Identifying potential imaging markers for diffusion property changes in a mouse model of amyotrophic lateral sclerosis: Application of the continuous time random walk model to ultrahigh b-value diffusion-weighted MR images of spinal cord tissue.

Diffusion MRI (dMRI) explores tissue microstructures by analyzing diffusion-weighted signal decay measured at different b-values. While relatively low b-values are used for most dMRI models, high b-value diffusion-weighted imaging (DWI) techniques have gained interest given that the non-Gaussian water diffusion behavior observed at high b-values can yield potentially valuable information. In this study, we investigated anomalous diffusion behaviors associated with degeneration of spinal cord tissue using a continuous time random walk (CTRW) model for DWI data acquired across an extensive range of ultrahigh b-values. The diffusion data were acquired in situ from the lumbar level of spinal cords of wild-type and age-matched transgenic SOD1G93A mice, a well-established animal model of amyotrophic lateral sclerosis (ALS) featuring progressive degeneration of axonal tracts in this tissue. Based on the diffusion decay behaviors at low and ultrahigh b-values, we applied the CTRW model using various combinations of b-values and compared diffusion metrics calculated from the CTRW model between the experimental groups. We found that diffusion-weighted signal decay curves measured with ultrahigh b-values (up to 858,022 s/mm2 in this study) were well represented by the CTRW model. The anomalous diffusion coefficient obtained from lumbar spinal cords was significantly higher in SOD1G93A mice compared with control mice (14.7 × 10-5 ± 5.54 × 10-5  vs. 7.87 × 10-5 ± 2.48 × 10-5  mm2 /s, p = 0.01). We believe this is the first study to illustrate the efficacy of the CTRW model for analyzing anomalous diffusion regimes at ultrahigh b-values. The CTRW modeling of ultrahigh b-value dMRI can potentially present a novel approach for noninvasively evaluating alterations in spinal cord tissue associated with ALS pathology.

Diffusion in Sephadex Gel Structures: Time Dependency Revealed by Multi-Sequence Acquisition over a Broad Diffusion Time Range.

It has been increasingly reported that in biological tissues diffusion-weighted MRI signal attenuation deviates from mono-exponential decay, especially at high b-values. A number of diffusion models have been proposed to characterize this non-Gaussian diffusion behavior. One of these models is the continuous-time random-walk (CTRW) model, which introduces two new parameters: a fractional order time derivative α and a fractional order spatial derivative ß. These new parameters have been linked to intravoxel diffusion heterogeneities in time and space, respectively, and are believed to depend on diffusion times. Studies on this time dependency are limited, largely because the diffusion time cannot vary over a board range in a conventional spin-echo echo-planar imaging sequence due to the accompanying T2 decays. In this study, we investigated the time-dependency of the CTRW model in Sephadex gel phantoms across a broad diffusion time range by employing oscillating-gradient spin-echo, pulsed-gradient spin-echo, and pulsed-gradient stimulated echo sequences. We also performed Monte Carlo simulations to help understand our experimental results. It was observed that the diffusion process fell into the Gaussian regime at extremely short diffusion times whereas it exhibited a strong time dependency in the CTRW parameters at longer diffusion times.

Evaluation of early microstructural changes in the R6/1 mouse model of Huntington's disease by ultra-high field diffusion MR imaging.

