(Mathematical) Necessary conditions for the selection of gradient vectors in DTI

A set of necessary conditions for the choice of diffusion gradient vectors to make the linear equations nonsingular for the estimation of the diffusion matrix are given in a coordinate free manner. The conditions assert that the initial step in the design of a DTI experiment with six or more acquisitions must be to select six valid diffusion gradients first and then add new ones.     

Understanding changes in DTI metrics in patients with different stages of neurocysticercosis

Diffusion tensor imaging (DTI) was performed on 25 patients with neurocysticercosis (NCC). The aim of this study was to investigate the changes in DTI measures during the evolutionary course of NCC lesions from vesicular to calcified stage in the brain. DTI measures were quantified from the NCC lesions of all patients. On the basis of conventional imaging findings, NCC lesions were classified into vesicular, vesicular colloidal, granular nodular and calcified stages. Significant inverse correlation was observed between the evolutionary stage of NCC lesion and mean diffusivity (MD; r=−0.748, P<0.001) and spherical anisotropy (CS; r=−0.585, P<.001) values. Significant direct correlations were observed between evolutionary stages of NCC lesion and mean fractional anisotropy (FA; r=0.575, P<0.001), linear anisotropy (CL; r=0.478, p<0.001) and planar anisotropy (CP; r=0.561, p<0.001) values. Successive decrease in MD values calculated from NCC lesions was observed, moving from vesicular to granular nodular stage. On FA, CL and CP maps, a significant increase in signal intensity value was observed in calcified as compared to other stages. We conclude that DTI measures may indicate the evolutionary changes in NCC from vesicular to calcified stage.     

Manifestation and post hoc correction of gradient cross-term artifacts in DTI

Cross-terms between imaging and diffusion gradients, unaccounted for during tensor calculations, can lead to erroneous estimation of diffusivity and fractional anisotropy (FA) in regions of isotropic and anisotropic diffusion. Cross-term of magnitude 136.8±1.6 s/mm(2), artificially introduced in the slice-encode direction, caused an increase in FA in isotropic phantom from 0.0546±0.0001 to 0.0996±0.0001, while the change in chimpanzee brain depended on the orientation of the white matter (WM). Mean diffusivity (MD) remained unchanged in isotropic phantom, but increased by ～20% in the WM due to cross-terms. A bias was observed in the principal eigenvectors in both phantom and chimpanzee brain, resulting in significant increase in midline crossing fibers along the bias than perpendicular to it in tractography in chimpanzee brain. Post hoc correction of these artifacts was achieved by estimating the cross-term factors using calibration scans on an isotropic phantom and modifying the b-matrix before tensor calculation. Upon correction, the FA and MD values closely resembled the values obtained from sequence without cross-terms, and the bias in principal eigenvectors was eliminated. Customized sequences involving large b-values, high-resolution imaging, or long diffusion or echo times should therefore be evaluated and any residual cross-terms corrected before implementation.     

Optimal estimation of the diffusion coefficient from non-averaged and averaged noisy magnitude data

The magnitude operation changes the signal distribution in MRI images from Gaussian to Rician. This introduces a bias that must be taken into account when estimating the apparent diffusion coefficient. Several estimators are known in the literature. In the present paper, two novel schemes are proposed. Both are based on simple least squares fitting of the measured signal, either to the median (MD) or to the maximum probability (MP) value of the Probability Density Function (PDF). Fitting to the mean (MN) or a high signal-to-noise ratio approximation to the mean (HS) is also possible. Special attention is paid to the case of averaged magnitude images. The PDF, which cannot be expressed in closed form, is analyzed numerically. A scheme for performing maximum likelihood (ML) estimation from averaged magnitude images is proposed. The performance of several estimators is evaluated by Monte Carlo (MC) simulations. We focus on typical clinical situations, where the number of acquisitions is limited. For non-averaged data the optimal choice is found to be MP or HS, whereas uncorrected schemes and the power image (PI) method should be avoided. For averaged data MD and ML perform equally well, whereas uncorrected schemes and HS are inadequate. MD provides easier implementation and higher computational efficiency than ML. Unbiased estimation of the diffusion coefficient allows high resolution diffusion tensor imaging (DTI) and may therefore help solving the problem of crossing fibers encountered in white matter tractography.     

