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.

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.

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.

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.

Analysis of YFP(J16)-R6/2 reporter mice and postmortem brains reveals early pathology and increased vulnerability of callosal axons in Huntington's disease.

Cumulative evidence indicates that the onset and severity of Huntington's disease (HD) symptoms correlate with connectivity deficits involving specific neuronal populations within cortical and basal ganglia circuits. Brain imaging studies and pathological reports further associated these deficits with alterations in cerebral white matter structure and axonal pathology. However, whether axonopathy represents an early pathogenic event or an epiphenomenon in HD remains unknown, nor is clear the identity of specific neuronal populations affected. To directly evaluate early axonal abnormalities in the context of HD in vivo, we bred transgenic YFP(J16) with R6/2 mice, a widely used HD model. Diffusion tensor imaging and fluorescence microscopy studies revealed a marked degeneration of callosal axons long before the onset of motor symptoms. Accordingly, a significant fraction of YFP-positive cortical neurons in YFP(J16) mice cortex were identified as callosal projection neurons. Callosal axon pathology progressively worsened with age and was influenced by polyglutamine tract length in mutant huntingtin (mhtt). Degenerating axons were dissociated from microscopically visible mhtt aggregates and did not result from loss of cortical neurons. Interestingly, other axonal populations were mildly or not affected, suggesting differential vulnerability to mhtt toxicity. Validating these results, increased vulnerability of callosal axons was documented in the brains of HD patients. Observations here provide a structural basis for the alterations in cerebral white matter structure widely reported in HD patients. Collectively, our data demonstrate a dying-back pattern of degeneration for cortical projection neurons affected in HD, suggesting that axons represent an early and potentially critical target for mhtt toxicity.

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.

Differentiating low- and high-grade pediatric brain tumors using a continuous-time random-walk diffusion model at high b-values.

PURPOSE: To demonstrate that a continuous-time random-walk (CTRW) diffusion model can improve diagnostic accuracy of differentiating low- and high-grade pediatric brain tumors. METHODS: Fifty-four children with histopathologically confirmed brain tumors underwent diffusion MRI scans at 3Twith 12 b-values (0-4000 s/mm(2) ). The diffusion imageswere fit to a simplified CTRW model to extract anomalous diffusion coefficient, Dm , and temporal and spatial heterogeneity parameters, α and ß, respectively. Using histopathology results as reference, a k-means clustering algorithm and a receiver operating characteristic (ROC) analysis were employed to determine the sensitivity, specificity, and diagnostic accuracy of the CTRW parameters in differentiating tumor grades. RESULTS: Significant differences between the low- and high-grade tumors were observed in the CTRW parameters (p-values<0.001). The k-means analysis showed that the combination of three CTRW parameters produced higher diagnostic accuracy (85% vs. 75%) and specificity (83% vs. 54%) than the apparent diffusion coefficient (ADC) from a mono-exponential model. The ROC analysis revealed that any combination of the CTRW parameters gave a larger area under the curve (0.90-0.96) than using ADC (0.80). CONCLUSION: With its sensitivity to intravoxel heterogeneity, the simplified CTRW model is useful for non-invasive grading of pediatric brain tumors, particularly when surgical biopsy is not feasible. Magn Reson Med 76:1149-1157, 2016. © 2015 Wiley Periodicals, Inc.

Anomalous T2 relaxation in normal and degraded cartilage.