Diffusion MRI (dMRI) has been able to detect early structural changes related to neurological symptoms present in Huntington's disease (HD). However, there is still a knowledge gap to interpret the biological significance at early neuropathological stages. The purpose of this study is two-fold: (i) establish if the combination of Ultra-High Field Diffusion MRI (UHFD-MRI) techniques can add a more comprehensive analysis of the early microstructural changes observed in HD, and (ii) evaluate if early changes in dMRI microstructural parameters can be linked to cellular biomarkers of neuroinflammation. Ultra-high field magnet (16.7T), diffusion tensor imaging (DTI), and neurite orientation dispersion and density imaging (NODDI) techniques were applied to fixed ex-vivo brains of a preclinical model of HD (R6/1 mice). Fractional anisotropy (FA) was decreased in deep and superficial grey matter (GM) as well as white matter (WM) brain regions with well-known early HD microstructure and connectivity pathology. NODDI parameters associated with the intracellular and extracellular compartment, such as intracellular ventricular fraction (ICVF), orientation dispersion index (ODI), and isotropic volume fractions (IsoVF) were altered in R6/1 mice GM. Further, histological studies in these areas showed that glia cell markers associated with neuroinflammation (GFAP & Iba1) were consistent with the dMRI findings. dMRI can be used to extract non-invasive information of neuropathological events present in the early stages of HD. The combination of multiple imaging techniques represents a better approach to understand the neuropathological process allowing the early diagnosis and neuromonitoring of patients affected by HD.

Assessing neuraxial microstructural changes in a transgenic mouse model of early stage Amyotrophic Lateral Sclerosis by ultra-high field MRI and diffusion tensor metrics.

OBJECTIVE: Cell structural changes are one of the main features observed during the development of amyotrophic lateral sclerosis (ALS). In this work, we propose the use of diffusion tensor imaging (DTI) metrics to assess specific ultrastructural changes in the central nervous system during the early neurodegenerative stages of ALS. METHODS: Ultra-high field MRI and DTI data at 17.6T were obtained from fixed, excised mouse brains, and spinal cords from ALS (G93A-SOD1) mice. RESULTS: Changes in fractional anisotropy (FA) and linear, planar, and spherical anisotropy ratios (CL, CP, and CS, respectively) of the diffusion eigenvalues were measured in white matter (WM) and gray matter (GM) areas associated with early axonal degenerative processes (in both the brain and the spinal cord). Specifically, in WM structures (corpus callosum, corticospinal tract, and spinal cord funiculi) as the disease progressed, FA, CL, and CP values decreased, whereas CS values increased. In GM structures (prefrontal cortex, hippocampus, and central spinal cord) FA and CP decreased, whereas the CL and CS values were unchanged or slightly smaller. Histological studies of a fluorescent mice model (YFP, G93A-SOD1 mouse) corroborated the early alterations in neuronal morphology and axonal connectivity measured by DTI. CONCLUSIONS: Changes in diffusion tensor shape were observed in this animal model at the early, nonsymptomatic stages of ALS. Further studies of CL, CP, and CS as imaging biomarkers should be undertaken to refine this neuroimaging tool for future clinical use in the detection of the early stages of ALS.

Multicomponent diffusion analysis reveals microstructural alterations in spinal cord of a mouse model of amyotrophic lateral sclerosis ex vivo.

The microstructure changes associated with degeneration of spinal axons in amyotrophic lateral sclerosis (ALS) may be reflected in altered water diffusion properties, potentially detectable with diffusion-weighted (DW) MRI. Prior work revealed the classical mono-exponential model fails to precisely depict decay in DW signal at high b-values. In this study, we aim to investigate signal decay behaviors at ultra-high b-values for non-invasive assessment of spinal cord alterations in the transgenic SOD1G93A mouse model of ALS. A multiexponential diffusion analysis using regularized non-negative least squares (rNNLS) algorithm was applied to a series of thirty DW MR images with b-values ranging from 0 to 858,022 s/mm2 on ex vivo spinal cords of transgenic SOD1G93A and age-matched control mice. We compared the distributions of measured diffusion coefficient fractions between the groups. The measured diffusion weighted signals in log-scale showed non-linear decay behaviors with increased b-values. Faster signal decays were observed with diffusion gradients applied parallel to the long axis of the spinal cord compared to when oriented in the transverse direction. Multiexponential analysis at the lumbar level in the spinal cord identified ten subintervals. A significant decrease of diffusion coefficient fractions was found in the ranges of [1.63×10-8,3.70×10-6] mm2/s (P = 0.0002) and of [6.01×10-6,4.20×10-5] mm2/s (P = 0.0388) in SOD1G93A mice. Anisotropic diffusion signals persisted at ultra-high b-value DWIs of the mouse spinal cord and multiexponential diffusion analysis offers the potential to evaluate microstructural alterations of ALS-affected spinal cord non-invasively.