Geometric analysis of the b-dependent effects of Rician signal noise on diffusion tensor imaging estimates and determining an optimal b value

The optimal diffusion weighting (DW) factor, b, for use in diffusion tensor imaging (DTI) studies remains uncertain. In this study, the geometric relations of DW quantities are examined, in particular, the effects of Rician noise in the measured magnetic resonance signal. This geometric analysis is used to make theoretical predictions for selecting a b value to reduce the influence of noise. It is shown that the optimal b value for DTI studies in healthy human parenchyma is approximately b=1200 s mm⁻², with a simple relation given as well for a given expected apparent diffusion coefficient. Monte-Carlo simulations on sets of realistic DTI measures are then performed, verifying the optimal DW for minimizing estimate errors. The effects of noise on various DTI parameters such as anisotropy indices (fractional anisotropy and scaled relative anisotropy), mean diffusivity, radial diffusivity, eigenvalues and the direction of the first eigenvector are investigated as well.     

Brain tissue classification based on DTI using an improved Fuzzy C-means algorithm with spatial constraints

We present an effective method for brain tissue classification based on diffusion tensor imaging (DTI) data. The method accounts for two main DTI segmentation obstacles: random noise and magnetic field inhomogeneities. In the proposed method, DTI parametric maps were used to resolve intensity inhomogeneities of brain tissue segmentation because they could provide complementary information for tissues and define accurate tissue maps. An improved fuzzy c-means with spatial constraints proposal was used to enhance the noise and artifact robustness of DTI segmentation. Fuzzy c-means clustering with spatial constraints (FCM_S) could effectively segment images corrupted by noise, outliers, and other imaging artifacts. Its effectiveness contributes not only to the introduction of fuzziness for belongingness of each pixel but also to the exploitation of spatial contextual information. We proposed an improved FCM_S applied on DTI parametric maps, which explores the mean and covariance of the feature spatial information for automated segmentation of DTI. The experiments on synthetic images and real-world datasets showed that our proposed algorithms, especially with new spatial constraints, were more effective.     

Optimal real-time estimation in diffusion tensor imaging

Diffusion tensor imaging (DTI) constitutes the most used paradigm among the diffusion-weighted magnetic resonance imaging (DW-MRI) techniques due to its simplicity and application potential. Recently, real-time estimation in DW-MRI has deserved special attention, with several proposals aiming at the estimation of meaningful diffusion parameters during the repetition time of the acquisition sequence. Specifically focusing on DTI, the underlying model of the noise present in the acquired data is not taken into account, leading to a suboptimal estimation of the diffusion tensor. In this paper, we propose an optimal real-time estimation framework for DTI reconstruction in single-coil acquisitions. By including an online estimation of the time-changing noise variance associated to the acquisition process, the proposed method achieves the sequential best linear unbiased estimator. Results on both synthetic and real data show that our method outperforms those so far proposed, reaching the best performance of the existing proposals by processing a substantially lower number of diffusion images.     

Noise estimation in parallel MRI: GRAPPA and SENSE

Parallel imaging methods allow to increase the acquisition rate via subsampled acquisitions of the k-space. SENSE and GRAPPA are the most popular reconstruction methods proposed in order to suppress the artifacts created by this subsampling. The reconstruction process carried out by both methods yields to a variance of noise value which is dependent on the position within the final image. Hence, the traditional noise estimation methods – based on a single noise level for the whole image – fail. In this paper we propose a novel methodology to estimate the spatial dependent pattern of the variance of noise in SENSE and GRAPPA reconstructed images. In both cases, some additional information must be known beforehand: the sensitivity maps of each receiver coil in the SENSE case and the reconstruction coefficients for GRAPPA.     

Effects of random subject rotation on optimised diffusion gradient sampling schemes in diffusion tensor MRI

The choice of the number (N) and orientation of diffusion sampling gradients required to measure accurately the water diffusion tensor remains contentious. Monte Carlo studies have suggested that between 20 and 30 uniformly distributed sampling orientations are required to provide robust estimates of water diffusions parameters. These simulations have not, however, taken into account what effect random subject motion, specifically rotation, might have on optimised gradient schemes, a problem which is especially relevant to clinical diffusion tensor MRI (DT-MRI). Here this question is investigated using Monte Carlo simulations of icosahedral sampling schemes and in vivo data. These polyhedra-based schemes, which have the advantage that large N can be created from optimised subsets of smaller N, appear to be ideal for the study of restless subjects since if scanning needs to be prematurely terminated it should be possible to identify a subset of images that have been acquired with a near optimised sampling scheme. The simulations and in vivo data show that as N increases, the rotational variance of fractional anisotropy (FA) estimates becomes progressively less dependent on the magnitude of subject rotation (), while higher FA values are progressively underestimated as increases. These data indicate that for large subject rotations the B-matrix should be recalculated to provide accurate diffusion anisotropy information.     

