Two-point water-fat imaging with partially-opposed-phase (POP) acquisition: An asymmetric Dixon method

Authors

  • Qing-San Xiang

    Corresponding author
    1. Department of Radiology, University of British Columbia, Vancouver, Canada
    2. Department of Physics and Astronomy, University of British Columbia, Vancouver, Canada
    • Dept. of Radiology, BC Children's Hospital, 4480 Oak Street, Vancouver, BC, Canada V6H 3V4
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  • Presented in part at the 11th Annual Meeting of ISMRM, Toronto, Canada, 2003.

Abstract

A novel two-point water-fat imaging method is introduced. In addition to the in-phase acquisition, water and fat magnetization vectors are sampled at partially-opposed-phase (POP) rather than exactly antiparallel as in the original Dixon method. This asymmetric sampling encodes more valuable phase information for identifying water and fat. From the magnitudes of the two complex images, a big and a small chemical component are first robustly obtained pixel by pixel and then used to form two possible error phasor candidates. The true error phasor is extracted from the two error phasor candidates through a simple procedure of regional iterative phasor extraction (RIPE). Finally, least-squares solutions of water and fat are obtained after the extracted error phasor is smoothed and removed from the complex images. For noise behavior, the effective number of signal averages NSA* is typically in the range of 1.87–1.96, very close to the maximum possible value of 2. Compared to earlier approaches, the proposed method is more efficient in data acquisition and straightforward in processing, and the final results are more robust. At both 1.5T and 0.3T, well separated and identified in vivo water and fat images covering a broad range of anatomical regions have been obtained, supporting the clinical utility of the method. Magn Reson Med, 2006. © 2006 Wiley-Liss, Inc.

Water-fat imaging can be considered as the simplest type of chemical shift imaging for a binary system with only two spectral components. Although it is able to provide diagnostically valuable information about the content of water and lipids within each voxel, in the clinical setting water-fat imaging is by far mostly used to achieve high-quality fat suppression. The fat image is typically discarded, while the water image serves as a fat-suppressed image. Water-fat imaging is a superior alternative to less effective fat-suppression techniques such as RF presaturation and short-tau inversion-recovery (STIR), where image uniformity, contrast, and flexibility of sequence parameters are compromised.

Early approaches to water-fat imaging included the original Dixon method (1) and the quadrature sampling (QS) method (2–4). The Dixon method acquires two images in which the water and fat magnetization vectors are sampled when they are parallel and antiparallel, i.e., at sampling angles of (0°, 180°) (also known as “in-phase” and “opposed-phase”). Water and fat images are produced by adding and subtracting the two acquired images. The QS method acquires a single image and samples the two vectors when they are perpendicular, i.e., at a sampling angle of (90°), and outputs water and fat images as the real and imaginary parts of the complex data sampled. Theoretically, both approaches should work if there is no phase error. However, phase errors are inevitable in practice, and thus both methods have to be further developed with effective phase correction for successful clinical application.

Simple methods have been proposed to achieve phase correction when phase errors are limited to a certain range (5, 6). For a more general solution, Yeung and Komos (7) added a “third point” at (−180°) to the double acquisitions at (0°, 180°) in Dixon's method, and attempted to achieve phase correction by using phase unwrapping. This idea was later termed the “three-point Dixon” method and led to many variations, including sampling schemes of either (−180°, 0°, 180°) or (0°, 180°, 360°) as described by Glover (10) and others (8–11). More studies on this subject can be found in a recently published book (12). Although these variations have achieved some success in separating water and fat, they have three fundamental drawbacks. First of all, considerable redundancy exists in the sampled data (13–17) (also see Appendix A). Second, the processing algorithm typically relies on general phase unwrapping, which itself is very challenging and can be unreliable (18–20), especially when there are disconnected tissues in the field of view (FOV). Finally, even when the water and fat can be separated, there is no simple way to automatically distinguish them, i.e., to identify which is water and which is fat. This is analogous to the difficulty of unambiguously determining how the two hands on a clock rotate if they are only observed at parallel and antiparallel. This can be a nuisance for multislice scans (9) or disconnected tissues in the FOV where inconsistent water and fat segments can be displayed.

Although the development of the QS method has been relatively slow, the asymmetric sampling in the QS method has the potential to enable not only separation but also identification of the water and fat vectors given their definite leading and lagging phase relationship, provided that phase errors can be corrected. A method known as direct phase encoding (DPE), which uses asymmetric sampling, has been proposed (17). DPE eliminates the above-mentioned redundancy problem with a general asymmetric sampling scheme of (α0, α0 +α, α0 + 2α), and enables even pixel-level analytical water-fat separation and identification. DPE is more robust than methods based on symmetric sampling and phase unwrapping, as evidenced by a large number of clinical cases (21, 22). Asymmetric sampling also allows a pure phase encoding of the chemical shift with a (−90°, 90°, 270°) scheme (23). Since all three images are of the QS type, from their magnitudes T2* effects can be extracted and corrected. Successful implementations of asymmetric sampling with accelerated phase mapping and array coils have been reported (24, 25). Chemical shift imaging with spectrum modeling, combining optimized asymmetric sampling with least-squares solution and region-growing, has been proposed (26). Another approach using three or more acquisitions has been described that uses a pixel-independent iterative least-squares estimation (27), and was improved recently by a region-growing algorithm (28). New applications of the traditional symmetric two-point Dixon method have been published (29, 30).

