Exploring diffusion across permeable barriers at high gradients. II. Localization regime

Measuring the velocity autocorrelation function using diffusion NMR

Molecular self-diffusion in the presence of barriers results in time-dependent displacements that are controlled by barrier characteristics, such as thickness, arrangement, and permeability, which manifests itself in the form of the ensemble-average velocity autocorrelation function (VAF). We describe a direct method to measure the VAF based on a combination of diffusion-weighted nuclear magnetic resonance (NMR) measurements in which two time-shifted diffusion encodings are separated by a longitudinal storage period. The VAF estimated from simulated data is shown to agree with the known expression for impermeable parallel planes. Simulations of diffusion in periodically spaced, permeable planes and connected, box-shaped pores are also presented. We find that scaling of the VAF faster than t-1/2 is indicative of barrier permeation or exchange between domains and that this can be captured by the proposed method. As an experimental proof-of-concept, we present data from an ex vivo neonatal mouse spinal cord studied using a permanent magnet NMR MOUSE system. We report a transition from t-1/2 to t-3/2 scaling at t ≈ 10 ms, consistent perhaps with transmembrane water exchange. Compared to other NMR-based approaches, this method can potentially access several orders of magnitude in time (ms - s), revealing a wealth of VAF behaviors with one experimental paradigm.

Restriction-induced time-dependent transcytolemmal water exchange: Revisiting the Kӓrger exchange model

The Kӓrger model and its derivatives have been widely used to incorporate transcytolemmal water exchange rate, an essential characteristic of living cells, into analyses of diffusion MRI (dMRI) signals from tissues. The Kӓrger model consists of two homogeneous exchanging components coupled by an exchange rate constant and assumes measurements are made with sufficiently long diffusion time and slow water exchange. Despite successful applications, it remains unclear whether these assumptions are generally valid for practical dMRI sequences and biological tissues. In particular, barrier-induced restrictions to diffusion produce inhomogeneous magnetization distributions in relatively large-sized compartments such as cancer cells, violating the above assumptions. The effects of this inhomogeneity are usually overlooked. We performed computer simulations to quantify how restriction effects, which in images produce edge enhancements at compartment boundaries, influence different variants of the Kӓrger-model. The results show that the edge enhancement effect will produce larger, time-dependent estimates of exchange rates in e.g., tumors with relatively large cell sizes (>10 μm), resulting in overestimations of water exchange as previously reported. Moreover, stronger diffusion gradients, longer diffusion gradient durations, and larger cell sizes, all cause more pronounced edge enhancement effects. This helps us to better understand the feasibility of the Kärger model in estimating water exchange in different tissue types and provides useful guidance on signal acquisition methods that may mitigate the edge enhancement effect. This work also indicates the need to correct the overestimated transcytolemmal water exchange rates obtained assuming the Kärger-model.

The Diffusion Exchange Ratio (DEXR): A minimal sampling of diffusion exchange spectroscopy to probe exchange, restriction, and time-dependence

Water exchange is increasingly recognized as an important biological process that can affect the study of biological tissue using diffusion MR. Methods to measure exchange, however, remain immature as opposed to those used to characterize restriction, with no consensus on the optimal pulse sequence (s) or signal model (s). In general, the trend has been towards data-intensive fitting of highly parameterized models. We take the opposite approach and show that a judicious sub-sample of diffusion exchange spectroscopy (DEXSY) data can be used to robustly quantify exchange, as well as restriction, in a data-efficient manner. This sampling produces a ratio of two points per mixing time: (i) one point with equal diffusion weighting in both encoding periods, which gives maximal exchange contrast, and (ii) one point with the same total diffusion weighting in just the first encoding period, for normalization. We call this quotient the Diffusion EXchange Ratio (DEXR). Furthermore, we show that it can be used to probe time-dependent diffusion by estimating the velocity autocorrelation function (VACF) over intermediate to long times (∼2-500ms). We provide a comprehensive theoretical framework for the design of DEXR experiments in the case of static or constant gradients. Data from Monte Carlo simulations and experiments acquired in fixed and viable ex vivo neonatal mouse spinal cord using a permanent magnet system are presented to test and validate this approach. In viable spinal cord, we report the following apparent parameters from just 6 data points: τk=17±4ms, fNG=0.72±0.01, Reff=1.05±0.01μm, and κeff=0.19±0.04μm/ms, which correspond to the exchange time, restricted or non-Gaussian signal fraction, an effective spherical radius, and permeability, respectively. For the VACF, we report a long-time, power-law scaling with ≈t-2.4, which is approximately consistent with disordered domains in 3-D. Overall, the DEXR method is shown to be highly efficient, capable of providing valuable quantitative diffusion metrics using minimal MR data.