A combined DTI and structural MRI study in medicated-naïve chronic schizophrenia

Disconnection in white matter (WM) pathway and alterations in gray matter (GM) structure have been hypothesized as pathogenesis in schizophrenia. However, the relationship between the abnormal WM integrity and the alteration of GM in anatomically connected areas remains uncertain. Moreover, the potential influence of antipsychotic medication on WM anisotropy and cortical morphology was not excluded in previous studies. In this study, a total number of 34 subjects were enrolled, including 17 medicated-naïve chronic schizophrenia patients and 17 healthy controls. Tract-based spatial statistics (TBSS) were applied to investigate the level of WM integrity. The FreeSurfer surface-based analysis was used to determine GM volume, cortical thickness and the surface area of GM regions which corresponded to abnormal WM fiber tracts. We observed that patients possessed lower fractional anisotropy (FA) values in the left inferior fronto-occipital fasciculus (IFOF) and left inferior longitudinal fasciculus (ILF), along with smaller GM volume and cortical thinning in temporal lobe than the healthy controls, which reflected the underlying WM and GM disruption that contributed to the disease. In the patient population, the lower connectivity of ILF and IFOF was positively associated with cortical thickness in left lateral orbitofrontal cortex, superior temporal gyrus and lingual gyrus in males, and positively correlated with GM volume in left lateral orbitofrontal cortex in females. On the other hand, it was negatively correlated with cortical area of middle temporal gyrus in males and temporal pole in females respectively, but not when genders were combined. These findings suggested that abnormal WM integrity and anatomical correspondence of GM alterations in schizophrenia were interdependent on gender-separated analysis in patients with schizophrenia. Moreover, combining TBSS and FreeSurfer might be a useful method to provide significant insight into interacting processes related to WM fiber tracts and GM changes in schizophrenia.     

Exploiting dynamical coherence: A geometric approach to parameter estimation in nonlinear models

Parameter estimation in nonlinear models is a common task, and one for which there is no general solution at present. In the case of linear models, the distribution of forecast errors provides a reliable guide to parameter estimation, but in nonlinear models the facts that predictability may vary with location in state space, and that the distribution of forecast errors is expected not to be Normal, means that parameter estimation based on least squares methods will result in systematic errors. A new approach to parameter estimation is presented which focuses on the geometry of trajectories of the model rather than the distribution of distances between model forecast and the observation at a given lead time. Specifically, we test a number of candidate trajectories to determine the duration for which they can shadow the observations, rather than evaluating a forecast error statistic at any specific lead time(s). This yields insights into both the parameters of the dynamical model and those of the observational noise model. The advances reported here are made possible by extracting more information from the dynamical equations, and thus improving the balance between information gleaned from the structural form of the equations and that from the observations. The technique is illustrated for both flows and maps, applied in 2-, 3-, and 8-dimensional dynamical systems, and shown to be effective in a case of incomplete observation where some components of the state are not observed at all. While the demonstration of effectiveness is strong, there remain fundamental challenges in the problem of estimating model parameters when the system that generated the observations is not a member of the model class. Parameter estimation appears ill defined in this case.     

Moderating registration misalignment in voxelwise comparisons of DTI data: a performance evaluation of skeleton projection

An evaluative methodology and five accompanying performance measures were developed to quantitatively assess the performance of the skeleton projection algorithm constituting the heart of tract-based spatial statistics (TBSS). The performance measures were designed to quantify the accuracy of skeleton projection in its indented task of alleviating any residual misalignment that may remain after image registration. A ground truth fractional anisotropy (FA) image was slightly warped using a realistic warp field that served to model post-registration residual misalignment of varying magnitudes. Skeleton projection was then used to register the warped FA image to the ground truth. Performing skeleton projection was found to yield up to 50% better correspondence between the values of FA compared to smoothing, despite the fact that less than 10% of post-registration misalignment was corrected. The align-max-with-max strategy underlying TBSS was posited as a potential explanation for this high correspondence in the values of FA, at the expense of lesser alignment between anatomically concordant voxels.     

