Impact of Outliers on Diffusion Tensor and Q-Ball Imaging: Clinical Implications and Correction Strategies

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1 CME JOURNAL OF MAGNETIC RESONANCE IMAGING 33: (2011) Original Research Impact of Outliers on Diffusion Tensor and Q-Ball Imaging: Clinical Implications and Correction Strategies Michael A. Sharman, MS, 1,2 * Julien Cohen-Adad, PhD, 3,4 Maxime Descoteaux, PhD, 5 Arnaud Messé, MS, 4,6 Habib Benali, PhD, 4,6,7 and Stéphane Lehericy, MD, PhD 1,2 Purpose: To measure the impact of corrupted images often found to occur in diffusion-weighted magnetic resonance imaging (DW-MRI). To propose a robust method for the correction of outliers, applicable to diffusion tensor imaging (DTI) and q-ball imaging (QBI). Materials and Methods: Monte Carlo simulations were carried out to measure the impact of outliers on DTI and QBI reconstruction in a single voxel. Methods to correct outliers based on q-space interpolation and direction removal were then implemented and validated in real image data. Results: Corruption in a single voxel led to clear variations in DTI and QBI metrics. In real data, the method of q-space interpolation was successful in identifying corrupted voxels and restoring them to values consistent with those of uncorrupted images. Conclusion: For images containing few gradient directions, where outlier removal was either impossible due to limited volumes or resulted in large changes in DTI/QBI metrics, q-space interpolation proved to be the method of choice for image restoration. A simple decision support 1 UMR-S975, CRICM-INSERM-UPMC, Paris, Île-de-France, France. 2 Centre for Neuroimaging Research (CENIR), Hôpital Pitié-Salpêtrière, Paris France. 3 Athinoula A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital, Harvard Medical School, Charlestown, Massachusetts, USA. 4 UMR-S678, INSERM-UPMC, Paris, France. 5 Computer Science Department, Université de Sherbrooke, Québec, Canada. 6 IFR-49, Paris, France. 7 Unité de Neuroimagerie Fonctionnelle, CRIUGM, Université de Montréal, Canada. Contract grant sponsor: European Union (EU) Framework Project 6; Contract grant sponsor: GENEPARK: Genomic Biomarkers for Parkinson s Disease, Action Line: LIFESCIHEALTH Life sciences, genomics and biotechnology for health; Contract grant number: LSH Development of innovative methods for diagnosis of nervous system disorders. *Address reprint requests to: M.A.S., Centre for Neuroimaging Research (CENIR), Bâtiment Paul Castaigne, Niveau-1, Pitié-Salpêtrière Hospital, boulevard de l Hôpital, Paris Cedex michael.sharman@etu.upmc.fr Received August 9, 2010; Accepted February 24, DOI /jmri View this article online at wileyonlinelibrary.com. system is proposed to assist clinicians in the correction of their corrupted DW data. Key Words: diffusion; tensor; q-ball; imaging; magnetic resonance J. Magn. Reson. Imaging 2011;33: VC 2011 Wiley-Liss, Inc. DIFFUSION-WEIGHTED MAGNETIC RESONANCE IMAGING (DW-MRI) is a useful tool for noninvasive mapping of white matter architecture. Corrupted images are often found to occur within volumes of acquired data, particularly when using ultrafast sequences such as echo planar imaging (EPI) (1). Among these corruptions are those characterized by large, unpredictable signal variations that may originate from patient motion (2), physiological-related fluctuations (3,4), spiking artifact (5), and magnetic gradients causing patient table vibrations (6,7). The most common method of processing DW-MRI data is diffusion tensor imaging (DTI), wherein acquired signal values are fitted to a diffusion tensor model (8). A variety of clinically relevant metrics characterizing the anisotropy of water diffusion at the individual voxel level may be determined from DTI, including fractional anisotropy (FA), mean diffusivity (MD), and the eigenvectors and eigenvalues associated with the tensor, especially the principal eigenvector (PEV). These measures are used to infer pathological processes affecting white matter (9). Corrupted slices within acquired DW-MRI data impact the estimation of these metrics and may subsequently bias group studies, particularly in cases where only subtle differences are expected (10 12). Other groups have previously addressed the issue of data corruption in DW-MRI. Detection methods include identifying outliers following diffusion tensor reconstruction (4,13), or finding local maxima on the Laplacian of DW signals across diffusion gradient samples (q-space) (14). Correction methods include discarding corrupted voxels (4,15) and fitting the corrupted signal using linear regression methods (6). Despite these investigations, no study has determined VC 2011 Wiley-Liss, Inc. 1491

