Topgaard D. Diffusion tensor distribution imaging. NMR Biomed. 2019 :e4066.Abstract
Conventional diffusion MRI yields voxel-averaged parameters that suffer from ambiguities for heterogeneous anisotropic materials such as brain tissue. Using principles from solid-state NMR spectroscopy, we have previously introduced the shape of the diffusion encoding tensor as a separate acquisition dimension that disentangles isotropic and anisotropic contributions to the observed diffusivities, thereby allowing for unconstrained data inversion into diffusion tensor distributions with "size," "shape," and orientation dimensions. Here we combine our recent non-parametric data inversion algorithm and data acquisition protocol with an imaging pulse sequence to demonstrate spatial mapping of diffusion tensor distributions using a previously developed composite phantom with multiple isotropic and anisotropic components. We propose a compact format for visualizing two-dimensional arrays of the distributions, new scalar parameters quantifying intra-voxel heterogeneity, and a binning procedure giving maps of all relevant parameters for each of the components resolved in the multidimensional distribution space.
Lasič S, Lundell H, Topgaard D, Dyrby TB. Effects of imaging gradients in sequences with varying longitudinal storage time-Case of diffusion exchange imaging. Magn Reson Med. 2018;79 (4) :2228-2235.Abstract
PURPOSE: To illustrate the potential bias caused by imaging gradients in correlation MRI sequences using longitudinal magnetization storage (LS) and examine the case of filter exchange imaging (FEXI) yielding maps of the apparent exchange rate (AXR). METHODS: The effects of imaging gradients in FEXI were observed on yeast cells. To analyze the AXR bias, signal evolution was calculated by applying matrix exponential operators. RESULTS: A sharp threshold for the slice thickness was identified, below which the AXR is increasingly underestimated. The bias can be understood in terms of an extended low-pass diffusion filtering during the LS interval, which is more pronounced at lower exchange rates. For a total exchange rate constant larger than 1 s, the AXR bias is expected to be negligible when slices thicker than 2.5 mm are used. CONCLUSION: In correlation experiments like FEXI, relying on LS with variable duration, imaging gradients may cause disrupting effects that cannot be easily mitigated and should be carefully considered for unbiased results. In typical clinical applications of FEXI, the imaging gradients are expected to cause a negligible AXR bias. However, the AXR bias may be significant in preclinical settings or whenever thin imaging slices are used. Magn Reson Med 79:2228-2235, 2018. © 2017 International Society for Magnetic Resonance in Medicine.
Nilsson M, Larsson J, Lundberg D, Szczepankiewicz F, Witzel T, Westin C-F, Bryskhe K, Topgaard D. Liquid crystal phantom for validation of microscopic diffusion anisotropy measurements on clinical MRI systems. Magn Reson Med. 2018;79 (3) :1817-1828.Abstract
PURPOSE: To develop a phantom for validating MRI pulse sequences and data processing methods to quantify microscopic diffusion anisotropy in the human brain. METHODS: Using a liquid crystal consisting of water, detergent, and hydrocarbon, we designed a 0.5-L spherical phantom showing the theoretically highest possible degree of microscopic anisotropy. Data were acquired on the Connectome scanner using echo-planar imaging signal readout and diffusion encoding with axisymmetric b-tensors of varying magnitude, anisotropy, and orientation. The mean diffusivity, fractional anisotropy (FA), and microscopic FA (µFA) parameters were estimated. RESULTS: The phantom was observed to have values of mean diffusivity similar to brain tissue, and relaxation times compatible with echo-planar imaging echo times on the order of 100 ms. The estimated values of µFA were at the theoretical maximum of 1.0, whereas the values of FA spanned the interval from 0.0 to 0.8 as a result of varying orientational order of the anisotropic domains within each voxel. CONCLUSIONS: The proposed phantom can be manufactured by mixing three widely available chemicals in volumes comparable to a human head. The acquired data are in excellent agreement with theoretical predictions, showing that the phantom is ideal for validating methods for measuring microscopic diffusion anisotropy on clinical MRI systems. Magn Reson Med 79:1817-1828, 2018. © 2017 The Authors Magnetic Resonance in Medicine published by Wiley Periodicals, Inc. on behalf of International Society for Magnetic Resonance in Medicine. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes.
