Sensitivity analyses of probabilistic and deterministic DTI tractography methodologies for studying arm muscle architecture.

PURPOSE: To determine the sensitivity profiles of probabilistic and deterministic DTI tractography methods in estimating geometric properties in arm muscle anatomy. METHODS: Spin-echo diffusion-weighted MR images were acquired in the dominant arm of 10 participants. Both deterministic and probabilistic tractography were performed in two different muscle architectures of the parallel-structured biceps brachii (and the pennate-structured flexor carpi ulnaris. Muscle fascicle geometry estimates and number of fascicles were evaluated with respect to tractography turning angle, polynomial fitting order, and SNR. The DTI tractography estimated fascicle lengths were compared with measurements obtained from conventional cadaveric dissection and ultrasound modalities. RESULTS: The probabilistic method generally estimated fascicle lengths closer to ranges reported by conventional methods than the deterministic method, most evident in the biceps brachii (p > 0.05), consisting of longer, arc-like fascicles. For both methods, a wide turning angle (50º-90°) generated fascicle lengths that were in close agreement with conventional methods, most evident in the flexor carpi ulnaris (p > 0.05), consisting of shorter, feather-like fascicles. The probabilistic approach produced at least two times more fascicles than the deterministic approach. For both approaches, second-order fitting yielded about double the complete tracts as third-order fitting. In both muscles, as SNR decreased, deterministic tractography produced less fascicles but consistent geometry (p > 0.05), whereas probabilistic tractography produced a consistent number but altered geometry of fascicles (p < 0.001). CONCLUSION: Findings from this study provide best practice recommendations for implementing DTI tractography in skeletal muscle and will inform future in vivo studies of healthy and pathological muscle structure.

Characterization of Atypical Corticospinal Tract Microstructure and Hand Impairments in Early-Onset Hemiplegic Cerebral Palsy: Preliminary Findings.

Unilateral brain injuries occurring before at or shortly after full-term can result in hemiplegic cerebral palsy (HCP). HCP affects one side of the body and can be characterized in the hand with measures of weakness and a loss of independent hand control resulting in mirror movements. Hand impairment severity is extremely heterogeneous across individuals with HCP and the neural basis for this variability is unclear. We used diffusion MRI and tractography to investigate the relationship between structural morphology of the supraspinal corticospinal tract (CST) and the severity of two typical hand impairments experienced by individuals with HCP, grasp weakness and mirror movements. Results from nine children with HCP and eight children with typical development show that there is a significant hemispheric association between CST microstructure and hand impairment severity that may be explained by atypical development and fiber distribution of motor pathways. Further analysis in the non-lesioned (dominant) hemisphere shows significant differences for CST termination in the cortex between participants with HCP and those with typical development. These findings suggest that structural disparities at the cellular level in the seemingly unaffected hemisphere after early unilateral brain injury may be the cause of heterogeneous hand impairments seen in this population.Clinical Relevance- Quantitative measurement of the variability in hand function in individuals with HCP is necessary to represent the distinct impairments experienced by each person. Further understanding of the structural neural morphology underlying distal upper extremity motor deficits after early unilateral brain injury will help lead to the development of more specific targeted interventions that increase functional outcomes.

Motivators and barriers to research participation for individuals with cerebral palsy and their families.

OBJECTIVE(S): Our objective was to investigate the motivators and barriers associated with the individual or family decision to participate in cerebral palsy research. Based on this information, we offer suggestions to increase the likelihood of participation in future CP studies. METHODS: A digital survey was administered to stakeholders affected by cerebral palsy across the US. Our analysis focused on variables related to personal interests, travel, and study-specific elements. Statistical tests investigated the effects of responder type, cerebral palsy type, and Gross Motor Function Classification System level on travel and study-specific element variables. Recommendations were informed by responses reflecting the majority of respondents. RESULTS: Based on 233 responses, we found that respondents highly valued research participation (on average 88.2/100) and compensation (on average 62.3/100). Motivators included the potential for direct benefit (62.2%) and helping others (53.4%). The primary barriers to participation were schedule limitations (48.9%) and travel logistics (32.6%). Schedule limitations were especially pertinent to caregivers, while individuals with more severe cerebral palsy diagnoses reported the necessity of additional items to comfortably travel. CONCLUSIONS: Overall, we encourage the involvement of stakeholders affected by cerebral palsy in the research process. Researchers should consider offering flexible study times, accommodating locations, and compensation for time and travel expenses. We recommend a minimum compensation of $15/hour and a maximum time commitment of 4 hours/day to respect participants' time and increase likelihood of research participation. Future studies should track how attitudes toward research change with time and experience.