2 1492 Sharman et al. outlier impact in advanced reconstruction methods, like q-ball imaging (QBI) (16). In contrast to DTI, QBI makes no prior model assumptions regarding molecular diffusion in a voxel. Rather, q-space is sampled with high angular resolution diffusion imaging (HARDI) (16). The Funk-Radon Transform (FRT) is then used to reconstruct diffusion orientation distribution function (ODF), representing the angular distribution of the spin displacement probability density function, and hence the voxel s underlying fiber orientations (16). Recent advances in QBI have led to fast and robust ODF reconstruction through the incorporation of spherical harmonics (SH), which provide a basis for complex functions on the sphere (17). Following the reconstruction of the ODF it is possible to calculate the generalized fractional anisotropy (gfa), a metric with similar implications to the FA in DTI as far as quantifying the anisotropic nature of the ODF (16). It is also possible to derive a set of principal diffusion vectors (PDVs), an analog of PEV in DTI, generalized to the number of diffusion directions revealed by the diffusion ODF (16). Diffusion ODF reconstruction in QBI has been shown to be successful in modeling the structure of complex fiber bundles, including crossing fiber configurations (18,19). The purpose of this study was to measure the impact of outliers arising within DW-MRI and to propose a robust method to correct for them, considering both DTI and QBI reconstructions. Monte Carlo simulations were first performed to evaluate the fundamental impact of corrupted data on diffusion in a single voxel for both single-fiber and crossing-fiber scenarios. Second, strategies to correct corrupted data in real DW-MRI images were implemented and validated. The validation consisted of comparing the complete removal of corrupted volumes with the interpolation of those volumes in q-space for datasets with differing numbers of gradient directions. The method of q-space interpolation involved three distinct processing steps: data filtering, outlier acceptance/rejection, and correction via interpolation, similar to that presented previously (14). The overall focus was to address the impact of outliers in standard clinical analysis, which thus far remains unaddressed in previous work. From the results obtained in this study a decision support system for processing DW-MRI data was proposed to assist clinicians in the treatment of corruption-affected DW-MRI data. MATERIALS AND METHODS Simulations Model Datasets Four datasets were prepared for the simulation part of the study, with different diffusion sampling schemes of 12, 16, 32, and 64 gradient directions. Orientations were generated using an electrostatic repulsion algorithm (20) and diffusion parameters were based on the literature. The trace of the diffusion tensor D was set to Tr(D) ¼ mm 2 /s (20), as the sum of the eigenvalues l 1 ¼ mm 2 /s, l 2 ¼ mm 2 /s, and l 3 ¼ mm 2 /s. This value was verified in real data by performing region of interest (ROI) sampling of the same eigenvalues in brain areas of high anisotropy (eg, pyramidal tract). These eigenvalues were used to determine a starting FA value of 0.64 (21). Signal attenuation was then derived using the Stejskal-Tanner equation (22) modified as shown in Eqs. [1] and [2] for a single voxel: ADC ¼ 1 b ln I k ¼ gk T I Dg k ½1Š D xx D xy D xz gk T Dg k ¼ðg k1 g k2 g k3 Þ4 D yx D yy D yz 5 g 3 k1 4 g k2 5 ½2Š D zx D zy D zz g k3 The b-value (b k ) in Eq. [1] was set to 1000 s/mm 2 (4,23,24), the mean apparent diffusion coefficient (ADC) to mm 2 /s, and nondiffusionweighted signal intensity (I 0 ) to 1000 (24). Diffusion signal values for each model dataset were simulated such as to create scenarios for single and crossing fibers, mimicking varying brain anisotropy. For single fibers, diffusion was assumed oriented in the z-direction, and a single set of signal values I k used to derive the diffusion tensor. For crossing fibers, two sets of signal values were derived (one set (I k1 ) oriented in x and the other (I k2 )inz direction) and combined in a pairwise sum such that I k ¼ 0.5I k1 þ0.5i k2. Beginning with the starting value for FA, the diffusion tensor D (D xx, D yy, D zz, D xy, D yz, D xz ) was computed using a least squares fitting method and used to derive a set of signal values (I k ) for gradient directions (g k ) in each model dataset (k ¼ 12, 16, 32, and 64). Rician noise was then added to these values for signal-to-noise ratios (SNRs) of 10:1, 20:1, and 100:1 (24). To ensure a realistic range of SNR, these values were verified against real data. Since noise was assumed to follow a Rayleigh distribution, the SNR was verified by dividing the standard deviation of mean signal s in a selected brain region by the standard deviation s m of a region in background. Monte Carlo Method Signal corruption was carried out with Monte Carlo simulations on four datasets (12, 16, 32, and 64 directions) for DTI and two (32 and 64 directions) for QBI, while assessing the effects of several parameters on FA (DTI), gfa (QBI), and angular confidence (AC) (both DTI and QBI). Parameters varied as part of the simulations were: 1) number of corrupted directions; 2) corruption severity; and 3) amount of SNR. To evaluate the impact of parameter 1, 10,000 iterations were performed (23,24) wherein for each random combination of directions were corrupted within each dataset. This began with a single direction and progressively increased up to half of the total number of directions in the dataset (12, 16, 32, or 64). Parameter 2 was investigated at each iteration by multiplying the signal (I k ) in each gradient direction by I k *0.1 (high corruption), I k *0.5 (medium corruption), and I k *0.8 (low

3 Impact of Outliers on DTI and QBI 1493 corruption), similar to (4). Parameter 3 was analyzed by setting SNR equal to 10:1, 20:1, and 100:1. Estimation of DTI and Its Metrics Following corruption, the diffusion tensor D was reconstructed with a rearranged version of Eq. [2] with I k representing the matrix of corrupted signal values at gradient directions (g k ). The tensor was then diagonalized to determine its eigenvalues (l) and revised FA. Revised FA values for each iteration were compared with precorrupted values to find FA standard deviation (s) and mean absolute change in FA error (Err-FA) using Eqs. [3] and [4] (25), where N is the number of iterations: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P N i¼1 FA i 1 P u 2 N t N i¼1 FA i sðfaþ ¼ N 1 ErrFA ¼ 1 X N N FA i¼1 i FA original For single fiber simulation cases, the 95% AC interval was calculated, measuring the extent to which the direction of tensor PEV changed. For each iteration, tensor amplitude was mapped on a discrete sphere (724 samples), from which the first maximum was extracted (ie, tensor PEV, and denoted by y). Mean PEV was then computed by averaging all of the y from the iterations (y i ), and the angular error (Erry) was computed from the dot product of y i and y original as shown in Eq. [5]: Errh ¼ 1 N X N 180 i¼1 cos 1 h i h original p All Erry values were subsequently ordered to calculate AC, the angle corresponding to the threshold of the 95% confidence interval for 10,000 iterations (24). Estimation of Diffusion ODF (QBI) and Its Metrics Single shell HARDI was applied to reconstruct the ODF (26), where the sphere (or shell) of q-space corresponded to a b-value of 1000 s/mm 2. The HARDI signal is expressed as a spherical harmonics (SH) series of order L and the FRT solved using the Funk Hecke theorem. ODF reconstruction C in direction y, c is summarized in Eq. [6], where Y is the SH of order k and degree m, c are SH coefficients for the HARDI signal, and P k is an order k Legendre polynomial: Cðh; uþ ¼ XL X k k¼0 m¼ k 2pP k ð0þck m Y k m ðh; uþ ½6Š A regularization parameter of and maximum estimation order L ¼ 6 were consistently applied to estimate the ODF (27). The generalized FA (gfa) is defined as the standard deviation of the ODF divided by its root mean square, shown in Eqs. [7] and [8] (16). An independent quantitative measure, gfa ranges between 0 and 1. ½3Š ½4Š ½5Š sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n P vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n i¼1 ð gfa ¼ Cðu iþ hciþ 2 u ðn 1Þ P ðc0 0 n i¼1 Cðu iþ 2 ¼ t1 P Þ2 L P k k¼0 m¼ k ðcm k Þ2 hci¼ 1 Xn n Cðu i¼1 iþ Here, u i is the direction of interest at iteration i. As with the FA in DTI, gfa values were recalculated following each iteration i and compared with their original corresponding gfa values to determine changes as a result of applying each parameter, ie, Err-gFA (refer to Eqs. [3] and [4], substituting gfa for FA). Principal diffusion in the ODF may correspond to multiple directions u *, referred to as principal diffusion vectors (PDVs), determined from the ODF using Eq. [9] (16). u ¼ arg max cðuþ A similar method of analysis as that described for PEV was used to determine the angular