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.

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.

Unveiling early cortical and subcortical neuronal degeneration in ALS mice by ultra-high field diffusion MRI.

Amyotrophic lateral sclerosis (ALS) is a neurodegenerative disease primarily characterized by the progressive impairment of motor functions. However, a significant portion of affected patients develops severe cognitive dysfunction, developing a widespread white (WM) and gray matter (GM) microstructural impairment. The objective of this study is to determine if Gaussian and non-Gaussian diffusion models gathered by ultra-high field diffusion MRI (UHFD-MRI) are an appropriate tool to detect early structural changes in brain white and gray matter in a preclinical model of ALS. ALS brains (G93A-SOD1mice) were scanned in a 16.7 T magnet. Diffusion tensor imaging (DTI) and neurite orientation dispersion and density imaging (NODDI) have shown presymptomatic decrease in axonal organization by Fractional Anisotropy (FA) and neurite content by Intracellular Volume Fraction (ICVF) across deep WM (corpus callosum) as well as superficial (cortex) and deep (hippocampus) GM. Additional diffusion kurtosis imaging (DKI) analysis demonstrated broader and earlier GM reductions in mean kurtosis (MK), possibly related to the decrease in neuronal complexity. Histological validation was obtained by an ALS fluorescent mice reporter (YFP, G93A-SOD1 mice). The combination of DTI, NODDI, and DKI models have proved to provide a more complete assessment of the early microstructural changes in the ALS brain, particularly in areas associated with high cognitive functions. This comprehensive approach should be considered as a valuable tool for the early detection of neuroimaging markers.

Detection of axonal degeneration in a mouse model of Huntington's disease: comparison between diffusion tensor imaging and anomalous diffusion metrics.

OBJECTIVE: The goal of this work is to study the changes in white matter integrity in R6/2, a well-established animal model of Huntington's disease (HD) that are captured by ex vivo diffusion imaging (DTI) using a high field MRI (17.6 T). MATERIALS AND METHODS: DTI and continuous time random walk (CTRW) models were used to fit changes in the diffusion-weighted signal intensity in the corpus callosum of controls and in R6/2 mice. RESULTS: A significant 13% decrease in fractional anisotropy, a 7% increase in axial diffusion, and a 33% increase in radial diffusion were observed between R6/2 and control mice. No change was observed in the CTRW beta parameter, but a significant decrease in the alpha parameter (- 21%) was measured. Histological analysis of the corpus callosum showed a decrease in axonal organization, myelin alterations, and astrogliosis. Electron microscopy studies demonstrated ultrastructural changes in degenerating axons, such as an increase in tortuosity in the R6/2 mice. CONCLUSIONS: DTI and CTRW diffusion models display quantitative changes associated with the microstructural alterations observed in the corpus callosum of the R6/2 mice. The observed increase in the diffusivity and decrease in the alpha CTRW parameter providing support for the use of these diffusion models for non-invasive detection of white matter alterations in HD.

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.

Machine-Learning Prediction of Drug-Induced Cardiac Arrhythmia: Analysis of Gene Expression and Clustering.

A marked delay in the electrical repolarization of heart ventricles is characterized by prolongation of the Q-T wave (QT) interval on a surface electrocardiogram. Such a delay can lead to potentially life-threatening cardiac arrhythmia (torsades de pointes). Such prolongation is also a widely accepted cardiac safety biomarker in drug development. Current preclinical drug-safety assays include patch clamp analysis to evaluate drug-related blockade of cardiac repolarizing ion currents. Recently reported patch clamp assay results have shown predictive sensitivities and specificities in the ranges of 64%-82% and 75%-88%, respectively. In this project, we use a support vector machine classifier to find mean sensitivities and specificities of 85% and 90%, respectively, across 77 drug subclassifications. Clustering by gene expression profile similarities shows that drugs known to prolong the QT interval do not always form distinct groups, but the number of groups is limited. The most common biological network links associated with these groups involve genes linked with fatty acid metabolism, G proteins, intracellular glutathione, immune responses, apoptosis, mitochondrial function, electron transport, and mitogen-activated protein kinases. These results suggest that machine-learning analysis of gene expression and clustering may augment cardiac safety predictions for improving drug-safety assessments.

