Analytical and numerical solutions of the space and time fractional bloch-torrey equation

Fractional order dynamics in physics, particularly when applied to diffusion, leads to an extension of the concept of Brown-ian motion through a generalization of the Gaussian probability function to what is termed anomalous diffusion. As MRI is applied with increasing temporal and spatial resolution, the spin dynamics are being examined more closely; such examinations extend our knowledge of biological materials through a detailed analysis of relaxation time distribution and water diffusion heterogeneity. Here the dynamic models become more complex as they attempt to correlate new data with a multiplicity of tissue compartments where processes are often anisotropic. Anomalous diffusion in the human brain using fractional order calculus has been investigated. Recently, a new diffusion model was proposed by solving the Bloch-Torrey equation using fractional order calculus with respect to time and space (see R.L. Magin et al., J. Magnetic Resonance, 190 (2008) 255-270). However effective numerical methods and supporting error analyses for the fractional Bloch-Torrey equation are still limited. In this paper, the space and time fractional Bloch-Torrey equation (ST-FBTE) is considered. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we derive an analytical solution for the ST-FBTE with initial and boundary conditions on a finite domain. Secondly, we propose an implicit numerical method (INM) for the ST-FBTE, and the stability and convergence of the INM are investigated. We prove that the implicit numerical method for the ST-FBTE is unconditionally stable and convergent. Finally, we present some numerical results that support our theoretical analysis.

Fractional models for the migration of biological cells in complex spatial domains

Travelling wave phenomena are observed in many biological applications. Mathematical theory of standard reaction-diffusion problems shows that simple partial differential equations exhibit travelling wave solutions with constant wavespeed and such models are used to describe, for example, waves of chemical concentrations, electrical signals, cell migration, waves of epidemics and population dynamics. However, as in the study of cell motion in complex spatial geometries, experimental data are often not consistent with constant wavespeed. Non-local spatial models have successfully been used to model anomalous diffusion and spatial heterogeneity in different physical contexts. In this paper, we develop a fractional model based on the Fisher-Kolmogoroff equation and analyse it for its wavespeed properties, attempting to relate the numerical results obtained from our simulations to experimental data describing enteric neural crest-derived cells migrating along the intact gut of mouse embryos. The model proposed essentially combines fractional and standard diffusion in different regions of the spatial domain and qualitatively reproduces the behaviour of neural crest-derived cells observed in the caecum and the hindgut of mouse embryos during in vivo experiments.

Effects of hyperbaric oxygen and inducible nitric oxide synthase inhibitor treatment on femoral head osteonecrosis in a rat model

Introduction: Apoptosis is the final destiny of many cells in the body, though this process has been observed in some pathological processes. One of these pathological processes is femoral head non-traumatic osteonecrosis. Among many pro/anti-apoptotic factors, nitric oxide has recently been an area of further interest. Osteocyte apoptosis and its relation to pro-apoptotic action invite further research, and the inducible form of nitric oxide synthase (iNOS)—which produces a high concentration of nitric oxide—has been flagged. The aim of this study was to investigate the effect of hyperbaric oxygen (HBO) and inducible NOS suppressor (Aminoguanidine) in prevention of femoral head osteonecrosis in an experimental model of osteonecrosis in spontaneous hypertensive rats (SHRs). Methods: After animal ethic approval 34 SHR rats were divided into four groups. Ten rats were allocated to the control group without any treatment, and eight rats were allocated to three treatment groups namely: HBO, Aminoguanidine (AMG), and the combination of HBO and AMG treatments (HBO+AMG). The HBO group received 250 kPa of oxygen via hyperbaric chamber for 30 days started at their 5th week of life; the AMG group received 1mg/ml of AMG in drinking water from the fifth week till the 17th week of life; and the last group received a combination of these treatments. Rats were sacrificed at the end of the 17th week of life and both femurs were analysed for evidence of osteonecrosis using Micro CT scan and H&E staining. Also, osteocyte apoptosis and the presence of two different forms of NOS (inducible (iNOS) and endothelial (eNOS)) were analysed by immunostaining and apoptosis staining (Hoechst and TUNEL). Results: Bone morphology of metaphyseal and epiphyseal area of all rats were investigated and analysed. Micro CT findings revealed significantly higher mean fractional trabecular bone volume (FBV) of metaphyseal area in untreated SHRs compared with all other treatments (HBO, P<0.05, HBO+AMG, P<0.005, and AMG P<0.001). Bone surface to volume ratio also significantly increased with HBO+AMG and AMG treatments when compared with the control group (18.7 Vs 20.8, P<0.05, and 18.7 Vs 21.1, P<0.05). Epiphyseal mean FBV did not change significantly among groups. In the metaphyseal area, trabecular thickness and numbers significantly decreased with AMG treatment, while trabecular separation significantly increased with both AMG and HBO+AMG treatment. Histological ratio of no ossification and osteonecrosis was 37.5%, 43.7%, 18.7% and 6.2% of control, HBO, HBO+AMG and AMG groups respectively with only significant difference observed between HBO and AMG treatment (P<0.01). High concentration of iNOS was observed in the region of osteonecrosis while there was no evidence of eNOS activity around that region. In comparison with the control group, the ratio of osteocyte apoptosis significantly reduced in AMG treatment (P<0.005). We also observed significantly fewer apoptotic osteocytes in AMG group comparing with HBO treatment (P<0.05). Conclusion: None of our treatments prevents osteonecrosis at the histological or micro CT scan level. High concentration of iNOS in the region of osteonecrosis and significant reduction of osteocyte apoptosis with AMG treatment were supportive of iNOS modulating osteocyte apoptosis in SHRs.