Development of DTI Based Probabilistic Tractography Methods to Characterize Arm Muscle Architecture in Individuals Post Hemiparetic Stroke.

A hemiparetic stroke may lead to changes in muscle structure that further exacerbate motor impairments of the paretic limb. Cadaveric measurements have previously been used to study structural parameters in skeletal muscles but has several limitations, including ex vivo fixation. Here, we present novel application of diffusion tensor imaging (DTI) based probabilistic tractography methods, in comparison to the traditional deterministic approach, with respect to cadaveric dissection to quantify in vivo muscle fascicles in the biceps brachii. Preliminary results show that probabilistic tractography yields longer fascicle lengths that are more consistent with cadaveric measurements, albeit with higher variability, while deterministic tractography identifies shorter fascicle lengths, but with less variability. Results suggest that DTI tractography techniques can capture fascicles consistent with previously published cadaveric measurements and can identify interlimb differences in fascicle lengths in an individual with stroke.Clinical Relevance- The methods proposed here describe a non-invasive way to quantify heterogeneous musculoskeletal parameters such as across upper arm muscles in individuals with hemiparetic stroke. This will expand the current knowledge of macro- and micro-structural muscle changes that occur after stroke and may lead to more effective rehabilitation strategies to prevent such changes in individuals with stroke.

Vascular health and diffusion properties of normal appearing white matter in midlife.

In this study, we perform a region of interest diffusion tensor imaging and advanced diffusion complexity analysis of normal appearing white matter to determine the impact of vascular health on these diffusivity metrics in midlife adults. 77 participants (26 black, 35 female) at year 30 visit in the Coronary Artery Risk Development in Young Adults longitudinal study were scanned with an advanced diffusion-weighted imaging and fluid-attenuated inversion recovery protocol. Fractional anisotropy and non-linear diffusion complexity measures were estimated. Cumulative measures across 30 years (9 study visits) of systolic blood pressure, body mass index, glucose, smoking and cholesterol were calculated as the area under the curve from baseline up to year 30 examination. Partial correlation analyses assessed the association between cumulative vascular health measures and normal appearing white matter diffusion metrics in these participants. Midlife normal appearing white matter diffusion properties were significantly associated (P < 0.05) with cumulative exposure to vascular risk factors from young adulthood over the 30-year time period. Higher cumulative systolic blood pressure exposure was associated with increased complexity and decreased fractional anisotropy. Higher cumulative body mass index exposure was associated with decreased fractional anisotropy. Additionally, in the normal appearing white matter of black participants (P < 0.05), who exhibited a higher cumulative vascular risk exposure, fractional anisotropy was lower and complexity was higher in comparison to normal appearing white matter in white participants. Higher burden of vascular risk factor exposure from young adulthood to midlife is associated with changes in the diffusion properties of normal appearing white matter in midlife. These changes which may reflect axonal disruption, increased inflammation and/or increased glial proliferation, were primarily observed in both anterior and posterior normal appearing white matter regions of the corpus callosum. These results suggest that microstructural changes in normal appearing white matter are sensitive to vascular health during young adulthood and are possibly therapeutic targets in interventions focused on preserving white matter health across life.

Blood pressure, executive function, and network connectivity in middle-aged adults at risk of dementia in late life.

Midlife blood pressure is associated with structural brain changes, cognitive decline, and dementia in late life. However, the relationship between early adulthood blood pressure exposure, brain structure and function, and cognitive performance in midlife is not known. A better understanding of these relationships in the preclinical stage may advance our mechanistic understanding of vascular contributions to late-life cognitive decline and dementia and may provide early therapeutic targets. To identify resting-state functional connectivity of executive control networks (ECNs), a group independent components analysis was performed of functional MRI scans of 600 individuals from the Coronary Artery Risk Development in Young Adults longitudinal cohort study, with cumulative systolic blood pressure (cSBP) measured at nine visits over the preceding 30 y. Dual regression analysis investigated performance-related connectivity of ECNs in 578 individuals (mean age 55.5 ± 3.6 y, 323 female, 243 Black) with data from the Stroop color-word task of executive function. Greater connectivity of a left ECN to the bilateral anterior gyrus rectus, right posterior orbitofrontal cortex, and nucleus accumbens was associated with better executive control performance on the Stroop. Mediation analyses showed that while the relationship between cSBP and Stroop performance was mediated by white matter hyperintensities (WMH), resting-state connectivity of the ECN mediated the relationship between WMH and executive function. Increased connectivity of the left ECN to regions involved in reward processing appears to compensate for the deleterious effects of WMH on executive function in individuals across the burden of cumulative systolic blood pressure exposure in midlife.