In vivo diffusion MRI detects early spinal cord axonal pathology in a mouse model of amyotrophic lateral sclerosis.

Diffusion magnetic resonance imaging (MRI) exhibits contrast that identifies macro- and microstructural changes in neurodegenerative diseases. Previous studies have shown that MR diffusion tensor imaging (DTI) can observe changes in spinal cord white matter in animals and humans affected with symptomatic amyotrophic lateral sclerosis (ALS). The goal of this preclinical work was to investigate the sensitivity of DTI for the detection of signs of tissue damage before symptoms appear. High-field MRI data were acquired using a 9.4-T animal scanner to examine the spinal cord of an ALS mouse model at pre- and post-symptomatic stages (days 80 and 120, respectively). The MRI results were validated using yellow fluorescent protein (YFP) via optical microscopy of spinal cord tissue slices collected from the YFP,G93A-SOD1 mouse strain. DTI maps of diffusion-weighted imaging (DWI) signal intensity, mean diffusivity (MD), fractional anisotropy (FA), axial diffusivity (AD) and radial diffusivity (RD) were computed for axial slices of the lumbar region of the spinal cord. Significant changes were observed in FA (6.7% decrease, p < 0.01), AD (19.5% decrease, p < 0.01) and RD (16.1% increase, p < 0.001) at postnatal day 80 (P80). These differences were correlated with changes in axonal fluorescence intensity and membrane cellular markers. This study demonstrates the value of DTI as a potential tool to detect the underlying pathological progression associated with ALS, and may accelerate the discovery of therapeutic strategies for patients with this disease.

Ultra-High Field Diffusion MRI Reveals Early Axonal Pathology in Spinal Cord of ALS mice.

BACKGROUND: Amyotrophic lateral sclerosis (ALS) is a disease characterized by a progressive degeneration of motor neurons leading to paralysis. Our previous MRI diffusion tensor imaging studies detected early white matter changes in the spinal cords of mice carrying the G93A-SOD1 mutation. Here, we extend those studies using ultra-high field MRI (17.6 T) and fluorescent microscopy to investigate the appearance of early structural and connectivity changes in the spinal cords of ALS mice. METHODS: The spinal cords from presymptomatic and symptomatic mice (80 to 120 days of age) were scanned (ex-vivo) using diffusion-weighted MRI. The fractional anisotropy (FA), axial (AD) and radial (RD) diffusivities were calculated for axial slices from the thoracic, cervical and lumbar regions of the spinal cords. The diffusion parameters were compared with fluorescence microscopy and membrane cellular markers from the same tissue regions. RESULTS: At early stages of the disease (day 80) in the lumbar region, we found, a 19% decrease in FA, a 9% decrease in AD and a 35% increase in RD. Similar changes were observed in cervical and thoracic spinal cord regions. Differences between control and ALS mice groups at the symptomatic stages (day 120) were larger. Quantitative fluorescence microscopy at 80 days, demonstrated a 22% reduction in axonal area and a 22% increase in axonal density. Tractography and quantitative connectome analyses measured by edge weights showed a 52% decrease in the lumbar regions of the spinal cords of this ALS mice group. A significant increase in ADC (23.3%) in the ALS mice group was related to an increase in aquaporin markers. CONCLUSIONS: These findings suggest that the combination of ultra-high field diffusion MRI with fluorescent ALS mice reporters is a useful approach to detect and characterize presymptomatic white matter micro-ultrastructural changes and axonal connectivity anomalies in ALS.

Diffusion Tensor Imaging of Tendons and Ligaments at Ultra-High Magnetic Fields.