A RBF meshless approach for modeling a fractal mobile/immobile transport model

A fractional differential equation is used to describe a fractal model of mobile/immobile transport with a power law memory function. This equation is the limiting equation that governs continuous time random walks with heavy tailed random waiting times. In this paper, we firstly propose a finite difference method to discretize the time variable and obtain a semi-discrete scheme. Then we discuss its stability and convergence. Secondly we consider a meshless method based on radial basis functions (RBFs) to discretize the space variable. In contrast to conventional FDM and FEM, the meshless method is demonstrated to have distinct advantages: calculations can be performed independent of a mesh, it is more accurate and it can be used to solve complex problems. Finally the convergence order is verified from a numerical example which is presented to describe a fractal model of mobile/immobile transport process with different problem domains. The numerical results indicate that the present meshless approach is very effective for modeling and simulating fractional differential equations, and it has good potential in the development of a robust simulation tool for problems in engineering and science that are governed by various types of fractional differential equations.

Bucket attack on numeric set watermarking model and safeguards

Numeric set watermarking is a way to provide ownership proof for numerical data. Numerical data can be considered to be primitives for multimedia types such as images and videos since they are organized forms of numeric information. Thereby, the capability to watermark numerical data directly implies the capability to watermark multimedia objects and discourage information theft on social networking sites and the Internet in general. Unfortunately, there has been very limited research done in the field of numeric set watermarking due to underlying limitations in terms of number of items in the set and LSBs in each item available for watermarking. In 2009, Gupta et al. proposed a numeric set watermarking model that embeds watermark bits in the items of the set based on a hash value of the items’ most significant bits (MSBs). If an item is chosen for watermarking, a watermark bit is embedded in the least significant bits, and the replaced bit is inserted in the fractional value to provide reversibility. The authors show their scheme to be resilient against the traditional subset addition, deletion, and modification attacks as well as secondary watermarking attacks. In this paper, we present a bucket attack on this watermarking model. The attack consists of creating buckets of items with the same MSBs and determine if the items of the bucket carry watermark bits. Experimental results show that the bucket attack is very strong and destroys the entire watermark with close to 100% success rate. We examine the inherent weaknesses in the watermarking model of Gupta et al. that leave it vulnerable to the bucket attack and propose potential safeguards that can provide resilience against this attack.