This paper describes a new two-point technique with asymmetric sampling. Like the original Dixon method, it acquires only two images and therefore is more time-efficient compared to methods that use three or more acquisitions. However, the water and fat vectors are sampled at “in-phase” and “partially-opposed-phase (POP),” i.e., the second image is a POP acquisition at a sampling angle close to but not equal to 180°, such as 135°. Similarly to QS (2–4) and DPE (17), the asymmetry in the POP acquisition allows water and fat to not only be separated but also identified by making use of the available leading or lagging phase relationship. The principle of the POP method will be described first, followed by specific implementations and human imaging applications, demonstrating its utility in clinical settings.

THEORY

Pixel-Level Water-Fat Separation With Two General Dixon Magnitude Images

In his original work, Dixon (1) acquired two images, I1 and I2, with water and fat vectors at parallel and antiparallel, respectively known as the “in-phase” and “opposed-phase” images. In order to handle phase errors present in I1 and I2, their magnitudes M1 =|I1| and M2 =|I2|, were added and subtracted to achieve water and fat separation.

In the present work this idea is extended to a general scenario in which the two images, I1 and I2, are acquired with water and fat vectors at arbitrary rotation angles, θ1 and θ2. These angles can be readily chosen by adjusting the parameters of a variety of pulse sequences, such as spin-echo, gradient-echo, combination of spin-echo and gradient-echo, rapid acquisition with relaxation enhancement (RARE) or fast-spin-echo, and steady-state free precession (SSFP) (1, 11, 17, 27). Therefore, two triangles can be formed as shown in Fig. 1, with their two unknown sides representing the two chemical components W and F to be found. Simultaneous solution of these two triangles using the “cosine theorem” yields the two roots below for the two desired quantities W and F:

equation image(1)
equation image(2)

where B and S are the “big” and “small” chemical components within each pixel, as indicated by the + and − signs. The water-fat solutions (W, F) must be either (B, S) or (S, B) depending on whether water or fat is dominating. It is easy to see that when (θ1, θ2)=(0°, 180°) the B and S solutions reduce to the more familiar situation described in the original paper by Dixon (1), namely B = (M1 + M2)/2 and S = (M1M2)/2. Note that these solutions provide only the big and small components (B, S). A binary ambiguity still exists regarding which component is water and which is fat, since at each pixel the water and fat components are only separated but not yet identified. However, the tasks of water-fat imaging should include making this pixel-level W-F separation consistent in the entire image, and ultimately achieving automatic identification of the two components. These tasks require the use of phase information.

Figure 1.

Geometric interpretation of water-fat separation with generalized two-point Dixon acquisitions. The two triangles represent two images sampling the water and fat vectors with two arbitrary rotation angles of θ1 and θ2, respectively. In both triangles, one side is known as M1 or M2 along with the angle facing it. Solving the two triangles simultaneously, the two unknown sides can be determined as either (W,F) = (B,S) or (W,F) = (S,B) as described by Eqs. [1] and [2].

In-Phase and POP Acquisitions

Clearly, there is an infinite number of possible sampling angles that can be chosen to achieve the above pixel-level W-F separation (ignoring noise considerations), as long as cosθ1 ≠ cosθ2. Some of these options have been investigated. They include (θ1, θ2) = (90°, 180°) (31) and (θ1, θ2) = (45°, 180°) (32), both of which use asymmetric sampling that allows further water and fat identification after phase error correction. However, they do not use an in-phase acquisition and therefore have limitations for more general phase correction. In-phase acquisition is particularly useful for phase correction as its phase represents the pure phase errors without contribution from the water-fat combination. This is because, for the in-phase case, the water and fat vectors are by definition pointing in the same direction. Therefore, it can be used easily and robustly to remove a common component of phase errors from all acquisitions, reducing the remaining phase errors only to a smaller relative phase error. This is especially the case for long-TE gradient-echo acquisitions in which the common phase error can be quite large as it is accumulated during the entire TE. To further optimize two-point water-fat imaging, in this paper a new sampling scheme of (θ1, θ2) = (0, α) is proposed (33), where α is an angle close to but not equal to 180°. This means that the two complex images, I1 and I2, are acquired at in-phase and POP, which can be written as

equation image(3)
equation image(4)

where P1 and P2 are two unknown phasors (complex numbers with magnitudes of unity) representing phase errors associated with the two image acquisitions. Here the term “phase error” is used to describe any undesired extra phase, excluding contributions from noise. The physical sources of such phase errors include main magnetic field inhomogeneity, receiver coil phase nonuniformity, eddy currents, and data acquisition window off-centering.

According to Eqs. [1] and [2], the (B, S) solutions for (θ1, θ2) = (0, α) become

equation image(5)
equation image(6)

The operations of taking absolute values are necessary to ensure the B and S solutions to have only non-negative real values. In theory, B and S represent the magnitudes of chemical components with the smallest possible value of zero. However, in practice, they can have negative or even complex values as calculated from Eqs. [1] and [2] due to noise and artifacts if absolute values are not used. This is more likely for S due to the subtraction in Eq. [6].

