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We discuss the situations under which Brownian yet non-Gaussian (BnG) diffusion can be observed in the model of a particle's motion in a random landscape of diffusion coefficients slowly varying in space (quenched disorder). Our conclusion is that such behavior is extremely unlikely in the situations when the particles, introduced into the system at random at t = 0, are observed from the preparation of the system on. However, it indeed may arise in the case when the diffusion (as described in Ito interpretation) is observed under equilibrated conditions. This paradigmatic situation can be translated into the model of the diffusion coefficient fluctuating in time along a trajectory, i.e. into a kind of the 'diffusing diffusivity' model.

Recent advances in single particle tracking and supercomputing techniques demonstrate the emergence of normal or anomalous, viscoelastic diffusion in conjunction with non-Gaussian distributions in soft, biological, and active matter systems. We here formulate a stochastic model based on a generalised Langevin equation in which non-Gaussian shapes of the probability density function and normal or anomalous diffusion have a common origin, namely a random parametrisation of the stochastic force. We perform a detailed analysis demonstrating how various types of parameter distributions for the memory kernel result in exponential, power law, or power-log law tails of the memory functions. The studied system is also shown to exhibit a further unusual property: the velocity has a Gaussian one point probability density but non-Gaussian joint distributions. This behaviour is reflected in the relaxation from a Gaussian to a non-Gaussian distribution observed for the position variable. We show that our theoretical results are in excellent agreement with stochastic simulations.

Recent works show that glass-forming liquids display Fickian non-Gaussian Diffusion, with non-Gaussian displacement distributions persisting even at very long times, when linearity in the mean square displacement (Fickianity) has already been attained. Such non-Gaussian deviations temporarily exhibit distinctive exponential tails, with a decay length λ growing in time as a power-law. We herein carefully examine data from four different glass-forming systems with isotropic interactions, both in two and three dimensions, namely, three numerical models of molecular liquids and one experimentally investigated colloidal suspension. Drawing on the identification of a proper time range for reliable exponential fits, we find that a scaling law λ(t)∝tα, with α≃1/3, holds for all considered systems, independently from dimensionality. We further show that, for each system, data at different temperatures/concentration can be collapsed onto a master-curve, identifying a characteristic time for the disappearance of exponential tails and the recovery of Gaussianity. We find that such characteristic time is always related through a power-law to the onset time of Fickianity. The present findings suggest that FnGD in glass-formers may be characterized by a “universal” evolution of the distribution tails, independent from system dimensionality, at least for liquids with isotropic potential.

Diffusion MRI (dMRI) has become one of the most important imaging modalities for noninvasively probing tissue microstructure. Diffusional Kurtosis MRI (DKI) quantifies the degree of non-Gaussian diffusion, which in turn has been shown to increase sensitivity towards, e.g., disease and orientation mapping in neural tissue. However, the specificity of DKI is limited as different sources can contribute to the total intravoxel diffusional kurtosis, including: variance in diffusion tensor magnitudes (Kiso), variance due to diffusion anisotropy (Kaniso), and microscopic kurtosis (μK) related to restricted diffusion, microstructural disorder, and/or exchange. Interestingly, μK is typically ignored in diffusion MRI signal modelling as it is assumed to be negligible in neural tissues. However, recently, Correlation Tensor MRI (CTI) based on Double-Diffusion-Encoding (DDE) was introduced for kurtosis source separation, revealing non negligible μK in preclinical imaging. Here, we implemented CTI for the first time on a clinical 3T scanner and investigated the sources of total kurtosis in healthy subjects. A robust framework for kurtosis source separation in humans is introduced, followed by estimation of μK (and the other kurtosis sources) in the healthy brain. Using this clinical CTI approach, we find that μK significantly contributes to total diffusional kurtosis both in grey and white matter tissue but, as expected, not in the ventricles. The first μK maps of the human brain are presented, revealing that the spatial distribution of μK provides a unique source of contrast, appearing different from isotropic and anisotropic kurtosis counterparts. Moreover, group average templates of these kurtosis sources have been generated for the first time, which corroborated our findings at the underlying individual-level maps. We further show that the common practice of ignoring μK and assuming the multiple Gaussian component approximation for kurtosis source estimation introduces significant bias in the estimation of other kurtosis sources and, perhaps even worse, compromises their interpretation. Finally, a twofold acceleration of CTI is discussed in the context of potential future clinical applications. We conclude that CTI has much potential for future in vivo microstructural characterizations in healthy and pathological tissue.