A finite volume scheme with preconditioned Lanczos method for two-dimensional space-fractional reaction–diffusion equations

Fractional differential equations have been increasingly used as a powerful tool to model the non-locality and spatial heterogeneity inherent in many real-world problems. However, a constant challenge faced by researchers in this area is the high computational expense of obtaining numerical solutions of these fractional models, owing to the non-local nature of fractional derivatives. In this paper, we introduce a finite volume scheme with preconditioned Lanczos method as an attractive and high-efficiency approach for solving two-dimensional space-fractional reaction–diffusion equations. The computational heart of this approach is the efficient computation of a matrix-function-vector product f(A)bf(A)b, where A A is the matrix representation of the Laplacian obtained from the finite volume method and is non-symmetric. A key aspect of our proposed approach is that the popular Lanczos method for symmetric matrices is applied to this non-symmetric problem, after a suitable transformation. Furthermore, the convergence of the Lanczos method is greatly improved by incorporating a preconditioner. Our approach is show-cased by solving the fractional Fisher equation including a validation of the solution and an analysis of the behaviour of the model.

A Novel High Order Space-Time Spectral Method for the Time Fractional Fokker--Planck Equation

The fractional Fokker-Planck equation is an important physical model for simulating anomalous diffusions with external forces. Because of the non-local property of the fractional derivative an interesting problem is to explore high accuracy numerical methods for fractional differential equations. In this paper, a space-time spectral method is presented for the numerical solution of the time fractional Fokker-Planck initial-boundary value problem. The proposed method employs the Jacobi polynomials for the temporal discretization and Fourier-like basis functions for the spatial discretization. Due to the diagonalizable trait of the Fourier-like basis functions, this leads to a reduced representation of the inner product in the Galerkin analysis. We prove that the time fractional Fokker-Planck equation attains the same approximation order as the time fractional diffusion equation developed in [23] by using the present method. That indicates an exponential decay may be achieved if the exact solution is sufficiently smooth. Finally, some numerical results are given to demonstrate the high order accuracy and efficiency of the new numerical scheme. The results show that the errors of the numerical solutions obtained by the space-time spectral method decay exponentially.

Bivariate genome-wide association study of genetically correlated neuroimaging phenotypes from DTI and MRI through a seemingly unrelated regression model

Large multisite efforts (e.g., the ENIGMA Consortium), have shown that neuroimaging traits including tract integrity (from DTI fractional anisotropy, FA) and subcortical volumes (from T1-weighted scans) are highly heritable and promising phenotypes for discovering genetic variants associated with brain structure. However, genetic correlations (rg) among measures from these different modalities for mapping the human genome to the brain remain unknown. Discovering these correlations can help map genetic and neuroanatomical pathways implicated in development and inherited risk for disease. We use structural equation models and a twin design to find rg between pairs of phenotypes extracted from DTI and MRI scans. When controlling for intracranial volume, the caudate as well as related measures from the limbic system - hippocampal volume - showed high rg with the cingulum FA. Using an unrelated sample and a Seemingly Unrelated Regression model for bivariate analysis of this connection, we show that a multivariate GWAS approach may be more promising for genetic discovery than a univariate approach applied to each trait separately.

Gaussian Schell-model beams and general shape invariance

We present two six-parameter families of anisotropic Gaussian Schell-model beams that propagate in a shape-invariant manner, with the intensity distribution continuously twisting about the beam axis. The two families differ in the sense or helicity of this beam twist. The propagation characteristics of these shape-invariant beams are studied, and the restrictions on the beam parameters that arise from the optical uncertainty principle are brought out. Shape invariance is traced to a fundamental dynamical symmetry that underlies these beams. This symmetry is the product of spatial rotation and fractional Fourier transformation.

Application of a model to assess aflatoxin risk in peanuts.

