Multi-site study of additive genetic effects on fractional anisotropy of cerebral white matter: Comparing meta and megaanalytical approaches for data pooling

Combining datasets across independent studies can boost statistical power by increasing the numbers of observations and can achieve more accurate estimates of effect sizes. This is especially important for genetic studies where a large number of observations are required to obtain sufficient power to detect and replicate genetic effects. There is a need to develop and evaluate methods for joint-analytical analyses of rich datasets collected in imaging genetics studies. The ENIGMA-DTI consortium is developing and evaluating approaches for obtaining pooled estimates of heritability through meta-and mega-genetic analytical approaches, to estimate the general additive genetic contributions to the intersubject variance in fractional anisotropy (FA) measured from diffusion tensor imaging (DTI). We used the ENIGMA-DTI data harmonization protocol for uniform processing of DTI data from multiple sites. We evaluated this protocol in five family-based cohorts providing data from a total of 2248 children and adults (ages: 9-85) collected with various imaging protocols. We used the imaging genetics analysis tool, SOLAR-Eclipse, to combine twin and family data from Dutch, Australian and Mexican-American cohorts into one large "mega-family". We showed that heritability estimates may vary from one cohort to another. We used two meta-analytical (the sample-size and standard-error weighted) approaches and a mega-genetic analysis to calculate heritability estimates across-population. We performed leave-one-out analysis of the joint estimates of heritability, removing a different cohort each time to understand the estimate variability. Overall, meta- and mega-genetic analyses of heritability produced robust estimates of heritability.

White matter integrity measured by fractional anisotropy correlates poorly with actual individual fiber anisotropy

Fractional anisotropy (FA), a very widely used measure of fiber integrity based on diffusion tensor imaging (DTI), is a problematic concept as it is influenced by several quantities including the number of dominant fiber directions within each voxel, each fiber's anisotropy, and partial volume effects from neighboring gray matter. High-angular resolution diffusion imaging (HARDI) can resolve more complex diffusion geometries than standard DTI, including fibers crossing or mixing. The tensor distribution function (TDF) can be used to reconstruct multiple underlying fibers per voxel, representing the diffusion profile as a probabilistic mixture of tensors. Here we found that DTIderived mean diffusivity (MD) correlates well with actual individual fiber MD, but DTI-derived FA correlates poorly with actual individual fiber anisotropy, and may be suboptimal when used to detect disease processes that affect myelination. Analysis of the TDFs revealed that almost 40% of voxels in the white matter had more than one dominant fiber present. To more accurately assess fiber integrity in these cases, we here propose the differential diffusivity (DD), which measures the average anisotropy based on all dominant directions in each voxel.

A novel measure of fractional anisotropy based on the tensor distribution function

Fractional anisotropy (FA), a very widely used measure of fiber integrity based on diffusion tensor imaging (DTI), is a problematic concept as it is influenced by several quantities including the number of dominant fiber directions within each voxel, each fiber's anisotropy, and partial volume effects from neighboring gray matter. With High-angular resolution diffusion imaging (HARDI) and the tensor distribution function (TDF), one can reconstruct multiple underlying fibers per voxel and their individual anisotropy measures by representing the diffusion profile as a probabilistic mixture of tensors. We found that FA, when compared with TDF-derived anisotropy measures, correlates poorly with individual fiber anisotropy, and may sub-optimally detect disease processes that affect myelination. By contrast, mean diffusivity (MD) as defined in standard DTI appears to be more accurate. Overall, we argue that novel measures derived from the TDF approach may yield more sensitive and accurate information than DTI-derived measures.

Estimating and forecasting generalized fractional Long memory stochastic volatility models

In recent years fractionally differenced processes have received a great deal of attention due to its flexibility in financial applications with long memory. This paper considers a class of models generated by Gegenbauer polynomials, incorporating the long memory in stochastic volatility (SV) components in order to develop the General Long Memory SV (GLMSV) model. We examine the statistical properties of the new model, suggest using the spectral likelihood estimation for long memory processes, and investigate the finite sample properties via Monte Carlo experiments. We apply the model to three exchange rate return series. Overall, the results of the out-of-sample forecasts show the adequacy of the new GLMSV model.