Relative Error Phasor P and Its Two Possible Candidates (Pu, Pv)

By taking the magnitude of Eq. [3] we have

equation image(7)

Because the phase error P1 in Eq. [3] is typically a smooth spatial function, it is possible to remove it from I2 without significantly introducing associated noise. After I1 is smoothed, e.g., with a 9 × 9 sliding window, its phasor Pmath image can be removed from I2 by multiplying the complex conjugate (Pmath image)*, resulting in

equation image(8)

where P=P2(Pmath image)* represents a relative error phasor between P2 and the phasor of the smoothed I1 denoted as Pmath image. Note that P1S has very much reduced noise compared to P1 due to smoothing. Clearly, W and F can be easily solved from Eqs. [7] and [8] if P can be found. Although this phasor P is not completely known, it must be restricted to one of the following two possibilities, Pu and Pv, since the correct assignments of the (B, S) solutions must be either (B, S) = (W, F) or (B, S) = (F, W):

equation image(9)
equation image(10)

where Pu and Pv are the two possible error phasors constructed from the (B, S) solution pair. The remaining task is to determine, at each pixel, which one of the two phasor candidates (Pu, Pv) represents the true error phasor P.

To avoid meaningless random phasors in regions of background noise only, Eqs. [9] and [10] are used along with a tissue mask to set Pu = Pv = 0 for nontissue pixels. Such a tissue mask is readily obtained by comparing all pixels in the image M1 = | I1 |, with the standard deviation (SD) σn of a noise field n(x, y) obtained as the difference between M1 and the sum of B and S as described in Eqs. [5] and [6], respectively. This difference is nonzero due to the absolute value operations in Eqs. [5] and [6], and its value essentially reflects the noise level in the data that is quantified by the SD defined as equation image where <n> is the mean value of n(x, y) over the entire FOV. If and only if the value of a pixel in M1 is greater than a threshold, say six times the SD σn, it is considered to be tissue. Although this threshold value is not critical to the final result, a rather high value is chosen to ensure that the surviving pixels are indeed tissue; otherwise, more processing time may be needed to incorporate pixels with random phase.

Regional Iterative Phasor Extraction (RIPE)

In this work it is shown that a simple RIPE procedure is able to take the two possible phasors (Pu, Pv) as input, and output the desired true phasor P. RIPE can be considered as a type of “cellular automata” (34) where interesting long-range global structures can be developed from only local iterations following a set of simple rules. In fact, cellular automata have been employed to study a similar two-state lattice, called the Ising model in physics (35), as well as to solve the problem of phase unwrapping (36) that is closely related to water-fat imaging. RIPE includes four operations, as follows:

Step 1: Initialization

The phasor iteration begins with an initial state P0. For typical cases, good results can be obtained by setting P0 as a simple average of the two phasor candidates of Pu and Pv. To ensure robustness for more challenging cases associated with extraordinary phase error, noise, and artifacts, a more sophisticated way to set up the initial state P0 is described below:

The FOV is divided into an array of small non-overlapping square regions, each with a size of K × K (e.g., K = 8) pixels. Within each small region, various configurations chosen from (Pu, Pv) at each pixel are tested. Specifically, the phasor is flipped at each pixel between Pu and Pv, and the phasor that makes the total regional magnetization larger is kept. A configuration is selected as Ps if it maximizes the total regional magnetization MS, defined as the magnitude of total phasor PS summed over all pixels in the region. This configuration corresponds to a minimum regional phasor dephasing to reflect the fact that the true phase error P is nearly a constant in the small region. For the unselected complementary phasor configuration, its total magnetization MC is also computed, and a parameter C is defined as the refocusing contrast between the two configurations:

equation image(11)

The final initial state P0 is taken as the selected phasor PS weighted by the contrast C that is significantly greater at a tissue interface between water and fat than that in areas with only a single component, such as in brain tissue. In other words, the initial state P0 is found with more confidence near tissue interfaces, where more weight is given to establish “seeds” in subsequent phasor iteration.

Multiscale phasor initialization can also be performed by repeating the above procedure with various region sizes K and averaging the results from all levels as the final P0.

Step 2: Smoothing

The initial weighted phasor P0 is smoothed. After each subsequent iteration (see step 4) the smoothing is performed again. In general, at each step of the iteration the resulting phasor from the previous step, Pn, is smoothed in complex form, i.e., the real and imaginary parts are smoothed separately to yield the resultant Pmath image:

equation image(12)

A sliding window is used for smoothing. Typically, a larger window gives better results, as long as the local relative phase error does not vary too much within the window. A window of 37 × 37 pixels was experimentally found to work well for the data acquired on our clinical scanners. A fast smoothing algorithm was developed to speed up this operation (see Appendix B).

Step 3: Phasor Updating

After the smoothed phasor Pmath image is obtained, it is compared with the two phasor candidates (Pu, Pv) and a new phasor Pn+1 is updated according to

equation image(13)

In other words, the phasor is updated for every pixel by choosing from either Pu or Pv the one that is closer to the smoothed value of Pmath image, and is set to zero if there is a tie. Other operations, such as the “dot product” (17), can be used here as well to make a mathematically equivalent judgment as to whether Pu or Pv is closer to Pmath image. To monitor the state of the phasor evolution, an integer variable N is used as a counter to record the total number of changed pixels at each step of iteration. Pixels that are different between steps n and n + 1 are counted.

Step 4: Repeating

If the number of changed pixels N has not stabilized, operations 2 and 3 are repeated. Here stabilization means that N reduces to zero or a small constant as the number of iteration n increases. In noisy data, it is possible to see N falling into a periodic pattern, although such a case is rare. When N is eventually stabilized, it is usually equal to or close to zero, meaning that no or very few pixels are undetermined. At this time, the phasor iteration can be terminated and the result is output as Pstable. Typically, N is quickly stabilized after only few iterations. For example, the resulting sequence of N for the data presented in Fig. 5 is (27816, 323, 40, 9, 2, 2, 10).

