Steps on the Path to Clinical Translation: A workshop by the British and Irish Chapter of the ISMRM.

The British and Irish Chapter of the International Society for Magnetic Resonance in Medicine (BIC-ISMRM) held a workshop entitled "Steps on the path to clinical translation" in Cardiff, UK, on 7th September 2022. The aim of the workshop was to promote discussion within the MR community about the problems and potential solutions for translating quantitative MR (qMR) imaging and spectroscopic biomarkers into clinical application and drug studies. Invited speakers presented the perspectives of radiologists, radiographers, clinical physicists, vendors, imaging Contract/Clinical Research Organizations (CROs), open science networks, metrologists, imaging networks, and those developing consensus methods. A round-table discussion was held in which workshop participants discussed a range of questions pertinent to clinical translation of qMR imaging and spectroscopic biomarkers. Each group summarized their findings via three main conclusions and three further questions. These questions were used as the basis of an online survey of the broader UK MR community.

Comparative analysis of signal models for microscopic fractional anisotropy estimation using q-space trajectory encoding.

Microscopic diffusion anisotropy imaging using diffusion-weighted MRI and multidimensional diffusion encoding is a promising method for quantifying clinically and scientifically relevant microstructural properties of neural tissue. Several methods for estimating microscopic fractional anisotropy (µFA), a normalized measure of microscopic diffusion anisotropy, have been introduced but the differences between the methods have received little attention thus far. In this study, the accuracy and precision of µFA estimation using q-space trajectory encoding and different signal models were assessed using imaging experiments and simulations. Three healthy volunteers and a microfibre phantom were imaged with five non-zero b-values and gradient waveforms encoding linear and spherical b-tensors. Since the ground-truth µFA was unknown in the imaging experiments, Monte Carlo random walk simulations were performed using axon-mimicking fibres for which the ground truth was known. Furthermore, parameter bias due to time-dependent diffusion was quantified by repeating the simulations with tuned waveforms, which have similar power spectra, and with triple diffusion encoding, which, unlike q-space trajectory encoding, is not based on the assumption of time-independent diffusion. The truncated cumulant expansion of the powder-averaged signal, gamma-distributed diffusivities assumption, and q-space trajectory imaging, a generalization of the truncated cumulant expansion to individual signals, were used to estimate µFA. The gamma-distributed diffusivities assumption consistently resulted in greater µFA values than the second order cumulant expansion, 0.1 greater when averaged over the whole brain. In the simulations, the generalized cumulant expansion provided the most accurate estimates. Importantly, although time-dependent diffusion caused significant overestimation of µFA using all the studied methods, the simulations suggest that the resulting bias in µFA is less than 0.1 in human white matter.

Clinically feasible diffusion MRI in muscle: Time dependence and initial findings in Duchenne muscular dystrophy.

PURPOSE: To characterize the diffusion time-dependence in muscle in healthy adult volunteers, boys with Duchenne's muscular dystrophy (DMD), and age-matched controls in a clinically feasible acquisition time for pediatric applications. METHODS: Diffusion data were acquired using a pulsed gradient stimulated echo diffusion preparation at 5 different diffusion times (70, 130, 190, 250, and 330 ms), at 4 different b-values (0, 200, 400, 600, and 800 s/mm2 ) and 6 directions (orthogonal x, y, and z and diagonal xy, xz, and yz) and processed to obtain standard diffusion indices (mean diffusivity [MD] and fractional anisotropy [FA]) at each diffusion time. RESULTS: Time-dependent diffusion was seen in muscle in healthy adult volunteers, boys with DMD, and age-matched controls. Boys with DMD showed reduced MD and increased FA values in comparison to age matched controls across a range of diffusion times. A diffusion time of Δ = 190 ms had the largest effect size. CONCLUSIONS: These results could be used to optimize diffusion imaging in this disease further and imply that these diffusion indices may become an important biomarker in monitoring progression in DMD in the future.