The Diffusion Exchange Ratio (DEXR): A minimal sampling of diffusion exchange spectroscopy to probe exchange, restriction, and time-dependence

Water exchange is increasingly recognized as an important biological process that can affect the study of biological tissue using diffusion MR. Methods to measure exchange, however, remain immature as opposed to those used to characterize restriction, with no consensus on the optimal pulse sequence(s) or signal model(s). In general, the trend has been towards data-intensive fitting of highly parameterized models. We take the opposite approach and show that a judicious sub-sample of diffusion exchange spectroscopy (DEXSY) data can be used to robustly quantify exchange, as well as restriction, in a data-efficient manner. This sampling produces a ratio of two points per mixing time: (i) one point with equal diffusion weighting in both encoding periods, which gives maximal exchange contrast, and (ii) one point with the same total diffusion weighting in just the first encoding period, for normalization. We call this quotient the Diffusion EXchange Ratio (DEXR). Furthermore, we show that it can be used to probe time-dependent diffusion by estimating the velocity autocorrelation function (VACF) over intermediate to long times (~ 2-500 ms). We provide a comprehensive theoretical framework for the design of DEXR experiments in the case of static or constant gradients. Data from Monte Carlo simulations and experiments acquired in fixed and viable ex vivo neonatal mouse spinal cord using a permanent magnet system are presented to test and validate this approach. In viable spinal cord, we report the following apparent parameters from just 6 data points:τ k = 17 ± 4 m s ,f N G = 0.71 ± 0.01 ,R e f f = 1.10 ± 0.01 μ m , andκ eff = 0.21 ± 0.06 μ m / m s , which correspond to the exchange time, restricted or non-Gaussian signal fraction, an effective spherical radius, and permeability, respectively. For the VACF, we report a long-time, power-law scaling with ≈ t - 2.4 , which is approximately consistent with disordered domains in 3-D. Overall, the DEXR method is shown to be highly efficient, capable of providing valuable quantitative diffusion metrics using minimal MR data.

Maximizing SNR per unit time in diffusion MRI with multiband T-Hex spirals

PURPOSE

The characterization of tissue microstructure using diffusion MRI (dMRI) signals is rapidly evolving, with increasing sophistication of signal representations and microstructure models. However, this progress often requires signals to be acquired with very high b-values (e.g., b > 30 ms/μm2 ), along many directions, and using multiple b-values, leading to long scan times and extremely low SNR in dMRI images. The purpose of this work is to boost the SNR efficiency of dMRI by combining three particularly efficient spatial encoding techniques and utilizing a high-performance gradient system (Gmax  ≤ 300 mT/m) for efficient diffusion encoding.

METHODS

Spiral readouts, multiband imaging, and sampling on tilted hexagonal grids (T-Hex) are combined and implemented on a 3T MRI system with ultra-strong gradients. Image reconstruction is performed through an iterative cg-SENSE algorithm incorporating static off-resonance distributions and field dynamics as measured with an NMR field camera. Additionally, T-Hex multiband is combined with a more conventional EPI-readout and compared with state-of-the-art blipped-CAIPIRINHA sampling. The advantage of the proposed approach is furthermore investigated for clinically available gradient performance and diffusion kurtosis imaging.