de Almeida Martins JP, Topgaard D. Multidimensional correlation of nuclear relaxation rates and diffusion tensors for model-free investigations of heterogeneous anisotropic porous materials. Sci Rep. 2018;8 (1) :2488.Abstract
Despite their widespread use in non-invasive studies of porous materials, conventional MRI methods yield ambiguous results for microscopically heterogeneous materials such as brain tissue. While the forward link between microstructure and MRI observables is well understood, the inverse problem of separating the signal contributions from different microscopic pores is notoriously difficult. Here, we introduce an experimental protocol where heterogeneity is resolved by establishing 6D correlations between the individual values of isotropic diffusivity, diffusion anisotropy, orientation of the diffusion tensor, and relaxation rates of distinct populations. Such procedure renders the acquired signal highly specific to the sample's microstructure, and allows characterization of the underlying pore space without prior assumptions on the number and nature of distinct microscopic environments. The experimental feasibility of the suggested method is demonstrated on a sample designed to mimic the properties of nerve tissue. If matched to the constraints of whole body scanners, this protocol could allow for the unconstrained determination of the different types of tissue that compose the living human brain.
Palmgren M, Hernebring M, Eriksson S, Elbing K, Geijer C, Lasič S, Dahl P, Hansen JS, Topgaard D, Lindkvist-Petersson K. Quantification of the intracellular life time of water molecules to measure transport rates of human aquaglyceroporins. J Membr Biol. 2017;250 (6) :629-639.Abstract
Orthodox aquaporins are transmembrane channel proteins that facilitate rapid diffusion of water, while aquaglyceroporins facilitate the diffusion of small uncharged molecules such as glycerol and arsenic trioxide. Aquaglyceroporins play important roles in human physiology, in particular for glycerol metabolism and arsenic detoxification. We have developed a unique system applying the strain of the yeast Pichia pastoris, where the endogenous aquaporins/aquaglyceroporins have been removed and human aquaglyceroporins AQP3, AQP7, and AQP9 are recombinantly expressed enabling comparative permeability measurements between the expressed proteins. Using a newly established Nuclear Magnetic Resonance approach based on measurement of the intracellular life time of water, we propose that human aquaglyceroporins are poor facilitators of water and that the water transport efficiency is similar to that of passive diffusion across native cell membranes. This is distinctly different from glycerol and arsenic trioxide, where high glycerol transport efficiency was recorded.
Eriksson S, Elbing K, Söderman O, Lindkvist-Petersson K, Topgaard D, Lasič S. NMR quantification of diffusional exchange in cell suspensions with relaxation rate differences between intra and extracellular compartments. PLoS One. 2017;12 (5) :e0177273.Abstract
Water transport across cell membranes can be measured non-invasively with diffusion NMR. We present a method to quantify the intracellular lifetime of water in cell suspensions with short transverse relaxation times, T2, and also circumvent the confounding effect of different T2 values in the intra- and extracellular compartments. Filter exchange spectroscopy (FEXSY) is specifically sensitive to exchange between compartments with different apparent diffusivities. Our investigation shows that FEXSY could yield significantly biased results if differences in T2 are not accounted for. To mitigate this problem, we propose combining FEXSY with diffusion-relaxation correlation experiment, which can quantify differences in T2 values in compartments with different diffusivities. Our analysis uses a joint constrained fitting of the two datasets and considers the effects of diffusion, relaxation and exchange in both experiments. The method is demonstrated on yeast cells with and without human aquaporins.