error (Erry) and AC of the sets of PDVs for single fiber QBI simulations. Real Data Data Acquisition and Processing MRI examination was performed on 10 healthy subjects using a 3 T system (Siemens TRIO 32 channel TIM using a 12-channel head coil) with a standard twice refocused EPI sequence. A total of 60 axial slices were obtained for each diffusion-weighted volume using the following parameters: TR/TE 12,000/86 msec, one average, flip angle 90, slice thickness 2 mm no gap, in-plane voxel size 2 mm. Diffusion weighting was along 64 directions, with five interspersed b-zero images, and b-value ¼ 1000 s/mm 2. Data were acquired within European Union (EU) Framework Project 6 GENEPARK. Correction for head motion and eddy currents was performed using FSL (FMRIB, UK). The diffusion tensor was then fitted using the method of least squares. Selection of Corrupted Images The nature of the artifact with which image correction was concerned was one of large, unpredictable losses of signal within a few slices of one or more DW volumes, as shown in Fig. 1. This artifact was found to arise naturally from the scanner, possibly as a result of unpredictable patient table vibrations. To identify corrupted images, a visual assessment of the DW-MRI acquisition was undertaken, as per standard practice in a clinical setting. Automated detection methods are also possible (5). Scans were considered corrupted where clear losses of signal were evident in one or more slices (14). Such images were put aside for corrective processing. Correction by Q-Space Interpolation Correcting corrupt images by q-space interpolation involves interpolating signal values from a corrupted ½7Š ½8Š ½9Š

4 1494 Sharman et al. Figure 1. Examples of (a) uncorrupted, axial; and (b,c) corrupted axial, coronal; DW images. Arrows denote areas of signal loss. volume s nearest good neighbors in q-space using a q-space interpolation algorithm. How well the algorithm performs depends on the proximity, quantity, and orientation of good neighbors. More neighbors will, for example, only be useful if these directions retain the integrity of the corrupt values they are trying to correct. A series of trials indicated that five neighbors produced reliable results for a range of dataset sizes, and that more than this showed little improvement. Q-space interpolation requires an overdetermined dataset and problems may arise if the ratio of corrupted to uncorrupted volumes is too high (15). In addition, corruption is sometimes associated with a specific direction or group of directions (6). As a means of demonstrating whether such factors would limit the functioning of q-space interpolation in any particular instance, a directional algorithm was devised. The directional algorithm was developed in a similar way to the Monte Carlo simulations previously described. Numerous iterations of random direction combinations were marked as corrupt in different sized (12, 16, 32, and 64 directions) model datasets, and the five nearest neighbors of these directions grouped to find corrupted directions appearing as neighbors. Requiring at least two uncorrupted neighbors to resolve any corrupted direction was set as a minimum criterion. A likelihood of functioning curve was hence produced (Fig. 7) showing the percentage chance that q-space interpolation functioned with a given number of corrupted directions. Despite previous findings regarding table vibrations, DW-MRI data acquired by our scanner were found not to be affected by any directional corruption patterns. Prior to q-space interpolation, spatial and q-space smoothing methods were applied to each voxel in the image to avoid picking outliers that were actually noise. Image slices were spatially smoothed with a 2D Gaussian filter (s ¼ 0.5, filter width ¼ 2 voxels). The use of spatial smoothing is supported by the fact that, when a voxel is hampered by an artifact, its neighbors are also hampered (15), as illustrated in Fig. 1. A q-space smoothing filter was then applied to the closest five neighbors to the direction of interest, using a weighted mean to limit the attenuation of potential outliers. The rationale for q-space smoothing is that the DW signal, for which attenuation is driven by white matter architecture, is usually smooth with low transitions (28). Following smoothing, q-space interpolation was performed. In this step the user was asked to reference prior image volumes found visually or otherwise to be clearly corrupted. These were then excluded from the interpolation process. The recorded signals of all voxels