Stakeholder Perspectives on Engaging With Cerebral Palsy Research Studies After Onset of COVID-19 in the United States.

OBJECTIVE: To investigate the effect of the coronavirus disease 2019 (COVID-19) pandemic on perspectives toward participation in cerebral palsy (CP) research. DESIGN: An online survey with questions relating to the comfort levels of research participation was filled out by people who had CP or had a child with CP. SETTING: The online survey was administered through Research Electronic Data Capture platform. PARTICIPANTS: A total of 233 (n=233) individuals with CP (42.5%; n=99) or with a child with CP (57.1%; n=133) consented and at least partially completed the online survey (n=210 complete; n=23 partially complete). All participants resided in the United States. INTERVENTIONS: Not applicable. MAIN OUTCOME MEASURES: Readiness to participate was analyzed in the context of the time point for research participation during COVID-19 and whether or not the study offered direct benefits to participants. RESULTS: Participants were consistently willing to participate sooner in studies that offered direct benefit than in those that did not. Adults responding for themselves had sooner time points for studies without direct benefit compared with parents answering for a child (P=.030). Gross Motor Function Classification System level, but not age or CP type, affected the time point for studies without direct benefit (P=.017). Personal values influenced selected time point for studies without direct benefit (P=.007), whereas environmental factors affected the time point for studies with direct benefit (P=.002). Local COVID-19 incidence rates were not associated with time points for either research type; however, respondents expected precautions to be taken if they chose to participate. CONCLUSIONS: As the pandemic evolves, researchers should consider the perspectives of potential participants as well as ethical and safety factors when reinitiating in-person CP research.

Cytosolic diffusivity and microscopic anisotropy of N-acetyl aspartate in human white matter with diffusion-weighted MRS at 7 T.

Metabolite diffusion measurable in humans in vivo with diffusion-weighted spectroscopy (DW-MRS) provides a window into the intracellular morphology and state of specific cell types. Anisotropic diffusion in white matter is governed by the microscopic properties of the individual cell types and their structural units (axons, soma, dendrites). However, anisotropy is also markedly affected by the macroscopic orientational distribution over the imaging voxel, particularly in DW-MRS, where the dimensions of the volume of interest (VOI) are much larger than those typically used in diffusion-weighted imaging. One way to address the confound of macroscopic structural features is to average the measurements acquired with uniformly distributed gradient directions to mimic a situation where fibers present in the VOI are orientationally uniformly distributed. This situation allows the extraction of relevant microstructural features such as transverse and longitudinal diffusivities within axons and the related microscopic fractional anisotropy. We present human DW-MRS data acquired at 7 T in two different white matter regions, processed and analyzed as described above, and find that intra-axonal diffusion of the neuronal metabolite N-acetyl aspartate is in good correspondence to simple model interpretations, such as multi-Gaussian diffusion from disperse fibers where the transverse diffusivity can be neglected. We also discuss the implications of our approach for current and future applications of DW-MRS for cell-specific measurements.

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.

Studying neurons and glia non-invasively via anomalous subdiffusion of intracellular metabolites.

Cells in the central nervous system, neurons and glia, display a wide range of structural features. Molecular diffusion properties in the intracellular space of these cells reflect this structural diversity, deviating from standard Gaussian dynamics and resulting in anomalous subdiffusion. By tracking the displacement of intracellular metabolites, diffusion-weighted magnetic resonance spectroscopy allows for in vivo compartment-specific and cell-preferential morphological analysis of neurons and glia in the human brain. Suggestive of different intracellular environments between tissue type, the neuronal and glial intracellular space in gray matter is significantly more subdiffusive than in white matter. An important difference is found between the subdiffusion of choline, a predominantly glial metabolite, in gray and white matter, potentially reflecting differences in structural complexity between fibrous and protoplasmic astrocytes. The exclusively intracellular metabolite subdiffusive dynamics, taken together with water intra- and extracellular displacement, provide new insight of differing extracellular gray and white matter properties and exchange between tissue compartments.

Infarct location is associated with quality of life after mild ischemic stroke.

