Recovery of chemical Estimates by Field Inhomogeneity Neighborhood Error Detection (REFINED): Fat/Water Separation at 7T

I.Abstract

Purpose

To reduce swaps in fat-water separation methods, a particular issue on 7T small animal scanners due to field inhomogeneity, using image postprocessing innovations that detect and correct errors in the B0 field map.

Materials and Methods

Fat-water decompositions and B0 field maps were computed for images of mice acquired on a 7T Bruker BioSpec scanner, using a computationally efficient method for solving the Markov Random Field formulation of the multi-point Dixon model. The B0 field maps were processed with a novel hole-filling method, based on edge strength between regions, and a novel k-means method, based on field-map intensities, which were iteratively applied to automatically detect and reinitialize error regions in the B0 field maps. Errors were manually assessed in the B0 field maps and chemical parameter maps both before and after error correction.

Results

Partial swaps were found in 6% of images when processed with FLAWLESS. After REFINED correction, only 0.7% of images contained partial swaps, resulting in an 88% decrease in error rate. Complete swaps were not problematic.

Conclusion

Ex post facto error correction is a viable supplement to a priori techniques for producing globally smooth B0 field maps, without partial swaps. With our processing pipeline, it is possible to process image volumes rapidly, robustly, and almost automatically.

II. Introduction

Magnetic resonance imaging (MRI) fat quantification can be used for monitoring several physiological and disease processes, including fatty infiltration of the liver (hepatic steatosis)(1, 2) and heart(3, 4), as well as for automatic analysis of adipose(5) and brown adipose(6) volumes. Mouse models are ideal for studying the effects of genes, diet, exercise and therapeutics on lipid stores, so we are developing MRI methods for performing fat quantification on such models.

Dixon methods have been useful for monitoring changes in the lipid content of tissues. This class of methods uses multiple acquisitions with varying echo time shifts to elicit varying phase differences between fat and water(7). The chemical parameters are then estimated by modeling the signal as a function of the phase differences caused by the chemical and field inhomogeneity shifts(8). However, since these phase differences are modeled as complex exponentials, the multiplication of the two complex exponentials results in a periodic cost function with multiple minima per period(9, 10). This creates an ambiguity in the estimation of the chemical parameter maps that can lead to a local “swap” of fat and water, where the fat estimate appears in the water parameter map, and vice versa.

Many attempts have been made to resolve the ambiguity using the a priori knowledge of B0 field map (ψ) smoothness, based on the observation that real magnetic fields do not have discontinuities. Region-growing schemes(11–14) have been used to generate initial guesses at each pixel. However, these algorithms are prone to error propagation, since the initialization at a pixel depends on the previously solved pixels.

Another approach to using the a priori knowledge of field smoothness is to formulate the cost function with a Markov Random Field (MRF) prior(Equation 1):

$$\psi =\begin{array}{l}\mathit{arg}\hfill \\ \mathit{min}\hfill \end{array}\left(\underset{\mathit{data}\mathit{cost}}{\underbrace{{\mid S_{\mathit{computed}}(\psi ,TE)-S_{\mathit{signal}}(TE)\mid }^2}}+\mu \underset{\mathit{smoothness}\mathit{cost}}{\underbrace{\sum _{j\in N}{w}_{q,j}{\mid {\psi }^q-{\psi }^j\mid }^2}}\right)$$ (1)

where μ is a weighting constant (empirically determined), and w is the inverse of the Euclidean distance between the current pixel of interest (q) and a neighboring pixel (j). This formulation can then be solved using a number of network algorithms such as Iterated Conditional Modes (ICM)(15, 10), Graph Cuts(16, 17), or Loopy Belief Propagation(18). The MRF formulation enforces local smoothness, but this does not necessarily ensure correct decomposition of fat and water because of what we call “clumping errors”: regions that have sufficient cohesiveness to result in partial swaps (part of the image) or complete swaps (the entire image).

Partial swaps of water and fat occur when portions of the chemical parameter maps are swapped relative to one another. This happens because each of these regions has sufficient spatial cohesiveness to overcome the smoothness penalties at the borders between the regions (Figure 1). Partial swaps leave demarcations in the B0 field map wherever a swap occurs. In contrast, complete swaps occur when the MRF formulation results in a globally smooth B0 field map, but the estimation in every pixel converges to an incorrect minimum, resulting in a chemical swap at every pixel. Complete swaps may not leave discontinuities in the B0 field map to indicate that an error has been made.

Figure 1.