RESULTS

High fidelity in vivo images with b-values up to 40 ms/μm2 are obtained. The approach provides superior SNR efficiency over other state-of-the-art multiband diffusion readout schemes.

CONCLUSION

The demonstrated gains hold promise for the widespread dissemination of advanced microstructural scans, especially in clinical populations.

Particle-environment interactions in arbitrary dimensions: A unifying analytic framework to model diffusion with inert spatial heterogeneities

Inert interactions between randomly moving entities and spatial disorder play a crucial role in quantifying the diffusive properties of a system, with examples ranging from molecules advancing along dendritic spines to antipredator displacements of animals due to sparse vegetation. Despite the ubiquity of such phenomena, a general framework to model the movement explicitly in the presence of spatial heterogeneities is missing. Here, we tackle this challenge and develop an analytic theory to model inert particle-environment interactions in domains of arbitrary shape and dimensions. We use a discrete space formulation, which allows us to model the interactions between an agent and the environment as perturbed dynamics between lattice sites. Interactions from spatial disorder, such as impenetrable and permeable obstacles or regions of increased or decreased diffusivity, as well as many others, can be modelled using our framework. We provide exact expressions for the generating function of the occupation probability of the diffusing particle and related transport quantities such as first-passage, return, and exit probabilities and their respective means. We uncover a surprising property, the disorder indifference phenomenon of the mean first-passage time in the presence of a permeable barrier in quasi-1D systems. We demonstrate the widespread applicability of our formalism by considering three examples that span across scales and disciplines. (1) We explore an enhancement strategy of transdermal drug delivery. (2) We represent the movement decisions of an animal undergoing thigomotaxis, the tendency to remain at the peripheries of its enclosure, using a spatially disordered environment. (3) We illustrate the use of spatial heterogeneities to model inert interactions between particles by modeling the search for a promoter region on the DNA by transcription factors during gene transcription.

Incorporating interface permeability into the diffusion MRI signal representation while using impermeable Laplace eigenfunctions

Objective. The complex-valued transverse magnetization due to diffusion-encoding magnetic field gradients acting on a permeable medium can be modeled by the Bloch-Torrey partial differential equation. The diffusion magnetic resonance imaging (MRI) signal has a representation in the basis of the Laplace eigenfunctions of the medium. However, in order to estimate the permeability coefficient from diffusion MRI data, it is desirable that the forward solution can be calculated efficiently for many values of permeability.Approach. In this paper we propose a new formulation of the permeable diffusion MRI signal representation in the basis of the Laplace eigenfunctions of the same medium where the interfaces are made impermeable.Main results.We proved the theoretical equivalence between our new formulation and the original formulation in the case that the full eigendecomposition is used. We validated our method numerically and showed promising numerical results when a partial eigendecomposition is used. Two diffusion MRI sequences were used to illustrate the numerical validity of our new method.Significance.Our approach means that the same basis (the impermeable set) can be used for all permeability values, which reduces the computational time significantly, enabling the study of the effects of the permeability coefficient on the diffusion MRI signal in the future.

Cumulant expansion with localization: A new representation of the diffusion MRI signal

Diffusion MR is sensitive to the microstructural features of a sample. Fine-scale characteristics can be probed by employing strong diffusion gradients while the low b-value regime is determined by the cumulants of the distribution of particle displacements. A signal representation based on the cumulants, however, suffers from a finite convergence radius and cannot represent the 'localization regime' characterized by a stretched exponential decay that emerges at large gradient strengths. Here, we propose a new representation for the diffusion MR signal. Our method provides not only a robust estimate of the first three cumulants but also a meaningful extrapolation of the entire signal decay.

