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Keywords:

  • multipeak water–fat separation;
  • IDEAL;
  • hierarchical decomposition;
  • MRI;
  • human imaging

Abstract

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

We describe a generalized version of hierarchical IDEAL that can flexibly handle arbitrary chemical species at arbitrary echo times. The proposed work is fast and robust, and it has three key features: (1) multiresolution approach, which allows the method to handle images with disjoint regions, makes it less susceptible to local optima, and reduces the ambiguity of the separation; (2) direct phase estimation, which bypasses the phase wrapping issue, and (3) efficient algebraic formulation, which enables fast calculation and insensitivity to spatially varying phase across the image, from sources such as partial echo acquisition, receiver coils, motion, and flow. Representative results at 1.5 T and 3 T are presented from human ankle, wrist, and a water/oil/silicone phantom. Magn Reson Med, 2013. © 2012 Wiley Periodicals, Inc.


INTRODUCTION

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

A number of MRI techniques have been proposed to separate chemical species based on images acquired at multiple echo times (1–7). These techniques are able to separate signals much more cleanly compared to traditional techniques, such as fat saturation. An important step in some of these techniques is the generation of a B0 field map, and several common approaches have been reported (8–12). One approach is region growing, which takes advantage of the similarity in off-resonance between neighboring voxels (8, 9). Potential disadvantages of region growing are that low signal-to-noise regions may affect the phase unwrapping process, and disjoint regions in the image are potentially problematic for region growing. Another approach is to enforce a smoothness constraint on the estimated field map (3, 11, 12). This can be achieved with low-pass spatial filtering of the field map (3), or an explicit smoothness term in the optimization (11, 12). However, the optimization can be trapped easily in local optima. The graph-cut algorithm has been proposed recently to overcome such problems (11). A third approach is multiresolution, which gradually hones in on the solution in a coarse-to-fine manner (8, 10, 13). Although multiresolution optimization does not guarantee the global optimum, it tends to improve robustness against local optima and it implicitly enforces spatial continuity.

Previously, we described a multiresolution technique called “hierarchical IDEAL” (13), which has been applied successfully to several thousand images to date (14). The original technique took advantage of the discrete Fourier transform, and could only separate two species (water and fat) using three specific echo times chosen to maximize the effective number of signal averages for all fat/water ratios (15).

This work presents a generalization of hierarchical IDEAL that handles arbitrary echo times and species. Three features of the proposed method are noteworthy: (1) efficient algebraic formulation, (2) direct phase estimation, and (3) multiresolution approach. We demonstrate the proposed method using data from arbitrary echo times and with two or more species.

THEORY

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

Signal Model and Algebra

The signal model relating the detected NMR signal to the chemical species is (3, 16):

  • equation image(1)

sm(x) is the intensity of a voxel at position x and at the mth echo time, TEm; ρn(x) is the complex-valued intensity of the nth species at position x. Each species potentially consists of multiple spectral peaks p, as defined by relative amplitudes αn,p and chemical-shift frequencies Δfn,p (16). This model assumes a single decay constant Rmath image(x) for each spatial position x. Finally, the signal is modulated by off-resonant precession from the B0 inhomogeneity, Ψ(x). The least-squares fitting error of the Rmath image-corrected signals for a given voxel is as follows, incorporating arbitrary species, arbitrary echo times, and a multipeak spectral model (16):

  • equation image(2)

The superscript + denotes the Moore-Penrose pseudoinverse; diag(x) denotes a diagonal matrix with the elements given by the vector x. A is an M x N matrix capturing the multi-peak spectral model with elements am,n defined as:

  • equation image(3)

d(Rmath image, Ψ) is an Mx1 vector correcting for the Rmath image decay and off-resonant precession that occur after the first echo time, TE1.

  • equation image(4)

Equation [2] can be expanded further as follows:

  • equation image(5)

The open circle ○ denotes an element-wise multiplication; the asterisks indicate complex conjugation, and the summation is performed over the M x M matrix elements. The three factors in Eq. [5] are noteworthy. (IAA+) is a fixed factor dependent only on the echo times and the spectral model. (ssH) is a spatially varying factor, which is the outer product of the intensities across all the echo images at a given spatial location. (ddH) is the only factor that varies with the optimization parameters.