B0 field map with clumping errors. When clumping errors are made, the swapped regions (arrows) appear piecewise constant, while the unswapped region (any part of the mouse caudal to the apex of the visible lung and the phantom) also appears similarly piecewise constant. The variation between regions is much higher than the variation within each region. We found it useful to categorize two types of partial swap regions: those that are circumscribed by other regions (“holes”, solid arrows), and those that touch the edge (“edge groups”, dashed arrows).

Clumping errors have been addressed previously by the FLAWLESS method(10), which is a computationally efficient, ICM-based way of solving the MRF formulation. The extrapolation-initialization method, used by FLAWLESS to extrapolate initial guesses for slices of 3-D volumes based on the medians of previously solved B0 field maps, resolved almost all complete swaps, but did not solve all partial swaps.

So, we proposed that instead of seeking more complicated initialization and constraints, that we can find and correct the errors after they are made. The purpose of this paper is to explore the feasibility of automatically finding and correcting partial swaps in 7T small animal imaging by analyzing the B0 field maps.

III. Algorithms

We started by running FLAWLESS(19) on each image volume. Then, we used a combination approach, as described below, to detect and correct the swaps found in the results.

First, we developed a hole-filling algorithm to detect and correct errors that appeared as holes. If a B0 field map output of FLAWLESS contained partial swaps, the spatial gradient between unswapped and swapped regions was much larger than the gradient within each region (see example in Figure 1 and Figure 2a). Therefore, error regions in the animal were detected by thresholding and flood-filling a gradient map. Once detected, we replaced these error regions with the median of the rest of the B0 field map. We used the median instead of the mean because the median insensitive to possible bias due to the remaining partial swaps that do not look like holes, which are included in the operation, since they are not differentiated by this error detection algorithm. The resulting B0 field map was used as an initial guess for a new run of ICM. This procedure is described in detail in Table 1, and is illustrated in Figure 2.

Figure 2.

Hole-Filling. (a) is the B0 field map output of FLAWLESS, with several discontinuities in the rostral third of the animal; some of these regions are swapped in the chemical parameter maps. (b) is a thresholded gradient map of the B0 field map. If a flood-filling procedure is performed on (b), the result is (c), which represents the error regions found in (a) that look like holes. The locations in (a) corresponding to those mapped in (c), are replaced with the median of the rest of the B0 field map. Rerunning ICM using (d) as an initial guess results in (e), with the holes in the B0 field map corrected.

Second, we developed a histogram analysis to cluster and correct swapped regions. As shown in Figure 1, B0 field maps with errors were comprised of regions where the variation within a region was much smaller than the variation between regions. These swaps could not be detected by the hole-filling, because they touched the edge of the animal, and often occurred starting around the lungs, because the MRF smoothness cost could not easily propagate across the thin amount of tissue between the lungs and the outside of the mouse, especially if motion artifact was present. These errors were detected by segmenting a histogram of the B0 field map with k-means clustering(20). Then, all of the groups with group centers that were significantly different from the center of the majority pixel population had their values set to the median of the majority population. The resulting map was used to initialize a new run of the ICM algorithm. This procedure is described in detail in Table 2, and is illustrated in Figure 3.

Figure 3.

k-means error correction. When partial swaps occur in the B0 field map (a), the k-means algorithm segments them into clusters separate from the majority population (see arrow in label image, (b)). Replacing the values in these clusters by the median of the majority population generates an initial guess (c) that is globally smooth. Rerunning the ICM algorithm with this new initial guess yields a B0 field map that is no longer swapped (d).

We propose to process the B0 field maps by iteratively applying hole-filling and k-means correction because we do not expect either method to be completely sufficient for detecting all possible errors. We believe that the best results are likely to arise from the procedure described in Table 3, which we term Recovery of chemical Estimates by Field Inhomogeneity Neighborhood Error Detection (REFINED).

Table 3.

REFINED Error Correction Procedure for Increased Robustness

Run FLAWLESS, including the extrapolation-initialization method Correct B0 field map using hole-filling algorithm Correct B0 field map using k-means algorithm Correct B0 field map using hole-filling algorithm again

IV. Methods

We acquired 12 animal volume data sets with a total of 134 images. The animals were all genetically wild-type Bl/6 mice undergoing dietary perturbations such as long-term high-fat diet, long-term low-fat diet, short-term starvation, and short-term high-fat refeeding after starvation. The images were acquired as part of various ongoing studies of lipid deposition under IACUC approval.

