Dmitry S. Novikov

Revealing mesoscopic structural universality with diffusion

Significance A major challenge in the physical and life sciences is to see the invisible by quantifying the structure of materials and living tissues below nominal imaging resolution. Measurement of molecular diffusion performed at a macroscopic level is a common noninvasive structural probe. Interpreting its results involves identifying which structural features, out of a myriad of parameters, most strongly affect the diffusion coefficient. Here we address this ill-posed problem by classifying media into a relatively small number of universality classes, based on the nature of long-range structural correlations. We show that the time dependence of a macroscopic diffusion coefficient distinguishes between these classes. Using this approach, we identify dominant restrictions to MRI-measured water diffusion in muscles and in cortical gray matter. Measuring molecular diffusion is widely used for characterizing materials and living organisms noninvasively. This characterization relies on relations between macroscopic diffusion metrics and structure at the mesoscopic scale commensurate with the diffusion length. Establishing such relations remains a fundamental challenge, hindering progress in materials science, porous media, and biomedical imaging. Here we show that the dynamical exponent in the time dependence of the diffusion coefficient distinguishes between the universality classes of the mesoscopic structural complexity. Our approach enables the interpretation of diffusion measurements by objectively selecting and modeling the most relevant structural features. As an example, the specific values of the dynamical exponent allow us to identify the relevant mesoscopic structure affecting MRI-measured water diffusion in muscles and in brain, and to elucidate the structural changes behind the decrease of diffusion coefficient in ischemic stroke.Molecular diffusion measurements are widely used to probe m icrostructure in materials and living organisms noninvasively. The precise relation of diffusion metrics t o microstructure remains a major challenge: In complex samples, it is often unclear which structural features are m ost relevant and can be quantified. Here we classify the structural complexity in terms of the long time tail exponen t in the molecular velocity autocorrelation function. The specific values of the dynamical exponent let us identify the relevant tissue microanatomy affecting water diffusion measured with MRI in muscles and in brain, and the m icrostructural changes in ischemic stroke. Our framework presents a systematic way to identify the most rel evant part of structural complexity using transport measured with a variety of techniques.

Random walk with barriers.

Restrictions to molecular motion by barriers (membranes) are ubiquitous in porous media, composite materials and biological tissues. A major challenge is to characterize the microstructure of a material or an organism nondestructively using a bulk transport measurement. Here we demonstrate how the long-range structural correlations introduced by permeable membranes give rise to distinct features of transport. We consider Brownian motion restricted by randomly placed and oriented membranes (d − 1 dimensional planes in d dimensions) and focus on the disorder-averaged diffusion propagator using a scattering approach. The renormalization group solution reveals a scaling behavior of the diffusion coefficient for large times, with a characteristically slow inverse square root time dependence for any d. Its origin lies in the strong structural fluctuations introduced by the spatially extended random restrictions, representing a novel universality class of the structural disorder. Our results agree well with Monte Carlo simulations in two dimensions. They can be used to identify permeable barriers as restrictions to transport, and to quantify their permeability and surface area.

Monte Carlo study of a two-compartment exchange model of diffusion

Multisite exchange models have been applied frequently to quantify measurements of transverse relaxation and diffusion in living tissues. Although the simplicity of such models is attractive, the precise relationship of the model parameters to tissue properties may be difficult to ascertain. Here, we investigate numerically a two‐compartment exchange (Kärger) model as applied to diffusion in a system of randomly packed identical parallel cylinders with permeable walls, representing cells with permeable membranes, that may serve particularly as a model for axons in the white matter of the brain. By performing Monte Carlo simulations of restricted diffusion, we show that the Kärger model may provide a reasonable coarse‐grained description of the diffusion‐weighted signal in the long time limit, as long as the cell membranes are sufficiently impermeable, i.e. whenever the residence time in a cell is much longer than the time it takes to diffuse across it. For larger permeabilities, the exchange time obtained from fitting to the Kärger model overestimates the actual exchange time, leading to an underestimated value of cell membrane permeability. Copyright

One diffusion acquisition and different white matter models: how does microstructure change in human early development based on WMTI and NODDI?

White matter microstructural changes during the first three years of healthy brain development are characterized using two different models developed for limited clinical diffusion data: White Matter Tract Integrity (WMTI) metrics from Diffusional Kurtosis Imaging (DKI) and Neurite Orientation Dispersion and Density Imaging (NODDI). Both models reveal a non-linear increase in intra-axonal water fraction and in tortuosity of the extra-axonal space as a function of age, in the genu and splenium of the corpus callosum and the posterior limb of the internal capsule. The changes are consistent with expected behavior related to myelination and asynchrony of fiber development. The intra- and extracellular axial diffusivities as estimated with WMTI do not change appreciably in normal brain development. The quantitative differences in parameter estimates between models are examined and explained in the light of each models assumptions and consequent biases, as highlighted in simulations. Finally, we discuss the feasibility of a model with fewer assumptions.

Mesoscopic structure of neuronal tracts from time-dependent diffusion

Interpreting brain diffusion MRI measurements in terms of neuronal structure at a micrometer level is an exciting unresolved problem. Here we consider diffusion transverse to a bundle of fibers, and show theoretically, as well as using Monte Carlo simulations and measurements in a phantom made of parallel fibers mimicking axons, that the time dependent diffusion coefficient approaches its macroscopic limit slowly, in a (ln t)/t fashion. The logarithmic singularity arises due to short range disorder in the fiber packing. We identify short range disorder in axonal fibers based on histological data from the splenium, and argue that the time dependent contribution to the overall diffusion coefficient from the extra-axonal water dominates that of the intra-axonal water. This dominance may explain the bias in measuring axon diameters in clinical settings. The short range disorder is also reflected in the asymptotically linear frequency dependence of the diffusion coefficient measured with oscillating gradients, in agreement with recent experiments. Our results relate the measured diffusion to the mesoscopic structure of neuronal tissue, uncovering the sensitivity of diffusion metrics to axonal arrangement within a fiber tract, and providing an alternative interpretation of axonal diameter mapping techniques.