The combined effect of nonlinear filtration and window size on the accuracy of tissue displacement estimation using detected echo signals

In cardiac elastography, the regional strain and strain rate imaging is based on displacement estimation of tissue sections within the heart muscle carried out with various block-matching techniques (cross-correlation, sum of absolute differences, sum of squared differences, etc.). The accuracy of these techniques depends on a combination of ultrasonic imaging parameters such as ultrasonic frequency of interrogation, signal-to-noise ratio, size of a kernel used in a block-matching algorithm, type of data and speckle decorrelation. In this paper, we discuss the possibility to enhance the accuracy of the displacement estimation via nonlinear filtering of B-mode images before block-matching operation. The combined effect of a filter algorithm and a kernel size on the accuracy of the displacement estimation is analyzed using a 36-frame sequence of grayscale B-mode images of a human heart acquired by an ultrasound system operating at 1.77 MHz. It is shown that the nonlinear filtering of images enables to obtain the desired accuracy (less than one pixel) of the displacement estimation with smaller kernels than without filtering. These results are obtained for two filters--an adaptive anisotropic diffusion filter and a nonlinear Gaussian filter chain.     

Euclidian distance-weighted smoothing for quantitative MRI: application to intervoxel anisotropy index mapping with DTI

During the computation of intervoxel anisotropy features, the inclusion of both eigenvalues and eigenvectors reduces the effect of noise, but spatial averaging blurs the resulting maps. We propose a new adaptive technique that uses data-dependent weights in the averaging process so that the influence of each neighbor in the local window is proportional to the similarity of characteristics of the neighbor considered to those of the reference central voxel. This likeness criterion is based on the multidimensional Euclidian distance using the entire available multispectral information contained in the diffusion-weighted images. This solution is controlled by a single parameter beta that results from a compromise between edge-preserving and noise-smoothing abilities. This Euclidian distance-weighted technique is a generic solution for filtering noise during parametric reconstruction. It was applied to map anisotropy using an intervoxel lattice index (LI) from experimental images of mouse brain in vivo and achieves noise reduction without distorting small anatomical structures. We also show how to employ in the discrimination scheme the images not used in the estimation of the considered feature.     

Alteration of diffusion tensor parameters in postmortem brain

In autopsy of humans, there is usually an interval of hours to days between death and tissue fixation, during which the cadaver is stored below room temperature to retard tissue autolysis. We have attempted to model this process and evaluate the alteration in diffusion indices of the postmortem brain in pigs, which were kept at 4°C. The pigs were scanned prior to death and at 3, 6, 9, 12, 18, 24, 30, 36, 42, 48 and 72 h postmortem. Regions of interest were placed in the corpus callosum, internal capsule, periventricular and subcortical white matter anteriorly and posteriorly. There was a slight increase in fractional anisotropy (FA) in the first 3 h postmortem. The FA remained stable up to 72 h postmortem. There was a marked decrease in trace, eigenmajor (λmajor), eigenmedium (λmedium) and eigenminor (λminor), particularly in the first 3 h following death. This study supports the utility of measuring diffusion anisotropy if the time elapsed between death and tissue fixation is within 3 days. However, trace and eigenvalues decreased markedly within the first few hours postmortem. Therefore trace and eigenvalues obtained from ex vivo studies cannot be extrapolated to in vivo studies.     

Analyzing diffusion tensor images with ghosting artifacts: the effects of direct and indirect normalization

The current study aims to assess the applicability of direct or indirect normalization for the analysis of fractional anisotropy (FA) maps in the context of diffusion-weighted images (DWIs) contaminated by ghosting artifacts. We found that FA maps acquired by direct normalization showed generally higher anisotropy than indirect normalization, and the disparities were aggravated by the presence of ghosting artifacts in DWIs. The voxel-wise statistical comparisons demonstrated that indirect normalization reduced the influence of artifacts and enhanced the sensitivity of detecting anisotropy differences between groups. This suggested that images contaminated with ghosting artifacts can be sensibly analyzed using indirect normalization.     

An empirical characterization of the quality of DTI data and the efficacy of dyadic sorting

Metrics calculated from images acquired using the diffusion tensor imaging (DTI) technique possess a systematic bias that depends on signal-to-noise ratio (SNR). Dyadic sorting provides a simple method for remediating some of this bias within a region(s) of interest (ROI). Although this bias and its removal using dyadic sorting have been studied previously within a theoretical framework, one can employ precise geometric knowledge of microstructures to perform an empirical comparison between expected DTI results and those measured with a scanner. In this project, the biasing effect of low SNR (approximately 1-10) on DTI eigenvalues was measured directly using water-filled capillary structures of two different sizes, and the magnitude of the corrective effect of dyadically sorting eigenvector-eigenvalue pairs was characterized. Multiple DTI series were acquired for determining DTI metrics at eight unique SNR values, using T(R) to vary signal intensity via T(1) contrast. Differences between the second and third eigenvalues, which should be equal for prolate geometry, ranged from approximately 23% to 45% and from 19% to 41% for large and small inner diameter capillaries after sorting eigenvalues by magnitude, and ranged from approximately 1% to 18% and from 1% to 4% after dyadic sorting. A high-resolution DTI series was used to observe the effect of ROI size on dyadic sorting. For restriction of diffusion on the scale of the small capillary at SNR approximately 18, an ROI with > or =50 pixels is adequate to determine fractional anisotropy to 99% accuracy, while larger ROI are required to resolve the two smaller eigenvalues to the same accuracy ( approximately 330-390 pixels). At low values of SNR, the iteration of dyadic sorting is suggested to achieve good accuracy. A method for the incorporation of empirical measurements into a bias-correction map, which would be useful for characterizing uncertainty and for reducing systematic bias in DTI data, is introduced.     