Practical computation of the diffusion MRI signal based on Laplace eigenfunctions: permeable interfaces

The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses in a heterogeneous medium such as brain tissue can be modeled by the Bloch-Torrey partial differential equation. The spatial integral of the solution of this equation in realistic geometry provides a gold-standard reference model for the diffusion MRI signal arising from different tissue micro-structures of interest. A closed form representation of this reference diffusion MRI signal, called matrix formalism, which makes explicit the link between the Laplace eigenvalues and eigenfunctions of the tissue geometry and its diffusion MRI signal, was derived 20 years ago. In addition, once the Laplace eigendecomposition has been computed and saved, the diffusion MRI signal can be calculated for arbitrary diffusion-encoding sequences and b-values at negligible additional cost. In a previous publication, we presented a simulation framework that we implemented inside the MATLAB-based diffusion MRI simulator SpinDoctor that efficiently computes the matrix formalism representation for biological cells subject to impermeable membrane boundary conditions. In this work, we extend our simulation framework to include geometries that contain permeable cell membranes. We describe the new computational techniques that allowed this generalization and we analyze the effects of the magnitude of the permeability coefficient on the eigendecomposition of the diffusion and Bloch-Torrey operators. This work is another step in bringing advanced mathematical tools and numerical method development to the simulation and modeling of diffusion MRI.

Design and characterization of a 3D-printed axon-mimetic phantom for diffusion MRI

PURPOSE

To introduce and characterize inexpensive and easily produced 3D-printed axon-mimetic diffusion MRI phantoms in terms of pore geometry and diffusion kurtosis imaging metrics.

METHODS

Phantoms were 3D-printed with a composite printing material that, after the dissolution of the polyvinyl alcohol, exhibits microscopic fibrous pores. Confocal microscopy and synchrotron phase-contrast micro-CT imaging were performed to visualize and assess the pore sizes. Diffusion MRI scans of four identical phantoms and phantoms with varying print parameters in water were performed at 9.4 T. Diffusion kurtosis imaging was fit to both data sets and used to assess the reproducibility between phantoms and effects of print parameters on diffusion kurtosis imaging metrics. Identical scans were performed 25 and 76 days later, to test their stability.

RESULTS

Segmentation of pores in three microscopy images yielded a mean, median, and SD of equivalent pore diameters of 7.57 μm, 3.51 μm, and 12.13 μm, respectively. Phantoms had T₁ /T₂ = 2 seconds/180 ms, and those with identical parameters showed a low coefficient of variation (~10%) in mean diffusivity (1.38 × 10^-3 mm² /s) and kurtosis (0.52) metrics and radial diffusivity (1.01 × 10^-3 mm² /s) and kurtosis (1.13) metrics. Printing temperature and speed had a small effect on diffusion kurtosis imaging metrics (< 16%), whereas infill density had a larger and more variable effect (> 16%). The stability analysis showed small changes over 2.5 months (< 7%).

CONCLUSION

Three-dimension-printed axon-mimetic phantoms can mimic the fibrous structure of axon bundles on a microscopic scale, serving as complex, anisotropic diffusion MRI phantoms.

The localization regime in a nutshell

High diffusion-sensitizing magnetic field gradients have been more and more often applied nowadays to achieve a better characterization of the microstructure. As the resulting spin-echo signal significantly deviates from the conventional Gaussian form, various models have been employed to interpret these deviations and to relate them with the microstructural properties of a sample. In this paper, we argue that the non-Gaussian behavior of the signal is a generic universal feature of the Bloch-Torrey equation. We provide a simple yet rigorous description of the localization regime emerging at high extended gradients and identify its origin as a symmetry breaking at the reflecting boundary. We compare the consequent non-Gaussian signal decay to other diffusion NMR regimes such as slow-diffusion, motional-narrowing and diffusion-diffraction regimes. We emphasize limitations of conventional perturbative techniques and advocate for non-perturbative approaches which may pave a way to new imaging modalities in this field.

The present and the future of microstructure MRI: From a paradigm shift to normal science

The aspiration of imaging tissue microstructure with MRI is to uncover micrometer-scale tissue features within millimeter-scale imaging voxels, in vivo. This kind of super-resolution has fueled a paradigm shift within the biomedical imaging community. However, what feels like an ongoing revolution in MRI, has been conceptually experienced in physics decades ago; from this point of view, our current developments can be seen as Thomas Kuhn's "normal science" stage of progress. While the concept of model-based quantification below the nominal imaging resolution is not new, its possibilities in neuroscience and neuroradiology are only beginning to be widely appreciated. This disconnect calls for communicating the progress of tissue microstructure MR imaging to its potential users. Here, a number of recent research developments are outlined in terms of the overarching concept of coarse-graining the tissue structure over an increasing diffusion length. A variety of diffusion models and phenomena are summarized on the phase diagram of diffusion MRI, with the unresolved problems and future directions corresponding to its unexplored domains.