Background In patients with mild ischemic stroke, small but eloquent infarcts may have devastating effects, particularly on health-related quality of life. Aim This study investigates the association between acute infarct location and three-month health-related quality of life in patients with mild ischemic stroke. Methods We evaluated consecutively enrolled patients from a single center between August 2012 and July 2013. Our primary outcome at three months was impairment in any health-related quality of life domain (upper extremity, lower extremity, executive function, and general concerns) defined by a T-score <45. We analyzed the association between acute infarct locations and impaired health-related quality of life at three months in univariate and multivariable analysis. Results Among 229 patients (mean age 64.9 years, 55% male, 29.7% black, and median initial NIHSS score 1), impaired health-related quality of life was noted in 84 (36.2%) patients at three months. In univariate analysis, patients with subcortical infarcts (56.0% vs. 39.3%, p = 0.02) and brainstem infarcts (21.4% vs. 10.3%, p = 0.02) were more likely to have impaired health-related quality of life. In multivariable analysis, patients with subcortical and/or brainstem infarcts had increased odds of impaired health-related quality of life (adjusted OR 2.54, 95% CI 1.29-5.01, p = 0.01). Conclusions After mild ischemic stroke, subcortical and brainstem infarct locations predict impairment in health-related quality of life.

Comparison of perfusion signal acquired by arterial spin labeling-prepared intravoxel incoherent motion (IVIM) MRI and conventional IVIM MRI to unravel the origin of the IVIM signal.

PURPOSE: Applications of intravoxel incoherent motion (IVIM) imaging in the brain are scarce, whereas it has been successfully applied in other organs with promising results. To better understand the cerebral IVIM signal, the diffusion properties of the arterial blood flow within different parts of the cerebral vascular tree (i.e., different generations of the branching pattern) were isolated and measured by employing an arterial spin labeling (ASL) preparation module before an IVIM readout. METHODS: ASL preparation was achieved by T1 -adjusted time-encoded pseudo-continuous ASL (te-pCASL). The IVIM readout module was achieved by introducing bipolar gradients immediately after the excitation pulse. The results of ASL-IVIM were compared with those of conventional IVIM to improve our understanding of the signal generation process of IVIM. RESULTS: The pseudo-diffusion coefficient D* as calculated from ASL-IVIM data was found to decrease exponentially for postlabeling delays (PLDs) between 883 ms and 2176 ms, becoming relatively stable for PLDs longer than 2176 ms. The fast compartment of the conventional IVIM-experiment shows comparable apparent diffusion values to the ASL signal with PLDs between 1747 ms and 2176 ms. At the longest PLDs, the observed D* values (4.0 ± 2.8 × 10-3 mm2 /s) are approximately 4.5 times higher than the slow compartment (0.90 ± 0.05 × 10-3 mm2 /s) of the conventional IVIM experiment. CONCLUSION: This study showed much more complicated diffusion properties of vascular signal than the conventionally assumed single D* of the perfusion compartment in the two-compartment model of IVIM (biexponential behavior). Magn Reson Med 79:723-729, 2018. © 2017 International Society for Magnetic Resonance in Medicine.

Upper Extremity Motor Impairments and Microstructural Changes in Bulbospinal Pathways in Chronic Hemiparetic Stroke.

Following hemiparetic stroke, precise, individuated control of single joints is often replaced by highly stereotyped patterns of multi-joint movement, or abnormal limb synergies, which can negatively impact functional use of the paretic arm. One hypothesis for the expression of these synergies is an increased dependence on bulbospinal pathways such as the rubrospinal (RubST) tract and especially the reticulospinal (RetST) tracts, which co-activate multiple muscles of the shoulder, elbow, wrist, and fingers. Despite indirect evidence supporting this hypothesis in humans poststroke, it still remains unclear whether it is correct. Therefore, we used high-resolution diffusion tensor imaging (DTI) to quantify white matter microstructure in relation to severity of arm synergy and hand-related motor impairments. DTI was performed on 19 moderately to severely impaired chronic stroke individuals and 15 healthy, age-matched controls. In stroke individuals, compared to controls, there was significantly decreased fractional anisotropy (FA) and significantly increased axial and radial diffusivity in bilateral corona radiata and body of the corpus callosum. Furthermore, poststroke, the contralesional (CL) RetST FA correlated significantly with both upper extremity (UE) synergy severity (r = -0.606, p = 0.003) and hand impairment (r = -0.609, p = 0.003). FA in the ipsilesional RubST significantly correlated with hand impairment severity (r = -0.590, p = 0.004). For the first time, we separately evaluate RetST and RubST microstructure in chronic stroke individuals with UE motor impairment. We demonstrate that individuals with the greatest UE synergy severity and hand impairments poststroke have the highest FA in the CL RetST a pattern consistent with increased myelination and suggestive of neuroplastic reorganization. Since the RetST pathway microstructure, in particular, is sensitive to abnormal joint coupling and hand-related motor impairment in chronic stroke, it could help test the effects of specific, and novel, anti-synergy neurorehabilitation interventions for recovery from hemiparesis.

