterministic methods.

These advantages are evident particularly in Figures 5, 6, and 7. In Figure 5, the large swap at the edge of the field of view is resolved with the method presented here due to the flexibility of the graph-cuts optimization to shift any number of periods to find the solution that minimizes the regularized cost function. Figure 6 shows that at the tissue interface, where the signal-to-noise ratio can be low, accounting for noise in the statistical reconstruction presented here allows for improved depiction of the vasculature near that interface. The importance of balancing data fidelity and smoothness of the ΔB₀ map is evident in Figure 7 at the second toe, where over-smoothing of the field maps in G and J leads to full (E) or partial (H) swap of the water and fat signal. Because the regularized method presented here takes the data fidelity into account during the optimization, with the appropriate choice of regularization parameter, μ, the appropriate solution is found.

Note that the theory outlined in this work is also applicable to three or more echoes. In this work, the measurement matrix, G_v, was of size 2 × N_C, B_v(ω) is size 2 × 2, and C is size 2 × 2 because the scope is limited to dual-echo Dixon imaging. In the general case, the sizes of G_v, B_v(ω), and C would be N_E × N_C, N_E × N_C, and N_E × 2, where N_E is the number of echoes. Some work using the constrained-phase signal model for greater than two echoes has been done (17), and while the results for three echoes were comparable, a signal-to-noise ratio improvement is expected with the constrained-phase reconstruction due to the constraint that the fat and water signals be real-valued.

There are some limitations to this technique. First, to obtain acceptable performance, the regularization parameter, μ, must be tuned. One approach is to tune the regularization parameter manually through the use of several test datasets for a certain anatomy and protocol, and then prospectively apply that regularization value to similar datasets. There are also methods to find the optimal parameter with other algorithms (32), however, these methods have not yet been developed for use with constrained-phase Dixon imaging. Second, the computation time needed for a full 3D volume is still clinically infeasible as indicated in the reported times. One possible solution is to perform the ΔB₀ estimation on a down-sampled dataset under the assumption that the loss in resolution will not adversely affect the estimation of a smooth field (22). If this avenue is taken, the particular resolution needed for the field map and initial phase estimation will be investigated quantitatively in the future. Alternatively, reconstruction times may be reduced by performing the first unwrapping steps (without refinement) with graph cuts and then using a descent-based method for the refinement of the field map (e.g. preconditioned gradient descent) (15). Third, the smoothness of the ϕ₀ map is assumed to be implicitly inherited from the smoothness of the ΔB₀ map in this work; however, there may be an advantage (e.g. in regions containing air) to jointly promoting smoothness in both the ΔB₀ map and the ϕ₀ map.

In conclusion, a dual-echo Dixon image reconstruction with a constrained-phase signal model and graph-cuts guided optimization has been derived, implemented, and demonstrated in both phantom and in vivo images at both 1.5T and 3.0T. In comparison to existing dual-echo Dixon methods, the maximum likelihood method presented here takes noise into account and has the ability to resolve large wraps in the ΔB₀ map. As a result, this methods provides improved performance in the presence of very large ΔB₀ inhomogeneities.

Appendix A Dual-Echo Multi-Coil Phase-Constrained Reconstruction

To find a solution to

$${\widehat{X}}_v(\omega ,{\varphi }_0)=\arg \underset{X_v\in {\mathrm{ℝ}}^2}{\min }[\mathrm{Re}\{\text{tr}[(A_v(\omega ,{\varphi }_0)X_v{S}_v-G_v){\mathrm{\Lambda }}^{-1}{(A_v(\omega ,{\varphi }_0)X_v{S}_v-G_v)}^{\ast }]\}]$$

it is useful to first expand the expression to be minimized.

$$\mathcal{L}(X_v,{\omega }_v,{\varphi }_{0,v})\equiv \text{tr}[X_v^T\mathrm{Re}\{A_v^{\ast }(\omega ,{\varphi }_0)A_v(\omega ,{\varphi }_0)\}{X}_v{S}_v{\mathrm{\Lambda }}^{-1}{S}_v^{\ast }]-2\mathrm{Re}\{\text{tr}[X_v^T{A}_v^{\ast }(\omega ,{\varphi }_0)G_v{\mathrm{\Lambda }}^{-1}{S}_v^{\ast }]\}$$ [A.1a]

$$\equiv X_v^T\mathrm{Re}\{A_v^{\ast }(\omega ,{\varphi }_0)A_v(\omega ,{\varphi }_0)\}{X}_v{S}_v{\mathrm{\Lambda }}^{-1}{S}_v^{\ast }]-2X_v^T\mathrm{Re}\{A_v^{\ast }(\omega ,{\varphi }_0)G_v{\mathrm{\Lambda }}^{-1}{S}_v^{\ast }\}$$ [A.1b]

where terms irrelevant to the optimization have been dropped, and we have invoked the circular permutation property of the trace (33).