PURPOSE: To compare the ordinary monoexponential model with three anomalous relaxation models-the stretched Mittag-Leffler, stretched exponential, and biexponential functions-using both simulated and experimental cartilage relaxation data. METHODS: Monte Carlo simulations were used to examine both the ability of identifying a given model under high signal-to-noise ratio (SNR) conditions and the accuracy and precision of parameter estimates under more modest SNR as would be encountered clinically. Experimental transverse relaxation data were analyzed from normal and enzymatically degraded cartilage samples under high SNR and rapid echo sampling to compare each model. RESULTS: Both simulation and experimental results showed improvement in signal representation with the anomalous relaxation models. The stretched exponential model consistently showed the lowest mean squared error in experimental data and closely represents the signal decay over multiple decades of the decay time (e.g., 1-10 ms, 10-100 ms, and >100 ms). The stretched exponential parameter αse showed an inverse correlation with biochemically derived cartilage proteoglycan content. CONCLUSION: Experimental results obtained at high field suggest potential application of αse as a measure of matrix integrity. Simulation reflecting more clinical imaging conditions, indicate the ability to robustly estimate αse and distinguish between normal and degraded tissue, highlighting its potential as a biomarker for human studies. Magn Reson Med 76:953-962, 2016. © 2015 Wiley Periodicals, Inc.

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.

Simultaneous, multidirectional acquisition of displacement fields in magnetic resonance elastography of the in vivo human brain.

BACKGROUND: To implement a multidirectional motion encoding scheme for magnetic resonance elastography (MRE) of the human brain with reduced acquisition time, and investigate its performance relative to a conventional MRE scheme. METHODS: The sample interval modulation (SLIM) scheme was implemented in a multishot, variable density spiral MRE sequence. The brains of seven healthy volunteers were investigated with both SLIM-MRE and conventional MRE acquisitions in a single imaging session on a clinical 3 Tesla MRI scanner with 50 Hz vibration. Following extraction of displacement fields, complex shear modulus property maps were estimated for each encoding concept. RESULTS: The SLIM-MRE and conventional MRE acquisitions produced deformation fields that were nearly identical and exhibited an average correlation coefficient of 0.95 (all p < 0.05). Average properties of white matter differed between the two acquisitions by less than 5% for all volunteers, which is better than reproducibility estimates for conventional MRE alone. CONCLUSION: The use of SLIM provides very similar quantitative property estimates compared with the conventional MRE encoding scheme. The SLIM acquisition is 2.5 times faster than the conventional acquisition, and may speed the adoption of MRE in clinical settings.

Assessment of glenoid chondral healing: comparison of microfracture to autologous matrix-induced chondrogenesis in a novel rabbit shoulder model.

BACKGROUND: Management of glenohumeral arthrosis in young patients is a considerable challenge, with a growing need for non-arthroplasty alternatives. The objectives of this study were to develop an animal model to study glenoid cartilage repair and to compare surgical repair strategies to promote glenoid chondral healing. METHODS: Forty-five rabbits underwent unilateral removal of the entire glenoid articular surface and were divided into 3 groups--untreated defect (UD), microfracture (MFx), and MFx plus type I/III collagen scaffold (autologous matrix-induced chondrogenesis [AMIC])--for the evaluation of healing at 8 weeks (12 rabbits) and 32 weeks (33 rabbits) after injury. Contralateral shoulders served as unoperated controls. Tissue assessments included 11.7-T magnetic resonance imaging (long-term healing group only), equilibrium partitioning of an ionic contrast agent via micro-computed tomography (EPIC-µCT), and histologic investigation (grades on International Cartilage Repair Society II scoring system). RESULTS: At 8 weeks, x-ray attenuation, thickness, and volume did not differ by treatment group. At 32 weeks, the T2 index (ratio of T2 values of healing to intact glenoids) was significantly lower for the MFx group relative to the AMIC group (P = .01) whereas the T1ρ index was significantly lower for AMIC relative to MFx (P = .01). The micro-computed tomography-derived repair tissue volume was significantly higher for MFx than for UD. Histologic investigation generally suggested inferior healing in the AMIC and UD groups relative to the MFx group, which exhibited improvements in both integration of repair tissue with subchondral bone and tidemark formation over time. DISCUSSION: Improvements conferred by AMIC were limited to magnetic resonance imaging outcomes, whereas MFx appeared to promote increased fibrous tissue deposition via micro-computed tomography and more hyaline-like repair histologically. The findings from this novel model suggest that MFx promotes biologic resurfacing of full-thickness glenoid articular injury.