Neural Plasticity in Moderate to Severe Chronic Stroke Following a Device-Assisted Task-Specific Arm/Hand Intervention.

Currently, hand rehabilitation following stroke tends to focus on mildly impaired individuals, partially due to the inability for severely impaired subjects to sufficiently use the paretic hand. Device-assisted interventions offer a means to include this more severe population and show promising behavioral results. However, the ability for this population to demonstrate neural plasticity, a crucial factor in functional recovery following effective post-stroke interventions, remains unclear. This study aimed to investigate neural changes related to hand function induced by a device-assisted task-specific intervention in individuals with moderate to severe chronic stroke (upper extremity Fugl-Meyer < 30). We examined functional cortical reorganization related to paretic hand opening and gray matter (GM) structural changes using a multimodal imaging approach. Individuals demonstrated a shift in cortical activity related to hand opening from the contralesional to the ipsilesional hemisphere following the intervention. This was driven by decreased activity in contralesional primary sensorimotor cortex and increased activity in ipsilesional secondary motor cortex. Additionally, subjects displayed increased GM density in ipsilesional primary sensorimotor cortex and decreased GM density in contralesional primary sensorimotor cortex. These findings suggest that despite moderate to severe chronic impairments, post-stroke participants maintain ability to show cortical reorganization and GM structural changes following a device-assisted task-specific arm/hand intervention. These changes are similar as those reported in post-stroke individuals with mild impairment, suggesting that residual neural plasticity in more severely impaired individuals may have the potential to support improved hand function.

A multimodal MRI approach to identify and characterize microstructural brain changes in neuropsychiatric systemic lupus erythematosus.

Systemic lupus erythematosus (SLE) is an autoimmune disease with multi-organ involvement and results in neurological and psychiatric (NP) symptoms in up to 40% of the patients. To date, the diagnosis of neuropsychiatric systemic lupus erythematosus (NPSLE) poses a challenge due to the lack of neuroradiological gold standards. In this study, we aimed to better localize and characterize normal appearing white matter (NAWM) changes in NPSLE by combining data from two quantitative MRI techniques, diffusion tensor imaging (DTI) and magnetization transfer imaging (MTI). 9 active NPSLE patients (37 ± 13 years, all females), 9 SLE patients without NP symptoms (44 ± 11 years, all females), and 14 healthy controls (HC) (40 ± 9 years, all females) were included in the study. MTI, DTI and fluid attenuated inversion recovery (FLAIR) images were collected from all subjects on a 3 T MRI scanner. Magnetization transfer ratio (MTR), mean diffusivity (MD), fractional anisotropy (FA), radial diffusivity (RD), axial diffusivity (AD) maps and white matter lesion maps based on the FLAIR images were created for each subject. MTR and DTI data were then co-analyzed using tract-based spatial statistics and a cumulative lesion map to exclude lesions. Significantly lower MTR and FA and significantly higher AD, RD and MD were found in NPSLE compared to HC in NAWM regions. The differences in DTI measures and in MTR, however, were only moderately co-localized. Additionally, significant differences in DTI measures, but not in MTR, were found between NPSLE and SLE patients, suggesting that the underlying microstructural changes detected by MD are linked to the onset of NPSLE. The co-analysis of the anatomical distribution of MTI and DTI measures can potentially improve the diagnosis of NPSLE and contribute to the understanding of the underlying microstructural damage.

Parsimonious continuous time random walk models and kurtosis for diffusion in magnetic resonance of biological tissue.

In this paper, we provide a context for the modeling approaches that have been developed to describe non-Gaussian diffusion behavior, which is ubiquitous in diffusion weighted magnetic resonance imaging of water in biological tissue. Subsequently, we focus on the formalism of the continuous time random walk theory to extract properties of subdiffusion and superdiffusion through novel simplifications of the Mittag-Leffler function. For the case of time-fractional subdiffusion, we compute the kurtosis for the Mittag-Leffler function, which provides both a connection and physical context to the much-used approach of diffusional kurtosis imaging. We provide Monte Carlo simulations to illustrate the concepts of anomalous diffusion as stochastic processes of the random walk. Finally, we demonstrate the clinical utility of the Mittag-Leffler function as a model to describe tissue microstructure through estimations of subdiffusion and kurtosis with diffusion MRI measurements in the brain of a chronic ischemic stroke patient.

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.

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.

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.