Elevated left and reduced right orbitomedial prefrontal fractional anisotropy in adults with bipolar disorder revealed by tract-based spatial statistics

Context - Diffusion tensor imaging (DTI) studies in adults with bipolar disorder (BD) indicate altered white matter (WM) in the orbitomedial prefrontal cortex (OMPFC), potentially underlying abnormal prefrontal corticolimbic connectivity and mood dysregulation in BD. Objective - To use tract-based spatial statistics (TBSS) to examine WM skeleton (ie, the most compact whole-brain WM) in subjects with BD vs healthy control subjects. Design - Cross-sectional, case-control, whole-brain DTI using TBSS. Setting - University research institute. Participants - Fifty-six individuals, 31 having a DSM-IV diagnosis of BD type I (mean age, 35.9 years [age range, 24-52 years]) and 25 controls (mean age, 29.5 years [age range, 19-52 years]). Main Outcome Measures - Fractional anisotropy (FA) longitudinal and radial diffusivities in subjects with BD vs controls (covarying for age) and their relationships with clinical and demographic variables. Results - Subjects with BD vs controls had significantly greater FA (t > 3.0, P = .05 corrected) in the left uncinate fasciculus (reduced radial diffusivity distally and increased longitudinal diffusivity centrally), left optic radiation (increased longitudinal diffusivity), and right anterothalamic radiation (no significant diffusivity change). Subjects with BD vs controls had significantly reduced FA (t > 3.0, P = .05 corrected) in the right uncinate fasciculus (greater radial diffusivity). Among subjects with BD, significant negative correlations (P < .01) were found between age and FA in bilateral uncinate fasciculi and in the right anterothalamic radiation, as well as between medication load and FA in the left optic radiation. Decreased FA (P < .01) was observed in the left optic radiation and in the right anterothalamic radiation among subjects with BD taking vs those not taking mood stabilizers, as well as in the left optic radiation among depressed vs remitted subjects with BD. Subjects having BD with vs without lifetime alcohol or other drug abuse had significantly decreased FA in the left uncinate fasciculus. Conclusions - To our knowledge, this is the first study to use TBSS to examine WM in subjects with BD. Subjects with BD vs controls showed greater WM FA in the left OMPFC that diminished with age and with alcohol or other drug abuse, as well as reduced WM FA in the right OMPFC. Mood stabilizers and depressed episode reduced WM FA in left-sided sensory visual processing regions among subjects with BD. Abnormal right vs left asymmetry in FA in OMPFC WM among subjects with BD, likely reflecting increased proportions of left-sided longitudinally aligned and right-sided obliquely aligned myelinated fibers, may represent a biologic mechanism for mood dysregulation in BD.

Generalized Fractional Evolution Equation

2000 Mathematics Subject Classification: Primary 46F25, 26A33; Secondary: 46G20

A Brief Story about the Operators of the Generalized Fractional Calculus

2000 Mathematics Subject Classification: 26A33, 33C60, 44A20

Generalized Fractional Calculus, Special Functions and Integral Transforms: What is the Relation?

Виржиния С. Кирякова - В този обзор илюстрираме накратко наши приноси към обобщенията на дробното смятане (анализ) като теория на операторите за интегриране и диференциране от произволен (дробен) ред, на класическите специални функции и на интегралните трансформации от лапласов тип. Показано е, че тези три области на анализа са тясно свързани и взаимно индуцират своето възникване и по-нататъшно развитие. За конкретните твърдения, доказателства и примери, вж. Литературата.