Differentiating $\mathcal{L}$ with respect to X_v and setting it equal to zero allows the least squares solution of X_v to be found.

$${\nabla }_{X_v}\mathcal{L}=2\mathrm{Re}\{A_v^{\ast }(\omega ,{\varphi }_0)A_v(\omega ,{\varphi }_0)\}{X}_v{S}_v{\mathrm{\Lambda }}^{-1}{S}_v^{\ast }-2\mathrm{Re}\{A_v^{\ast }(\omega ,{\varphi }_0)G_v{\mathrm{\Lambda }}^{-1}{S}_v^{\ast }\}=0$$ [A.2]

The estimate for X_v, ${\widehat{X}}_v(\omega ,{\varphi }_0)$, is then given by

$${\widehat{X}}_v(\omega ,{\varphi }_0)=Q^{-1}\mathrm{Re}\{A_v^{\ast }(\omega ,{\varphi }_0)F_v\}{Z}_v^{-1}$$ [A.3]

where $Q=\mathrm{Re}\{A_v^{\ast }(\omega ,{\varphi }_0)A_v(\omega ,{\varphi }_0)\}=\mathrm{Re}\left\{C^{\ast }C\right\}$, $F_v=G_v{\mathrm{\Lambda }}^{-1}{S}_v^{\ast }$, and $Z_v=S_v{\mathrm{\Lambda }}^{-1}{S}_v^{\ast }$

Appendix B Simplification of the Two-Dimensional Cost Function

$$\left[{\widehat{\omega }}_v,{\widehat{\varphi }}_{0,v}\right]=\arg \underset{[{\omega }_v,{\varphi }_{0,v}]\in \mathrm{ℝ}}{\min }\left[\mathrm{Re}\left\{\text{tr}\left((A_v(\omega ,{\varphi }_0){\widehat{X}}_v(\omega ,{\varphi }_0)S_v-G_v){\mathrm{\Lambda }}^{-1}{(A_v(\omega ,{\varphi }_0){\widehat{X}}_v(\omega ,{\varphi }_0)S_v-G_v)}^{\ast }\right)\right\}\right]$$ [B.1]

where ${\widehat{X}}_v(\omega ,{\varphi }_0)=Q^{-1}\mathrm{Re}\{A_v^{\ast }(\omega ,{\varphi }_0)F_v\}{Z}_v^{-1}$ from Equation [A.3]. The two-dimensional cost function can be expressed as

$$\left[{\widehat{\omega }}_v,{\widehat{\varphi }}_{0,v}\right]\equiv \arg \underset{[{\omega }_v,{\varphi }_{0,v}]\in \mathrm{ℝ}}{\min }[{\mathcal{L}}_1({\omega }_v,{\varphi }_{0,v})-{\mathcal{L}}_2({\omega }_v,{\varphi }_{0,v})]$$ [B.2]

where

$${\mathcal{L}}_1({\omega }_v,{\varphi }_{0,v})=\mathrm{Re}\left\{\text{tr}\left(A_v(\omega ,{\varphi }_0){\widehat{X}}_v(\omega ,{\varphi }_0)Z_v{\widehat{X}}_v^T(\omega ,{\varphi }_0)A_v^{\ast }(\omega ,{\varphi }_0)\right)\right\}$$ [B.3]

$${\mathcal{L}}_2({\omega }_v,{\varphi }_{0,v})=2\mathrm{Re}\left\{\text{tr}\left(F_v{\widehat{X}}_v^T(\omega ,{\varphi }_0)A_v^{\ast }(\omega ,{\varphi }_0)\right)\right\}$$ [B.4]

