Probing restrictive diffusion dynamics at short time scales

Diffusion imaging gradients serve to spectrally filter the temporally evolving diffusion tensor. In this framework, the design of diffusion sensitizing gradients is reduced to the problem of adequately sampling q-space in the spectral domain. The practical limitations imposed by the requirement for delta-function type diffusion-sensitizing gradients to adequately sample q-space, can be relaxed if these impulse gradients are replaced with chirped oscillatory gradients. It is well known that in many systems of interest, dispersion of velocity will itself produce a peak in the velocity correlation function near w=0, while restricted diffusion will manifest itself in the dispersion spectrum at higher frequencies. In this paper, chirped diffusion-sensitizing gradients are proposed and analytically shown to yield an efficient sampling of q-space in a manner that asymptotically approaches that using delta-function diffusion-sensitizing gradient. The challenge is the consequent reduction in diffusion sensitivity as one probes higher frequency dynamics. This problem is addressed by restricting the gradient power to a spectral bandwidth corresponding to the diffusion spectral range of the underlying restrictive geometry. Simultaneous imaging of diffusion and flow at microscopic resolution and at temporally resolvable diffusion time scales thus becomes possible in vivo. Simulations and experiments validate the proposed approach.     

Dispersion measurements using time-of-flight remote detection MRI

Remote detection nuclear magnetic resonance and magnetic resonance imaging can be used to study fluid flow and dispersion in a porous medium from a purely Eulerian point of view (i.e., in a laboratory frame of reference). Information about fluid displacement is obtained on a macroscopic scale in a long-time regime, while local velocity distributions are averaged out. It is shown how these experiments can be described using the common flow propagator formalism and how experimental data can be analyzed to obtain effective porosity, flow velocity inside the porous medium, fluid dispersion and flow tracing of fluid.     

Measurement of pulsatile flow using MRI and a Bayesian technique of probability analysis

This work shows that complete spatial information of periodic pulsatile fluid flows can be rapidly obtained by Bayesian probability analysis of flow encoded magnetic resonance imaging data. These data were acquired as a set of two-dimensional images (complete two-dimensional sampling of k-space or reciprocal position space) but with a sparse (six point) and nonuniform sampling of q-space or reciprocal displacement space. This approach enables more precise calculation of fluid velocity to be achieved than by conventional two q-sample phase encoding of velocities, without the significant time disadvantage associated with the complete flow measurement required for Fourier velocity imaging. For experimental comparison with the Bayesian analysis applied to nonuniformly sampled q-space data, a Fourier velocity imaging technique was used with one-dimensional spatial encoding within a selected slice and a uniform sampling of q-space using 64 values of the pulsed gradients to encode fluid flow. Because the pulsatile flows were axially symmetric within the resolution of the experiment, the radial variation of fluid velocity, in the direction of the pulsed gradients, was reconstructed from one-dimensional spatial projections of the velocity by exploiting the central slice theorem. Data were analysed for internal consistency using linearised flow theories. The results show that nonuniform q-space sampling followed by Bayesian probability analysis is at least as accurate as the combined uniform q-space sampling with Fourier velocity imaging and projection reconstruction method. Both techniques give smaller errors than a two-point sampling of q-space (the conventional flow encoding experiment).     

Definition of displacement probability and diffusion time in q-space magnetic resonance measurements that use finite-duration diffusion-encoding gradients

