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Badia M, El-Moudny S, Benhamou M, Ossmani ME. Study of cage effect and subdiffusion in Pickering emulsions from Molecular Dynamics simulations. J Mol Liq 2017. [DOI: 10.1016/j.molliq.2017.05.050] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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2
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Janakiraman D, Sebastian KL. Path-integral formulation for Lévy flights: evaluation of the propagator for free, linear, and harmonic potentials in the over- and underdamped limits. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:061105. [PMID: 23367891 DOI: 10.1103/physreve.86.061105] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/02/2012] [Indexed: 06/01/2023]
Abstract
Lévy flights can be described using a Fokker-Planck equation, which involves a fractional derivative operator in the position coordinate. Such an operator has its natural expression in the Fourier domain. Starting with this, we show that the solution of the equation can be written as a Hamiltonian path integral. Though this has been realized in the literature, the method has not found applications as the path integral appears difficult to evaluate. We show that a method in which one integrates over the position coordinates first, after which integration is performed over the momentum coordinates, can be used to evaluate several path integrals that are of interest. Using this, we evaluate the propagators for (a) free particle, (b) particle subjected to a linear potential, and (c) harmonic potential. In all the three cases, we have obtained results for both overdamped and underdamped cases.
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Affiliation(s)
- Deepika Janakiraman
- Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore 560012, India
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3
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Özarslan E, Shepherd TM, Koay CG, Blackband SJ, Basser PJ. Temporal scaling characteristics of diffusion as a new MRI contrast: findings in rat hippocampus. Neuroimage 2012; 60:1380-93. [PMID: 22306798 PMCID: PMC3303993 DOI: 10.1016/j.neuroimage.2012.01.105] [Citation(s) in RCA: 34] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/15/2011] [Revised: 01/18/2012] [Accepted: 01/20/2012] [Indexed: 11/21/2022] Open
Abstract
Features of the diffusion-time dependence of the diffusion-weighted magnetic resonance imaging (MRI) signal provide a new contrast that could be altered by numerous biological processes and pathologies in tissue at microscopic length scales. An anomalous diffusion model, based on the theory of Brownian motion in fractal and disordered media, is used to characterize the temporal scaling (TS) characteristics of diffusion-related quantities, such as moments of the displacement and zero-displacement probabilities, in excised rat hippocampus specimens. To reduce the effect of noise in magnitude-valued MRI data, a novel numerical procedure was employed to yield accurate estimation of these quantities even when the signal falls below the noise floor. The power-law dependencies characterize the TS behavior in all regions of the rat hippocampus, providing unique information about its microscopic architecture. The relationship between the TS characteristics and diffusion anisotropy is investigated by examining the anisotropy of TS, and conversely, the TS of anisotropy. The findings suggest the robustness of the technique as well as the reproducibility of estimates. TS characteristics of the diffusion-weighted signals could be used as a new and useful marker of tissue microstructure.
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Affiliation(s)
- Evren Özarslan
- Section on Tissue Biophysics and Biomimetics, PPITS, NICHD, National Institutes of Health, Bethesda, MD 20892, USA.
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4
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Cushman JH, O'Malley D, Park M. Anomalous diffusion as modeled by a nonstationary extension of Brownian motion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:032101. [PMID: 19391995 DOI: 10.1103/physreve.79.032101] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/31/2008] [Revised: 01/10/2009] [Indexed: 05/27/2023]
Abstract
If the mean-square displacement of a stochastic process is proportional to t;{beta} , beta not equal1 , then it is said to be anomalous. We construct a family of Markovian stochastic processes with independent nonstationary increments and arbitrary but a priori specified mean-square displacement. We label the family as an extended Brownian motion and show that they satisfy a Langevin equation with time-dependent diffusion coefficient. If the time derivative of the variance of the process is homogeneous, then by computing the fractal dimension it can be shown that the complexity of the family is the same as that of the Brownian motion. For two particles initially separated by a distance x , the finite-size Lyapunov exponent (FSLE) measures the average rate of exponential separation to a distance ax . An analytical expression is developed for the FSLEs of the extended Brownian processes and numerical examples presented. The explicit construction of these processes illustrates that contrary to what has been stated in the literature, a power-law mean-square displacement is not necessarily related to a breakdown in the classical central limit theorem (CLT) caused by, for example, correlation (fractional Brownian motion or correlated continuous-time random-walk schemes) or infinite variance (Levy motion). The classical CLT, coupled with nonstationary increments, can and often does give rise to power-law moments such as the mean-square displacement.