The MRI acquisitions were performed on a Bruker BioSpec 7T/30cm system with a 2-D asymmetric RARE protocol. The asymmetry was set to acquire three 3-D volumes per animal with fat/water phase differences of π/6, 5π/6, and 3π/2, which corresponded to 7.93e-5, 3.96e-4 and 7.14e-4 second delays between the Hahn echo and gradient echo(7). The repetition times (TR), echo times (TE), and spatial resolutions were variable.

Each image volume was analyzed first with FLAWLESS, and then with FLAWLESS plus REFINED. All analysis was performed in MATLAB (MathWorks, Inc., Natick, MA), on a 3.2GHz computer with 8GB of RAM (iBUYPOWER, Inc., Los Angeles, CA).

Parameters in the algorithm were optimized manually. Thresholds (see Table 1 and Table 2) were manually tuned by examining intermediate and final results. The number of clusters for k-means was optimized by counting the number of relevant swaps as a function of cluster number, and choosing the number of clusters that yielded the lowest error rate. An error was considered to be relevant only if a discontinuity in the B0 field map occurred inside of the main body of the animal. Errors in irrelevant regions, such as small areas outside of the main body and air spaces in the animal (e.g., lungs) were not counted.

Table 1.

Hole-Filling Correction

Filter B0 field map output of FLAWLESS with Prewitt filter filter with horizontal and vertical Prewitt filters, independently take L2 magnitude of edge strength at each pixel (square root of sum of squares of horizontal and vertical edge strengths) Threshold edge strength to only include edges with strength greater than half the fat-water frequency shift Ensures edges associated with swaps are included, while edges associated with internal field variations are excluded Use eroded animal mask to exclude animal/air border Use flood-filling algorithm to make a binary map of errors that look like holes Replace error regions with median of non-error regions Use result as an initial guess for another run of ICM Repeat steps 1–6 until no holes are found

Table 2.

k-Means Correction

Segment B0 field map into k groups using k-means Sort the group centers by ascending B0 field map value Find the group with the largest population of pixels this group is assumed to be correctly decomposed For each group where the center differs from the center of the largest group by more than fat-water frequency shift, replace with the median of the largest group Blur the result and use as an initial guess for another run of ICM Repeat steps 1–5 until B0 field map stops changing

Validation of the REFINED algorithm was done by manually inspecting each B0 field map and chemical decomposition for swaps, since no standard exists for automatic detection of errors. The utility of REFINED was determined by comparing the fractions of images with relevant swaps

V. Results

Figure 2e shows the result of the hole-filling correction on a representative image. This algorithm was able to find discontinuous regions that were entirely contained within the animal. By reinitializing these areas, we were able to correct these B0 field maps. However, the hole-filling algorithm was not capable of finding error regions that touched the edge of the animal.

Figure 3d shows the results of using the k-means correction scheme on the result of the hole-filling algorithm. The optimal number of clusters was empirically found to be 6. However, having high quality data with little motion artifact allowed this parameter to vary from 4 to 8 with little difference in results.

Using the REFINED procedure (Table 3), the example chemical decompositions shown in Figure 1, which contained swaps, were corrected to form parameter maps that look like those found in Figure 4.

Figure 4.

Post error correction parameter maps. The swapping errors, which can be seen in the top row of B0 field maps in Figure 5, are corrected. Areas too small to be reached by the MRF regularization (dotted arrow) and spaces inside the animal (solid arrows) were not counted as relevant errors.

Figure 5 shows the result of each step in the error correction process. We found that the most effective way to correct swaps was to alternate between the two error corrections methods, as described in Table 3. As shown in Figure 5, both hole-filling and k-means can be necessary, but the order in which they are needed can vary.

Figure 5.

Step-By-Step results of error correction. The top row shows the B0 field maps after each correction step in REFINED for an example image. Most of the errors were not holes, and therefore were unaffected by the first hole-filling step. The k-means step improved the reconstruction to the point where only holes were left. These were caught by the second hole-filling. Conversely, the second row shows the results for each step for a different example image. Here, the first hole-filling caught all of the errors, except a few edge errors, which were fixed by the k-means step. All three error correction steps in REFINED were sometimes needed; sometimes they were not, but we found the procedure outlined in Table 3 to be sufficient to prevent most errors.

