Computer simulation and six-dimensional spin density and velocity NMR microimaging of lacunar systems: A comparative analysis of percolation properties

Observation of anomalous diffusion in excised tissue by characterizing the diffusion-time dependence of the MR signal

This report introduces a novel method to characterize the diffusion-time dependence of the diffusion-weighted magnetic resonance (MR) signal in biological tissues. The approach utilizes the theory of diffusion in disordered media where two parameters, the random walk dimension and the spectral dimension, describe the evolution of the average propagators obtained from q-space MR experiments. These parameters were estimated, using several schemes, on diffusion MR spectroscopy data obtained from human red blood cell ghosts and nervous tissue autopsy samples. The experiments demonstrated that water diffusion in human tissue is anomalous, where the mean-square displacements vary slower than linearly with diffusion time. These observations are consistent with a fractal microstructure for human tissues. Differences observed between healthy human nervous tissue and glioblastoma samples suggest that the proposed methodology may provide a novel, clinically useful form of diffusion MR contrast.

Fractal analysis methods for solid alkane monolayer domains at SiO2/air interfaces

A systematic evaluation of various fractal analysis methods is essential for studying morphologies of finite and noisy experimental patterns such as domains of long chain alkanes at SiO(2)/air interfaces. The derivation of trustworthy fractal dimensions crucially relies on the definition of confidence intervals for the assumed scaling range. We demonstrate that the determination of the intervals can be improved largely by comparing the scaling behavior of different morphological measures (area, boundary, curvature). We show that the combination of area and boundary data from coarse-grained structures obtained with the box-counting method reveals clear confidence limits and thus credible morphological data. This also holds for the Minkowski density method. It also reveals the confidence range. Its main drawback, the larger swing-in period at the lower cutoff compared to the box-counting method, is compensated by more details on the scaling behavior of area, boundary, and curvature. The sandbox method is less recommendable. It essentially delivers the same data as box-counting, but it is more susceptible to finite size effects at the lower cutoff. It is found that the domain morphology depends on the surface coverage of alkanes. The individual domains at low surface coverage have a fractal dimension of approximately 1.7, whereas at coverages well above 50% the scaling dimension is 2 with a large margin of uncertainty at approximately 50% coverage. This change in morphology is attributed to a crossover from a growth regime dominated by diffusion-limited aggregation of individual domains to a regime where the growth is increasingly affected by annealing and the interaction of solid growth fronts which approach each other and thus compete for the alkane supply.

NMR acceleration mapping in percolation model objects

An NMR microscopy technique is described that permits direct mapping of local accelerations. The method is tested with water flow through a random site percolation model object and compared with computational fluid dynamics simulations. A general formalism, the "polygon rule," is reported for the design of gradient pulse sequences for phase encoding of higher order motions, or, in other words, for compensation of phase shifts by lower motional orders.

NMR visualization of displacement correlations for flow in porous media

The temporal correlations of velocities for both water and a water-glycerol mixture flowing through a random packings of monodisperse spherical particles have been investigated using two-dimensional nuclear magnetic resonance methods. By combining various flow rates, fluid viscosities, and bead sizes, a wide range of flow parameters has been covered, the dimensionless Peclet number ranging from 100 to 100 000. The velocity exchange spectroscopy (VEXSY) technique has been employed to measure the correlation between velocities during two intervals separated from each other by a mixing time tau(m). This time is made both large and small compared with the time constant tau(c), required for a fluid element possessing the average flow velocity to cover a distance equal to the characteristic size in the system, the bead diameter. The two-dimensional conditional probability of displacement resulting from the VEXSY method reveals the existence of different "subensembles" of molecules, including a slow moving pool whose displacement is dominated by Brownian motion, an intermediate ensemble whose velocities change little over the mixing time, and a fast flowing ensemble which loses correlation due to mechanical dispersion. We find that that the approach to asymptotic dispersion, as tau(c)/tau(m) increases, depends strongly on the Peclet number, the deviation of the velocity autocorrelation function from a monoexponential Ornstein-Uhlenbeck process becoming more pronounced with increasing Peclet number.