Elevations of diffusion anisotropy are associated with hyper-acute stroke: a serial imaging study

Diffusion tensor imaging (DTI) studies of human ischemic stroke within 24 h of symptom onset have reported variable findings of changes in diffusion anisotropy. Serial DTI within 24 h may clarify these heterogeneous results. We characterized longitudinal changes of diffusion anisotropy by analyzing discrete ischemic white matter (WM) and gray matter (GM) regions during the hyperacute (2.5-7 h) and acute (21.5-29 h) scanning phases of ischemic stroke onset in 13 patients. Mean diffusivity (MD), fractional anisotropy (FA) and T2-weighted signal intensity were measured for deep and subcortical WM and deep and cortical GM areas in lesions outlined by a > or =30% decrease in MD. Average reductions of approximately 40% in relative (r) MD were observed in all four brain regions during both the hyperacute and acute phases post stroke. Overall, 9 of 13 patients within 7 h post symptom onset showed elevated FA in at least one of the four tissues, and within the same cohort, 11 of 13 patients showed reduced FA in at least one of the ischemic WM and GM regions at 21.5-29 h after stroke. The fractional anisotropy in the lesion relative to the contralateral side (rFA, mean+/-S.D.) was significantly elevated in some patients in the deep WM (1.10+/-0.11, n=4), subcortical WM (1.13+/-0.14, n=4), deep GM (1.07+/-0.06, n=1) and cortical GM (1.22+/-0.13, n=5) hyperacutely (< or =7 h); however, reductions of rFA at approximately 24 h post stroke were more consistent (rFA= 0.85+/-0.12).     

Effect of B-value in revealing postinfarct myocardial microstructural remodeling using MR diffusion tensor imaging

Nonmonoexponential diffusion behavior has been previously reported to exist in some biological tissues, making quantification of diffusion tensor imaging (DTI) indices dependent on diffusion sensitivity of b-value. This study aims to investigate the effect of b-value in revealing postinfarct myocardial microstructural remodeling in ex vivo hearts. DTI scans were performed on heart samples 1, 3, 5, and 7 days after infarction induction as well as intact controls with b-values of 500 to 2500 s/mm². DTI indices, including fractional anisotropy (FA), and mean and directional diffusivities, were measured in infarct, adjacent and remote regions with zero and each non-zero b-values respectively using conventional DTI analysis. Experimental results showed that these DTI indices decreased gradually with b-values in all regions and groups. Optimal b-values were found to vary with targeted DTI indices, and could strengthen DTI ability in revealing myocardium degradation with using conventional DTI approach. Specifically, FA showed the most sensitive detection of fiber integrity degradation at moderate b-values (≈ 1500 to 2000 s/mm²), and the greatest ability of mean and directional diffusivities in monitoring diffusivity alteration occurred at relatively small b-values (≤ 1500 s/mm²) during the necrotic and fibrotic phases. These findings may provide useful information for DTI protocol parameter optimization in assessing heart microstructures at other pathological or in vivo states in the future.     

Diffusion tensor imaging in horizontal gaze palsywith progressive scoliosis

Horizontal gaze palsy with progressive scoliosis (HGPPS) is a rare disorder characterized by absence of conjugate horizontal eye movements, preservation of vertical gaze and convergence, progressive scoliosis developing in childhood and adolescence. It is caused by mutations in the ROBO3 gene which are critical for the crossing of long ascending medial lemniscal and descending corticospinal tracts in the medulla. Diffusion tensor imaging on a 14-year-old boy with HGPPS revealed ipsilateral ascending and descending connectivity in the brainstem without any crossing over of the major tracts although normal interhemispheric connections in the corpus callosum was demonstrable. Absent decussation of smaller sized superior cerebellar peduncles but with normal crossing over of the middle cerebellar peduncle was also observed. Tractography is a valuable investigative modality to assess neuronal connections in the brain and is a useful adjunct to the structural magnetic resonance imaging in confirming the diagnosis of HGPPS.