Changes in white matter connectivity following therapy for anomia post stroke

Background. The majority of studies investigating the neural mechanisms underlying treatment-induced recovery in aphasia have focused on the cortical regions associated with language processing. However, the integrity of the white matter connecting these regions may also be crucial to understanding treatment mechanisms. Objective. This study investigated the integrity of the arcuate fasciculus (AF) and uncinate fasciculus (UF) before and after treatment for anomia in people with aphasia. Method. Eight people with aphasia received 12 treatment sessions to improve naming; alternating between phonologically-based and semantic-based tasks, with high angular resolution diffusion imaging conducted pre and post treatment. The mean generalized fractional anisotropy (GFA), a measure of fiber integrity, and number of fibers in the AF and UF were compared pre and post treatment, as well as with a group of 14 healthy older controls. Results. Pre treatment, participants with aphasia had significantly fewer fibers and lower mean GFA in the left AF compared with controls. Post treatment, mean GFA increased in the left AF to be statistically equivalent to controls. Additionally, mean GFA in the left AF pre and post treatment positively correlated with maintenance of the phonologically based treatment. No differences were found in the right AF, or the UF in either hemisphere, between participants with aphasia and controls, and no changes were observed in these tracts following treatment. Conclusions. Anomia treatments may improve the integrity of the white matter connecting cortical language regions. These preliminary results add to the understanding of the mechanisms underlying treatment outcomes in people with aphasia post stroke.

How does angular resolution affect diffusion imaging measures?

A key question in diffusion imaging is how many diffusion-weighted images suffice to provide adequate signal-to-noise ratio (SNR) for studies of fiber integrity. Motion, physiological effects, and scan duration all affect the achievable SNR in real brain images, making theoretical studies and simulations only partially useful. We therefore scanned 50 healthy adults with 105-gradient high-angular resolution diffusion imaging (HARDI) at 4T. From gradient image subsets of varying size (6 ≤ N ≤ 94) that optimized a spherical angular distribution energy, we created SNR plots (versus gradient numbers) for seven common diffusion anisotropy indices: fractional and relative anisotropy (FA, RA), mean diffusivity (MD), volume ratio (VR), geodesic anisotropy (GA), its hyperbolic tangent (tGA), and generalized fractional anisotropy (GFA). SNR, defined in a region of interest in the corpus callosum, was near-maximal with 58, 66, and 62 gradients for MD, FA, and RA, respectively, and with about 55 gradients for GA and tGA. For VR and GFA, SNR increased rapidly with more gradients. SNR was optimized when the ratio of diffusion-sensitized to non-sensitized images was 9.13 for GA and tGA, 10.57 for FA, 9.17 for RA, and 26 for MD and VR. In orientation density functions modeling the HARDI signal as a continuous mixture of tensors, the diffusion profile reconstruction accuracy rose rapidly with additional gradients. These plots may help in making trade-off decisions when designing diffusion imaging protocols.

Multicontrast connectometry: A new tool to assess cerebellum alterations in early relapsing-remitting multiple sclerosis.

BACKGROUND: Cerebellar pathology occurs in late multiple sclerosis (MS) but little is known about cerebellar changes during early disease stages. In this study, we propose a new multicontrast "connectometry" approach to assess the structural and functional integrity of cerebellar networks and connectivity in early MS. METHODS: We used diffusion spectrum and resting-state functional MRI (rs-fMRI) to establish the structural and functional cerebellar connectomes in 28 early relapsing-remitting MS patients and 16 healthy controls (HC). We performed multicontrast "connectometry" by quantifying multiple MRI parameters along the structural tracts (generalized fractional anisotropy-GFA, T1/T2 relaxation times and magnetization transfer ratio) and functional connectivity measures. Subsequently, we assessed multivariate differences in local connections and network properties between MS and HC subjects; finally, we correlated detected alterations with lesion load, disease duration, and clinical scores. RESULTS: In MS patients, a subset of structural connections showed quantitative MRI changes suggesting loss of axonal microstructure and integrity (increased T1 and decreased GFA, P < 0.05). These alterations highly correlated with motor, memory and attention in patients, but were independent of cerebellar lesion load and disease duration. Neither network organization nor rs-fMRI abnormalities were observed at this early stage. CONCLUSION: Multicontrast cerebellar connectometry revealed subtle cerebellar alterations in MS patients, which were independent of conventional disease markers and highly correlated with patient function. Future work should assess the prognostic value of the observed damage. Hum Brain Mapp 36:1609-1619, 2015. © 2014 Wiley Periodicals, Inc.