in the image were compared to their nearest remaining neighbors, using the method of robust variance (29) to identify outliers. The method of robust variance computes a t-distribution for each cluster of neighbors x i : t ¼ x i medðx i Þ P 1 N i¼1 ðx i xþ 2 N ½10Š A P-value is calculated from the cumulative distribution function (CDF) of the t-distribution, which finds the probability that a single observation from the t-distribution with n degrees of freedom will fall in the interval (-1, x], where x is a random sample from the population (30). A two-tailed test was assumed. The above process resulted in the creation of a map of image voxels requiring correction, which was then applied to the original, unsmoothed, and corrupted image to produce a new corrected image. Corrected slices were assessed visually for quality. Correction by Direction Removal Correction of corrupted data by q-space interpolation was compared with that of direction removal, a method previously investigated by several authors (23,25). This method involved removing any direction found to contain one or more corrupted slices, and then reconstructing the diffusion tensor/odf with those that remained. To quantify the extent to which removing directions changes scalar metrics, increasing numbers of direction combinations were progressively removed from four real datasets (12, 16, 32, and 64 directions) in a similar way to the Monte Carlo simulations previously described. For DTI, directions

5 Impact of Outliers on DTI and QBI 1495 were removed up to a minimum of six remaining. For QBI, directions were removed up to a minimum of 30 remaining. FA (DTI) and gfa (QBI) were reconstructed within an ROI consisting of 16 voxels in the pyramidal tract. Regional ROI means were compared before and after direction removal to determine error. Method Performance: Manually Corrupted Data To determine corrective performance, both methods were trialled on manually corrupted data in order to control the values of scalar metrics (FA, gfa). Here, a previously uncorrupted image was corrupted by multiplying signal values in four of its volumes by 0.1. Four scenarios were then compared via a series of t- tests: 1) uncorrupted data (original, prior to manual corruption); 2) manually corrupted data (corruption of signal I k *0.1); 3) q-space interpolation; and 4) direction removal. The resulting FA and gfa maps from each scenario were normalized into standard space using SPM8 (Wellcome Trust Centre for Neuroimaging, London, UK). The four scenarios were then compared by obtaining per-slice mean and variance values from a normalized ROI template (10). Method Performance: Real Corrupted Data Both methods were also tested on real corrupted data. Here, corruption was not simulated, but rather arose naturally from the scanner during acquisition. A single subject image with corruption in four directions was selected and its data subsampled to arrive at 12, 16, and 32 direction datasets. Uniform distribution of subsampled directions was assured by downsampling to directions resembling those generated using the electrostatic repulsion algorithm (20). The directional algorithm was used to verify that corrupted directions were not closely aligned in q-space. Following the implementation of the two methods, resulting data were compared with the corrupted scenario. Unlike the manually corrupted test, no uncorrupted data existed with which to verify the success of the correction methods. The FA and gfa maps from 10 additional (uncorrupted) subjects were therefore obtained instead and their values extracted for the same ROI. Values of their metrics were averaged and verified against the literature for similar brain regions (10 12). These were then used as uncorrupted data with which to compare data corrected by the two methods. RESULTS Simulations Impact of Number of Corrupted Directions / Corruption Severity: DTI Increasing the number of corrupted directions in all simulated datasets resulted in impacts on FA, for both single and crossing fiber cases, as shown in Fig. 2. For the single fiber case, an increase in Err-FA occurred progressively with increasing numbers of corrupted directions for larger direction sets, ie, 32, 64 directions (Fig. 2c,d), a trend that became more sharply linear as direction set size increased. For crossing fiber cases and for the smaller single fiber direction sets, ie, 12, 16 directions (Fig. 2a,b,e h), Err- FA was observed to increase slightly for the first few corrupted directions in the set. In results not reported, we observed similar