Equation [5] yields two important advantages for the proposed method. First, it is computationally efficient, as the first two factors can be precalculated, leaving each iteration to construct (ddH), perform the element-wise multiplication in Eq. [5], and sum the M × M matrix elements. Further efficiency is gained with the Hermitian matrix symmetry. Second, Eq. [5] can be applied to a volume of interest (VOI) by integrating the second term across the VOI:

  • equation image(6)

Equation [6] is preferred over first averaging the signals within the VOI and then calculating (ssH) for the averaged signal. The averaging approach is susceptible to signal cancellation from spatially varying phase, which can arise from partial echo acquisition, flow, motion, receiver coil phase, and other sources. By combining Eqs. [5] and [6], the proposed method is completely immune to spatially varying phase across the image.

Direct Phase Estimation

Estimating the off-resonant frequency Ψ in Eq. [2] is subject to phase wrapping, as different off-resonant frequencies Ψ that yield the same values for equation image are considered equivalent in the optimization. As a result, the optimization algorithm may drive toward incorrect frequency periods that corrupt the final field map (8, 11). The proposed work optimizes the phase increment directly (referred to as phasor in prior work (7)) and defines an echo time resolution ΔTE such that all echo-time spacings are multiples of ΔTE. In that case, Eq. [4] simplifies to (17):

  • equation image(7)
  • equation image(8)

The optimization directly determines the best-fit complex-valued d, as opposed to the real-valued Rmath image and Ψ. In practice, ΔTE can also be defined to be short enough to avoid phase wrapping over this period.

Multiresolution Optimization

The multiresolution approach starts with the full image, estimates the global correction factor d (Eq. [8]), subdivides the image into multiple overlapping regions, and then repeats the estimation process using the result from the prior level as a starting point (Fig. 1). The final resolution of the subdivision is controlled by the number of hierarchical levels, and the zoom factor during subdivision (i.e., ratio of region side length before and after subdivision) (see Fig. 1). In our experience, seven hierarchical levels and a zoom factor of 2/3 provides a good balance between spatial resolution and speed. With these parameters, there are 4096 (= 46) subdivided regions at the finest resolution for a two-dimensional (2D) image, with the side of each region covering 8.8% (=(2/3)6) of the field of view.

thumbnail image

Figure 1. Multiresolution decomposition starts with by setting the VOI to be the full image (gray area at level 1), performing the optimization on the VOI to determine the complex-valued d (top right), and then recursively dividing the VOI into multiple overlapping regions to repeat the optimization at a finer spatial resolution. A zoom factor of 2/3 (or 67%) is shown here, indicating that at each subdivision, the side length of a region shrinks to 2/3 of its former value. The number of hierarchical levels is controlled by the user. At the last decomposition level, the optimization yields an optimized complex-valued d for each VOI (gray area). These factors are interpolated across the VOIs at the finest level to determine the spatial map of d(x).

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The optimization was driven by the Nelder–Mead simplex algorithm (18). For each region, the optimization proceeds by first precalculating Eq. [6]. Then, Eq. [5] is minimized by adjusting d. The magnitude of d yields the reciprocal of the Tmath image decay during ΔTE, whereas the phase of d is the negative of the off-resonant precession during ΔTE.

After the optimization at the last level, the optimized d values from the overlapping regions are interpolated to form the final spatial map of d(x), using a Hanning window kernel. The d(x) map is used in Eq. [1] to solve for the maps of the chemical species.