Quasi-diffusion magnetic resonance imaging (QDI): A fast, high b-value diffusion imaging technique.

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 â€‹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 â€‹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.

Validation and noise robustness assessment of microscopic anisotropy estimation with clinically feasible double diffusion encoding MRI.

PURPOSE: Double diffusion encoding (DDE) MRI enables the estimation of microscopic diffusion anisotropy, yielding valuable information on tissue microstructure. A recent study proposed that the acquisition of rotationally invariant DDE metrics, typically obtained using a spherical "5-design," could be greatly simplified by assuming Gaussian diffusion, facilitating reduced acquisition times that are more compatible with clinical settings. Here, we aim to validate the new minimal acquisition scheme against the standard DDE 5-design, and to quantify the proposed method's noise robustness to facilitate future clinical use. THEORY AND METHODS: DDE MRI experiments were performed on both ex vivo and in vivo rat brains at 9.4 T using the 5-design and the proposed minimal design and taking into account the difference in the number of acquisitions. The ensuing microscopic fractional anisotropy (µFA) maps were compared over a range of b-values up to 5000 s/mm2 . Noise robustness was studied using analytical calculations and numerical simulations. RESULTS: The minimal protocol quantified µFA at an accuracy comparable to the estimates obtained by means of the more theoretically robust DDE 5-design. µFA's sensitivity to noise was found to strongly depend on compartment anisotropy and tensor magnitude in a nonlinear manner. When µFA < 0.75 or when mean diffusivity is particularly low, very high signal-to-noise ratio is required for precise quantification of µFA. CONCLUSION: Our work supports using DDE for quantifying microscopic diffusion anisotropy in clinical settings but raises hitherto overlooked precision issues when measuring µFA with DDE and typical clinical signal-to-noise ratio.

Measuring diffusion exchange across the cell membrane with DEXSY (Diffusion Exchange Spectroscopy).

INTRODUCTION: To combine numerical simulations, in vitro and in vivo experiments to evaluate the feasibility of measuring diffusion exchange across the cell membrane with diffusion exchange spectroscopy (DEXSY). METHODS: DEXSY acquisitions were simulated over a range of permeabilities in nerve tissue and yeast substrates. In vitro measurements were performed in a yeast substrate and in vivo measurements in mouse tumor xenograft models, all at 9.4 T. RESULTS: Diffusion exchange was observed in simulations over a physiologically relevant range of cell permeability values. In vitro and in vivo measures also provided evidence of diffusion exchange, which was quantified with the Diffusion Exchange Index (DEI). CONCLUSIONS: Our findings provide preliminary evidence that DEXSY can be used to make in vivo measurements of diffusion exchange and cell membrane permeability.

Time-dependent diffusion MRI as a probe of microstructural changes in a mouse model of Duchenne muscular dystrophy.

Dystrophic muscles show a high variability of fibre sizes and altered sarcolemmal integrity, which are typically assessed by histology. Time-dependent diffusion MRI is sensitive to tissue microstructure and its investigation through age-related changes in dystrophic and healthy muscles may help the understanding of the onset and progression of Duchenne muscular dystrophy (DMD). We investigated the capability of time-dependent diffusion MRI to quantify age and disease-related changes in hind-limb muscle microstructure between dystrophic (mdx) and wild-type (WT) mice of three age groups (7.5, 22 and 44 weeks). Diffusion time-dependent apparent diffusion coefficients (ADCs) of the gastrocnemius and tibialis anterior muscles were determined versus age and diffusion-gradient orientation at six diffusion times (Δ; range: 25-350 ms). Mean muscle ADCs were compared between groups and ages, and correlated with T2 , using Student's t test, one-way analysis of variance and Pearson correlation, respectively. Muscle fibre sizes and sarcolemmal integrity were evaluated by histology and compared with diffusion measurements. Hind-limb muscle ADC showed characteristic restricted diffusion behaviour in both mdx and WT animals with decreasing ADC values at longer Δ. Significant differences in ADC were observed at long Δ values (≥ 250 ms; p < 0.05, comparison between groups; p < 0.01, comparison between ages) with ADC increased by 5-15% in dystrophic muscles, indicative of reduced diffusion restriction. No significant correlation was found between T2 and ADC. Additionally, muscle fibre size distributions showed higher variability and lower mean fibre size in mdx than WT animals (p < 0.001). The extensive Evans Blue Dye uptake shown in dystrophic muscles revealed substantial sarcolemmal damage, suggesting diffusion measurements as more consistent with altered permeability rather than changes in muscle fibre sizes. This study shows the potential of diffusion MRI to non-invasively discriminate between dystrophic and healthy muscles with enhanced sensitivity when using long Δ.