MR diffusion kurtosis imaging (DKI) was proposed recently to study the deviation of water diffusion from Gaussian distribution. Mean kurtosis, the directionally averaged kurtosis, has been shown to be useful in assessing pathophysiological changes, thus yielding another dimension of information to characterize water diffusion in biological tissues. In this study, orthogonal transformation of the 4th order diffusion kurtosis tensor was introduced to compute the diffusion kurtoses along the three eigenvector directions of the 2nd order diffusion tensor. Such axial ( K //) and radial ( K ┴) kurtoses measured the kurtoses along the directions parallel and perpendicular, respectively, to the principal diffusion direction. DKI experiments were performed in normal adult ( N = 7) and formalin-fixed rat brains ( N = 5). DKI estimates were documented for various white matter (WM) and gray matter (GM) tissues, and compared with the conventional diffusion tensor estimates. The results showed that kurtosis estimates revealed different information for tissue characterization. For example, K // and K ┴ under formalin fixation condition exhibited large and moderate increases in WM while they showed little change in GM despite the overall dramatic decrease of axial and radial diffusivities in both WM and GM. These findings indicate that directional kurtosis analysis can provide additional microstructural information in characterizing neural tissues.

Many studies on biological and soft matter systems report the joint presence of a linear mean-squared displacement and a non-Gaussian probability density exhibiting, for instance, exponential or stretched-Gaussian tails. This phenomenon is ascribed to the heterogeneity of the medium and is captured by random parameter models such as 'superstatistics' or 'diffusing diffusivity'. Independently, scientists working in the area of time series analysis and statistics have studied a class of discrete-time processes with similar properties, namely, random coefficient autoregressive models. In this work we try to reconcile these two approaches and thus provide a bridge between physical stochastic processes and autoregressive models. We start from the basic Langevin equation of motion with time-varying damping or diffusion coefficients and establish the link to random coefficient autoregressive processes. By exploring that link we gain access to efficient statistical methods which can help to identify data exhibiting Brownian yet non-Gaussian diffusion.

The developed magnetic resonance electrical properties tomography (MREPT) techniques visualize the internal conductivity distribution at Larmor frequency by measuring the B1 transceive phase data. In biological tissues, electrical conductivity is influenced by ion concentrations and mobility. To visualize the anisotropic conductivity tensor of biological tissues, we use the Einstein–Smoluchowski equation, which links the diffusion coefficient to particle mobility. By assuming a correlation between ion mobility and water diffusivity, we aim to decompose the internal isotropic conductivity at Larmor frequency into anisotropic conductivity tensors within the intra- and extra-neurite compartments. The multi-compartment spherical mean technique (MC-SMT), utilizing both high and low b-value diffusion-weighted imaging (DWI) data, characterizes the diffusion of water molecules within and across the intra- and extra-neurite compartments by analyzing the microstructural intricacies and the foundational architectural arrangement of the brain’s tissues. By analyzing the relationships between the measured DWI data, the microscopic diffusion signal, and the fiber orientation distribution function, we predict the DWI data for the intra- and extra-neurite compartments using spherical harmonic decomposition. Using the predicted DWI data for the intra- and extra-neurite compartments, we develop a conductivity tensor imaging method that operates specifically within the separated compartments. Human brain experiments, involving four healthy volunteers and a brain tumor patient, were performed to assess and confirm the reliability of the proposed method. •Intra-neurite volume fraction and diffusion patterns reconstructed using DWI data with multiple b-values.•Prediction of intra- and extra-neurite DWI data.•Conductivity tensor imaging for intra- and extra-neurite compartments using the predicted DWI data.