Rayleigh-Bénard percolation transition of thermal convection in porous media: computational fluid dynamics, NMR velocity mapping, NMR temperature mapping

Stationary thermal convection and heat conduction were studied in random-site percolation clusters in a Rayleigh-Bénard configuration experimentally by NMR microscopy techniques and numerically with the aid of a finite volume method. The porosity of the percolation clusters, and the temperature difference applied to the convection cell were varied. Two-dimensional percolation networks were generated with the aid of a random-number algorithm. The resulting clusters were used as templates for the fabrication of model objects. The convective velocity distribution of silicon oil or ethylene glycol filled into the pore space was mapped and evaluated in the form of histograms. The flow patterns visualized in the simulated and the measured velocity maps show good coincidence. In the histograms, two velocity regimes can be distinguished and attributed to local convection rolls responsible for the low-velocity part, and cluster-spanning flow loops characterized by a high-velocity cut-off, respectively. The maximum velocity as a function of the porosity and the overall temperature difference is shown to be indicative for the hydro-thermodynamic Rayleigh-Bénard instability and the geometrical percolation threshold. The coincidence of the Rayleigh-Bénard instability (modified for porous media) and the percolation transition (modified for closed loops) gives rise to a new critical phenomenon termed the Rayleigh-Bénard percolation transition. It occurs at a certain combination of the porosity and the overall temperature difference in the cell. Temperature maps were recorded with the aid of a relaxation-based NMR technique. The consequence of different thermal conductivities in the matrix and in the fluid is that horizontal-temperature gradients arise even in the absence of flow. This leads to a superposition of uncritical convective flow driven by the horizontal-temperature gradients whenever closed-loop pathways are possible and the critical Rayleigh-Bénard convection based on vertical-temperature gradients.

Maps of electric current density and hydrodynamic flow in porous media: NMR experiments and numerical simulations

The electric current density in percolation clusters was mapped with the aid of a NMR microscopy technique monitoring the spatial distribution of spin precession phase shifts caused by the currents. A test structure and a quasi-two-dimensional random-site percolation model object filled with an electrolyte solution were examined and compared with numerical simulations based on potential theory. The current density maps permit the evaluation of histograms and of volume-averaged current densities as a function of the probe volume radius as relationships characterizing transport in the clusters. The current density maps are juxtaposed to velocity maps acquired in flow NMR experiments in the same objects. It is demonstrated that electric current and hydrodynamic flow lead to transport patterns deviating in a characteristic way due to the different dependencies of the transport resistances on the pore channel width.

Diffusion on random-site percolation clusters: theory and NMR microscopy experiments with model objects

Quasi-two-dimensional random-site percolation model objects were fabricated based on computer-generated templates. Samples consisting of two compartments, a reservoir of H2O gel attached to a percolation model object, which was initially filled with D2O, were examined with nuclear magnetic resonance microscopy for rendering proton spin density maps. The propagating proton/deuteron interdiffusion profiles were recorded and evaluated with respect to anomalous diffusion parameters. The deviation of the concentration profiles from those expected for unobstructed diffusion directly reflects the anomaly of the propagator for diffusion on a percolation cluster. The fractal dimension of the random walk d(w) evaluated from the diffusion measurements on the one hand and the fractal dimension d(f) deduced from the spin density map of the percolation object on the other permits one to experimentally compare dynamical and static exponents. Approximate calculations of the propagator are given on the basis of the fractional diffusion equation. Furthermore, the ordinary diffusion equation was solved numerically for the corresponding initial and boundary conditions for comparison. The anomalous diffusion constant was evaluated and is compared to the Brownian case. Some ad hoc correction of the propagator is shown to pay tribute to the finiteness of the system. In this way, anomalous solutions of the fractional diffusion equation could experimentally be verified.

Rayleigh-Bénard percolation transition study of thermal convection in porous media: numerical simulation and NMR experiments

Thermal convection was studied as a function of the porosity in random-site percolation model objects in a Rayleigh-Bénard configuration. NMR velocity mapping experiments and numerical simulations using the finite-volume method are compared. Velocity histograms were evaluated and can be described by power laws in a wide range. The maximum velocity as a function of the porosity indicates a combined percolation/Rayleigh-Bénard transition.