Error Phasor Removal and Least-Squares Solutions of (W, F)

The output of the phasor iteration Pstable is selected at each pixel from the two phasor candidates (Pu, Pv). Ideally, it represents the true error phasor P in Eq. [8] with a smooth spatial variation. In reality, however, it is not so smooth, for the following reasons: First, the Pstable map can have small “holes” at those pixels that are replaced by zeros in Eq. [13]. Second, the noise in the (B, S) solutions may have an impact on the phasor candidates (Pu, Pv), and result in some roughness of Pstable. These issues can be handled by using a second-pass (W, F) solution similar to that used in the DPE method (17). The phasor Pstable is first smoothed (e.g., 13 × 13 sliding window) and renormalized to unit magnitude as defined below:

equation image(14)

The smoothed and renormalized phasor Pmath image is then removed from J2 in Eq. [8], resulting in a phase-corrected complex POP image Jmath image,

equation image(15)

Combining the above equation and Eq. [7], we have the following linear system:

equation image(16)

where the three matrices are respectively defined as

equation image(17)
equation image(18)
equation image(19)

where the operators Re(Jmath image) and Im(Jmath image) respectively return the real and imaginary parts of the complex number in the parentheses. Equation [16] describes an overdetermined system, since there are three equations but only two unknowns after the error phasor P in Eq. [8] is determined, smoothed, and removed. Therefore, a least-squares solution for (WLS, FLS) can be found as the final water and fat images:

equation image(20)

where L is the “pseudo-inverse” matrix given by

equation image(21)

where the superscripts “T” and “−1” are respectively the transpose and inverse matrix operations. Finally, the least-squares solution XLS can be explicitly written as

equation image(22)

Analysis of Noise Behavior

The noise variances σ2W and σ2F in the final water and fat images, given by the linear solutions of (WLS, FLS), depend on the “pseudo-inverse” matrix L and can be calculated as

equation image(23)
equation image(24)

where Lij are elements of matrix L in Eq. [21], and σ02 is the noise variance in the real or imaginary parts of the phase-corrected complex images, and assumed to be a constant. This assumption is based on the facts that noise variance in magnitude is equal to or less than those in the real and imaginary parts of a complex image (37), and that the phasor removal in Eqs. [8] and [15] affect the noise variance very little, particularly due to the use of smoothed error phasors Pmath image and Pmath image in which noise has been highly suppressed. It is then straightforward to show that both noise variances σmath image and σmath image in Eqs. [23] and [24] are given by

equation image(25)

Consequently, the effective number of signal averages NSA* (10, 17) for both water and fat images is

equation image(26)

The NSA* as a function of angle α is plotted in Fig. 2, where a rather wide central plateau can be seen, suggesting a desirable insensitivity of NSA* to the sampling angle α. At α = 135°, the NSA* equals 1.957, which is very close to the maximum possible value of 2. Compared to traditional (0°, 180°) sampling, the small price paid in NSA* is justified by the ability to acquire additional phase information.

Figure 2.

The effective number of signal averages (NSA*) plotted as a function of α. In a region near α = 180°, the NSA* varies slowly and stays approximately equal to the maximum possible value of 2. In particular, at α = 135° the NSA* is 1.957, and in the interval between 120° and 240° the NSA* is higher than 1.875.

MATERIALS AND METHODS

Pulse Sequences and Experiments

The proposed technique was implemented on 1.5T clinical scanners (TwinSpeed, GE Healthcare, Waukesha, WI, USA, and Edge, Picker International, Cleveland, OH, USA) at local hospitals in a user-friendly manner such that technologists could perform the entire procedure, including complex image acquisition and water-fat image reconstruction, without supervision. The technique was also tested on a 0.3T whole-body scanner (Centauri, XinAo MDT; Langfang, Hebei, China) by acquiring spin-echo complex images with a (0°, 135°) sampling scheme. Standard spin-echo pulse sequences were modified to acquire interleaved complex images. The modified sequences were similar to those for a typical scan of two averages, except that the 180° refocusing RF pulse for one of the excitations was shifted in time (e.g., by 0.84 ms at 1.5T), and the data were not averaged but reconstructed into two separate complex images corresponding to (0°, 135°) sampling. It is important to interleave the two acquisitions because otherwise misregistration artifacts due to subject motion may be seen (27). The sequences were also designed to be compatible with array coils. For each coil element, the two complex images were first phase-corrected according to Eqs. [7] and [8]. After this phase alignment, images from all coil elements were subsequently averaged with normalized M1 weighting wj, given by wj=(M1)ji(M1)i, to form a combined set of two complex images, leading to a single set of combined water and fat images. With the proposed image combination, although noise is regulated by magnitude weighting to give a somewhat different signal-to-noise ratio (SNR) compared to other methods (24, 27), solution reliability and processing efficiency are the two most important considerations here. The complex images from a single coil usually contain regions of very low SNR. In these regions the phase correction and (W, F) solutions can become unreliable, which will have a negative effect on the final result when these unreliable solutions are combined. Combining the complex images as proposed in this paper can eliminate this problem easily. Also, since the subsequent phase correction and (W, F) solution are performed only once, instead of as many times as the number of coil elements, there is a significant advantage for processing efficiency. Ignoring the small amount of time spent on the phase alignment, the total processing time is essentially independent of the number of coils used (24).

