Grassmann Variables and Pseudoclassical Nuclear Magnetic Resonance

The concept of a propagator is useful and is a well-known object in diffusion NMR experiments. Here, we investigate the related concept; the propagator for the magnetisation or the Green's function of the Torrey-Bloch equations. The magnetisation propagator is constructed by defining functions such as the Hamiltonian and Lagrangian and using these to define a path integral. It is shown that the equations of motion derived from the Lagrangian produce complex-valued trajectories (classical paths) and it is conjectured that the end-points of these trajectories are real-valued. The complex nature of the trajectories also suggests that the spin degrees of freedom are also encoded into the trajectories and this idea is explored by explicitly modeling the spin or precessing magnetisation by anticommuting Grassmann variables. A pseudoclassical Lagrangian is constructed by combining the diffusive (bosonic) Lagrangian with the Grassmann (fermionic) Lagrangian, and performing the path integral over the Grassmann variables recovers the original Lagrangian that was used in the construction of the propagator for the magnetisation. The trajectories of the pseudoclassical model also provide some insight into the nature of the end-points.Comment: 25 page

Grassmann variables and pseudoclassical Nuclear Magnetic Resonance

The concept of a propagator is useful and is a well-known object in diffusion NMR experiments. Here, we investigate the related concept; the propagator for the magnetization or the Green's function of the Torrey-Bloch equations. The magnetization propagator is constructed by defining functions such as the Hamiltonian and Lagrangian and using these to define a path integral. It is shown that the equations of motion derived from the La-grangian produce complex-valued trajectories (classical paths) and it is conjectured that the end-points of these trajectories are real-valued. The complex nature of the trajectories also suggests that the spin degrees of freedom are also encoded into the trajectories and this idea is explored by explicitly modeling the spin or precessing magnetization by anticommuting Grassmann variables. A pseudoclassical Lagrangian is constructed by combining the dif-fusive (bosonic) Lagrangian with the Grassmann (fermionic) Lagrangian, and performing the path integral over the Grassmann variables recovers the original Lagrangian that was used in the construction of the propagator for the magnetization. The trajectories of the pseudoclassical model also provide some insight into the nature of the end-points

The effect of collagen fibril orientation on the biphasic mechanics of articular cartilage

The highly inhomogeneous distribution of collagen fibrils may have important effects on the biphasic mechanics of articular cartilage. However, the effect of the inhomogeneity of collagen fibrils has mainly been investigated using simplified three-layered models, which may have underestimated the effect of collagen fibrils by neglecting their realistic orientation. The aim of this study was to investigate the effect of the realistic orientation of collagen fibrils on the biphasic mechanics of articular cartilage. Five biphasic material models, each of which included a different level of complexity of fibril reinforcement, were solved using two different finite element software packages (Abaqus and FEBio). Model 1 considered the realistic orientation of fibrils, which was derived from diffusion tensor magnetic resonance images. The simplified three-layered orientation was used for Model 2. Models 3–5 were three control models. The realistic collagen orientations obtained in this study were consistent with the literature. Results from the two finite element implementations were in agreement for each of the conditions modelled. The comparison between the control models confirmed some functions of collagen fibrils. The comparison between Models 1 and 2 showed that the widely-used three-layered inhomogeneous model can produce similar fluid load support to the model including the realistic fibril orientation; however, an accurate prediction of the other mechanical parameters requires the inclusion of the realistic orientation of collagen fibrils.</p

Deuterium brain imaging at 7T during D 2 O dosing

PurposeTo characterize the (2H) deuterium MR signal measured from human brain at 7T in participants loading with D2O to ˜1.5% enrichment over a six-week period.Methods2H spectroscopy and imaging measurements were used to track the time-course of 2H enrichment within the brain during the initial eight-hour loading period in two participants. Multi-echo gradient echo (MEGE) images were acquired at a range of TR values from four participants during the steady-state loading period and used for mapping 2H T1 and T2* relaxation times. Co-registration to higher resolution 1H images allowed T1 and T2* relaxation times of deuterium in HDO in cerebrospinal fluid (CSF), gray matter (GM), and white matter (WM) to be estimated.Results2H concentrations measured during the eight-hour loading were consistent with values estimated from cumulative D2O dose and body mass. Signal changes measured from three different regions of the brain during loading showed similar time-courses. After summing over echoes, gradient echo brain images acquired in 7.5 minutes with a voxel volume of 0.36 ml showed an SNR of ˜16 in subjects loaded to 1.5%. T1-values for deuterium in HDO were significantly shorter than corresponding values for 1H in H2O, while T2* values were similar. 2H relaxation times in CSF were significantly longer than in GM or WM.ConclusionDeuterium MR Measurements at 7T were used to track the increase in concentration of 2H in brain during heavy water loading. 2H T1 and T2* relaxation times from water in GM, WM, and CSF are reported

Quantifying T2 relaxation time changes within lesions defined by apparent diffusion coefficient in grey and white matter in acute stroke patients