Injuries to tendons and ligaments are a common problem limiting daily activities and athletic participation across all age groups. Conventional magnetic resonance imaging (MRI) is reliable for detecting complete tears in tendons and ligaments, but it has difficulty identifying low-grade injuries due to poor contrast and low intensity signal. We describe recent MRI advances using ultra-high magnetic fields and very short time echoes which overcome many of the limitations of the low signal and the short T2 of connective tissues. Using diffusion and relaxometry measurements at 11.7 T, we measured the high field relaxation times, mean diffusivity, and the fractional anisotropy of rabbit semitendinosus tendons and medial collateral ligaments. We found that ultra-high field diffusion tensor imaging and tractography provide repeatable and quantitative maps of fiber organization, which could improve injury diagnosis and therapeutic treatments and assist in presurgical planning.

Diffusion tensor imaging identifies presymptomatic axonal degeneration in the spinal cord of ALS mice.

Extensive pathological evidence indicates that axonal degeneration represents an early and critical event in amyotrophic lateral sclerosis (ALS). Unfortunately, few MRI studies have focused in the early detection of white matter (WM) alterations in the spinal cord region. To unveil these WM changes, we performed high resolution diffusion tensor imaging (DTI) and correlated the results with histological analysis of adjacent slices taken from the spinal cords of presymptomatic mice. The DTI studies demonstrated a significant reduction in fractional anisotropy (FA) as well as axial diffusivities (AD) and an increase in radial diffusivity (RD), predominantly at lower segments of the spinal cord. Increases in FA and a reduction in AD and RD were observed in spinal cord (SC) gray matter (GM). Diffusion changes are associated with early and progressive alterations in axonal connectivity following a distal to proximal progression. Histological data tagging neuronal, axonal and glial cell markers demonstrated presymptomatic alterations in spinal cord WM and GM. This study demonstrates that DTI methods are optimal preclinical imaging tools to detect structural anomalies in WM and GM spinal cord during early stages of the disease.

Bioengineering and Cybernetics: A Modern Caduceus.

Like the caduceus, a medical symbol of entwined serpents, bioengineering and cybernetics have interwoven together ideas and concepts for over 50 years. Half a century is a long time, and whether we are talking about an academic discipline, our lives, or an old car, achieving 50 is a number that brings pause to the conversation. In books, wine, or collectibles, 50 years is termed vintage, which carries the connotation of depth and maturity. Certainly, in the case of the discipline of bioengineering, 50 years is a milestone of growth and development. By all academic measures (number of departments, current enrollment and graduates, size of faculty, and impact factor for its publications), bioengineering is a mature discipline. Presently, there are almost 100 ABET-certified bioengineering degree programs in the United States alone.

Concurrent 3D acquisition of diffusion tensor imaging and magnetic resonance elastography displacement data (DTI-MRE): Theory and in vivo application.

PURPOSE: To introduce a newly developed technique (DTI-MRE) for the simultaneous acquisition of diffusion tensor imaging (DTI) and 3D-vector field magnetic resonance elastography (MRE) data, and to demonstrate its feasibility when applied in vivo to the mouse brain. METHODS: In DTI-MRE, simultaneous encoding is achieved by using a series of diffusion/motion-sensitizing gradients (dMSGs) with specific timing and directions. By adjusting the duration of the dMSGs with the diffusion time and with the mechanical vibration frequency, the shear wave motion and diffusion are encoded into the MR phase and MR magnitude signals, respectively. The dMSGs are applied in a noncollinear and noncoplanar manner that optimizes the capture of both the DTI signal attenuation and the three-dimensional MRE displacements. In this work, the feasibility of the DTI-MRE technique was demonstrated on in vivo mouse brains (n=3) using a 9.4T animal MRI scanner. The DTI-MRE derived parameters (MD, mean diffusivity; FA, fractional anisotropy; MRE displacement fields; and shear modulus |G|) were compared with those acquired using conventional, separate MRE and diffusion methods. RESULTS: The averaged (MD, FA, and |G|) values for three mice are (0.580 ± 0.050 µm2 /ms, 0.43 ± 0.02, and 4.80 ± 0.06 kPa) and (0.583 ± 0.035 µm2 /ms, 0.46 ± 0.02, and 4.91 ± 0.19 kPa) for DTI-MRE, and conventional DTI and 3D-vector field MRE measurements, respectively. All derived parameters (MD, FA, |G|, and displacement) obtained using the combined DTI-MRE method and conventional methods were significantly correlated with P < 0.05. CONCLUSION: Simultaneous acquisition of DTI and 3D-vector field MRE is feasible in vivo and reduces the scan time by up to 50% compared with conventional, separate acquisitions, while providing an immediate co-registration of maps of diffusion properties and stiffness. Magn Reson Med 77:273-284, 2017. © 2016 Wiley Periodicals, Inc.