Hundreds of examinations have been performed in the past few years, covering virtually all anatomic regions, including the head, neck, shoulder, arms, wrists, hands, chest, spine, abdomen, pelvis, thighs, legs, calves, ankles, and feet in axial, sagittal, coronal, and oblique slice orientations, at field strengths of 1.5T and 0.3T. Each examination included 13–28 slices, typically with T1-weighted contrast using TR/TE = 600–800/8–24 ms. A double-echo gradient-echo sequence was also used to acquire two complex images at TE1 = 2.1 ms and TE2 = 4.6 ms with TR = 100 ms. Assuming an approximate frequency difference between water and fat of 217 Hz, this resulted in POP and in-phase acquisitions at nominal angles of 165° and 360° that were used in subsequent phase correction and (W, F) solution. Four-element array coils were used for improved SNR.

Data Processing

Water and fat images were obtained from the two complex images acquired with (0, α) sampling by steps summarized in Fig. 3. Programs were written in C-language to carry out fully automatic processing without human intervention. Since RIPE processing uses large window smoothing at each step, a fast algorithm was developed for time efficiency (see Appendix B).

Figure 3.

Flowchart of obtaining water and fat images from the two complex images acquired with (0, α) sampling, where α is an angle close to but not equal to 180°, e.g., α = 135°, forming a POP acquisition.

NSA* Study With Statistical Measurements

The theoretical NSA* in Eq. [26] was tested with statistical measurements of noise variance in the water and fat images obtained with the proposed method. The measurements were conducted on data generated by computer simulations, in order to isolate the pure effect of noise and to have full control over other factors that can introduce artifacts in the results. Figure 4a shows the structure of a digital phantom used and three rectangular ROIs. There are three regions in the phantom, with water and fat signal intensities defined as W = 1000, F = 0 in the lower region, W = 1000, F = 2000 in the middle region, and W = 0, F = 2000 in the upper region. Two complex images I1 and I2 were generated with water and fat vectors at in-phase and representative POP angles at 90°, 120°, and 135°. Linear and quadratic phase errors were applied to all complex images, which were chosen to be more challenging than those under typical experimental conditions (phase error > 4π across the FOV). Gaussian noise was also added to both real and imaginary parts of all complex images (37). As an example, Fig. 4b shows the phase of the complex image with POP acquisition at 135°, where nonlinear phase errors in the object as well as random noise in the background can be seen.

Figure 4.

Study of NSA* by statistical measurements. a: Structure of the water- and fat-containing phantom used and the three ROIs defined. b: Phase map of the POP acquisition with α = 135°, where nonlinear phase errors across the object and random noise in the background can be seen. c and d: Reconstructed water and fat images respectively. e–h: Noise fields obtained as subtraction between two repeated acquisitions and reconstructions. e and f: Noise fields for the intermediate (B, S) solutions, respectively. g and h: Noise fields for the final (W, F) solutions, respectively. The noise in the (W, F) solutions is quite different from that in the (B, S) solutions, with less amplitude and more uniform distribution.

The complex images I1 and I2 were processed by the proposed algorithm, leading to four output images: W and F as the final water and fat images, and B and S as the intermediate results of the big and small chemical components according to Eqs. [5] and [6]. Figure 4c and d show the successfully separated and identified water and fat images, respectively. To visualize the noise fields in the resultant images, the simulation was repeated, and the difference between the two results is displayed as Fig. 4e–h for the four output images (B, S) and (W, F). The noise levels in both B and S are seen to vary significantly from region to region in the phantom, but such variations are not visible in the W and F images shown as the uniform noise fields in Fig. 4e and h.

RESULTS

Figure 5 shows error phasors as intermediate results, and water-fat images as final results from a T1-weighted spin-echo axial head scan at 1.5 T with α = 135°. Figure 5a and b are imaginary parts of the two phasor candidates,Pu and Pv, obtained from Eqs. [9] and [10]. Figure 5c is the imaginary part of the resultant phasor Pstable after phasor iteration, which has more global continuity than Pu and Pv, representing the true relative error phasor P in Eq. [8].

Figure 5.

Error phasors and final results of a spin-echo axial head scan at 1.5 T with α = I°. a and b: Imaginary parts of the two error phasor candidates, Pu and Pv, given by Eqs. [9] and [10], respectively. c: Corresponding imaginary part of the final stabilized result Pstable. Each pixel in c is selected from either a or b by RIPE such that a global continuity is achieved, as indicated by the more uniform contrast in c. The final (d) water and (e) fat images obtained as real-valued least-squares solutions from Eq. [22], after the error phasor corresponding to (c) is removed after smoothing are shown. The quality of these images, in terms of water-fat separation and identification, is excellent.

Figure 5d and e are water and fat images obtained as real-valued least-squares solutions from Eq. [22], after the error phasor in Fig. 5c is removed after smoothing. These results show excellent water-fat separation and identification.

Figure 6 shows the results from a T1-weighted spin-echo coronal head scan at 1.5 T with α = 135°, demonstrating excellent (a) water and (b) fat separation and identification. The lesion is better visualized in the water image compared to an image (c) without fat suppression, in which strong signals from fatty tissue can be seen.

Figure 6.

With α = 135° at 1.5 T, (a) water and (b) fat images obtained from a spin-echo coronal head scan. The lesion is highlighted and better visualized in the water image as compared to an image without fat suppression (c), where the fatty tissues have very high signal intensities.

Figure 7 shows the results from a T1-weighted spin-echo axial scan at the level of the shoulder at 1.5 T with α = 135°. Excellent (a) water and (b) fat separation and identification were achieved. In this location, high-quality RF fat suppression is generally difficult to achieve because of the complicated geometry involved.

Figure 7.