METHODS

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

All human images were acquired after informed consent. Ankle images were acquired in sagittal view on a GE 3T Signa EXCITE HDx scanner (GE Healthcare, Waukesha, WI) with a single-channel coil, using a 3D spoiled-gradient-echo sequence with three echo times (2.184, 2.978, and 3.772 ms), one echo per repetition time, a 125 kHz bandwidth, a flip angle of 5°, a 256 × 256 × 8 matrix, an in-plane field of view of 18 cm, and a voxel size of 5 mm along the z direction. Wrist and phantom data were acquired on a Siemens Magnetom Espree 1.5T (Siemens AG Medical Solutions, Erlangen, Germany). Wrist images were acquired with a four-channel wrist coil, using a 3D spoiled-gradient-echo sequence, two separate acquisitions with either uniformly spaced echo times (1.81, 4.26, 6.71, 9.16, 11.61, and 14.06 ms) or nonuniformly spaced echo times (1.81, 4.3, 7, 9.5, and 14.5 ms), a 128 kHz bandwidth, a flip angle of 10°, a 128 × 128 × 8 matrix, an in-plane field of view of 17 cm, and a voxel size of 3 mm along the z direction. Images of a phantom consisting of water, oil, and silicone (GE Silicone I (GE012A); Momentive, Huntersville, NC) were acquired using a 2D spoiled-gradient echo sequence with a four-channel head coil, six echo times with one echo per repetition (initial echo time of 2.1 ms, and echo spacing 2.3 ms), a 128 kHz bandwidth, a flip angle of 20°, a 256 × 256 matrix, and an in-plane field of view of 20 cm. The chemical shifts of water, canola oil, and silicone peaks were 4.7 ppm, 1.3 ppm, and 0.1 ppm, respectively. Data acquired with phased-array coil were first coil-combined in complex value before hierarchical IDEAL.

Hierarchical IDEAL can be applied directly to 3D images. However, to speed up computation, it was applied on a slice-by-slice basis, with the phase map of one slice used as the initial value for the next slice. To improve robustness against strong B0 variations along the z direction, hierarchical IDEAL was applied in low resolution (with three hierarchical levels for speed) first through all the slices as a preprocessing step, followed by applying hierarchical IDEAL with seven hierarchical levels.

A multipeak fat spectrum (16) was used in the in vivo signal model, with spectral peaks at 0.9, 1.3, 2.1, 2.76, 4.31, and 5.3 ppm (water peak at 4.7 ppm), and relative amplitudes of 0.087, 0.694, 0.128, 0.004, 0.039, and 0.048, respectively.

RESULTS

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

Figure 2 shows a set of typical results of ankle images for hierarchical IDEAL. The top row shows, from left to right, the phase of d(x), the Rmath image map, and the square root of the fitting error εHε. The phase map shows a variation in the superior-to-inferior direction, indicating a B0 inhomogeneity gradient. The phase and Rmath image maps have a lower spatial resolution compared to the full images, as controlled by the number of hierarchical levels. The brightness level of the fitting error map has been increased to reveal the error, which is slightly above the background noise level. The edges in the error image represent the high spatial frequencies beyond the finest resolution of the Rmath image and phase maps. The edges can be reduced with more hierarchical levels, albeit at an exponentially increasing computational time due to the hierarchical nature of the decomposition. Figure 3 shows four slices from the ankle data set, with clean water–fat separation throughout the field of view.

thumbnail image

Figure 2. (Top left to right) Phase of d(x) to correct for the dephasing over ΔTE; the Rmath image map is calculated from the magnitude of d(x) by Eq. [8]; and the fitting error shows the square root of εHε from Eq. [5]. The fitting error map has been brightened to reveal the error, which is close to the noise level. The spatial resolution of the phase and Rmath image maps is controlled by the number of hierarchical levels. (Bottom left to right) corresponding water and fat images.

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thumbnail image

Figure 3. Four slices from a 3D set of ankle images after hierarchical IDEAL separation of water and fat components.

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Figure 4 shows one slice from the wrist data from two separate acquisitions with six uniformly spaced echo times (1.81, 4.26, 6.71, 9.16, 11.61, and 14.06 ms) and five nonuniformly spaced echo times (1.81, 4.3, 7, 9.5, and 14.5 ms). The images at the first four echo times were reconstructed separately to demonstrate the flexibility of Hierarchical IDEAL. The brightness level has been increased to reveal details for comparison. It demonstrates the feasibility to achieve similar water–fat separation from multiple echo times with arbitrary echo spacings.

thumbnail image

Figure 4. One slice of water and fat separated images from a 3D set of wrist images from two acquisitions with six uniformly spaced echo times (1.81, 4.26, 6.71, 9.16, 11.61, and 14.06 ms) and five nonuniformly spaced echo times (1.81, 4.3, 7, 9.5, and 14.5 ms). Images of four echoes were reconstructed from data at the first four echo times of these acquisitions. The brightness level has been increased to reveal details for comparison.