In q-space diffusion NMR, the probability P(r,td) of a molecule having a displacement r in a diffusion time td is obtained under the assumption that the diffusion-encoding gradient g has an infinitesimal duration. However, this assumption may not always hold, particularly in human MRI where the diffusion-encoding gradient duration delta is typically of the same order of magnitude as the time offset Delta between encoding gradients. In this case, finite-delta effects complicate the interpretation of displacement probabilities measured in q-space MRI, and the form by which the signal intensity relates to them. By considering the displacement-specific dephasing, , of a set of spins accumulating a constant displacement vector r in the total time Delta+delta during which diffusion is encoded, the probability recovered by a finite-delta q-space experiment can be interpreted. It is shown theoretically that a data analysis using a modified q-space index q=gammadeltaetag, with gamma the gyromagnetic ratio and eta=square root (Delta-delta/3)/(Delta+delta), recovers the correct displacement probability distribution if diffusion is multi-Gaussian free diffusion. With this analysis, we show that the displacement distribution P(r,texp) is measured at the experimental diffusion-encoding time texp=Delta+delta, and not at the reduced diffusion time tr=Delta-delta/3 as is generally assumed in the NMR and MRI literature. It is also shown that, by defining a probability P(y,Delta) that a time tdeltac then eta is not equal to square root (Delta-delta/3)/(Delta+delta) which implies that we can no longer obtain the correct displacement probability from the displacement distribution. In the case that /g/=18 mT/m and Delta-delta=5 ms, the parameter deltac in ms is given by "deltac=0.49a2+0.24" where a is the sphere's radius expressed in microm. Simulation of q-space restricted diffusion MRI experiments indicate that if eta=square root (Delta-delta/3)/(Delta+delta), the recovered displacement probability is always better than the Gaussian approximation, and the measured diffusion coefficient matches the diffusion coefficient at time texp=Delta+delta better than it matches the diffusion coefficient at time tr=Delta-delta/3. These results indicate that q-space MRI measurements of displacement probability distributions are theoretically possible in biological tissues using finite-duration diffusion-encoding gradients provided certain compartment size and diffusion encoding gradient duration constraints are met.     

Nonasymptotic properties of transport and mixing

We study relative dispersion of passive scalar in nonideal cases, i.e., in situations in which asymptotic techniques cannot be applied; typically when the characteristic length scale of the Eulerian velocity field is not much smaller than the domain size. Of course, in such a situation usual asymptotic quantities (the diffusion coefficients) do not give any relevant information about the transport mechanisms. On the other hand, we shall show that the Finite Size Lyapunov Exponent, originally introduced for the predictability problem, appears to be rather powerful in approaching the nonasymptotic transport properties. This technique is applied in a series of numerical experiments in simple flows with chaotic behaviors, in experimental data analysis of drifter and to study relative dispersion in fully developed turbulence. (c) 2000 American Institute of Physics.     

Mapping the intracellular fraction of water by varying the gradient pulse length in q-space diffusion MRI

Finite gradient pulse lengths are traditionally considered a nuisance in q-space diffusion NMR and MRI, since the simple Fourier relation between the acquired signal and the displacement probability is invalidated. Increasing the value of the pulse length leads to an apparently smaller value of the estimated compartment size. We propose that q-space data at different gradient pulse lengths, but with the same effective diffusion time, can be used to identify and quantify components with free or restricted diffusion from multiexponential echo decay curves obtained on cellular systems. The method is demonstrated with experiments on excised human brain white matter and a series of model systems with well-defined free, restricted, and combined free and restricted diffusion behavior. Time-resolved diffusion MRI experiments are used to map the spatial distribution of the intracellular fraction in a yeast cell suspension during sedimentation, and observe the disappearance of this fraction after a heat treatment.     

Synchronized EPI phase contrast velocimetry in a mixing reactor

Notwithstanding its widespread use in cardiovascular and functional MRI studies, Echo Planar Imaging (EPI) has only recently been subjected to systematic validation studies. Most velocity measurement studies employing such ultrafast MRI methods involve the use of phantoms characterized by rigid or deformable solid motion. The current implementation involves a rotating phantom (angular velocity up to 10.5 rpm) with a superimposed swirling liquid flow (with axial velocities ranging between 0.145 and 0.27 cm/s) of water doped with copper sulfate. The standard implementation of single-shot EPI with phase contrast velocity encoding allows the complete mapping of the Eulerian velocity field in slices perpendicular to the rotation axis following a subtractive procedure requiring the synchronized acquisition of each velocity component on each selected transverse slice during two revolutions of the rotor. The image acquisition time is 100 ms (per velocity component) at each 64 x 64 slice. In addition to acquiring full-field velocity data for future direct comparisons with other techniques, EPI is employed here for the first time to reconstruct the three-dimensional flow field between the blades of a partitioned pipe mixer.     

On Measuring and Modeling Local Average Particle Velocities for particle ensembles in classifying systems

In most multi-phase flow problems, the particulate phase is exposed to an external field which causes dispersion. Therefore, local velocity measurements of the disperse phase are no longer equivalent with respect to averaging in time and averaging in volume. While the local time-averaged velocity still characterizes the transport of the ensemble in the Eulerian sense, one has to be be careful in modeling this velocity average by considering the ensemble's composition. It is shown for different particle ensembles that the conventional particle velocity average M_1,0 calculated with respect to the dispersion relationship and a particle size number density distribution is far below the measured ensemble average; the deviation depends on the width of the particle size distribution. It is deduced that Eulerian particle velocity values referring to a certain time interval can be modeled by a ratio of velocity moments M_2,0/M_1,0 calculated with particle size number distributions referring to a certain probe volume. This relationship was confirmed by measurements.     