With α = 135° at 1.5 T, (a) water and (b) fat images obtained from a spin-echo axial scan at the level of the shoulder, where high-quality RF fat suppression is difficult to achieve due to the complicated geometry. The water-fat separation and identification are excellent. A rim of noise is noticeable round the images. This was due to an area extension of the error phasor caused by smoothing, and is not a problem at all since tissue regions are not affected.

Figure 8 shows (a) water and (b) fat images obtained at 1.5 T from a breath-held axial abdominal scan with a double-echo gradient-echo sequence using TE1 = 2.1 ms and TE2 = 4.6 ms, which are acceptable settings for a commercial product sequence, corresponding to approximate sampling angles of (θ1, θ2) = (165°, 360°). If we consider the 360° image as an in-phase acquisition, the POP acquisition was at a relative sampling angle of α = 195°. Four sets of complex images were obtained from four-element array coils. They were first phase-corrected separately using Eqs. [7] and [8], and then combined with magnitude weighting before RIPE and the final water-fat solution. The water-fat separation and identification are both excellent, demonstrating the robustness and flexibility of the proposed method in terms of pulse sequences and coil configurations.

Figure 8.

With α = 195° at 1.5 T, (a) water and (b) fat images obtained from a breath-hold axial abdominal scan with a double-echo gradient-echo sequence. Four-element array coils were used, resulting in four sets of complex images, which were first phase-corrected using Eqs. [7] and [8] and then combined with magnitude weighting before the phasor iteration and final water-fat solution. Both separation and identification of the water-fat components are excellent.

Figure 9 shows multislice (a) water and (b) fat images obtained from a spin-echo sagittal knee scan with α = 135° at 0.3 T. Four typical consecutive slices are shown with excellent results of water-fat separation and identification. RF fat suppression is difficult to achieve at this field strength due to the small frequency difference between the water and fat components.

Figure 9.

With α = 135° at 0.3 T, multislice (a) water and (b) fat images obtained from a spin-echo sagittal knee scan. Four consecutive slices are shown with excellent results of water-fat separation and identification. RF fat suppression is difficult to achieve at this field strength due to the small frequency difference between the water and fat components.

The processing speed is fast enough to support online reconstruction. It depends on the particular data set used, since phasor iteration is involved, but typically, on a personal computer with 2.8 GHz clock rate and 504MB-RAM memory, it takes about 0.8 s to process a slice with 256 × 256 matrix size.

To study the NSA*, noisy complex images from the phantom shown in Fig. 4 are used. The original noise variance was measured from the original complex images I1 and I2 to be σmath image = 201.9, and the measured noise variances for the four output images (B, S) and (W, F) in all three ROIs are summarized in Table 1. It can be seen that the noise variances for B and S images are highly variable depending on the ROI locations. In general, both B and S images become noisier when their signal intensities are closer in the ROI (e.g., ROI 2), and the S image is always noisier than the B image. These behaviors are consistent with those observed in a three-point water-fat imaging method (40). On the other hand, the noise levels in W and F images are generally lower than those in the B and S images. They are also quite uniform, independently of the ROI location and thus the relative signal intensities of the two chemical components, and change only with the POP angle α. Therefore, at each α value, the mean noise variance can be calculated for the W and F images, leading to measured NSA* values, NSA*_measured, that are in agreement with the theoretical values, NSA*_theory, within only 1.7%, 1.0%, and 3.9% for the three α angles of 135°, 120°, and 90°. Note that α angles beyond this range are less interesting for practical reasons and thus were not tested in this work. Relative errors on the order of only ∼1% are reasonable, suggesting a good agreement between measurements and theory. The minor difference between measurements and theory could be due to residual noise in P1 and Pstable after smoothing, or statistical fluctuations of the random noise field considering the limited number of pixels in the ROI.

Table 1. Noise Variance for the final Solutions, as well as the Intermediate Solutions, Measured from Three ROIs Defined in Figure 4a
 α = 90°α = 120°α = 135°
WFBSWFBSWFBS
  • Three representative POP angles of α = 90°, 120° and 135° were used, and the measured NSA* values agree with the theory within 3.9%, 1.0%, and 1.7%, respectively.

ROI 1138137203399107111114176101106107136
ROI 214914412961941110109275487109107165274
ROI 314013221240211310211817510998109134
Mean σ2140.0  108.7  105.0
NSA*_measured1.442  1.857  1.923
NSA*_theory1.500  1.875  1.957
Relative error–3.9%  –1.0%  –1.7%

DISCUSSION

In this paper the classic water-fat separation method described by Dixon (1) is extended to a general sampling scheme of (θ1, θ2) with two arbitrary angles. Using magnitude information only, pixel-level water-fat separation is achieved by solving for the big and small chemical components (B, S) easily and robustly in a manner similar to the original Dixon method (1). Phase information is further used to resolve the remaining binary ambiguity (i.e., to identify which is water and which is fat). In particular, the water-fat system is sampled at two asymmetric points (0, α), i.e., at in-phase and POP, which enables more effective phase encoding of the chemical shift information than the traditional symmetric sampling of (0°, 180°). This is because with POP, both phase errors and water-fat combination can make general phase contributions to the complex image, while with traditional symmetric sampling, the latter makes only a less informative polarity contribution. The relative phase error P is analytically determined to be either Pu or Pv, which are subsequently extracted through a straightforward procedure termed RIPE. Finally, the extracted error phasor Pstable is removed from the complex equations after smoothing, leading to least-squares water and fat solutions with improved SNR compared to first-pass (B, S) solutions.