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Affiliation(s)
- John H Cushman
- Department of Earth and Atmospheric Sciences and Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, USA.
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Ozarslan E, Basser PJ, Shepherd TM, Thelwall PE, Vemuri BC, Blackband SJ. Observation of anomalous diffusion in excised tissue by characterizing the diffusion-time dependence of the MR signal. JOURNAL OF MAGNETIC RESONANCE (SAN DIEGO, CALIF. : 1997) 2006; 183:315-23. [PMID: 16962801 DOI: 10.1016/j.jmr.2006.08.009] [Citation(s) in RCA: 43] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/09/2006] [Revised: 08/07/2006] [Accepted: 08/19/2006] [Indexed: 05/11/2023]
Abstract
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.
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Affiliation(s)
- Evren Ozarslan
- STBB/LIMB/NICHD, National Institutes of Health, Bethesda, MD 20892, USA.
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6
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NMR investigations of correlations between longitudinal and transverse displacements in flow through random structured media. Chem Phys 2002. [DOI: 10.1016/s0301-0104(02)00668-7] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
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7
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Valiullin R, Furó I. Low-temperature phase separation of a binary liquid mixture in porous materials studied by cryoporometry and pulsed-field-gradient NMR. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:031508. [PMID: 12366121 DOI: 10.1103/physreve.66.031508] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/15/2002] [Indexed: 05/23/2023]
Abstract
The low-temperature liquid-liquid phase separation of the partially miscible hexane-nitrobenzene mixture imbibed in porous glasses of different pore sizes from 7 to 130 nm has been studied using 1H NMR (nuclear magnetic resonance) cryoporometry and pulse field gradient NMR methods. The mixture was quenched below both its upper critical solution temperature (T(cr)) and the freezing point of nitrobenzene. The size distribution of frozen nitrobenzene domains was derived through their melting point suppression according to the Gibbs-Thompson relation. The obtained data reveal small initial droplets of nitrobenzene surrounded by hexane, which are created as the temperature is decreased below T(cr) and which thereafter coalesce by a droplet-diffusion mechanism. The inter-relation between the pore size and the found size distribution and shapes of nitrobenzene domains is discussed, as well as several aspects of molecular self-diffusion.
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Affiliation(s)
- Rustem Valiullin
- Division of Physical Chemistry, Department of Chemistry, Royal Institute of Technology, SE-10044 Stockholm, Sweden
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Weber M, Kimmich R. Maps of electric current density and hydrodynamic flow in porous media: NMR experiments and numerical simulations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:026306. [PMID: 12241284 DOI: 10.1103/physreve.66.026306] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/05/2001] [Revised: 03/15/2002] [Indexed: 05/23/2023]
Abstract
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.
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Affiliation(s)
- Markus Weber
- Sektion Kernresonanzspektroskopie, Universität Ulm, 89069 Ulm, Germany
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Stallmach F, Vogt C, Kärger J, Helbig K, Jacobs F. Fractal geometry of surface areas of sand grains probed by pulsed field gradient NMR. PHYSICAL REVIEW LETTERS 2002; 88:105505. [PMID: 11909371 DOI: 10.1103/physrevlett.88.105505] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/23/2001] [Indexed: 05/23/2023]
Abstract
Pulsed field gradient NMR self-diffusion studies of water were used to determine surface-to-volume ratios and specific surface areas of the grains forming a glacial sand deposit. Both quantities exhibit a noninteger power-law dependence as a function of the diameters of the grains. The associated fractal dimensions of the surface area ( D(s)) and of the pore volume ( D(v)) are found to be D(s)-D(v) = -0.70+/-0.05 and D(s) = 2.20+/-0.05. The results demonstrate that NMR studies with native pore fluids are suitable to investigate the fractal nature of natural, unconsolidated porous materials.
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Affiliation(s)
- Frank Stallmach
- Fakultät für Physik und Geowissenschaften, Universität Leipzig, Linnestrasse 5, 04105 Leipzig, Germany.
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Klemm A, Metzler R, Kimmich R. Diffusion on random-site percolation clusters: theory and NMR microscopy experiments with model objects. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:021112. [PMID: 11863508 DOI: 10.1103/physreve.65.021112] [Citation(s) in RCA: 23] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/05/2001] [Revised: 08/23/2001] [Indexed: 05/23/2023]
Abstract
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.