In vivo demonstration of microscopic anisotropy in the human kidney using multidimensional diffusion MRI.

PURPOSE: To demonstrate the feasibility of multidimensional diffusion MRI to probe and quantify microscopic fractional anisotropy (µFA) in human kidneys in vivo. METHODS: Linear tensor encoded (LTE) and spherical tensor encoded (STE) renal diffusion MRI scans were performed in 10 healthy volunteers. Respiratory triggering and image registration were used to minimize motion artefacts during the acquisition. Kidney cortex-medulla were semi-automatically segmented based on fractional anisotropy (FA) values. A model-free analysis of LTE and STE signal dependence on b-value in the renal cortex and medulla was performed. Subsequently, µFA was estimated using a single-shell approach. Finally, a comparison of conventional FA and µFA is shown. RESULTS: The hallmark effect of µFA (divergence of LTE and STE signal with increasing b-value) was observed in all subjects. A statistically significant difference between LTE and STE signal was found in the cortex and medulla, starting from b = 750 s/mm2 and b = 500 s/mm2 , respectively. This difference was maximal at the highest b-value sampled (b = 1000 s/mm2 ) which suggests that relatively high b-values are required for µFA mapping in the kidney compared to conventional FA. Cortical and medullary µFA were, respectively, 0.53 ± 0.09 and 0.65 ± 0.05, both respectively higher than conventional FA (0.19 ± 0.02 and 0.40 ± 0.02). CONCLUSION: The feasibility of combining LTE and STE diffusion MRI to probe and quantify µFA in human kidneys is demonstrated for the first time. By doing so, we show that novel microstructure information-not accessible by conventional diffusion encoding-can be probed by multidimensional diffusion MRI. We also identify relevant technical limitations that warrant further development of the technique for body MRI.

The effects of ageing on mouse muscle microstructure: a comparative study of time-dependent diffusion MRI and histological assessment.

The investigation of age-related changes in muscle microstructure between developmental and healthy adult mice may help us to understand the clinical features of early-onset muscle diseases, such as Duchenne muscular dystrophy. We investigated the evolution of mouse hind-limb muscle microstructure using diffusion imaging of in vivo and in vitro samples from both actively growing and mature mice. Mean apparent diffusion coefficients (ADCs) of the gastrocnemius and tibialis anterior muscles were determined as a function of diffusion time (Δ), age (7.5, 22 and 44 weeks) and diffusion gradient direction, applied parallel or transverse to the principal axis of the muscle fibres. We investigated a wide range of diffusion times with the goal of probing a range of diffusion lengths characteristic of muscle microstructure. We compared the diffusion time-dependent ADC of hind-limb muscles with histology. ADC was found to vary as a function of diffusion time in muscles at all stages of maturation. Muscle water diffusivity was higher in younger (7.5 weeks) than in adult (22 and 44 weeks) mice, whereas no differences were observed between the older ages. In vitro data showed the same diffusivity pattern as in vivo data. The highlighted differences in diffusion properties between young and mature muscles suggested differences in underlying muscle microstructure, which were confirmed by histological assessment. In particular, although diffusion was more restricted in older muscle, muscle fibre size increased significantly from young to adult age. The extracellular space decreased with age by only ~1%. This suggests that the observed diffusivity differences between young and adult muscles may be caused by increased membrane permeability in younger muscle associated with properties of the sarcolemma.