The most common modality of diffusion MRI used in the ageing and development studies is diffusion tensor imaging (DTI) providing two key measures, fractional anisotropy and mean diffusivity. Here, we investigated diffusional changes occurring between childhood (average age 10.3 years) and mitddle adult age (average age 54.3 years) with the help of diffusion kurtosis imaging (DKI), a recent novel extension of DTI that provides additional metrics quantifying non-Gaussianity of water diffusion in brain tissue. We performed voxelwise statistical between-group comparison of diffusion tensor and kurtosis tensor metrics using two methods, namely, the tract-based spatial statistics (TBSS) and the atlas-based regional data analysis. For the latter, fractional anisotropy, mean diffusivity, mean diffusion kurtosis, and other scalar diffusion tensor and kurtosis tensor parameters were evaluated for white matter fibres provided by the Johns-Hopkins-University Atlas in the FSL toolkit (http://fsl.fmrib.ox.ac.uk/fsl/fslwiki/Atlases). Within the same age group, all evaluated parameters varied depending on the anatomical region. TBSS analysis showed that changes in kurtosis tensor parameters beyond adolescence are more widespread along the skeleton in comparison to the changes of the diffusion tensor metrics. The regional data analysis demonstrated considerably larger between-group changes of the diffusion kurtosis metrics than of diffusion tensor metrics in all investigated regions. The effect size of the parametric changes between childhood and middle adulthood was quantified using Cohen's d. We used Cohen's d related to mean diffusion kurtosis to examine heterogeneous maturation of various fibres. The largest changes of this parameter (interpreted as reflecting the lowest level of maturation by the age of children group) were observed in the association fibres, cingulum (gyrus) and cingulum (hippocampus) followed by superior longitudinal fasciculus and inferior longitudinal fasciculus. The smallest changes were observed in the commissural fibres, forceps major and forceps minor. In conclusion, our data suggest that DKI is sensitive to developmental changes in local microstructure and environment, and is particularly powerful to unravel developmental differences in major association fibres, such as the cingulum and superior longitudinal fasciculus. [Display omitted]

To enable application of non-Gaussian diffusion magnetic resonance imaging (dMRI) techniques in large-scale clinical trials and facilitate translation to clinical practice there is a requirement for fast, high contrast, techniques that are sensitive to changes in tissue structure which provide diagnostic signatures at the early stages of disease. Here we describe a new way to compress the acquisition of multi-shell b-value diffusion data, Quasi-Diffusion MRI (QDI), which provides a probe of subvoxel tissue complexity using short acquisition times (1–4 ​min). We also describe a coherent framework for multi-directional diffusion gradient acquisition and data processing that allows computation of rotationally invariant quasi-diffusion tensor imaging (QDTI) maps. QDI is a quantitative technique that is based on a special case of the Continuous Time Random Walk model of diffusion dynamics and assumes the presence of non-Gaussian diffusion properties within tissue microstructure. QDI parameterises the diffusion signal attenuation according to the rate of decay (i.e. diffusion coefficient, D in mm2 s−1) and the shape of the power law tail (i.e. the fractional exponent, α). QDI provides analogous tissue contrast to Diffusional Kurtosis Imaging (DKI) by calculation of normalised entropy of the parameterised diffusion signal decay curve, Hn, but does so without the limitations of a maximum b-value. We show that QDI generates images with superior tissue contrast to conventional diffusion imaging within clinically acceptable acquisition times of between 84 and 228 ​s. We show that QDI provides clinically meaningful images in cerebral small vessel disease and brain tumour case studies. Our initial findings suggest that QDI may be added to routine conventional dMRI acquisitions allowing simple application in clinical trials and translation to the clinical arena.

This work justifies further paradigmatic importance of the model of viscoelastic subdiffusion in random environments for the observed subdiffusion in cellular biological systems. Recently, we showed (2018, PCCP, 20, 24140) that this model displays several remarkable features, which makes it an attractive paradigm to explain the physical nature of subdiffusion occurring in biological cells. In particular, it combines viscoelasticity with distinct non-ergodic features. We extend this basic model to make it suitable for physical phenomena such as subdiffusion of lipids in disordered biological membranes upon including the inertial effects. For lipids, the inertial effects occur in the range of picoseconds, and a power-law decaying viscoelastic memory extends over the range of several nanoseconds. Thus, in the absence of disorder, diffusion would become normal on a time scale beyond this memory range. However, both experimentally and in some molecular-dynamical simulations, the time range of lipid subdiffusion extends far beyond the viscoelastic memory range. We study three 1d models of correlated quenched Gaussian disorder to explain the puzzle: singular short-range (exponentially correlated), smooth short-range (Gaussian-correlated), and smooth long-range (power-law correlated) disorder. For a moderate disorder strength, transient viscoelastic subdiffusion changes into the subdiffusion caused by the randomness of the environment. It is characterized by a time-dependent power-law exponent of subdiffusion α(t), which can show nonmonotonous behavior, in agreement with some recent molecular-dynamical simulations. Moreover, the spatial distribution of test particles in this disorder-dominated regime is shown to be a non-Gaussian, exponential power distribution with index χ = 1.45-2.3, which also correlates well with molecular-dynamical findings and experiments. Furthermore, this subdiffusion is nonergodic with single-trajectory averages showing a broad scatter, in agreement with experimental observations for viscoelastic subdiffusion of various particles in living cells.