trends as those shown for the single starting FA value of 0.64 for other different starting values. Standard deviation varied as corrupted directions increased, reflecting the number of possible combinations of corrupt directions, ie, certain combinations of directions were found to result in higher Err- FA and others lower Err-FA. This effect was seen to increase with the number of directions corrupted, and to be at zero when all of (or none of) the directions were corrupted. Generally, Err-FA was found to remain below a maximum value of 0.35*FA original. The effect on Err-FA of high (I k *0.1), medium (I k *0.5), and low (I k *0.8) corruption severity is presented in Fig. 2 for each direction set. More severe corruption was found to result in larger Err-FA. Standard deviation also varied with corruption severity, being higher for more severe corruption, reflecting the wider variety of possible combinations of the reconstructed tensor. Lower severity corruption resulted in a more linear trend in increasing Err-FA as the number of directions corrupted increased. Examples of tensor appearance with and without corruption are presented in Fig. 3. Variation across single fiber corruption severity was observed for AC. Similar trends to Err-FA were observed for AC, such as an increase in AC that was more pronounced for smaller (ie, 12, 16) direction sets than larger (ie, 32, 64) sets. As in the Err-FA results, increased severity of corruption resulted in larger changes in AC, with high corruption cases having the largest AC values of (12, 16 directions). Lower severity corruption also resulted in AC increasing more linearly as the number of directions corrupted increased. Impact of Number of Corrupted Directions / Corruption Severity: QBI Err-gFA increased linearly as the number of directions corrupted in both single and crossing fiber QBI cases increased, as shown in Fig. 4. As for DTI, less severe corruption resulted in smaller changes in Err-gFA, the observed curves becoming universally flatter. Standard deviation was also higher for higher corruption severity, representing the wider variety of diffusion ODFs reconstructed. The maximum error observed for mean gfa was 0.5*gFA original for the most severely corrupt 32-direction single and crossing fiber cases (Fig. 4a,c). A gradual increase in AC was found with increasing corruption severity and number of directions corrupted. The largest AC values of 60 were observed in high corruption 32- and 64-direction single fiber cases. Examples of diffusion ODF appearance for different corruption conditions are presented in Fig. 5. Impact of Acquisition Scheme: DTI/QBI Overall, the impact on Err-FA, Err-gFA, and AC of corrupting datasets containing more directions was less (per direction corrupted) than if the directions are fewer.

6 1496 Sharman et al. Figure 2. Impact on FA (DTI) of corruption severity and number of directions corrupt (SNR ¼ 20:1) in four direction sets and two different fiber cases: (a d) single fiber cases, 12, 16, 32, and 64 directions, respectively; (e h) crossing fiber cases, 12, 16, 32, and 64 directions, respectively. The maximum number of directions subjected to corruption in each set is equal to half the total number of gradient directions in the set.

7 Impact of Outliers on DTI and QBI 1497 Figure 3. Example simulated diffusion tensors for varying severity/directions. (a) 16-direction crossing fiber tensor with no directions corrupted; (b) 16-direction crossing fiber tensor with six directions corrupted at I k *0.5 (medium severity); (c) 64- direction single fiber tensor with no directions corrupted; (d) 64-direction single fiber tensor with 25 directions corrupted at I k *0.5 (medium severity). Color map presents varying isotropy. Figure 4. Impact on gfa (QBI) of corruption severity and number of directions corrupt (SNR ¼ 20:1) in two direction sets and two different fiber cases: (a,b) single fiber cases, 32 and 64 directions, respectively; (c,d) crossing fiber cases, 32 and 64 directions, respectively. The maximum number of directions subjected to corruption in each set is equal to half the total number of gradient directions in the set. Figure 5. Example simulated diffusion ODFs for varying severity/directions. (a) 64-direction single fiber ODF with no directions corrupted; (b) 64-direction single fiber ODF with 33 directions corrupted at I k *0.5 (medium severity); (c) 64-direction crossing fiber ODF with no directions corrupted; (d) 64-direction crossing fiber ODF with 33 directions corrupted at I k *0.5 (medium severity).