Lampinen B, Szczepankiewicz F, Mårtensson J, van Westen D, Sundgren PC, Nilsson M. Neurite density imaging versus imaging of microscopic anisotropy in diffusion MRI: A model comparison using spherical tensor encoding. Neuroimage. 2017;147 :517-531.Abstract
In diffusion MRI (dMRI), microscopic diffusion anisotropy can be obscured by orientation dispersion. Separation of these properties is of high importance, since it could allow dMRI to non-invasively probe elongated structures such as neurites (axons and dendrites). However, conventional dMRI, based on single diffusion encoding (SDE), entangles microscopic anisotropy and orientation dispersion with intra-voxel variance in isotropic diffusivity. SDE-based methods for estimating microscopic anisotropy, such as the neurite orientation dispersion and density imaging (NODDI) method, must thus rely on model assumptions to disentangle these features. An alternative approach is to directly quantify microscopic anisotropy by the use of variable shape of the b-tensor. Along those lines, we here present the 'constrained diffusional variance decomposition' (CODIVIDE) method, which jointly analyzes data acquired with diffusion encoding applied in a single direction at a time (linear tensor encoding, LTE) and in all directions (spherical tensor encoding, STE). We then contrast the two approaches by comparing neurite density estimated using NODDI with microscopic anisotropy estimated using CODIVIDE. Data were acquired in healthy volunteers and in glioma patients. NODDI and CODIVIDE differed the most in gray matter and in gliomas, where NODDI detected a neurite fraction higher than expected from the level of microscopic diffusion anisotropy found with CODIVIDE. The discrepancies could be explained by the NODDI tortuosity assumption, which enforces a connection between the neurite density and the mean diffusivity of tissue. Our results suggest that this assumption is invalid, which leads to a NODDI neurite density that is inconsistent between LTE and STE data. Using simulations, we demonstrate that the NODDI assumptions result in parameter bias that precludes the use of NODDI to map neurite density. With CODIVIDE, we found high levels of microscopic anisotropy in white matter, intermediate levels in structures such as the thalamus and the putamen, and low levels in the cortex and in gliomas. We conclude that accurate mapping of microscopic anisotropy requires data acquired with variable shape of the b-tensor.
Lampinen B, Szczepankiewicz F, van Westen D, Englund E, Sundgren PC, Lätt J, Ståhlberg F, Nilsson M. Optimal experimental design for filter exchange imaging: Apparent exchange rate measurements in the healthy brain and in intracranial tumors. Magn Reson Med. 2017;77 (3) :1104-1114.Abstract
PURPOSE: Filter exchange imaging (FEXI) is sensitive to the rate of diffusional water exchange, which depends, eg, on the cell membrane permeability. The aim was to optimize and analyze the ability of FEXI to infer differences in the apparent exchange rate (AXR) in the brain between two populations. METHODS: A FEXI protocol was optimized for minimal measurement variance in the AXR. The AXR variance was investigated by test-retest acquisitions in six brain regions in 18 healthy volunteers. Preoperative FEXI data and postoperative microphotos were obtained in six meningiomas and five astrocytomas. RESULTS: Protocol optimization reduced the coefficient of variation of AXR by approximately 40%. Test-retest AXR values were heterogeneous across normal brain regions, from 0.3 ± 0.2 s(-1) in the corpus callosum to 1.8 ± 0.3 s(-1) in the frontal white matter. According to analysis of statistical power, in all brain regions except one, group differences of 0.3-0.5 s(-1) in the AXR can be inferred using 5 to 10 subjects per group. An AXR difference of this magnitude was observed between meningiomas (0.6 ± 0.1 s(-1) ) and astrocytomas (1.0 ± 0.3 s(-1) ). CONCLUSIONS: With the optimized protocol, FEXI has the ability to infer relevant differences in the AXR between two populations for small group sizes. Magn Reson Med 77:1104-1114, 2017. © 2016 The Authors Magnetic Resonance in Medicine published by Wiley Periodicals, Inc. on behalf of International Society for Magnetic Resonance in Medicine. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.
Topgaard D. Multidimensional diffusion MRI. J Magn Reson. 2017;275 :98-113.Abstract
Principles from multidimensional NMR spectroscopy, and in particular solid-state NMR, have recently been transferred to the field of diffusion MRI, offering non-invasive characterization of heterogeneous anisotropic materials, such as the human brain, at an unprecedented level of detail. Here we revisit the basic physics of solid-state NMR and diffusion MRI to pinpoint the origin of the somewhat unexpected analogy between the two fields, and provide an overview of current diffusion MRI acquisition protocols and data analysis methods to quantify the composition of heterogeneous materials in terms of diffusion tensor distributions with size, shape, and orientation dimensions. While the most advanced methods allow estimation of the complete multidimensional distributions, simpler methods focus on various projections onto lower-dimensional spaces as well as determination of means and variances rather than actual distributions. Even the less advanced methods provide simple and intuitive scalar parameters that are directly related to microstructural features that can be observed in optical microscopy images, e.g. average cell eccentricity, variance of cell density, and orientational order - properties that are inextricably entangled in conventional diffusion MRI. Key to disentangling all these microstructural features is MRI signal acquisition combining isotropic and directional dimensions, just as in the field of multidimensional solid-state NMR from which most of the ideas for the new methods are derived.