The apparent diffusion coefficient (ADC) of cerebral water, as measured by diffusion MRI, rapidly decreases in ischaemia, highlighting a lesion in acute stroke patients. The MRI T 2 relaxation time changes in ischaemic brain such that T 2 in ADC lesions may be informative of the extent of tissue damage, potentially aiding in stratification for treatment. We have developed a novel user-unbiased method of determining the changes in T 2 in ADC lesions as a function of clinical symptom duration based on voxel-wise referencing to a contralateral brain volume. The spherical reference method calculates the most probable pre-ischaemic T 2 on a voxel-wise basis, making use of features of the contralateral hemisphere presumed to be largely unaffected. We studied whether T 2 changes in the two main cerebral tissue types, i.e. in grey matter (GM) and white matter (WM), would differ in stroke. Thirty-eight acute stroke patients were accrued within 9 h of symptom onset and scanned at 3 T for 3D T 1-weighted, multi b-value diffusion and multi-echo spin echo MRI for tissue type segmentation, quantitative ADC and absolute T 2 images, respectively. T 2 changes measured by the spherical reference method were 1.94  ±  0.61, 1.50  ±  0.52 and 1.40  ±  0.54 ms h−1 in the whole, GM, and WM lesions, respectively. Thus, T 2 time courses were comparable between GM and WM independent of brain tissue type involved. We demonstrate that T 2 changes in ADC-delineated lesions can be quantified in the clinical setting in a user unbiased manner and that T 2 change correlated with symptom onset time, opening the possibility of using the approach as a tool to assess severity of tissue damage in the clinical setting

A Comparison of T2 Relaxation-Based MRI Stroke Timing Methods in Hyperacute Ischemic Stroke Patients: A Pilot Study

Background: T2 relaxation-based magnetic resonance imaging (MRI) signals may provide onset time for acute ischemic strokes with an unknown onset. The ability of visual and quantitative MRI-based methods in a cohort of hyperacute ischemic stroke patients was studied.Methods: A total of 35 patients underwent 3T (3 Tesla) MRI (<9-hour symptom onset). Diffusion-weighted (DWI), apparent diffusion coefficient (ADC), T1-weighted (T1w), T2-weighted (T2w), and T2 relaxation time (T2) images were acquired. T2-weighted fluid attenuation inversion recovery (FLAIR) images were acquired for 17 of these patients. Image intensity ratios of the average intensities in ischemic and non-ischemic reference regions were calculated for ADC, DWI, T2w, T2 relaxation, and FLAIR images, and optimal image intensity ratio cut-offs were determined. DWI and FLAIR images were assessed visually for DWI/FLAIR mismatch.Results: The T2 relaxation time image intensity ratio was the only parameter with significant correlation with stroke duration (r = 0.49, P = .003), an area under the receiver operating characteristic curve (AUC = 0.77, P < .0001), and an optimal cut-off (T2 ratio = 1.072) that accurately identified patients within the 4.5-hour thrombolysis treatment window with sensitivity of 0.74 and specificity of 0.74. In the patients with the additional FLAIR, areas under the precision-recall-gain curve (AUPRG) and F1 scores showed that the T2 relaxation time ratio (AUPRG = 0.60, F1 = 0.73) performed considerably better than the FLAIR ratio (AUPRG = 0.39, F1 = 0.57) and the visual DWI/FLAIR mismatch (F1 = 0.25).Conclusions: Quantitative T2 relaxation time is the preferred MRI parameter in the assessment of patients with unknown onset for treatment stratification

Time Correlations in Fluid Transport Obtained by Sequential Rephasing Gradient Pulses

We present a basic experiment by which the evolution of the displacement probability density (propagator) of static or flowing fluid in N successive time intervals is obtained by single labeling, coupled with multiple rephasing events during the course of a pulsed field-gradient sequence. We term this type of sequence SERPENT: SEquential Rephasing by Pulsed field-gradients Encoding N Time-intervals. Realizations of the SERPENT experiment for the case N = 2 which include spin echo, stimulated echo, and Carr-Purcell pulse sequences are suggested. They have in common a spatial spin-labeling of the initial magnetization by a gradient of area q0, followed by successive rephasing via gradients q1 and q2 at times t = Δ1 and t = Δ2, respectively, where q0 + q1 + q2 = 0. A two-dimensional Fourier transform with respect to q1 and q2 gives directly the joint probability density W2(R1, Δ1; R2, Δ2) for displacements R1 and R2 in times Δ1 and Δ2, respectively. q1 and q2 may be in arbitrary directions. Assuming R1∥R2, the correlation coefficient ρR1,R2 then reflects the time-history of the fluctuating velocities. The behavior of the cross moment 〈R1(Δ1) · R2(Δ2)〉 can be obtained from either a full two-dimensional or a set of one-dimensional SERPENT measurements. Experimental results are presented for water flowing through a bed of packed glass beads. While Δ1 is appropriately chosen to sample the short-time velocity field within the system, increasing Δ2 clearly shows the loss of correlation when the average fluid element displacement exceeds the bead diameter. © 1999 Academic Press

Influence of cellulose on ion diffusivity in 1-ethyl-3-methyl-imidazolium acetate cellulose solutions

International audienceSolutions of microcrystalline cellulose in 1-ethyl-3-methyl-imidazolium acetate have been investigated using pulsedfield gradient 1H NMR. In all cases the geometrically larger cation was found to diffuse faster than the smaller anion. Arrhenius temperature analysis has been applied to the ion diffusivities giving activation energies. The diffusion and published viscosity data for these solutions were shown to follow the Stokes-Einstein relationship, giving hydrodynamic radii of 1.6 A&ring; (cation) and 1.8 A&ring; (anion). Theories for obstruction, free-volume and hydrodynamic effects on solvent diffusion have been applied. The Mackie-Meares and Maxwell-Fricke obstruction models provided a correct trend only when assuming a certain fraction of ions are bound to the polymer. From this fraction it was shown that the maximum dissolvable cellulose concentration is ˜27% w/w, which is consistent with the highest known prepared concentration of cellulose in this ionic liquid. The Phillies' hydrodynamic model is found to give the best description for the cellulose concentration dependence of the ion diffusivitie