8 1498 Sharman et al. Figure 6. Correction of an image slice using q-space interpolation. Most voxels in the corrupted image (a) have been successfully treated by the q-space interpolation algorithm, with significant restoration in the corrected image (b) of signal loss seen on the lower left part of the corrupted image. Impact of SNR Variation: DTI/QBI Varying SNR between 10:1 and 100:1 was shown to have a clear impact on Err-FA only when reduced numbers of directions (eg, 12, 16) were used to reconstruct the tensor. This effect appeared to decline as corruption became less severe (data not shown). For larger direction sets, as well as for Err-gFA in the case of QBI, the impact of noise on metrics was less apparent. Impacts of noise on DWI measurements have previously been investigated by several authors (23,31). Real Data Q-Space Interpolation Q-space interpolation was successful in restoring corrupted voxels to values consistent with those of uncorrupted images (Fig. 6). A small minority of voxels not well treated were due to cases where insufficient good neighbors existed to make a correct q- space interpolation. To determine likelihood of q- space interpolation functioning for a particular dataset, examples of the output of the directional algorithm are shown in Fig. 7. It can be seen that q-space interpolation functions successfully for all (12, 16, 32, and 64 directions) datasets for up to four corrupt directions. After this, likelihood falls at variable rates depending on dataset size. From these results it is recommended that data with greater than half of volumes corrupted be discarded, due to the low likelihood of correction. Direction Removal Direction removal also proved to be a successful method of treating corrupted images. To quantify the effects of the method on scalar metrics (FA, gfa), directions were progressively removed from uncorrupted data and the results are presented in Fig. 8. As seen here, removing directions from a corrupted dataset eventually results in substantial increases in Err-FA and Err-gFA, the per-direction effect of which is more pronounced depending on the size of the direction set. As for Fig. 7, these data suggest that images should be discarded if too many directions (eg, half the dataset) are found to be corrupted. Method Performance The performance of both correction methods was found to be similar for all direction sets except 12- direction data, where q-space interpolation was clearly the most effective method. Normalized ROI values for each of the tested scenarios (referred to here as Corrupted, Uncorrupted, Removed, and Interpolated) are presented in Fig. 9, with the mean FA/gFA values for the entire ROI displayed for each scenario. A very similar result was found for real corrupted data (results not shown). DISCUSSION The first aim of this study was to investigate the fundamental impact of outliers on both DTI and QBI metrics. While previous authors have addressed aspects

9 Impact of Outliers on DTI and QBI 1499 Figure 7. Limits of q-space interpolation. Notably illustrated are the maximum number of corrupted directions to ensure efficient use of the q-space interpolation method, for each of four direction sets tested. of this problem, particularly regarding DTI, the present approach was unique in having examined the impacts on QBI. It also provided an exhaustive assessment of the effects of varying multiple parameters simultaneously (number of gradient directions, number of directions corrupt, corruption severity, SNR, single/crossing fiber scenarios) for both DTI and QBI reconstructions. Regardless of the total number of directions used to resolve the diffusion tensor/odf, an impact on metrics arose as more gradient directions were subject to corruption. In addition, as severity increased, larger errors were observed in FA (Fig. 2), gfa (Fig. 4), and AC. The maximum error was a change of 35% of the original FA and gfa values that appeared in several cases (Fig. 2). Since DTI is incapable of truly describing crossing fiber cases, smaller direction sets (ie, 12, 16) showed similar patterns of Err-FA in crossing fiber and single fiber cases, as did larger sets (ie, 32, 64) (Fig. 2). For tensors comprised of fewer directions, the effect of corruption on shape was more severe than those with more. For QBI, the results of single and crossing fiber ErrgFA were similar (Fig. 4). This likely reflects the fact that gfa is derived from the generalized structure of the ODF, rather than on its principal directions, as done in DTI. Noise as measured by SNR had more of an effect on smaller direction sets (ie, 12, 16) sets than larger (ie, 32, 64) sets. The AC represents a relatively new metric in the domain of QBI reconstructions, which has been demonstrated as a necessary metric for QBI-based probabilistic tractography (31,32). AC values showed that the diffusion tensor PEV and the ODF PDV may be subject to an error that biased their orientations by as much as 60 for highly corrupt 32- and 64-direction data cases. In general, AC increased in magnitude with increasing numbers of directions corrupted. The most linear increase was observed where corruption severity was lower and number of directions in the dataset higher (ie, 32, 64), while for lower numbers of directions it was seen to level off (ie, 12, 16). In this work QBI was chosen as the HARDI reconstruction model because it is a well-studied technique, now popular in several clinical studies and a standard for computing gfa. It is important, however, to point out that QBI has an intrinsic smoothness in its reconstruction, due to Bessel kernel smoothing of the Funk-Radon Transform, which is an estimation of the true ODF. Hence, impact of outliers may be different when using sharper and more noise-sensitive HARDI reconstruction models such as spherical deconvolution (34,35), diffusion orientation transform (36), the normalized q-ball (37 40), among others. We believe that if a preprocessing Rician noise denoising of the DW data (41) is used or latest constrained regularized techniques are used (42), similar conclusions as those reported here could apply. Although Tuch (16) suggested that high b-values are theoretically necessary for q-ball imaging and HARDI, due to the dramatic loss in SNR at such b-values, many research groups agree that a b-value of around 1000 is a reasonable trade-off between SNR and angular contrast in the ODF profile reconstruction (17,19,32). We are Figure 8. Impact on FA/gFA of direction removal from real data, averaged over 10 subjects. (a) FA mean/standard deviation; (b) gfa mean/standard deviation. Error bars illustrate intersubject variability.