Practical computation of the diffusion MRI signal of realistic neurons based on Laplace eigenfunctions

The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses in a heterogeneous medium such as brain tissue can be modeled by the Bloch-Torrey partial differential equation. The spatial integral of the solution of this equation in realistic geometry provides a gold-standard reference model for the diffusion MRI signal arising from different tissue micro-structures of interest. A closed form representation of this reference diffusion MRI signal called matrix formalism, which makes explicit the link between the Laplace eigenvalues and eigenfunctions of the biological cell and its diffusion MRI signal, was derived 20 years ago. In addition, once the Laplace eigendecomposition has been computed and saved, the diffusion MRI signal can be calculated for arbitrary diffusion-encoding sequences and b-values at negligible additional cost. Up to now, this representation, though mathematically elegant, has not been often used as a practical model of the diffusion MRI signal, due to the difficulties of calculating the Laplace eigendecomposition in complicated geometries. In this paper, we present a simulation framework that we have implemented inside the MATLAB-based diffusion MRI simulator SpinDoctor that efficiently computes the matrix formalism representation for realistic neurons using the finite element method. We show that the matrix formalism representation requires a few hundred eigenmodes to match the reference signal computed by solving the Bloch-Torrey equation when the cell geometry originates from realistic neurons. As expected, the number of eigenmodes required to match the reference signal increases with smaller diffusion time and higher b-values. We also convert the eigenvalues to a length scale and illustrate the link between the length scale and the oscillation frequency of the eigenmode in the cell geometry. We give the transformation that links the Laplace eigenfunctions to the eigenfunctions of the Bloch-Torrey operator and compute the Bloch-Torrey eigenfunctions and eigenvalues. This work is another step in bringing advanced mathematical tools and numerical method development to the simulation and modeling of diffusion MRI.

Localization regime in diffusion NMR: Theory and experiments

In this work we investigate the emergence of the localization regime for diffusion NMR in various geometries: inside slabs, inside cylinders and outside rods arranged on a square array. At high gradients, the transverse magnetization is strongly attenuated in the bulk, whereas the macroscopic signal is formed by the remaining magnetization localized near boundaries of the sample. As a consequence, the signal is particularly sensitive to the microstructure. The theoretical analysis relies on recent mathematical advances on the study of the Bloch-Torrey equation. Experiments were conducted with hyperpolarized xenon-129 gas in 3D-printed phantoms and show an excellent agreement with numerical simulations and theoretical predictions. Our mathematical arguments and experimental evidence indicate that the localization regime with a stretched-exponential decay of the macroscopic signal is a generic feature of diffusion NMR that can be observed at moderately high gradients in most NMR scanners.

Quantifying brain microstructure with diffusion MRI: Theory and parameter estimation

We review, systematize and discuss models of diffusion in neuronal tissue, by putting them into an overarching physical context of coarse-graining over an increasing diffusion length scale. From this perspective, we view research on quantifying brain microstructure as occurring along three major avenues. The first avenue focusses on transient, or time-dependent, effects in diffusion. These effects signify the gradual coarse-graining of tissue structure, which occurs qualitatively differently in different brain tissue compartments. We show that transient effects contain information about the relevant length scales for neuronal tissue, such as the packing correlation length for neuronal fibers, as well as the degree of structural disorder along the neurites. The second avenue corresponds to the long-time limit, when the observed signal can be approximated as a sum of multiple nonexchanging anisotropic Gaussian components. Here, the challenge lies in parameter estimation and in resolving its hidden degeneracies. The third avenue employs multiple diffusion encoding techniques, able to access information not contained in the conventional diffusion propagator. We conclude with our outlook on future directions that could open exciting possibilities for designing quantitative markers of tissue physiology and pathology, based on methods of studying mesoscopic transport in disordered systems.