Machine learning based compartment models with permeability for white matter microstructure imaging.

Some microstructure parameters, such as permeability, remain elusive because mathematical models that express their relationship to the MR signal accurately are intractable. Here, we propose to use computational models learned from simulations to estimate these parameters. We demonstrate the approach in an example which estimates water residence time in brain white matter. The residence time τi of water inside axons is a potentially important biomarker for white matter pathologies of the human central nervous system, as myelin damage is hypothesised to affect axonal permeability, and thus τi. We construct a computational model using Monte Carlo simulations and machine learning (specifically here a random forest regressor) in order to learn a mapping between features derived from diffusion weighted MR signals and ground truth microstructure parameters, including τi. We test our numerical model using simulated and in vivo human brain data. Simulation results show that estimated parameters have strong correlations with the ground truth parameters (R2={0.88,0.95,0.82,0.99}) for volume fraction, residence time, axon radius and diffusivity respectively), and provide a marked improvement over the most widely used Kärger model (R2={0.75,0.60,0.11,0.99}). The trained model also estimates sensible microstructure parameters from in vivo human brain data acquired from healthy controls, matching values found in literature, and provides better reproducibility than the Kärger model on both the voxel and ROI level. Finally, we acquire data from two Multiple Sclerosis (MS) patients and compare to the values in healthy subjects. We find that in the splenium of corpus callosum (CC-S) the estimate of the residence time is 0.57±0.05s for the healthy subjects, while in the MS patient with a lesion in CC-S it is 0.33±0.12s in the normal appearing white matter (NAWM) and 0.19±0.11s in the lesion. In the corticospinal tracts (CST) the estimate of the residence time is 0.52±0.09s for the healthy subjects, while in the MS patient with a lesion in CST it is 0.56±0.05s in the NAWM and 0.13±0.09s in the lesion. These results agree with our expectations that the residence time in lesions would be lower than in NAWM because the loss of myelin should increase permeability. Overall, we find parameter estimates in the two MS patients consistent with expectations from the pathology of MS lesions demonstrating the clinical potential of this new technique.

Diffusion in hierarchical systems: A simulation study in models of healthy and diseased muscle tissue.

PURPOSE: To investigate the sensitivity of diffusion-MR signal to microstructural change in muscle tissue associated with pathology, and recommend optimal acquisition parameters. METHODS: We employ Monte-Carlo simulation of diffusing spins in hierarchical tissue to generate synthetic diffusion-weighted signal curves over a wide range of scan parameters. Curves are analyzed using entropy-a measure of curve complexity. Entropy change between a baseline and various microstructural scenarios is investigated. We find acquisitions that optimize entropy difference in each scenario. RESULTS: Permeability changes have a large effect on the diffusion-weighted signal curve. Muscle fiber atrophy is also important, although differentiating between mechanisms is challenging. Several acquisitions over a range of diffusion times is optimal for imaging microstructural change in muscle tissue. Sensitivity to permeability is optimized for high gradient strengths, but sensitivity to other scenarios is optimal at other values. CONCLUSIONS: The diffusion-attenuated signal is sensitive to the microstructural changes, but the changes are subtle. Taking full advantage of the changes to the overall curve requires a set of acquisitions over a range of diffusion times. Permeability causes the largest changes, but even the very subtle changes associated with fiber radius distribution change the curves more than noise alone. Magn Reson Med 78:1187-1198, 2017. © 2016 International Society for Magnetic Resonance in Medicine.

Assessment of non-Gaussian diffusion with singly and doubly stretched biexponential models of diffusion-weighted MRI (DWI) signal attenuation in prostate tissue.