Szczepankiewicz F, van Westen D, Englund E, Westin C-F, Ståhlberg F, Lätt J, Sundgren PC, Nilsson M. The link between diffusion MRI and tumor heterogeneity: Mapping cell eccentricity and density by diffusional variance decomposition (DIVIDE). Neuroimage. 2016;142 :522-532.Abstract
The structural heterogeneity of tumor tissue can be probed by diffusion MRI (dMRI) in terms of the variance of apparent diffusivities within a voxel. However, the link between the diffusional variance and the tissue heterogeneity is not well-established. To investigate this link we test the hypothesis that diffusional variance, caused by microscopic anisotropy and isotropic heterogeneity, is associated with variable cell eccentricity and cell density in brain tumors. We performed dMRI using a novel encoding scheme for diffusional variance decomposition (DIVIDE) in 7 meningiomas and 8 gliomas prior to surgery. The diffusional variance was quantified from dMRI in terms of the total mean kurtosis (MKT), and DIVIDE was used to decompose MKT into components caused by microscopic anisotropy (MKA) and isotropic heterogeneity (MKI). Diffusion anisotropy was evaluated in terms of the fractional anisotropy (FA) and microscopic fractional anisotropy (μFA). Quantitative microscopy was performed on the excised tumor tissue, where structural anisotropy and cell density were quantified by structure tensor analysis and cell nuclei segmentation, respectively. In order to validate the DIVIDE parameters they were correlated to the corresponding parameters derived from microscopy. We found an excellent agreement between the DIVIDE parameters and corresponding microscopy parameters; MKA correlated with cell eccentricity (r=0.95, p<10(-7)) and MKI with the cell density variance (r=0.83, p<10(-3)). The diffusion anisotropy correlated with structure tensor anisotropy on the voxel-scale (FA, r=0.80, p<10(-3)) and microscopic scale (μFA, r=0.93, p<10(-6)). A multiple regression analysis showed that the conventional MKT parameter reflects both variable cell eccentricity and cell density, and therefore lacks specificity in terms of microstructure characteristics. However, specificity was obtained by decomposing the two contributions; MKA was associated only to cell eccentricity, and MKI only to cell density variance. The variance in meningiomas was caused primarily by microscopic anisotropy (mean±s.d.) MKA=1.11±0.33 vs MKI=0.44±0.20 (p<10(-3)), whereas in the gliomas, it was mostly caused by isotropic heterogeneity MKI=0.57±0.30 vs MKA=0.26±0.11 (p<0.05). In conclusion, DIVIDE allows non-invasive mapping of parameters that reflect variable cell eccentricity and density. These results constitute convincing evidence that a link exists between specific aspects of tissue heterogeneity and parameters from dMRI. Decomposing effects of microscopic anisotropy and isotropic heterogeneity facilitates an improved interpretation of tumor heterogeneity as well as diffusion anisotropy on both the microscopic and macroscopic scale.
Topgaard D. NMR methods for studying microscopic diffusion anisotropy. In: Diffusion NMR of Confined Systems: Fluid Transport in Porous Solids and Heterogeneous Materials. Cambridge, UK: Royal Society of Chemistry ; 2016. Publisher's VersionAbstract

The structure of a porous material is imprinted on the pattern of translational diffusion of the liquids located in the pore space. Non-invasive NMR/MRI measurements of self-diffusion are thus an attractive means of studying the microscopic structure of non-transparent and sensitive materials such as the living human brain, e.g., to detect pathological conditions or to study normal brain development. Conventional methods give information that is averaged over the entire investigated sample or volume element, and fails to distinguish between properties such as pore size, shape, and orientation in case the material is heterogeneous. This chapter reviews recent progress in the field of advanced diffusion NMR/MRI methods to disentangle the confounding effects of the various pore space parameters, thereby enabling detailed characterization of anisotropic materials with complex architecture.