The effect of rotational angle and experimental parameters on the diffraction patterns and micro-structural information obtained from q-space diffusion NMR: implication for diffusion in white matter fibers

Diffusion NMR may provide, under certain experimental conditions, micro-structural information about confined compartments totally non-invasively. The influence of the rotational angle, the pulse gradient length and the diffusion time on the diffusion diffraction patterns and q-space displacement distribution profiles was evaluated for ensembles of long cylinders having a diameter of 9 and 20 microm. It was found that the diffraction patterns are sensitive to the rotational angle (alpha) and are observed only when diffusion is measured nearly perpendicular to the long axis of the cylinders i.e., when alpha= 90 degrees +/- 5 degrees under our experimental conditions. More importantly, we also found that the structural information extracted from the displacement distribution profiles and from the diffraction patterns are very similar and in good agreement with the experimental values for cylinders of 20 microm or even 9 microm, when data is acquired with parameters that satisfy the short gradient pulse (SGP) approximation (i.e., delta -->0) and the long diffusion time limit. Since these experimental conditions are hardly met in in vitro diffusion MRI of excised organs, and cannot be met in clinical MRI scanners, we evaluated the effect of the pulse gradient duration and the diffusion time on the structural information extracted from q-space diffusion MR experiments. Indeed it was found that, as expected, accurate structural information, and diffraction patterns are observed when Delta is large enough so that the spins reach the cylinders' boundaries. In addition, it was found that large delta results in extraction of a compartment size, which is smaller than the real one. The relevance of these results to q-space MRI of neuronal tissues and fiber tracking is discussed.     

Phase contrast MRI with improved temporal resolution by view sharing: k-space related velocity mapping properties.

Phase contrast techniques in combination with k-space segmented CINE imaging are widely used for the quantitative assessment of blood flow or tissue motion. The temporal resolution of the corresponding pulse sequences plays an important role concerning the potential of the method to fully detect time resolved flow or motion patterns. A further improvement of temporal or spatial resolution in phase contrast CINE MRI can be achieved by the application of view sharing. Based on simulations with point-spread-functions resulting from different cyclic flow or motion patterns an analysis of view sharing techniques in combination with phase contrast MRI is presented. Velocity mapping properties and the role of different k-space regions concerning the resulting values in the phase images and thus encoded velocities were investigated. It could be shown that the velocity induced phase shifts in phase contrast techniques are mainly encoded in the central sections of k-space which makes view sharing also suitable for velocity mapping. As a result the use of appropriate sampling and data acquisition schemes permits the assessment of flow or motion patterns with significantly improved temporal resolution without loss of functional information. In addition phantom measurements with an oscillation phantom were performed in order to validate the simulation results and to demonstrate the potential of view sharing techniques to accelerate phase contrast imaging and improve the detection of the underlying flow or motion dynamics.     

Pores resolving simulation of Darcy flows

A theoretical formulation and corresponding numerical solutions are presented for microscopic fluid flows in porous media with the domain sufficiently large to reproduce integral Darcy scale effects. Pore space geometry and topology influence flow through media, but the difficulty of observing the configurations of real pore spaces limits understanding of their effects. A rigorous direct numerical simulation (DNS) of percolating flows is a formidable task due to intricacies of internal boundaries of the pore space. Representing the grain size distribution by means of repelling body forces in the equations governing fluid motion greatly simplifies computational efforts. An accurate representation of pore-scale geometry requires that within the solid the repelling forces attenuate flow to stagnation in a short time compared to the characteristic time scale of the pore-scale flow. In the computational model this is achieved by adopting an implicit immersed-boundary method with the attenuation time scale smaller than the time step of an explicit fluid model. A series of numerical simulations of the flow through randomly generated media of different porosities show that computational experiments can be equivalent to physical experiments with the added advantage of nearly complete observability. Besides obtaining macroscopic measures of permeability and tortuosity, numerical experiments can shed light on the effect of the pore space structure on bulk properties of Darcy scale flows.     