Non-Gaussian diffusion dynamics was investigated in the two distinct water populations identified by a biexponential model of diffusion in prostate tissue. Diffusion-weighted MRI (DWI) signal attenuation was measured ex vivo in two formalin-fixed prostates at 9.4 T with diffusion times Δ = 10, 20 and 40 ms, and b values in the range 0.017-8.2 ms/µm(2) . A conventional biexponential model was compared with models in which either the lower diffusivity component or both of the components of the biexponential were stretched. Models were compared using Akaike's Information Criterion (AIC) and a leave-one-out (LOO) test of model prediction accuracy. The doubly stretched (SS) model had the highest LOO prediction accuracy and lowest AIC (highest information content) in the majority of voxels at Δ = 10 and 20 ms. The lower diffusivity stretching factor (α2 ) of the SS model was consistently lower (range ~0.3-0.9) than the higher diffusivity stretching factor (α1 , range ~0.7-1.1), indicating a high degree of diffusion heterogeneity in the lower diffusivity environment, and nearly Gaussian diffusion in the higher diffusivity environment. Stretched biexponential models demonstrate that, in prostate tissue, the two distinct water populations identified by the simple biexponential model individually exhibit non-Gaussian diffusion dynamics.

Contrast and stability of the axon diameter index from microstructure imaging with diffusion MRI.

The ActiveAx technique fits the minimal model of white matter diffusion to diffusion MRI data acquired using optimized protocols that provide orientationally invariant indices of axon diameter and density. We investigated how limitations of the available maximal gradient strength (Gmax) on a scanner influence the sensitivity to a range of axon diameters. Multishell high-angular-diffusion-imaging (HARDI) protocols for Gmax of 60, 140, 200, and 300 mT/m were optimized for the pulsed-gradient-spin-echo (PGSE) sequence. Data were acquired on a fixed monkey brain and Monte-Carlo simulations supported the results. Increasing Gmax reduces within-voxel variation of the axon diameter index and improves contrast beyond what is achievable with higher signal-to-noise ratio. Simulations reveal an upper bound on the axon diameter (∼10 µm) that pulsed-gradient-spin-echo measurements are sensitive to, due to a trade-off between short T2 and the long diffusion time needed to probe larger axon diameters. A lower bound (∼2.5 µm) slightly dependent on Gmax was evident, below which axon diameters are identifiable as small, but impossible to differentiate. These results emphasize the key-role of Gmax for enhancing contrast between axon diameter distributions and are, therefore, relevant in general for microstructure imaging methods and highlight the need for increased Gmax on future commercial systems.

Compartment models of the diffusion MR signal in brain white matter: a taxonomy and comparison.

This paper aims to identify the minimum requirements for an accurate model of the diffusion MR signal in white matter of the brain. We construct a taxonomy of multi-compartment models of white matter from combinations of simple models for the intra- and the extra-axonal spaces. We devise a new diffusion MRI protocol that provides measurements with a wide range of imaging parameters for diffusion sensitization both parallel and perpendicular to white matter fibres. We use the protocol to acquire data from two fixed rat brains, which allows us to fit, study and compare the different models. The study examines a total of 47 analytic models, including several well-used models from the literature, which we place within the taxonomy. The results show that models that incorporate intra-axonal restriction, such as ball and stick or CHARMED, generally explain the data better than those that do not, such as the DT or the biexponential models. However, three-compartment models which account for restriction parallel to the axons and incorporate pore size explain the measurements most accurately. The best fit comes from combining a full diffusion tensor (DT) model of the extra-axonal space with a cylindrical intra-axonal component of single radius and a third spherical compartment of non-zero radius. We also measure the stability of the non-zero radius intra-axonal models and find that single radius intra-axonal models are more stable than gamma distributed radii models with similar fitting performance.

Two-step anomalous diffusion tensor imaging.