Spin dephasing around randomly distributed vessels

We analyze the gradient echo signal in the presence of blood vessel networks. Both, diffusion and susceptibility effects are analytically emphasized within the Bloch-Torrey equation. Solving this equation, we present the first exact description of the local magnetization around a single vessel. This allows us to deduce the gradient echo signal of parallel vessels randomly distributed in a plane, which is valid for arbitrary mean vessel diameters in the range of physiological relevant blood volume fractions. Thus, the results are potentially relevant for gradient echo measurements of blood vessel networks with arbitrary vessel size.

Physical and numerical phantoms for the validation of brain microstructural MRI: A cookbook

Phantoms, both numerical (software) and physical (hardware), can serve as a gold standard for the validation of MRI methods probing the brain microstructure. This review aims to provide guidelines on how to build, implement, or choose the right phantom for a particular application, along with an overview of the current state-of-the-art of phantoms dedicated to study brain microstructure with MRI. For physical phantoms, we discuss the essential requirements and relevant characteristics of both the (NMR visible) liquid and (NMR invisible) phantom materials that induce relevant microstructural features detectable via MRI, based on diffusion, intra-voxel incoherent motion, magnetization transfer or magnetic susceptibility weighted contrast. In particular, for diffusion MRI, many useful phantoms have been proposed, ranging from simple liquids to advanced biomimetic phantoms consisting of hollow or plain microfibers and capillaries. For numerical phantoms, the focus is on Monte Carlo simulations of random walk, for which the basic principles, along with useful criteria to check and potential pitfalls are reviewed, in addition to a literature overview highlighting recent advances. While many phantoms exist already, the current review aims to stimulate further research in the field and to address remaining needs.

The Kärger vs bi-exponential model: Theoretical insights and experimental validations

We revise three common models accounting for water exchange in pulsed-gradient spin-echo measurements: a bi-exponential model with time-dependent water fractions, the Kärger model, and a modified Kärger model designed for restricted diffusion, e.g. inside cells. The three models are compared and applied to experimental data from yeast cell suspensions. The Kärger model and the modified Kärger model yield very close results and accurately fit the data. The bi-exponential model, although less rigorous, has a natural physical interpretation and suggests a new experimental modality to estimate the water exchange time.

Numerical study of a cylinder model of the diffusion MRI signal for neuronal dendrite trees

We study numerically how the neuronal dendrite tree structure can affect the diffusion magnetic resonance imaging (dMRI) signal in brain tissue. For a large set of randomly generated dendrite trees, synthetic dMRI signals are computed and fitted to a cylinder model to estimate the effective longitudinal diffusivity D(L) in the direction of neurites. When the dendrite branches are short compared to the diffusion length, D(L) depends significantly on the ratio between the average branch length and the diffusion length. In turn, D(L) has very weak dependence on the distribution of branch lengths and orientations of a dendrite tree, and the number of branches per node. We conclude that the cylinder model which ignores the connectivity of the dendrite tree, can still be adapted to describe the apparent diffusion coefficient in brain tissue.

Exploring diffusion across permeable barriers at high gradients. I. Narrow pulse approximation

The adaptive variation of the gradient intensity with the diffusion time at a constant optimal b-value is proposed to enhance the contribution of the nuclei diffusing across permeable barriers, to the pulsed-gradient spin-echo (PGSE) signal. An exact simple formula the PGSE signal is derived under the narrow pulse approximation in the case of one-dimensional diffusion across a single permeable barrier. The barrier contribution to the signal is shown to be maximal at a particular b-value. The exact formula is then extended to multiple permeable barriers, while the PGSE signal is shown to be sensitive to the permeability and to the inter-barrier distance. Potential applications of the protocol to survey diffusion in three-dimensional domains with permeable membranes are illustrated through numerical simulations.