q-Space high b value diffusion MRI of hemi-crush in rat spinal cord: evidence for spontaneous regeneration

The development of the damage following hemi-crush trauma in rat spinal cord was studied ex vivo using high b value (bmax = 1 x 10(7) s cm(-2)) q-space diffusion weighted MRI (DWI) at five days, ten days and six weeks post-trauma. Rat spinal cord trauma, produced by hemi-crush of 15s and 60s duration, was studied. The water signal decay in these diffusion experiments was found to be non mono-exponential and was analyzed using the q-space approach. The q-space MRI parameters were compared with T1 and T2 MR images, behavioral tests and histopathological osmium staining. A very good anatomical correlation was found between the q-space MRI parameters and the osmium staining. Interestingly, we found that in the 15s hemi-crush model significant recovery was observed in both the q-space MR images and the osmium staining six weeks post-trauma. However, in the 60s hemi-crush trauma model very little recovery was observed. These results paralleled those obtained from behavioral tests demonstrating that partial spontaneous recovery seems to occur in the 15s hemi-crush spinal cord model, which should be taken in consideration when using it to evaluate new therapies.     

Negative dispersion in bone: the role of interference in measurements of the apparent phase velocity of two temporally overlapping signals

In this study the attenuation coefficient and dispersion (frequency dependence of phase velocity) are measured using a phase sensitive (piezoelectric) receiver in a phantom in which two temporally overlapping signals are detected, analogous to the fast and slow waves typically found in measurements of cancellous bone. The phantom consisted of a flat and parallel Plexiglas plate into which a step discontinuity was milled. The phase velocity and attenuation coefficient of the plate were measured using both broadband and narrowband data and were calculated using standard magnitude and phase spectroscopy techniques. The observed frequency dependence of the phase velocity and attenuation coefficient exhibit significant changes in their frequency dependences as the interrogating ultrasonic field is translated across the step discontinuity of the plate. Negative dispersion is observed at specific spatial locations of the plate at which the attenuation coefficient rises linearly with frequency, a behavior analogous to that of bone measurements reported in the literature. For all sites investigated, broadband and narrowband data (3-7 MHz) demonstrate excellent consistency. Evidence suggests that the interference between the two signals simultaneously reaching the phase sensitive piezoelectric receiver is responsible for this negative dispersion.     

Determining pore sizes using an internal magnetic field

A concept is proposed to measure the pore size length scale by the internal magnetic field (B_i) in porous materials. The spatial distribution of the magnetic field inhomogeneity, a result of the magnetic susceptibility contrast between the porous material and the fluid, reflects the underlying pore geometry. Diffusion in B_i causes the initial decay of magnetization. At long times, the effect of B_i saturates when the diffusion length reaches a characteristic pore size. This method is independent of surface spin relaxation in determining pore sizes. Nuclear magnetic resonance experiments on packed glass beads and sedimentary rock samples will be presented.     

Laser velocimeter correlation measurements in subsonic and supersonic jets

This paper describes laser velocimeter cross-correlation measurements conducted in the flow field of a jet. The jet conditions were varied over a range of Mach numbers from 0·5 to 1·37, and in the case of the Mach 0·9 jet, tests were also run at an absolute exit temperature equal to 2·32 that of the ambient. From the families of cross-correlation curves, the integral length scales, the convection velocity and the integral time scale in the moving frame were determined. It was found that the axial and radial integral length scales changed almost linearly with the axial distance from the nozzle, at rates which were in agreement with Jones' [1]. The convection velocity and integral time scale also agreed with values previously obtained with hot wire anemometers. The effects of jet exit conditions on the variation of the length scales were not noticeable. However, in the case of the convection velocity and time scale, there was some dependence on the jet conditions.     