We extend the formalism of anomalous diffusion imaging to include directional anisotropy of fitted parameters. The resulting technique is termed anomalous diffusion tensor imaging (aDTI), and allows the directional properties of the distributed diffusion coefficient (α) and the anomalous diffusion exponent, (γ) to be analysed using the same analytical techniques as regular diffusion tensor imaging (DTI). Together, these parameters quantify the rate of diffusion (α) and the complexity of the diffusion environment (γ). We generated tensor images for the anomalous exponent tensor (Γ) and distributed diffusivity tensor (A) from in vivo human brain data and present images of eigenvalues, eigenvectors, Trace/3 (Tr), fractional anisotropy (FA) and tensor shape measures. In white matter, A is found to have a median Tr = 0.56 × 10(- 3) mm(2) s(- 1), FA = 0.58 and Γ Tr = 0.69, FA = 0.13. We observed that white matter shows a similar anisotropic geometry for the distributed diffusion tensor as for the regular diffusion tensor, whereas the anomalous exponent tensor exhibits a different shape characteristic which may be informative of tissue microstructure.

Clinical translation of quantitative magnetic resonance imaging biomarkers - An overview and gap analysis of current practice.

PURPOSE: This overview of the current landscape of quantitative magnetic resonance imaging biomarkers (qMR IBs) aims to support the standardisation of academic IBs to assist their translation to clinical practice. METHODS: We used three complementary approaches to investigate qMR IB use and quality management practices within the UK: 1) a literature search of qMR and quality management terms during 2011-2015 and 2016-2020; 2) a database search for clinical research studies using qMR IBs during 2016-2020; and 3) a survey to ascertain the current availability and quality management practices for clinical MRI scanners and associated equipment at research institutions across the UK. RESULTS: The analysis showed increased use of all qMR methods between the periods 2011-2015 and 2016-2020 and diffusion-tensor MRI and volumetry to be popular methods. However, the "translation ratio" of journal articles to clinical research studies was higher for qMR methods that have evidence of clinical translation via a commercial route, such as fat fraction and T2 mapping. The number of journal articles citing quality management terms doubled between the periods 2011-2015 and 2016-2020; although, its proportion relative to all journal articles only increased by 3.0%. The survey suggested that quality assurance (QA) and quality control (QC) of data acquisition procedures are under-reported in the literature and that QA/QC of acquired data/data analysis are under-developed and lack consistency between institutions. CONCLUSIONS: We summarise current attempts to standardise and translate qMR IBs, and conclude by outlining the ideal quality management practices and providing a gap analysis between current practice and a metrological standard.

Clinical quantitative MRI and the need for metrology.

MRI has been an essential diagnostic tool in healthcare for several decades. It offers unique insights into most tissues without the need for ionising radiation. Historically, MRI has been predominantly used qualitatively, images are formed to allow visual discrimination of tissues types and pathologies, rather than providing quantitative measurements. Increasingly, quantitative MRI (qMRI) is also finding clinical application, where images provide the basis for physical measurements of, e.g. tissue volume measures and represent aspects of tissue composition and microstructure. This article reviews some common current research and clinical applications of qMRI from the perspective of measurement science. qMRI not only offers additional information for radiologists, but also the opportunity for improved harmonisation and calibration between scanners and as such it is well-suited to large-scale investigations such as clinical trials and longitudinal studies. Realising these benefits, however, presents a new kind of technical challenge to MRI practioners. When measuring a parameter quantitatively, it is crucial that the reliability and reproducibility of the technique are well understood. Strictly speaking, a numerical result of a measurement is meaningless unless it is accompanied by a description of the associated measurement uncertainty. It is therefore necessary to produce not just estimates of physical properties in a quantitative image, but also their associated uncertainties. As the process of determining a physical property from the raw MR signal is complicated and multistep, estimation of uncertainty is challenging and there are many aspects of the MRI process that require validation. With the clinical implementation of qMRI techniques and its continued expansion, there is a clear and urgent need for metrology in this field.

Orientationally invariant indices of axon diameter and density from diffusion MRI.

