Revealing mesoscopic structural universality with diffusion

Abstract

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.A macroscopically uniform sample of a biological tissue, porous rock, or composite material appears incredibly complex at the mesoscopic scale. This scale, typically of the order ∼ 0:1 to 10 μm, is intermediate between the microscopic scale of molecular dimensions, where material properties such as the local diffusion coefficient originate, and the macroscopic sample dimensions or imaging resolution. Quantifying the mesoscopic complexity noninvasively is important in the physical sciences for characterizing artificial and natural samples, and in the life sciences for diagnosing diseases, such as stroke and Alzheimer's, that manifest themselves at a cellular level.Measuring molecular diffusion, e.g., of water, in such media has emerged as a universal noninvasive structural probe (1-8). Obtained with techniques ranging from single-molecule tracking (3) to diffusion-weighted MRI (dMRI) (4), macroscopic diffusion metrics are sensitive to the nominally invisible micronlevel sample architecture, thanks to the diffusion length, i.e., the rms molecular displacement LðtÞ = hδx 2 ðtÞi 1=2 , providing the mesoscopic length scale. However, a challenging ill-posed problem (1, 2, 9) has long been to quantitatively interpret a bulk diffusion measurement, i.e., to convert this manifestly sensitive metric into specific mesoscopic structural parameters, such as geometric properties of pores or biophysical parameters of cells.Characterizing structure below a nominally achievable imaging resolution requires a structural model that predicts the result of the bulk measurement; by comparing the measurement to the prediction, the model parameters may be quantified. At the most basic level, a model is a rough sketch which captures the most essential parts of the structural complexity while neglecting the rest. Given the inherently irregular, or disordered nature of most specimens, a key challenge is to adequately and parsimoniously represent structural disorder.Here we advocate that there are only a handful of qualitatively distinct ways to...

Improving the Realism of White Matter Numerical Phantoms: A Step toward a Better Understanding of the Influence of Structural Disorders in Diffusion MRI

“…A clear dependence of both radial and axial intracellular dispersive diffusivities with respect to the frequency of the employed OGSE sequence in the presence of beadings was observed. Moreover, results of MC molecular diffusion simulations in complex synthetic substrates mimicking the presence of beads showed a clear 1/ √ t dependence of the axial intracellular Apparent Diffusion Coefficient due to 1D short-range disorder introduced in the axial direction by the randomly placed beads, in good accordance with theoretical predictions in Burcaw et al [7] and Novikov et al in [11]. Indeed, beadings are supposed to primarily affect the diffusion process along the fibers and a 1/ √ t ∼ √ ω frequency-dependence of the parallel diffusivity in the intra-cellular space of beaded axons is expected.…”

“…The diameters follow a Gamma distribution, with a mean axonal diameter of 2.0 µm. The value of the scaling coefficient A reaches its maximum value of 9.09 µm 2 (all the values of A are given for a SNR of 30) in this configuration which corresponds exactly to the 2D short-range disorder geometry described in Burcaw et al [7] and Novikov et al [11] . From C1 to C2, the induction of global angular dispersion (fibers remain straight but are rotated to induce angular dispersion) of 3.5 • yields to a substantial diminution of the scaling coefficient from 9.09 down to 8.54 µm 2 .…”

“…However, in Budde and Frank [20], simulations were run on numerical phantoms with an hexagonal packing of fibers and periodic restrictions along the fiber. The periodicity of the employed simulation domain might affect the realism of the obtained diffusion signal since it does not reflect the 1D short-range disorder along white matter fibers nor the 2D short-range disorder in the plane transverse to fibers [7,11]. In our simulations, short-range disorder effects are expected to be accounted for, since the spacing between each beading is highly variable (as shown in Table 2, the variance of the spacing distribution is equal to one fourth of the spacing value) and the fibers are randomly placed in the phantom, thus mimicking transverse short-range disorder better representing actual brain white matter tissues.…”

“…An empirical law A ∼ l 2 c relating A to the correlation length l c was given in Burcaw et al [7] and Fieremans et al [8] but is not sufficient to catch the complexity of the scaling coefficient A. The value of A is a measure of the strength of the structural disorder [7,11], thus related to the geometrical properties characterizing the spatial organization of white matter at various scales. A possible approach to decipher the complex relationship between A and white matter features is to perform Monte-Carlo simulations of the diffusion process in diffusing media with increasing level of structural disorder.…”

Multi-Spherical Diffusion MRI: Exploring Diffusion Time Using Signal Sparsity

“…Recent literature is increasingly emphasizing the need for such a representation, where accounting for the diffusion time dependence of the extra-axonal diffusion signal [1,2] has already resulted in a more accurate estimation of the axon density and diameter [3]. To measure the four-dimensional dMRI signal it is necessary to go beyond a multi-shell q-space acquisition -which only varies gradient strength and direction -and also vary the diffusion time.…”