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Affiliation(s)
- Andreas Klemm
- Sektion Kernresonanzspektroskopie, Universität Ulm, 89069 Ulm, Germany
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12
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Shushin AI. Anomalous two-state model for anomalous diffusion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:051108. [PMID: 11735901 DOI: 10.1103/physreve.64.051108] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/30/2001] [Indexed: 05/23/2023]
Abstract
An anomalous two-state model (ATSM) with the anomalous long-tailed kinetics of transitions between states is proposed to describe the specific features of anomalous diffusion (AD) and AD-assisted transitions (ADAT) in the double-well potential. In the ATSM the system is assumed to undergo the conventional diffusion in both states but with different diffusion coefficients. The anomalous features of diffusion result from the modulation of the diffusion coefficient caused by transitions between ATSM states. The anomalous space-time evolution predicted by the ATSM is treated within the continuous time random walk theory. With the use of the proposed ATSM the transient behavior of the AD and the ADAT is analyzed in detail. We found a large variety of different (and sometimes peculiar) types of the space-time behavior of the free AD and ADAT. The free AD is found to be of subdiffusion or superdiffusion type for fairly long time depending on the relation between the parameters of the ATSM. The kinetics of the ADAT can be either conventional (exponential) or anomalous (of inverse power type) for different parameters of the model and time.
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Affiliation(s)
- A I Shushin
- Institute of Chemical Physics, Russian Academy of Sciences, 117977, GSP-1, Kosygin strasse 4, Moscow, Russia
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Klemm A, Kimmich R, Weber M. Flow through percolation clusters: NMR velocity mapping and numerical simulation study. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:041514. [PMID: 11308855 DOI: 10.1103/physreve.63.041514] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2000] [Indexed: 05/23/2023]
Abstract
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.
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Affiliation(s)
- A Klemm
- Sektion Kernresonanzspektroskopie, Universität Ulm, 89069 Ulm, Germany
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14
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Kimmich R, Klemm A, Weber M. Flow, diffusion, and thermal convection in percolation clusters: NMR experiments and numerical FEM/FVM simulations. Magn Reson Imaging 2001; 19:353-61. [PMID: 11445311 DOI: 10.1016/s0730-725x(01)00248-x] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
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.
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Affiliation(s)
- R Kimmich
- Sektion Kernresonanzspektroskopie, Universität Ulm, 89069, Ulm, Germany.
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Denisov SI, Horsthemke W. Anomalous diffusion of particles driven by correlated noise. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:7729-7734. [PMID: 11138045 DOI: 10.1103/physreve.62.7729] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/14/2000] [Indexed: 05/23/2023]
Abstract
We study the effect of an arbitrary stationary random force on the motion of damped particles. Using a Langevin description, we derive exact expressions for the dispersion of the particle position, of the particle velocity, and their cross dispersion. The particles can exhibit anomalous diffusion, and the connection between this behavior and the functional form of the noise correlations is investigated in detail. We also study anomalous diffusion for the special cases of overdamped and undamped particles.
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Affiliation(s)
- SI Denisov
- Department of Mechanics and Mathematics, Sumy State University, 2, Rimskiy-Korsakov Street, 40007 Sumy, Ukraine
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Rangarajan G, Ding M. Anomalous diffusion and the first passage time problem. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:120-133. [PMID: 11088443 DOI: 10.1103/physreve.62.120] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/18/2000] [Indexed: 05/23/2023]
Abstract
We study the distribution of the first passage time (FPT) in Levy type anomalous diffusion. Using the recently formulated fractional Fokker-Planck equation we obtain three results. (1) We derive an explicit expression for the FPT distribution in terms of Fox or H functions when the diffusion has zero drift. (2) For the nonzero drift case we obtain an analytical expression for the Laplace transform of the FPT distribution. (3) We express the FPT distribution in terms of a power series for the case of two absorbing barriers. The known results for ordinary diffusion (Brownian motion) are obtained as special cases of our more general results.
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Affiliation(s)
- G Rangarajan
- Department of Mathematics and Centre for Theoretical Studies, Indian Institute of Science, Bangalore 560 012, India
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Weis J, Smedby O, Hemmingsson A. Characterization of human head vasculature by percolation parameters. Magn Reson Imaging 1999; 17:411-5. [PMID: 10195584 DOI: 10.1016/s0730-725x(98)00185-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
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.
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Affiliation(s)
- J Weis
- Department of Diagnostic Radiology, University Hospital, Uppsala, Sweden.
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