Substitution for ${\widehat{X}}_v(\omega ,{\varphi }_0)$ and judicious use of the circular permutation property of the trace operator allows simplification of ${\mathcal{L}}_1$ and ${\mathcal{L}}_2$ to

$${\mathcal{L}}_1({\omega }_v,{\varphi }_{0,v})=D_v{(\omega ,{\varphi }_0)}^T{Q}^{-1}{D}_v(\omega ,{\varphi }_0)Z_v^{-1}$$ [B.5]

$${\mathcal{L}}_2({\omega }_v,{\varphi }_{0,v})=2D_v{(\omega ,{\varphi }_0)}^T{Q}^{-1}{D}_v(\omega ,{\varphi }_0)Z_v^{-1}$$ [B.6]

where $D_v(\omega ,{\varphi }_0)=\mathrm{Re}\{A_v^{\ast }(\omega ,{\varphi }_0)F_v\}$. Therefore, after substitution

$$\left[{\widehat{\omega }}_v,{\widehat{\varphi }}_{0,v}\right]\equiv \arg \underset{[{\omega }_v,{\varphi }_{0,v}]\in \mathrm{ℝ}}{\min }\left[-D_v{(\omega ,{\varphi }_0)}^T{Q}^{-1}{D}_v(\omega ,{\varphi }_0)Z_v^{-1}\right].$$ [B.7]

Appendix C Solving for ϕ_0,v

The derivative of Equation 5 can be found via the product rule. Note that Q and Z_v are independent of ϕ_0,v and can be treated as constants. Also recall that $D_v(\omega ,{\varphi }_0)=\mathrm{Re}\{A_v^{\ast }(\omega ,{\varphi }_0)F_v\}$ and $A_v(\omega ,{\varphi }_0)=e^{i{\varphi }_{0,v}}{B}_v(\omega )C$. Note that B_v(ω), C, and F_v are all independent of ϕ_0,v and that $Z_v^{-1}$ is a scalar. With the further definition of $H_v(\omega )=C^{\ast }{B}_v^{\ast }(\omega )F_v$, we have:

$${\nabla }_{{\varphi }_{0,v}}\mathcal{L}({\omega }_v,{\varphi }_{0,v})=-{\nabla }_{{\varphi }_{0,v}}[D_v{(\omega ,{\varphi }_0)}^T{Q}^{-1}{D}_v(\omega ,{\varphi }_0)]Z_v^{-1}$$ [C.1a]

$$=-{\nabla }_{{\varphi }_{0,v}}\left[\mathrm{Re}{\{e^{-i{\varphi }_{0,v}}{H}_v(\omega )\}}^T\right]Q^{-1}\mathrm{Re}\{e^{-i{\varphi }_{0,v}}{H}_v(\omega )\}{Z}_v^{-1}-\mathrm{Re}{\{e^{-i{\varphi }_{0,v}}{H}_v(\omega )\}}^T{Q}^{-1}{\nabla }_{{\varphi }_{0,v}}\left[\mathrm{Re}\{e^{-i{\varphi }_{0,v}}{H}_v(\omega )\}\right]Z_v^{-1}$$ [C.1b]

$$=-\mathrm{Im}{\{e^{-i{\varphi }_{0,v}}{H}_v(\omega )\}}^T{Q}^{-1}\mathrm{Re}\{e^{-i{\varphi }_{0,v}}{H}_v(\omega )\}{Z}_v^{-1}-\mathrm{Re}{\{e^{-i{\varphi }_{0,v}}{H}_v(\omega )\}}^T{Q}^{-1}\mathrm{Im}\{e^{-i{\varphi }_{0,v}}{H}_v(\omega )\}{Z}_v^{-1}$$ [C.1c]

Noting that Equation [C.1c] is an the expansion of a complex product of the form Im{K^T Re{L}K} = Im{K^T}Re{L}Re{K} + Re{K^T}Re{L}Im{K} and consolidating the phasors $e^{-i{\varphi }_{0,v}}$, the result is:

$${\nabla }_{{\varphi }_{0,v}}\mathcal{L}({\omega }_v,{\varphi }_{0,v})=-\mathrm{Im}\left\{e^{-2i{\varphi }_{0,v}}{H}_v{(\omega )}^T{Q}^{-1}{H}_v(\omega )\right\}{Z}_v^{-1}$$ [C.1d]

Setting ${\nabla }_{{\varphi }_{0,v}}\mathcal{L}({\omega }_v,{\varphi }_{0,v})=0$, the scalar $Z_v^{-1}$ may be divided out. For Im{·} = 0 to hold, the overall phase must equal nπ. Without loss of generality, assume n = 0.

$$0=\angle [e^{-2i{\varphi }_{0,v}}{H}_v{(\omega )}^T{Q}^{-1}{H}_v(\omega )].$$ [C.2]

Since $e^{-2i{\varphi }_{0,v}}$ has phase −2ϕ_0,v, the other terms must have phase 2ϕ_0,v for the equality to hold. Solving for ϕ _0,v gives

$${\widehat{\varphi }}_{0,v}=\frac{1}{2}\angle [H_v{(\omega )}^T{Q}^{-1}{H}_v(\omega )].$$ [C.3]