Flow through percolation clusters: NMR velocity mapping and numerical simulation study

Three- and (quasi-)two-dimensional percolation objects have been fabricated based on Monte Carlo generated templates. The object size was up to 12 cm (300 lattice sites) in each dimension. Random site, semicontinuous swiss-cheese, and semicontinuous inverse swiss-cheese percolation models above the percolation threshold were considered. The water-filled pore space was investigated by nuclear magnetic resonance (NMR) imaging and, after exerting a pressure gradient, by NMR velocity mapping. The spatial resolutions of the fabrication process and the NMR experiments were 400 microm and better than 300 microm, respectively. The experimental velocity resolution was 60 microm/s. The fractal dimension, the correlation length, and the percolation probability can be evaluated both from the computer generated templates and the corresponding NMR spin density maps. Based on velocity maps, the percolation backbones were determined. The fractal dimension of the backbones turned out to be smaller than that of the complete cluster. As a further relation of interest, the volume-averaged velocity was calculated as a function of the probe volume radius. In a certain scaling window, the resulting dependence can be represented by a power law, the exponent of which was not yet considered in the theoretical literature. The experimental results favorably compare to computer simulations based on the finite-element method (FEM) or the finite-volume method (FVM). This demonstrates that NMR microimaging as well as FEM/FVM simulations reliably reflect transport features in percolation clusters.

Flow, diffusion, and thermal convection in percolation clusters: NMR experiments and numerical FEM/FVM simulations

Percolation objects were fabricated based on computer-generated, two- or three-dimensional templates. Random-site, semi-continuous swiss cheese, and semi-continuous inverse swiss-cheese percolation models above the percolation threshold were considered. The water-filled pore space was investigated by NMR imaging and, in the presence of a pressure gradient, NMR velocity mapping. The fractal dimension, the correlation length, and the percolation probability were evaluated both from the computer-generated templates and the corresponding NMR spin density maps. Based on velocity maps, the percolation backbones were determined. The fractal dimension of the backbones turned out to be smaller than that of the complete cluster. As a further relation of interest, the volume-averaged velocity was calculated as a function of the probe volume radius. In a certain scaling window, the resulting dependence can be represented by a power law the exponent of which was not yet considered in the theoretical literature. The experimental results favorably compare to computer simulations based on the finite-element method (FEM) or the finite-volume method (FVM). Percolation theory suggests a relationship between the anomalous diffusion exponent and the fractal dimension of the cluster, i.e., between a dynamic and a structural parameter. We examined interdiffusion between two compartments initially filled with H2O and D2O, respectively, by proton imaging. The results confirm the theoretical expectation. As a third transport mechanism, thermal convection in percolation clusters of different porosities was studied with the aid of NMR velocity mapping. The velocity distribution is related to the convection roll size distribution. Corresponding histograms consist of a power law part representing localized rolls, and a high-velocity cut-off for cluster-spanning rolls. The maximum velocity as a function of the porosity clearly visualizes the percolation transition.

Characterization of human head vasculature by percolation parameters

A data reduction procedure, originally proposed for characterization of fractals and random percolation clusters, has been used to evaluate the vascular system of the human head. The motivation behind this study arose from the wish to study empirically transport properties of vascular systems and to find a suitable formalism for their description. MR angiographic data acquired by a standard 3D inflow method were used. The evaluated parameters refer to the backbone fractal dimensionality and the correlation length. The fractal dimensionality of the backbone was found to be 1.71 for the human head vasculature. This value fits the theoretical range of random percolation networks. It is concluded that concepts of percolation theory might have some value for characterizing the structure and transport properties of the vascular system.

Six-dimensional spin density/velocity NMR microscopy of percolation clusters

Using computer-simulated random-site percolation networks as templates, three-dimensional percolation cluster objects were fabricated. The pore space was filled with water and experimentally investigated with the aid of NMR microimaging. A pulse sequence for six-dimensional spin density/velocity NMR imaging was employed for the combined record of the three-dimensional spin-density distribution and the three-dimensional velocity vector field of water percolating through the pore space. An evaluation procedure for the NMR image data was established that reliably renders the characteristic parameters (fractal dimensionality, fractal dimensionality of the backbone, correlation length).