We found, as has been previously described(10), that the extrapolation-initialization technique was necessary – 30% of the 134 images had swaps when it was not used. With extrapolation-initialization, 8 images (6%) had relevant errors. When FLAWLESS and REFINED were combined, only 1 image (0.7%) had relevant errors. However, in 3 out of the 12 datasets, the automatically chosen starting slice contained clumping errors, and a manually chosen slice had to be used. If only the results using the automatically chosen starting slice were considered, the error rate of FLAWLESS jumped to 12% (16 images). With the addition of REFINED, the automatically chosen starting slice resulted in errors in 4 (3%) of the images. This represents a 88% improvement in error rate with the manually chosen starting slice, and a 75% improvement with the automatically chosen starting slice.

REFINED processing required additional computation time because it required additional runs of ICM. The average computation time per image was 49 seconds for FLAWLESS. With the addition of REFINED, this average was increased to 130 seconds per image. For a typical ten slice image volume, about 20 total minutes of computation time was needed. If swaps were found in the starting slice, the estimation process was quickly stopped and restarted on a different slice, effectively adding an extra slice’s worth of computation time.

VI. Discussion

The aim of these experiments was to explore the feasibility of achieving global smoothness of Dixon method B0 field maps by correcting partial swaps ex post facto. We were able to achieve an error rate of 0.7% by correcting the results of a reliable, previously described method (FLAWLESS). We have shown that fat-water decompositions can be performed reliably and automatically, with human interaction only being necessary for verifying the decomposition of the first slice to be processed.

REFINED allowed for correction of most of the B0 field map errors made by FLAWLESS. We found that performing a hole-filling correction, followed by a k-means correction, followed by another hole-filling correction was almost always sufficient to prevent errors from appearing in the final estimate. Leaving out any of these components increased the error rate. Changing the order also increased the error rate because it became difficult to tune the number of clusters used in the k-means algorithm.

k-means clustering was appropriate for clustering B0 field map values because a correctly solved B0 field map has a Gaussian, unimodal distribution(21). An incorrectly solved B0 field map has multiple modes. k-means was able to assign separate clusters to the error regions, as long as the error regions had sufficiently large populations.

The observed error rate with our methods was not zero, possibly due to several potential reasons. First, error regions at the edges of animals, which are not holes and do not have a sufficiently large number of pixels to be detected by the k-means step, are not detected, or corrected, by REFINED. Second, the ability to correct the B0 field maps also depends on the reinitialization. Since we reinitialize the error regions with the median of the unswapped region, if the physical field inhomogeneity gradient is large, then B0 field map correction could cause swaps. T2* is unlikely culprit, as our data was acquired with spin echoes. The multiple peak spectral model was not considered for the purpose of this work since the primary focus was on the B0 field map. Finally, if the source data contains artifacts such as ghosting from motion, then our algorithm will fail, as probably every Dixon method would. Ghosting destroys the phase of the source data, which might make it impossible to ensure proper and accurate decomposition of fat and water directly from the source images.

In principle, complete swaps can occur with REFINED processing, because neither of the error detection methods would be able to find such an error. However, extrapolation-initialization(10) prevented complete swaps, which meant that REFINED only had to detect partial swaps. If extrapolation-initialization was not used, REFINED was still able to correct partial swaps, but was unable to correct the complete swaps that appeared. If extrapolation-initialization is not used, more sophisticated methods, such as ones with anatomical knowledge, would be needed to detect complete swaps in rare cases where they occur, since the ability to fix swaps depends on the ability of the algorithm to locate errors. Most of the partial swaps were corrected by REFINED when extrapolation-initialization was not used. However, REFINED depends on the majority of the pixels being correctly decomposed, so if the swaps in the result of FLAWLESS make this assumption untrue, REFINED will not work. Once again, this would require much more a priori knowledge to correct automatically.

Additional computation time was required for REFINED error correction, but we note that the added robustness provided by the addition of REFINED allowed the fat quantification algorithm to run almost unattended. If additional robustness is necessary, it might be possible to correct the residual errors that appeared in some very few images by adding more iterations of the error correction algorithms. However, we found that the process described in Table 3 gave sufficient results, allowing us to stop the processing at this point.

Finally, our error correction scheme was applied to the results of FLAWLESS, where most of the errors were predictable clumping errors that were partial swaps. Our same correction algorithms can be applied to any algorithm that produces similar partial swaps. Most MRF-based algorithms like VARPRO-ICM(15) fall into this category. In principle, error corrections can be applied to the results of any fat-water decomposition algorithm, possibly with tuning to detect partial swap regions having different properties than those made by FLAWLESS. Finally, though REFINED was tested on mouse data in this report, the methods can in principle be used to increase the robustness of fat quantification in clinical data.