Simultaneous MR elastography and diffusion acquisitions: diffusion-MRE (dMRE).

PURPOSE: To present a new technique for concurrent MR elastography (MRE) and diffusion MRI: diffusion-MRE (dMRE). METHODS: In dMRE, shear wave motion and MR signal decay due to diffusion are encoded into the phase and magnitude components of the MR signal by using a pair of bipolar gradients for both motion-sensitization and diffusion encoding. The pulse sequence timing is adjusted so that the bipolar gradients are sensitive to both coherent and incoherent intravoxel motions. The shape, number, and duration of the gradient lobes can be adjusted to provide flexibility and encoding efficiency. In this proof-of-concept study, dMRE was validated using a tissue phantom composed of a gel bead embedded in a hydrated mixture of agarose and gelatin. The apparent diffusion coefficient (ADC) and shear stiffness measured using dMRE were compared with results obtained from separate, conventional spin-echo (SE) diffusion and SE-MRE acquisitions. RESULTS: The averaged ADC values (n = 3) for selected ROIs in the beads were (1.75 ± 0.16) µm(2) /ms and (1.74 ± 0.16) µm(2) /ms for SE-diffusion and dMRE methods, respectively. The corresponding shear stiffness values in the beads were (2.45 ± 0.23) kPa and (2.42 ± 0.20) kPa. CONCLUSION: Simultaneous MRE and diffusion acquisition is feasible and can be implemented with no observable interference between the two methods.

On random walks and entropy in diffusion-weighted magnetic resonance imaging studies of neural tissue.

PURPOSE: In diffusion-weighted MRI studies of neural tissue, the classical model assumes the statistical mechanics of Brownian motion and predicts a monoexponential signal decay. However, there have been numerous reports of signal decays that are not monoexponential, particularly in the white matter. THEORY: We modeled diffusion in neural tissue from the perspective of the continuous time random walk. The characteristic diffusion decay is represented by the Mittag-Leffler function, which relaxes a priori assumptions about the governing statistics. We then used entropy as a measure of the anomalous features for the characteristic function. METHODS: Diffusion-weighted MRI experiments were performed on a fixed rat brain using an imaging spectrometer at 17.6 T with b-values arrayed up to 25,000 s/mm(2). Additionally, we examined the impact of varying either the gradient strength, q, or mixing time, Δ, on the observed diffusion dynamics. RESULTS: In white and gray matter regions, the Mittag-Leffler and entropy parameters demonstrated new information regarding subdiffusion and produced different image contrast from that of the classical diffusion coefficient. The choice of weighting on q and Δ produced different image contrast within the regions of interest. CONCLUSION: We propose these parameters have the potential as biomarkers for morphology in neural tissue.

Classification of fractional order biomarkers for anomalous diffusion using q-space entropy.

In this study, we applied continuous random walk theory (CTRW) to develop a new model that characterizes anomalous diffusion in magnetic resonance imaging experiments. Furthermore, we applied a classification scheme based on information theoretic a techniques to characterize the degree of heterogeneity and complexity in biological tissues. From a CTRW approach, the Fourier transform of the generalized solution to the diffusion equation comes in the form of the Mittag-Leffler function. In this solution form, the relative stochastic uncertainty in the diffusion process can be computed with spectral entropy. We interrogated both white and gray matter regions of a fixed rat brain with diffusion - weighted magnetic resonance imaging experiments up to 26,000 s/mm² by independently weighting q and Δ. to investigate the effects on the diffusion phenomena. Our model fractional order parameters, α and ß, and entropy measure, H(q, Δ), differentiated between tissue types and extracted differing information within a region of interest based on the type of diffusion experiment performed. By combining fractional order modeling and information theory, new and powerful biomarkers are available to characterize tissue microstructure and provide contextual information about the anatomical complexity.

New Insights into the Fractional Order Diffusion Equation Using Entropy and Kurtosis.