Onset of fractional-order thermal convection in porous media.

The macroscopic description of buoyancy-driven thermal convection in porous media is governed by advection-diffusion processes, which in the presence of thermophysical heterogeneities fail to predict the onset of thermal convection and the average rate of heat transfer. This work extends the classical model of heat transfer in porous media by including a fractional-order advective-dispersive term to account for the role of thermophysical heterogeneities in shifting the thermal instability point. The proposed fractional-order model overcomes limitations of the common closure approaches for the thermal dispersion term by replacing the diffusive assumption with a fractional-order model. Through a linear stability analysis and Galerkin procedure, we derive an analytical formula for the critical Rayleigh number as a function of the fractional model parameters. The resulting critical Rayleigh number reduces to the classical value in the absence of thermophysical heterogeneities when solid and fluid phases have similar thermal conductivities. Numerical simulations of the coupled flow equation with the fractional-order energy model near the primary bifurcation point confirm our analytical results. Moreover, data from pore-scale simulations are used to examine the potential of the proposed fractional-order model in predicting the amount of heat transfer across the porous enclosure. The linear stability and numerical results show that, unlike the classical thermal advection-dispersion models, the fractional-order model captures the advance and delay in the onset of convection in porous media and provides correct scalings for the average heat transfer in a thermophysically heterogeneous medium.

Mechanical analysis of an axially symmetric cylindrical phantom with a spherical heterogeneity for MR elastography.

Cylindrical homogenous phantoms for magnetic resonance (MR) elastography in biomedical research provide one way to validate an imaging systems performance, but the simplified geometry and boundary conditions can cloak complexity arising at tissue interfaces. In an effort to develop a more realistic gel tissue phantom for MRE, we have constructed a heterogenous gel phantom (a sphere centrally embedded in a cylinder). The actuation comes from the phantom container, with the mechanical waves propagating toward the center, focusing the energy and thus allowing for the visualization of high-frequency waves that would otherwise be damped. The phantom was imaged and its stiffness determined using a 9.4 T horizontal MRI with a custom build piezo-elastic MRE actuator. The phantom was vibrated at three frequencies, 250, 500, and 750 Hz. The resulting shear wave images were first used to reconstruct material stiffness maps for thin (1 mm) axial slices at each frequency, from which the complex shear moduli µ were estimated, and then compared with forward modeling using a recently developed theoretical model which took µ as inputs. The overall accuracy of the measurement process was assessed by comparing theory with experiment for selected values of the shear modulus (real and imaginary parts). Close agreement is shown between the experimentally obtained and theoretically predicted wave fields.

Anisotropic fractional diffusion tensor imaging.

Traditional diffusion tensor imaging (DTI) maps brain structure by fitting a diffusion model to the magnitude of the electrical signal acquired in magnetic resonance imaging (MRI). Fractional DTI employs anomalous diffusion models to obtain a better fit to real MRI data, which can exhibit anomalous diffusion in both time and space. In this paper, we describe the challenge of developing and employing anisotropic fractional diffusion models for DTI. Since anisotropy is clearly present in the three-dimensional MRI signal response, such models hold great promise for improving brain imaging. We then propose some candidate models, based on stochastic theory.