Ning L, Westin C-F, Rathi Y. Estimation of Bounded and Unbounded Trajectories in Diffusion MRI. Front Neurosci. 2016;10 :129.Abstract
Disentangling the tissue microstructural information from the diffusion magnetic resonance imaging (dMRI) measurements is quite important for extracting brain tissue specific measures. The autocorrelation function of diffusing spins is key for understanding the relation between dMRI signals and the acquisition gradient sequences. In this paper, we demonstrate that the autocorrelation of diffusion in restricted or bounded spaces can be well approximated by exponential functions. To this end, we propose to use the multivariate Ornstein-Uhlenbeck (OU) process to model the matrix-valued exponential autocorrelation function of three-dimensional diffusion processes with bounded trajectories. We present detailed analysis on the relation between the model parameters and the time-dependent apparent axon radius and provide a general model for dMRI signals from the frequency domain perspective. For our experimental setup, we model the diffusion signal as a mixture of two compartments that correspond to diffusing spins with bounded and unbounded trajectories, and analyze the corpus-callosum in an ex-vivo data set of a monkey brain.
Ning L, Setsompop K, Michailovich O, Makris N, Shenton ME, Westin C-F, Rathi Y. A joint compressed-sensing and super-resolution approach for very high-resolution diffusion imaging. Neuroimage. 2016;125 :386-400.Abstract
Diffusion MRI (dMRI) can provide invaluable information about the structure of different tissue types in the brain. Standard dMRI acquisitions facilitate a proper analysis (e.g. tracing) of medium-to-large white matter bundles. However, smaller fiber bundles connecting very small cortical or sub-cortical regions cannot be traced accurately in images with large voxel sizes. Yet, the ability to trace such fiber bundles is critical for several applications such as deep brain stimulation and neurosurgery. In this work, we propose a novel acquisition and reconstruction scheme for obtaining high spatial resolution dMRI images using multiple low resolution (LR) images, which is effective in reducing acquisition time while improving the signal-to-noise ratio (SNR). The proposed method called compressed-sensing super resolution reconstruction (CS-SRR), uses multiple overlapping thick-slice dMRI volumes that are under-sampled in q-space to reconstruct diffusion signal with complex orientations. The proposed method combines the twin concepts of compressed sensing and super-resolution to model the diffusion signal (at a given b-value) in a basis of spherical ridgelets with total-variation (TV) regularization to account for signal correlation in neighboring voxels. A computationally efficient algorithm based on the alternating direction method of multipliers (ADMM) is introduced for solving the CS-SRR problem. The performance of the proposed method is quantitatively evaluated on several in-vivo human data sets including a true SRR scenario. Our experimental results demonstrate that the proposed method can be used for reconstructing sub-millimeter super resolution dMRI data with very good data fidelity in clinically feasible acquisition time.
Yolcu C, Memiç M, Şimşek K, Westin C-F, Özarslan E. NMR signal for particles diffusing under potentials: From path integrals and numerical methods to a model of diffusion anisotropy. Phys Rev E. 2016;93 (5-1) :052602.Abstract
We study the influence of diffusion on NMR experiments when the molecules undergo random motion under the influence of a force field and place special emphasis on parabolic (Hookean) potentials. To this end, the problem is studied using path integral methods. Explicit relationships are derived for commonly employed gradient waveforms involving pulsed and oscillating gradients. The Bloch-Torrey equation, describing the temporal evolution of magnetization, is modified by incorporating potentials. A general solution to this equation is obtained for the case of parabolic potential by adopting the multiple correlation function (MCF) formalism, which has been used in the past to quantify the effects of restricted diffusion. Both analytical and MCF results were found to be in agreement with random walk simulations. A multidimensional formulation of the problem is introduced that leads to a new characterization of diffusion anisotropy. Unlike the case of traditional methods that employ a diffusion tensor, anisotropy originates from the tensorial force constant, and bulk diffusivity is retained in the formulation. Our findings suggest that some features of the NMR signal that have traditionally been attributed to restricted diffusion are accommodated by the Hookean model. Under certain conditions, the formalism can be envisioned to provide a viable approximation to the mathematically more challenging restricted diffusion problems.