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Figure 5 shows images from the water/oil/silicone phantom, demonstrating the ability to separate more than two species. The white arrow in the figure points to slight residual fat signal in silicone image, which may be caused by inaccuracy in the spectral model.

thumbnail image

Figure 5. Representative slice from the water/oil/silicone phantom. From left to right: Image at the initial echo time (2.1 ms), water-only, fat-only, and silicone-only images. Arrow in silicone-only image points to residual fat signals (not obvious in image).

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Hierarchical IDEAL reconstruction time from coil-combined data is measured in Matlab (The MathWorks, Natick, MA) on a 64-bit Mac OS laptop with a 2.4-GHz Intel dual-core i5 processor. It took ∼6 s for the ankle data set (four slices, 256 × 256 images, three echoes), ∼10 s for wrist data (eight slices, 128 × 128, six echoes), and ∼2 s for the water/oil/silicone phantom (one slice, 256 × 256, six echoes).

DISCUSSION

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

We describe a generalization of Hierarchical IDEAL to handle arbitrary chemical species and echo times. The present method reduces exactly to the original Hierarchical IDEAL, when there are only two spectral species (water and a single-peak fat model) and three specific echo times.

The proposed method benefits from the synergistic combination of three features: the algebraic formulation, the direct phase estimation, and the multiresolution approach. The algebraic formulation (Eqs. [5] and [6]) allows for significant speed-up by separating factors that can be precalculated. This speed-up can be applied to other existing water–fat techniques based on the same least-squares metric. The algebraic formulation is beneficial to the multiresolution approach, as it allows spatial integration that is entirely immune to spatially varying phase, which can arise from a partial echo acquisition or other sources. The direct phase estimation also benefits the multiresolution approach, as each subdivided region undergoes a separate optimization. Optimizing the phase increment instead of frequency bypasses the algorithm driving toward different (aliased) frequency ranges for different regions, resulting in artifactual jumps in the final interpolated field map.

The multiresolution approach used in this work was chosen for several reasons. First, unlike region growing approaches, it can handle images with disjoint regions. Second, a multiresolution approach generally improves the robustness of the optimization (8), although it does not guarantee the global optimum. Third, it addresses the well-known issue of ambiguity in separating chemical species. When there is only a single species in water–fat imaging, there is an ambiguity of whether the signal belongs to water or fat. By starting from the full field of view, the algorithm is more likely to encounter all the chemical species, thereby avoiding this ambiguity. A multipeak spectral model also helps to reduce the ambiguity (19), but it may not suffice when there are few echo times. In general, the multiresolution approach can be adjusted to have a different starting and ending resolutions to account for different expected variations of the B0 homogeneity.

In this work, all results were shown with seven hierarchical levels and a zoom factor of 2/3. These parameters were chosen to balance speed with the needed resolution for the Rmath image and phase maps. For a 2D image with H hierarchical levels, the total number of optimization is given by (4H − 1)/3. Thus, it is advisable to keep H as low as possible in practice.

The method is available in Matlab (The Mathworks, Natick, MA) as part of the fat–water toolbox initiative in the 2012 ISMRM Workshop on Fat-water Separation.

Acknowledgements

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

The authors thank Houchun H. Hu and Samir D. Sharma for the ankle images used in this article. Also, they thank Houchun H. Hu and Diego Hernando for spearheading the fat–water toolbox initiative. Finally, they thank Michael S. Hansen, Brittany Yerby and Farid Sari-Sarraf for their contributions to and extensively testing of the initial version of hierarchical IDEAL. YJ thanks Mark A. Griswold for his support and guidance.

REFERENCES

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