Fractional order derivative operators offer a concise description to model multi-scale, heterogeneous and non-local systems. Specifically, in magnetic resonance imaging, there has been recent work to apply fractional order derivatives to model the non-Gaussian diffusion signal, which is ubiquitous in the movement of water protons within biological tissue. To provide a new perspective for establishing the utility of fractional order models, we apply entropy for the case of anomalous diffusion governed by a fractional order diffusion equation generalized in space and in time. This fractional order representation, in the form of the Mittag-Leffler function, gives an entropy minimum for the integer case of Gaussian diffusion and greater values of spectral entropy for non-integer values of the space and time derivatives. Furthermore, we consider kurtosis, defined as the normalized fourth moment, as another probabilistic description of the fractional time derivative. Finally, we demonstrate the implementation of anomalous diffusion, entropy and kurtosis measurements in diffusion weighted magnetic resonance imaging in the brain of a chronic ischemic stroke patient.

Wideband MR elastography for viscoelasticity model identification.

The growing clinical use of MR elastography requires the development of new quantitative standards for measuring tissue stiffness. Here, we examine a soft tissue mimicking phantom material (Ecoflex) over a wide frequency range (200 Hz to 7.75 kHz). The recorded data are fit to a cohort of viscoelastic models of varying complexity (integer and fractional order). This was accomplished using multiple sample sizes by employing geometric focusing of the shear wave front to compensate for the changes in wavelength and attenuation over this broad range of frequencies. The simple axisymmetric geometry and shear wave front of this experiment allows us to calculate the frequency-dependent complex-valued shear modulus of the material. The data were fit to several common models of linear viscoelasticity, including those with fractional derivative operators, and we identified the best possible matches over both a limited frequency band (often used in clinical studies) and over the entire frequency span considered. In addition to demonstrating the superior capability of the fractional order viscoelastic models, this study highlights the advantages of measuring the complex-valued shear modulus over as wide a range of frequencies as possible.

Application of sodium triple-quantum coherence NMR spectroscopy for the study of growth dynamics in cartilage tissue engineering.

We studied the tissue growth dynamics of tissue-engineered cartilage at an early growth stage after cell seeding for four weeks using sodium triple-quantum coherence NMR spectroscopy. The following tissue-engineering constructs were studied: 1) bovine chondrocytes cultured in alginate beads; 2) bovine chondrocytes cultured as pellets (scaffold-free chondrocyte pellets); and 3) human marrow stromal cells (HMSCs) seeded in collagen/chitosan based biomimetic scaffolds. We found that the sodium triple-quantum coherence spectroscopy could differentiate between different tissue-engineered constructs and native tissues based on the fast and slow components of relaxation rate as well as on the average quadrupolar coupling. Both fast (Tf ) and slow (Ts ) relaxation times were found to be longer in chondrocyte pellets and biomimetic scaffolds compared to chondrocytes suspended in alginate beads and human articular cartilage tissues. In all cases, it was found that relaxation rates and motion of sodium ions measured from correlation times were dependent on the amount of macromolecules, high cell density and anisotropy of the cartilage tissue-engineered constructs. Average quadrupolar couplings were found to be lower in the engineered tissue compared to native tissue, presumably due to the lack of order in collagen accumulated in the engineered tissue. These results support the use of sodium triple-quantum coherence spectroscopy as a tool to investigate anisotropy and growth dynamics of cartilage tissue-engineered constructs in a simple and reliable way.

Characterization of Anomalous Diffusion in Porous Biological Tissues Using Fractional Order Derivatives and Entropy.