10 1500 Sharman et al. Figure 9. Manual corruption FA and gfa values for four corrupted directions. For 12-direction FA data (a), Corrupted and both Uncorrupted and Interpolated were significantly different (P < 0.01), Removed and both Uncorrupted and Interpolated were significantly different (P < 0.01). Values for Corrupted and Removed were similar, as did those for Interpolated and Uncorrupted, confirming q-space interpolation as the correction method of choice. For 16, 32, and 64 direction FA (b), a statistically significant difference (P < 0.01) was observed between Corrupted and all of Interpolated, Removed and Uncorrupted. confident that the correction methods implemented in this work could equally be applied to high b-value scans, where SNRs are low and artifacts are common. Based on simulation findings, it should be clear that even minor corruption in DW data has the potential to affect directional parameters in a single voxel, biasing real data studies such as tractography. Corruption had a more pronounced effect on FA and gfa in real DTI and QBI datasets as the proportion of corrupted to total directions increased (Fig. 9). From real data analyses, it was furthermore clear that corruption impact on scalar metrics was modulated heavily by the directionality of corruption-affected signal values. For example, if diffusion signals parallel to the direction of orientation of the tensor PEV were chosen for corruption, this had a different impact on FA and gfa than those perpendicular. Simulations also revealed this effect to some extent, since the standard deviation of mean changes in Err-FA and Err-gFA provides an indication of the wide variation that occurs in these metrics with different direction combinations. The FA and gfa values compared here were obtained from the pyramidal tract, oriented rostrocaudally in the brain. If signal artifacts are direction dependent, as shown in previous work (6), the impact on anisotropic measurements of FA/gFA can be different depending on the type of tracts under analysis. In real data, q-space interpolation has the advantage over direction removal of maintaining uniformity of direction distribution on the sphere. Depending on the original size of the dataset in question, removal of too many directions from a particular component (x, y, or z) could eventually bias tensor estimation and hence scalar metrics. For this reason, we advocate q-space interpolation for correcting data containing lower numbers (eg, 12, 16) of gradient directions. A decision support system for clinical use with DTI and QBI reconstructions is sketched in Fig. 10. Following data acquisition for a known set of gradient directions (denoted by s), images must either be visually inspected or an automatic detection method Figure 10. Decision support tree for corrupted data. Once the number of gradient directions (s) and the number of those directions corrupted is known, a decision may be made on which method is most viable for data correction.

11 Impact of Outliers on DTI and QBI 1501 employed to determine whether corruption is evident. Volume numbers of affected directions should be noted. In accordance with results seen in Figs. 7 and 8, images with more than half their gradient directions corrupted should be discarded. Once corruption in a dataset has been identified, two correction methods are possible. Q-space interpolation will function provided sufficient uncorrupted neighbors are available to a corrupted direction, and direction removal will function provided that the number of directions required to be removed does not exceed the minimum necessary for reconstruction of the diffusion tensor (DTI) or ODF (QBI). For larger (>16 directions) datasets, direction removal is the preferred correction method, as it is the least time-intensive. Figure 8 shows the effect of direction removal on metrics, indicating that it is theoretically possible to remove directions up to the minimum number normally required to reconstruct the diffusion tensor/odf. For smaller (<16 directions) datasets, q-space interpolation is preferred. Running the directional algorithm (Fig. 7) can determine whether q-space interpolation is possible for a dataset. If so, q-space interpolation should be undertaken, after which further analysis can proceed. Interestingly, the fact that q-space interpolation and direction removal perform similarly in some cases to no correction serves as a demonstration of the (lack of) sensitivity of certain DWI metrics (particularly FA and gfa) to certain types of corruption. In conclusion, the simulations within this study found that minor corruption in DW data has the potential to affect directional parameters in a single voxel. 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