Time-of-flight remote detection MRI of thermally modified wood

We demonstrate that time-of-flight (TOF) remote detection (RD) magnetic resonance imaging (MRI) provides detailed information about physical changes in wood due to thermal modification that is not available with conventional nuclear magnetic resonance (NMR) based techniques. In the experiments, xenon gas was forced to flow through Pinus sylvestris pine wood samples, and the flow paths and dispersion of gas atoms were observed by measuring ¹²⁹Xe TOF RD MRI images from the samples. MRI sensitivity of xenon was boosted by the spin exchange optical pumping (SEOP) method. Two different samples were studied: a reference sample, dried at low temperature, and a modified sample, which was thermally modified at 240 °C after the drying. The samples were taken next to each other from the same wood plank in order to ensure the comparability of the results. The most important conclusion is that both the smaller dispersion observed in all the TOF RD experiments independent of each other and the decreased amount of flow paths shown by the time projection of z-encoded TOF RD MRI experiment imply that a large amount of pits connecting tracheid cells are closed in thermal modification. Closed pits may be one reason for reduced moisture content and improved dimensional stability of wood achieved in thermal modification. This is the first time biological samples have been investigated by TOF RD MRI.     

Effect of interstitial air pressure and particle size on the heat storage coefficient of a two phase system

An expression for the heat storage coefficient (HSC) of loose granular systems at interstitial air pressures is developed in terms of the characteristic pressure. The dependence of the HSC on pore and particle sizes and characteristic pressure is also discussed. To test the validity of the derived expression, the experiments are performed on a number of materials with various particle sizes and interstitial air pressures. The other reported data on the HSC are also compared with predicted values. A good agreement is found between them. This study suggests that the selection of proper particle size and partial evacuation of the pore improves the insulation characteristics of the materials.     

Dispersion of dilatation wave propagation in single-wall carbon nanotubes using nonlocal scale effects

Nonlocal continuum mechanics allows one to account for the small length scale effect that becomes significant when dealing with micro- or nano-structures. This paper investigates a model of wave propagation in single-wall carbon nanotubes (SWCNTs) with small scale effects are studied. The equation of motion of the dilatation wave is obtained using the nonlocal elastic theory. We show that a dispersive wave equation is obtained from a nonlocal elastic constitutive law, based on a mixture of a local and a nonlocal strain. The SWCNTs structures are treated within the multilayer thin shell approximation with the elastic properties taken to be those of the graphene sheet. The SWCNT was the (40,0) zigzag tube with an effective diameter of 3.13 nm. Nonlinear frequency equations of wave propagation in SWCNTs are described through the effect of small scale. The phase velocity and the group velocity are derived, respectively. The nonlinear dispersion relation is analyzed with different wave numbers versus scale coefficient. It can be observed from the results that the dispersion properties of the dilatation wave are induced by the small scale effects, which will disappear in local continuous models. The dispersion degree can be strengthened by increasing the scale coefficient and the wave number. Furthermore, the characteristics for the group velocity of the dilatation wave in carbon nanotubes can also be tuned by these factors.     

Permeability Coefficients from NMR q-Space Data: Models with Unevenly Spaced Semi-permeable Parallel Membranes

The NMR "q-space" experiment conducted on water provides information on the sizes of repeated structures on the micrometer-length scale in heterogeneous samples, including cell suspensions or tissues. Under some circumstances these plots display coherence peaks, and it has been implied theoretically that the position of the peaks will vary with the rate of molecular exchange across the membranes. This has been demonstrated (qualitatively) with human erythrocytes in suspension. Thus, in the quest for a quantitative approach to the interpretation of such data, we address here the "inverse problem," namely the estimate of the permeability coefficient of membranes from q-space experiments. The present work describes theoretical predictions of q-space plots from molecules diffusing in a simple system of parallel semi-permeable membranes arranged with separations that alternate between two different values; this was designed to (loosely) mimic the intra- and extracellular compartments in a suspension of cells or a tissue. The development of the theory was facilitated by symbolic computation, and the analysis of synthetic data was shown to be achievable by the use of a three-layer back-propagation artificial neural network.     

Dynamics of counterpropagating waves in parametrically driven systems: dispersion vs. advection

The dynamics of parametrically driven counterpropagating waves in a one-dimensional extended nearly conservative annular system are described by two coupled, damped, parametrically driven nonlinear Schrödinger (NLS) equations with opposite transport terms due to the group velocity, and small dispersion. The system is characterized by two length scales defined by a balance between (a) forcing and dispersion (the dispersive scale), and (b) forcing and advection at the group velocity (the transport scale). Both are large compared to the basic wavelength of the pattern. The dispersive scale plays an important role in the structure of solutions arising from secondary instabilities of frequency-locked spatially uniform standing waves (SW), and manifests itself both in traveling pulses or fronts and in extended spatio-temporal chaos, depending on the signs of the dispersion coefficient and nonlinearity.