In this high-resolution magnetic resonance imaging (MRI) study at 17.6 Tesla of a fixed rat brain, we used the continuous time random walk theory (CTRW) for Brownian motion to characterize anomalous diffusion. The complex mesoporus structure of biological tissues (membranes, organelles, and cells) perturbs the motion of the random walker (water molecules in proton MRI) introducing halts between steps (waiting times) and restrictions on step sizes (jump lengths). When such waiting times and jump lengths are scaled with probability distributions that follow simple inverse power laws (t-(1+α), |x|-(1+ß)) non-Gaussian motion gives rise to sub- and super- diffusion. In the CTRW approach, the Fourier transform yields a solution to the generalized diffusion equation that can be expressed by the Mittag-Leffler function (MLF), Eα (- Dα, ß|q|ßΔα). We interrogated both white and gray matter regions in a 1 mm slice of a fixed rat brain (190 µm in plane resolution) with diffusion weighted MRI experiments using b-values up to 25,000 s/mm2, by independently varying q and Δ. When fitting these data to our model, the fractional order parameters, α and ß, and the entropy measure, [Formula: see text], were found to provide excellent contrast between white and gray matter and to give results that were sensitive to the type of diffusion experiment performed.

Anomalous diffusion measured by a twice-refocused spin echo pulse sequence: analysis using fractional order calculus.

PURPOSE: To theoretically develop and experimentally validate a formulism based on a fractional order calculus (FC) diffusion model to characterize anomalous diffusion in brain tissues measured with a twice-refocused spin-echo (TRSE) pulse sequence. MATERIALS AND METHODS: The FC diffusion model is the fractional order generalization of the Bloch-Torrey equation. Using this model, an analytical expression was derived to describe the diffusion-induced signal attenuation in a TRSE pulse sequence. To experimentally validate this expression, a set of diffusion-weighted (DW) images was acquired at 3 Tesla from healthy human brains using a TRSE sequence with twelve b-values ranging from 0 to 2600 s/mm(2). For comparison, DW images were also acquired using a Stejskal-Tanner diffusion gradient in a single-shot spin-echo echo planar sequence. For both datasets, a Levenberg-Marquardt fitting algorithm was used to extract three parameters: diffusion coefficient D, fractional order derivative in space ß, and a spatial parameter µ (in units of µm). Using adjusted R-squared values and standard deviations, D, ß, and µ values and the goodness-of-fit in three specific regions of interest (ROIs) in white matter, gray matter, and cerebrospinal fluid, respectively, were evaluated for each of the two datasets. In addition, spatially resolved parametric maps were assessed qualitatively. RESULTS: The analytical expression for the TRSE sequence, derived from the FC diffusion model, accurately characterized the diffusion-induced signal loss in brain tissues at high b-values. In the selected ROIs, the goodness-of-fit and standard deviations for the TRSE dataset were comparable with the results obtained from the Stejskal-Tanner dataset, demonstrating the robustness of the FC model across multiple data acquisition strategies. Qualitatively, the D, ß, and µ maps from the TRSE dataset exhibited fewer artifacts, reflecting the improved immunity to eddy currents. CONCLUSION: The diffusion-induced signal attenuation in a TRSE pulse sequence can be described by an FC diffusion model at high b-values. This model performs equally well for data acquired from the human brain tissues with a TRSE pulse sequence or a conventional Stejskal-Tanner sequence.

Rayleigh-Lamb wave propagation on a fractional order viscoelastic plate.

A previous study of the authors published in this journal focused on mechanical wave motion in a viscoelastic material representative of biological tissue [Meral et al., J. Acoust. Soc. Am. 126, 3278-3285 (2009)]. Compression, shear and surface wave motion in and on a viscoelastic halfspace excited by surface and sub-surface sources were considered. It was shown that a fractional order Voigt model, where the rate-dependent damping component that is dependent on the first derivative of time is replaced with a component that is dependent on a fractional derivative of time, resulted in closer agreement with experiment as compared with conventional (integer order) models, such as those of Voigt and Zener. In the present study, this analysis is extended to another configuration and wave type: out-of-plane response of a viscoelastic plate to harmonic anti-symmetric Lamb wave excitation. Theoretical solutions are compared with experimental measurements for a polymeric tissue mimicking phantom material. As in the previous configurations the fractional order modeling assumption improves the match between theory and experiment over a wider frequency range. Experimental complexities in the present study and the reliability of the different approaches for quantifying the shear viscoelastic properties of the material are discussed.