The smoothing applied to the extracted error phasor Pstable in Eq. [14], as well as that applied to I1 leading to Eq.[8], suppressed noise efficiently and allowed much simplification in the noise analysis. Smoothing on Pstable also serves to fill in the “holes” on the tissue mask through an effective “phasor interpolation,” or to extend the tissue mask to cover pixels of very low SNR that do not survive masking. Smoothing could potentially introduce bias to the final W-F estimation, depending on how it is used and on the specific phase distribution in the smoothing window. However, this is a problem only if the central phase value differs significantly from the phase of the mean phasor in the window. Practically, since the phase map itself is typically smooth, moderate smoothing does not introduce significant bias, and thus is used by many other Dixon methods (17, 24, 25, 27–30) as well as IR phase correction (38) for SNR improvement. It is important to note that although RIPE uses many other heavier (e.g., 37 × 37) smoothing operations, they do not cause any bias at all. They only play a role in grouping the local pixels for error phasor extraction. The extracted error phasor Pstable is not smoothed, but is simply selected at each pixel from either Pu or Pv, as shown in Fig. 5a–c and Eq. [13].

Compared with early approaches (13–16) based on phase unwrapping with symmetric sampling at (0°, 180°), the proposed method has a number of advantages. Water and fat can not only be separated but also automatically identified, which is hardly possible when the two vectors were sampled only at parallel and antiparallel. The proposed method achieves the necessary phase correction by merely choosing between two candidate phasors, Pu and Pv, and thus avoids the necessity of performing the far more difficult task of general phase unwrapping. In general, it is much easier to determine a phasor than the corresponding phase. For a pixel, it is trivial to find the corresponding phasor if the phase is known, but it can be very challenging to determine the phase from a given phasor due to phase aliasing-induced ambiguity. In the problem of water-fat imaging, it is ultimately the correct phasor that needs to be determined, not the phase. Therefore, a method focused on phasor treatment is more suitable to this particular problem. This phasor treatment is similar to the polarity rather than phase treatment used in phase corrected inversion-recovery image reconstruction (38).

The mechanism of RIPE involves dynamics of cellular automata (32, 34), and a thorough analysis is beyond the scope of this paper. However, some desirable properties of the phasor iteration have been observed and can be briefly described. Consider the simple case of setting the initial state P0 as the mean value of Pu and Pv. This automatically applies a high weighting to areas where this initial value happens to be the true phasor P. This takes place near water-fat interfaces, i.e., transition areas between water- and fat-dominating regions. In these areas, water and fat may have equal intensity, i.e., B = S, and then P0 = Pu = Pv will receive a high weighting. In the case where Pu is very different from Pv, there will be a reduced magnitude for P0 due to “dephasing” between Pu and Pv. Therefore, the areas with correct initial state P0 are favored and would “grow” in subsequent steps of RIPE. The more sophisticated initialization method enhances this property. It should be pointed out that these areas will have zero magnitude (10) and random phase in the traditional opposed-phase image acquired at 180°. Thus useful phase information will be lost, making subsequent phase treatment difficult. In contrast, with POP acquisition, complete cancellation between water and fat vectors will not be possible, resulting in well defined and informative phase map everywhere on tissue.

Another desirable feature of the RIPE algorithm is that it is able to self-correct for minor local errors that occur in the phasor evolution. The final result is usually achieved with global phasor continuity, as a result of an evolutional balance from all pixels in the initial weighted phasor P0. This property is very desirable as compared to those observed in typical region-growing algorithms where an accidental error can often trigger a catastrophic regional mistake. RIPE is potentially useful for enhancing phase correction in other simple spectroscopic imaging methods (17, 26). Owing to its simplicity, it is straightforward to apply RIPE to data in 3D or higher dimensions.

As seen from Eqs. [1] and [2], more general (B, S) solutions can be obtained with two POP acquisitions as long as cosθ1 ≠ cosθ2, without having the in-phase acquisition. The subsequent phase correction with RIPE should be done on both the two POP images. After the two error phasors are determined and removed from the two POP images, a final (W, F) solution can be obtained. However, this general approach may not be optimal, especially for gradient-echo sequences with long TE values, since both P1 and P2 are accumulated during the entire TE and thus can be very strong. An in-phase acquisition is very helpful for easily and robustly removing P1 from P2, leaving the remaining phase error P only a small and relative one between P1 and P2.

Although the proposed technique has been successfully used for water-fat imaging on a large number of subjects, like any other technique it has an intrinsic limitation. In particular, if it is tested with a phantom containing only a single chemical component, such as an isolated bottle of water or oil, an incorrect answer could yield. This single peak difficulty is a common problem for many other methods, although it is usually not clearly stated. As briefly discussed in the DPE paper (17), this is a theoretical difficulty even for localized spectroscopy since the chemical shift information can hardly be useful without a reference signal. Fortunately, such a reference is typically available in vivo, where an isolated piece of tissue usually contains both water and fat, which can be used as a reference for each other, just like the simulated phantom data in which water and fat separation and identification were successfully achieved. In our experience, water and fat have been correctly identified in vivo with the proposed method.

The three-point DPE method (17) is able to separate and identify W and F at the pixel level, as well as the corresponding error phasor. In comparison, the two-point POP method achieves the same goal regionally when P0 is initialized by grouping pixels within a small neighborhood together, assuming that they share approximately the same phase error. The number of equations in that local region is then effectively increased without proportionally increasing the number of unknowns. This strategy works best near water-fat tissue interfaces where contrast between W and F changes, while in a uniform region there is little difference between a single pixel and its neighboring pixels.