Lasič S, Oredsson S, Partridge SC, Saal LH, Topgaard D, Nilsson M, Bryskhe K. Apparent exchange rate for breast cancer characterization. NMR Biomed. 2016;29 (5) :631-9.Abstract
Although diffusion MRI has shown promise for the characterization of breast cancer, it has low specificity to malignant subtypes. Higher specificity might be achieved if the effects of cell morphology and molecular exchange across cell membranes could be disentangled. The quantification of exchange might thus allow the differentiation of different types of breast cancer cells. Based on differences in diffusion rates between the intra- and extracellular compartments, filter exchange spectroscopy/imaging (FEXSY/FEXI) provides non-invasive quantification of the apparent exchange rate (AXR) of water between the two compartments. To test the feasibility of FEXSY for the differentiation of different breast cancer cells, we performed experiments on several breast epithelial cell lines in vitro. Furthermore, we performed the first in vivo FEXI measurement of water exchange in human breast. In cell suspensions, pulsed gradient spin-echo experiments with large b values and variable pulse duration allow the characterization of the intracellular compartment, whereas FEXSY provides a quantification of AXR. These experiments are very sensitive to the physiological state of cells and can be used to establish reliable protocols for the culture and harvesting of cells. Our results suggest that different breast cancer subtypes can be distinguished on the basis of their AXR values in cell suspensions. Time-resolved measurements allow the monitoring of the physiological state of cells in suspensions over the time-scale of hours, and reveal an abrupt disintegration of the intracellular compartment. In vivo, exchange can be detected in a tumor, whereas, in normal tissue, the exchange rate is outside the range experimentally accessible for FEXI. At present, low signal-to-noise ratio and limited scan time allows the quantification of AXR only in a region of interest of relatively large tumors. © 2016 The Authors. NMR in Biomedicine published by John Wiley & Sons Ltd.
Topgaard D. Director orientations in lyotropic liquid crystals: diffusion MRI mapping of the Saupe order tensor. Phys Chem Chem Phys. 2016;18 (12) :8545-53.Abstract
The macroscopic physical properties of a liquid crystalline material depend on both the properties of the individual crystallites and the details of their spatial arrangement. We propose a diffusion MRI method to estimate the director orientations of a lyotropic liquid crystal as a spatially resolved field of Saupe order tensors. The method relies on varying the shape of the diffusion-encoding tensor to disentangle the effects of voxel-scale director orientational order and the local diffusion anisotropy of the solvent. Proof-of-concept experiments are performed on water in lamellar and reverse hexagonal liquid crystalline systems with intricate patterns of director orientations.
Westin C-F, Knutsson H, Pasternak O, Szczepankiewicz F, Özarslan E, van Westen D, Mattisson C, Bogren M, O'Donnell LJ, Kubicki M, et al. Q-space trajectory imaging for multidimensional diffusion MRI of the human brain. Neuroimage. 2016;135 :345-62.Abstract
This work describes a new diffusion MR framework for imaging and modeling of microstructure that we call q-space trajectory imaging (QTI). The QTI framework consists of two parts: encoding and modeling. First we propose q-space trajectory encoding, which uses time-varying gradients to probe a trajectory in q-space, in contrast to traditional pulsed field gradient sequences that attempt to probe a point in q-space. Then we propose a microstructure model, the diffusion tensor distribution (DTD) model, which takes advantage of additional information provided by QTI to estimate a distributional model over diffusion tensors. We show that the QTI framework enables microstructure modeling that is not possible with the traditional pulsed gradient encoding as introduced by Stejskal and Tanner. In our analysis of QTI, we find that the well-known scalar b-value naturally extends to a tensor-valued entity, i.e., a diffusion measurement tensor, which we call the b-tensor. We show that b-tensors of rank 2 or 3 enable estimation of the mean and covariance of the DTD model in terms of a second order tensor (the diffusion tensor) and a fourth order tensor. The QTI framework has been designed to improve discrimination of the sizes, shapes, and orientations of diffusion microenvironments within tissue. We derive rotationally invariant scalar quantities describing intuitive microstructural features including size, shape, and orientation coherence measures. To demonstrate the feasibility of QTI on a clinical scanner, we performed a small pilot study comparing a group of five healthy controls with five patients with schizophrenia. The parameter maps derived from QTI were compared between the groups, and 9 out of the 14 parameters investigated showed differences between groups. The ability to measure and model the distribution of diffusion tensors, rather than a quantity that has already been averaged within a voxel, has the potential to provide a powerful paradigm for the study of complex tissue architecture.