Degeneracy in model parameter estimation for multi-compartmental diffusion in neuronal tissue

The ultimate promise of diffusion MRI (dMRI) models is specificity to neuronal microstructure, which may lead to distinct clinical biomarkers using noninvasive imaging. While multi‐compartment models are a common approach to interpret water diffusion in the brain in vivo, the estimation of their parameters from the dMRI signal remains an unresolved problem. Practically, even when q space is highly oversampled, nonlinear fit outputs suffer from heavy bias and poor precision. So far, this has been alleviated by fixing some of the model parameters to a priori values, for improved precision at the expense of accuracy. Here we use a representative two‐compartment model to show that fitting fails to determine the five model parameters from over 60 measurement points. For the first time, we identify the reasons for this poor performance. The first reason is the existence of two local minima in the parameter space for the objective function of the fitting procedure. These minima correspond to qualitatively different sets of parameters, yet they both lie within biophysically plausible ranges. We show that, at realistic signal‐to‐noise ratio values, choosing between the two minima based on the associated objective function values is essentially impossible. Second, there is an ensemble of very low objective function values around each of these minima in the form of a pipe. The existence of such a direction in parameter space, along which the objective function profile is very flat, explains the bias and large uncertainty in parameter estimation, and the spurious parameter correlations: in the presence of noise, the minimum can be randomly displaced by a very large amount along each pipe. Our results suggest that the biophysical interpretation of dMRI model parameters crucially depends on establishing which of the minima is closer to the biophysical reality and the size of the uncertainty associated with each parameter. Copyright

Novel White Matter Tract Integrity Metrics Sensitive to Alzheimer Disease Progression

BACKGROUND AND PURPOSE: Along with cortical abnormalities, white matter microstructural changes such as axonal loss and myelin breakdown are implicated in the pathogenesis of Alzheimer disease. Recently, a white matter model was introduced that relates non-Gaussian diffusional kurtosis imaging metrics to characteristics of white matter tract integrity, including the axonal water fraction, the intra-axonal diffusivity, and the extra-axonal axial and radial diffusivities. MATERIALS AND METHODS: This study reports these white matter tract integrity metrics in subjects with amnestic mild cognitive impairment (n = 12), Alzheimer disease (n = 14), and age-matched healthy controls (n = 15) in an effort to investigate their sensitivity, diagnostic accuracy, and associations with white matter changes through the course of Alzheimer disease. RESULTS: With tract-based spatial statistics and region-of-interest analyses, increased diffusivity in the extra-axonal space (extra-axonal axial and radial diffusivities) in several white matter tracts sensitively and accurately discriminated healthy controls from those with amnestic mild cognitive impairment (area under the receiver operating characteristic curve = 0.82–0.95), while widespread decreased axonal water fraction discriminated amnestic mild cognitive impairment from Alzheimer disease (area under the receiver operating characteristic curve = 0.84). Additionally, these white matter tract integrity metrics in the body of the corpus callosum were strongly correlated with processing speed in amnestic mild cognitive impairment (r = |0.80–0.82|, P < .001). CONCLUSIONS: These findings have implications for the course and spatial progression of white matter degeneration in Alzheimer disease, suggest the mechanisms by which these changes occur, and demonstrate the viability of these white matter tract integrity metrics as potential neuroimaging biomarkers of the earliest stages of Alzheimer disease and disease progression.

Denoising of diffusion MRI using random matrix theory

We introduce and evaluate a post-processing technique for fast denoising of diffusion-weighted MR images. By exploiting the intrinsic redundancy in diffusion MRI using universal properties of the eigenspectrum of random covariance matrices, we remove noise-only principal components, thereby enabling signal-to-noise ratio enhancements. This yields parameter maps of improved quality for visual, quantitative, and statistical interpretation. By studying statistics of residuals, we demonstrate that the technique suppresses local signal fluctuations that solely originate from thermal noise rather than from other sources such as anatomical detail. Furthermore, we achieve improved precision in the estimation of diffusion parameters and fiber orientations in the human brain without compromising the accuracy and spatial resolution.

In vivo observation and biophysical interpretation of time-dependent diffusion in human white matter.

The presence of micrometer-level restrictions leads to a decrease of diffusion coefficient with diffusion time. Here we investigate this effect in human white matter in vivo. We focus on a broad range of diffusion times, up to 600 ms, covering diffusion length scales up to about 30 μm. We perform stimulated echo diffusion tensor imaging on 5 healthy volunteers and observe a relatively weak time-dependence in diffusion transverse to major fiber tracts. Remarkably, we also find notable time-dependence in the longitudinal direction. Comparing models of diffusion in ordered, confined and disordered media, we argue that the time-dependence in both directions can arise due to structural disorder, such as axonal beads in the longitudinal direction, and the random packing geometry of fibers within a bundle in the transverse direction. These time-dependent effects extend beyond a simple picture of Gaussian compartments, and may lead to novel markers that are specific to neuronal fiber geometry at the micrometer scale.

Diffusion MRI noise mapping using random matrix theory.

To estimate the spatially varying noise map using a redundant series of magnitude MR images.