The proposed method with POP acquisition offers more flexibility in pulse sequence design. Although the angle α is mostly chosen to be 135° in this work, it does not have to be so. It does not even have to be in the range between 0° and 180°. It can be greater than 180°, say 225°, if required by sequence timing restrictions. Note that the resulting NSA* are the same for POP at 225° and 135° since NSA* is only a function of cos α as shown in Eq. [26] and Fig. 2. Obviously, 135° is better than 225° in terms of a smaller relative phase error P in Eq. [8], as well as a smaller Tmath image effect. An example of using POP at α = 195° is given in Fig. 8. The sampling scheme of (0, α) can be implemented with either spin-echo or gradient-echo (as those in this work) and other sequences. The in-phase acquisition at 0° is very useful in removing large phase errors accumulated during the entire TE for a gradient-echo sequence. Another advantage of the proposed method, as well as any other two-point method, is that it can be used on systems with other sources of phase errors that are not included in the traditional signal equations of Dixon acquisitions. One example is the effect of eddy currents, which can be time-dependent (12) and can potentially introduce extra phase errors into the signal equations, particularly on some low-field systems with strong eddy currents. This can be difficult for methods involving three or more points.

The choice of α is quite flexible. As can be seen in Fig. 2, the NSA* remains reasonably high when α is between 120° and 240°. Of course, to be a POP acquisition, α should not be too close to 180°, otherwise valuable phase information is lost. With 180° sampling, in regions where water and fat are equal, the phase is undefined, which may have negative effect on other pixels. Furthermore, the perfect symmetry between water and fat vectors allows only their separation—not their ultimate identification using chemical shift information (37, 39). The final water and fat images are given by Eq. [21] as a real-valued least-squares solution from the phase-corrected images M1 and J2C in Eqs. [7] and [15]. This solution is simpler than that described in the DPE method (17), in which three sets of solutions are optimally averaged, and can be shown to provide an optimal NSA* as well.

The proposed method is particularly suitable at low fields, e.g., 0.3 T as demonstrated in this work (Fig. 9), where the small frequency difference between water and fat makes RF fat suppression difficult. On the other hand, the longer Tmath image allows more time for TE increments, and the shorter T1 and lower intrinsic SNR justify the use of multiple acquisitions.

In summary, a novel two-point method for water-fat imaging with POP acquisition is introduced. The asymmetry between water and fat magnetization vectors in the POP acquisition provides an opportunity to not only separate but also identify the two chemical components. The phase correction involves a straightforward procedure termed RIPE. Excellent water-fat separation and identification in a large number of in vivo studies at 1.5T and 0.3 T suggest the clinical utility of the proposed method.

Acknowledgements

The author appreciates valuable discussions with Dr. Li An, helpful comments and suggestions on the manuscript from Dr. Xiaohong Joe Zhou and Dr. Elana Brief, important input and support for the revisions from Dr. Mark Henkelman, assistance in acquiring the gradient-echo data from Wayne Patola and Blaine Curry, assistance in testing the method on the 0.3 T system from Dr. Jianyu Lian and Ms. Dan Xu, and constant support and encouragement from colleagues, friends, and family.

APPENDIX A

Redundancy in (0°, 180°, 360°) and (−α, 0, α) Sampling

Assume that the phase error due to main field inhomogeneity is θ = γΔB0t, where t is the time increment between samples, and the temporally invariant phase offset due to other factors is ϕ.

Case A

For (0°, 180°, 360°) sampling, the three complex images are

equation image
equation image
equation image

Case B

For (−α, 0, α) sampling, where α is an arbitrary angle, we have

equation image
equation image
equation image

In both cases A and B, the third equation is not independent since its magnitude M3 and phase P3 are simply derivable from I1 and I2 as

equation image
equation image

The only exception is in background noise or at tissue interfaces where W = F, resulting in undetermined phase for some pixels. Therefore, the sampling schemes of (0°, 180°, 360°), as well as (−α, 0, α) including (−180°, 0°, 180−), are essentially redundant and should be avoided.

APPENDIX B

Fast Smoothing Algorithm

General smoothing involves the convolution of a kernel with an image, which can be slow particularly when the kernel size is large, as the processing time is usually proportional to the product of the number of pixels in the kernel and the number of pixels in the image. Smoothing can often be made faster if the image is fast Fourier transformed, low-pass-filtered, and transformed back to the image domain. In this paper a simple image domain fast smoothing algorithm is used for the purpose of phasor iteration with the following strategies:

First, the 2D convolution kernel is chosen to be a square boxcar, or a sliding window of M × M pixels. This allows the 2D convolution to be performed as two sequential 1D convolutions of size M along the X and Y directions, resulting in a significant reduction of processing time from M × M to 2M.

Second, for each 1D convolution, since the convolution kernel is a boxcar with values of either 1 or 0, when the summation is performed within the sliding window of M pixels wide, one does not have to repeat the summation at each pixel location. The summation for one pixel differs from that of the previous pixel by only two pixel values, (i.e., a newcomer that has just been included and an old pixel left from the other side of the window). Therefore, only one addition and one subtraction are needed to shift the convolution kernel by one pixel, instead of redundantly adding all pixels together within the entire sliding window after each shift of pixel location. Using the above strategy, the processing time becomes essentially independent of the kernel size, and is only proportional to the image size.

Another issue to be noted is that in this paper the fast smoothing algorithm is implemented noncyclically, i.e., the sliding window terminates at the end of FOV rather than wraps around, to be sensitive to the possible phase error discontinuity beyond FOV.

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