This paper proposes and tests a technique for imaging orientationally invariant indices of axon diameter and density in white matter using diffusion magnetic resonance imaging. Such indices potentially provide more specific markers of white matter microstructure than standard indices from diffusion tensor imaging. Orientational invariance allows for combination with tractography and presents new opportunities for mapping brain connectivity and quantifying disease processes. The technique uses a four-compartment tissue model combined with an optimized multishell high-angular-resolution pulsed-gradient-spin-echo acquisition. We test the method in simulation, on fixed monkey brains using a preclinical scanner and on live human brains using a clinical 3T scanner. The human data take about one hour to acquire. The simulation experiments show that both monkey and human protocols distinguish distributions of axon diameters that occur naturally in white matter. We compare the axon diameter index with the mean axon diameter weighted by axon volume. The index differs from this mean and is protocol dependent, but correlation is good for the monkey protocol and weaker, but discernible, for the human protocol where greater diffusivity and lower gradient strength limit sensitivity to only the largest axons. Maps of axon diameter and density indices from the monkey and human data in the corpus callosum and corticospinal tract reflect known trends from histology. The results show orientationally invariant sensitivity to natural axon diameter distributions for the first time with both specialist and clinical hardware. This demonstration motivates further refinement, validation, and evaluation of the precise nature of the indices and the influence of potential confounds.

Fractional Calculus Models of Magnetic Resonance Phenomena: Relaxation and Diffusion.

Applications of fractional calculus in magnetic resonance imaging (MRI) have increased over the last twenty years. From the mathematical, computational, and biophysical perspectives, fractional calculus provides new tools for describing the complexity of biological tissues (cells, organelles, membranes and macromolecules). Specifically, fractional order models capture molecular dynamics (transport, rotation, and vibration) by incorporating power law convolution kernels into the time and space derivatives appearing in the equations that govern nuclear magnetic resonance (NMR) phenomena. Hence, it is natural to expect fractional calculus models of relaxation and diffusion to be applied to problems in NMR and MRI. Early studies considered the fractal dimensions of multi-scale materials in the non-linear growth of the mean squared displacement, assumed power-law decays of the spectral density, and suggested stretched exponential signal relaxation to describe non-Gaussian behavior. Subsequently, fractional order generalization of the Bloch, and Bloch-Torrey equations were developed to characterize NMR (and MRI) relaxation and diffusion. However, even for simple geometries, analytical solutions of fractional order equations in time and space are difficult to obtain, and predictions of the corresponding changes in image contrast are not always possible. Currently, a multifaceted approach using coarse graining, simulation, and accelerated computation is being developed to identify 'imaging' biomarkers of disease. This review surveys the principal fractional order models used to describe NMR and MRI phenomena, identifies connections and limitations, and finally points to future applications of the approach.

Capturing complexity of the diffusion-weighted MR signal decay.

Diffusion-weighted MRI (dMRI) is a key component of clinical radiology. When analyzing diffusion-weighted images, radiologists often seek to infer microscopic tissue structure through measurements of the diffusion coefficient, D0 (mm2/s). This multi-scale problem is framed by the creation of diffusion models of signal decay based on physical laws, histological structure, and biophysical constraints. The purpose of this paper is to simplify the model building process by focusing on the observed decay in the effective diffusion coefficient as a function of diffusion weighting (b-value), D(b), that is often observed in complex biological tissues. We call this approach the varying diffusion curvature (VDC) model. Since this is a heuristic model, the exact functional form of this decay is not important, so here we examine a simple exponential function, D(b) = D0exp(-bD1), where D0 and D1 capture aspects of hindered and restricted diffusion, respectively. As an example of the potential of the VDC model, we applied it to dMRI data collected from normal and diseased human brain tissue using Stejskal-Tanner diffusion gradient pulses. In order to illustrate the connection between D0 and D1 and the sub-voxel structure we also analyzed dMRI data from families of Sephadex beads selected with increasing tortuosity. Finally, we applied the VDC model to dMRI simulations of nested muscle fiber phantoms whose permeability, atrophy, and fiber size distribution could be changed. These results demonstrate that the VDC model is sensitive to sub-voxel tissue structure and composition (porosity, tortuosity, and permeability), hence can capture tissue complexity in a manner that could be easily applied in clinical dMRI.