Ahlgren A, Knutsson L, Wirestam R, Nilsson M, Ståhlberg F, Topgaard D, Lasič S. Quantification of microcirculatory parameters by joint analysis of flow-compensated and non-flow-compensated intravoxel incoherent motion (IVIM) data. NMR Biomed. 2016;29 (5) :640-9.Abstract
The aim of this study was to improve the accuracy and precision of perfusion fraction and blood velocity dispersion estimates in intravoxel incoherent motion (IVIM) imaging, using joint analysis of flow-compensated and non-flow-compensated motion-encoded MRI data. A double diffusion encoding sequence capable of switching between flow-compensated and non-flow-compensated encoding modes was implemented. In vivo brain data were collected in eight healthy volunteers and processed using the joint analysis. Simulations were used to compare the performance of the proposed analysis method with conventional IVIM analysis. With flow compensation, strong rephasing was observed for the in vivo data, approximately cancelling the IVIM effect. The joint analysis yielded physiologically reasonable perfusion fraction maps. Estimated perfusion fractions were 2.43 ± 0.81% in gray matter, 1.81 ± 0.90% in deep gray matter, and 1.64 ± 0.72% in white matter (mean ± SD, n = 8). Simulations showed improved accuracy and precision when using joint analysis of flow-compensated and non-flow-compensated data, compared with conventional IVIM analysis. Double diffusion encoding with flow compensation was feasible for in vivo imaging of the perfusion fraction in the brain. The strong rephasing implied that blood flowing through the cerebral microvascular system was closer to the ballistic limit than the diffusive limit. © 2016 The Authors NMR in Biomedicine published by John Wiley & Sons Ltd.
de Almeida Martins JP, Topgaard D. Two-dimensional correlation of isotropic and directional diffusion using NMR. Phys Rev Lett. 2016;116 (8) :087601.Abstract

Diffusion nuclear magnetic resonance (NMR) is a powerful technique for studying porous media, but yields ambiguous results when the sample comprises multiple regions with different pore sizes, shapes, and orientations. Inspired by solid-state NMR techniques for correlating isotropic and anisotropic chemical shifts, we propose a diffusion NMR method to resolve said ambiguity. Numerical data inversion relies on sparse representation of the data in a basis of radial and axial diffusivities. Experiments are performed on a composite sample with a cell suspension and a liquid crystal.

Özarslan E, Westin C-F, Mareci TH. Characterizing magnetic resonance signal decay due to Gaussian diffusion: the path integral approach and a convenient computational method. Concepts Magn Reson Part A Bridg Educ Res. 2015;44 (4) :203-213.Abstract
The influence of Gaussian diffusion on the magnetic resonance signal is determined by the apparent diffusion coefficient (ADC) and tensor (ADT) of the diffusing fluid as well as the gradient waveform applied to sensitize the signal to diffusion. Estimations of ADC and ADT from diffusion-weighted acquisitions necessitate computations of, respectively, the b-value and b-matrix associated with the employed pulse sequence. We establish the relationship between these quantities and the gradient waveform by expressing the problem as a path integral and explicitly evaluating it. Further, we show that these important quantities can be conveniently computed for any gradient waveform using a simple algorithm that requires a few lines of code. With this representation, our technique complements the multiple correlation function (MCF) method commonly used to compute the effects of restricted diffusion, and provides a consistent and convenient framework for studies that aim to infer the microstructural features of the specimen.