Inter-site and inter-scanner diffusion MRI data harmonization.

We propose a novel method to harmonize diffusion MRI data acquired from multiple sites and scanners, which is imperative for joint analysis of the data to significantly increase sample size and statistical power of neuroimaging studies. Our method incorporates the following main novelties: i) we take into account the scanner-dependent spatial variability of the diffusion signal in different parts of the brain; ii) our method is independent of compartmental modeling of diffusion (e.g., tensor, and intra/extra cellular compartments) and the acquired signal itself is corrected for scanner related differences; and iii) inter-subject variability as measured by the coefficient of variation is maintained at each site. We represent the signal in a basis of spherical harmonics and compute several rotation invariant spherical harmonic features to estimate a region and tissue specific linear mapping between the signal from different sites (and scanners). We validate our method on diffusion data acquired from seven different sites (including two GE, three Philips, and two Siemens scanners) on a group of age-matched healthy subjects. Since the extracted rotation invariant spherical harmonic features depend on the accuracy of the brain parcellation provided by Freesurfer, we propose a feature based refinement of the original parcellation such that it better characterizes the anatomy and provides robust linear mappings to harmonize the dMRI data. We demonstrate the efficacy of our method by statistically comparing diffusion measures such as fractional anisotropy, mean diffusivity and generalized fractional anisotropy across multiple sites before and after data harmonization. We also show results using tract-based spatial statistics before and after harmonization for independent validation of the proposed methodology. Our experimental results demonstrate that, for nearly identical acquisition protocol across sites, scanner-specific differences can be accurately removed using the proposed method.

Comparison of fractional and geodesic anisotropy in diffusion tensor images of 90 monozygotic and dizygotic twins

We used diffusion tensor magnetic resonance imaging (DTI) to reveal the extent of genetic effects on brain fiber microstructure, based on tensor-derived measures, in 22 pairs of monozygotic (MZ) twins and 23 pairs of dizygotic (DZ) twins (90 scans). After Log-Euclidean denoising to remove rank-deficient tensors, DTI volumes were fluidly registered by high-dimensional mapping of co-registered MP-RAGE scans to a geometrically-centered mean neuroanatomical template. After tensor reorientation using the strain of the 3D fluid transformation, we computed two widely used scalar measures of fiber integrity: fractional anisotropy (FA), and geodesic anisotropy (GA), which measures the geodesic distance between tensors in the symmetric positive-definite tensor manifold. Spatial maps of intraclass correlations (r) between MZ and DZ twins were compared to compute maps of Falconer's heritability statistics, i.e. the proportion of population variance explainable by genetic differences among individuals. Cumulative distribution plots (CDF) of effect sizes showed that the manifold measure, GA, comparably the Euclidean measure, FA, in detecting genetic correlations. While maps were relatively noisy, the CDFs showed promise for detecting genetic influences on brain fiber integrity as the current sample expands.

Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain

Generalized fractional partial differential equations have now found wide application for describing important physical phenomena, such as subdiffusive and superdiffusive processes. However, studies of generalized multi-term time and space fractional partial differential equations are still under development. In this paper, the multi-term time-space Caputo-Riesz fractional advection diffusion equations (MT-TSCR-FADE) with Dirichlet nonhomogeneous boundary conditions are considered. The multi-term time-fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0, 1], [1, 2] and [0, 2], respectively. These are called respectively the multi-term time-fractional diffusion terms, the multi-term time-fractional wave terms and the multi-term time-fractional mixed diffusion-wave terms. The space fractional derivatives are defined as Riesz fractional derivatives. Analytical solutions of three types of the MT-TSCR-FADE are derived with Dirichlet boundary conditions. By using Luchko's Theorem (Acta Math. Vietnam., 1999), we proposed some new techniques, such as a spectral representation of the fractional Laplacian operator and the equivalent relationship between fractional Laplacian operator and Riesz fractional derivative, that enabled the derivation of the analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations. © 2012.