When exposed to hot (22-35 degrees C) and dry climatic conditions in the field during the final 4-6 weeks of pod filling, peanuts (Arachis hypogaea L.) can accumulate highly carcinogenic and immuno-suppressing aflatoxins. Forecasting of the risk posed by these conditions can assist in minimizing pre-harvest contamination. A model was therefore developed as part of the Agricultural Production Systems Simulator (APSIM) peanut module, which calculated an aflatoxin risk index (ARI) using four temperature response functions when fractional available soil water was <0.20 and the crop was in the last 0.40 of the pod-filling phase. ARI explained 0.95 (P <= 0.05) of the variation in aflatoxin contamination, which varied from 0 to c. 800 mu g/kg in 17 large-scale sowings in tropical and four sowings in sub-tropical environments carried out in Australia between 13 November and 16 December 2007. ARI also explained 0.96 (P <= 0.01) of the variation in the proportion of aflatoxin-contaminated loads (>15 mu g/kg) of peanuts in the Kingaroy region of Australia during the period between the 1998/99 and 2007/08 seasons. Simulation of ARI using historical climatic data from 1890 to 2007 indicated a three-fold increase in its value since 1980 compared to the entire previous period. The increase was associated with increases in ambient temperature and decreases in rainfall. To facilitate routine monitoring of aflatoxin risk by growers in near real time, a web interface of the model was also developed. The ARI predicted using this interface for eight growers correlated significantly with the level of contamination in crops (r=095, P <= 0.01). These results suggest that ARI simulated by the model is a reliable indicator of aflatoxin contamination that can be used in aflatoxin research as well as a decision-support tool to monitor pre-harvest aflatoxin risk in peanuts.

Fuzzy waste load allocation model: a multiobjective approach

Fuzzy Waste Load Allocation Model (FWLAM), developed in an earlier study, derives the optimal fractional levels, for the base flow conditions, considering the goals of the Pollution Control Agency (PCA) and dischargers. The Modified Fuzzy Waste Load Allocation Model (MFWLAM) developed subsequently is a stochastic model and considers the moments (mean, variance and skewness) of water quality indicators, incorporating uncertainty due to randomness of input variables along with uncertainty due to imprecision. The risk of low water quality is reduced significantly by using this modified model, but inclusion of new constraints leads to a low value of acceptability level, A, interpreted as the maximized minimum satisfaction in the system. To improve this value, a new model, which is a combination Of FWLAM and MFWLAM, is presented, allowing for some violations in the constraints of MFWLAM. This combined model is a multiobjective optimization model having the objectives, maximization of acceptability level and minimization of violation of constraints. Fuzzy multiobjective programming, goal programming and fuzzy goal programming are used to find the solutions. For the optimization model, Probabilistic Global Search Lausanne (PGSL) is used as a nonlinear optimization tool. The methodology is applied to a case study of the Tunga-Bhadra river system in south India. The model results in a compromised solution of a higher value of acceptability level as compared to MFWLAM, with a satisfactory value of risk. Thus the goal of risk minimization is achieved with a comparatively better value of acceptability level.

Characterization of anomalous relaxation using the time-fractional Bloch equation and multiple echo T2 *-weighted magnetic resonance imaging at 7T

PURPOSE To study the utility of fractional calculus in modeling gradient-recalled echo MRI signal decay in the normal human brain. METHODS We solved analytically the extended time-fractional Bloch equations resulting in five model parameters, namely, the amplitude, relaxation rate, order of the time-fractional derivative, frequency shift, and constant offset. Voxel-level temporal fitting of the MRI signal was performed using the classical monoexponential model, a previously developed anomalous relaxation model, and using our extended time-fractional relaxation model. Nine brain regions segmented from multiple echo gradient-recalled echo 7 Tesla MRI data acquired from five participants were then used to investigate the characteristics of the extended time-fractional model parameters. RESULTS We found that the extended time-fractional model is able to fit the experimental data with smaller mean squared error than the classical monoexponential relaxation model and the anomalous relaxation model, which do not account for frequency shift. CONCLUSIONS We were able to fit multiple echo time MRI data with high accuracy using the developed model. Parameters of the model likely capture information on microstructural and susceptibility-induced changes in the human brain.

Dynamics of Barrierless and Activated Chemical Reactions in a Dispersive Medium within the Fractional Diffusion Equation Approach

Barrierless chemical reactions have often been modeled as a Brownian motion on a one-dimensional harmonic potential energy surface with a position-dependent reaction sink or window located near the minimum of the surface. This simple (but highly successful) description leads to a nonexponential survival probability only at small to intermediate times but exponential decay in the long-time limit. However, in several reactive events involving proteins and glasses, the reactions are found to exhibit a strongly nonexponential (power law) decay kinetics even in the long time. In order to address such reactions, here, we introduce a model of barrierless chemical reaction where the motion along the reaction coordinate sustains dispersive diffusion. A complete analytical solution of the model can be obtained only in the frequency domain, but an asymptotic solution is obtained in the limit of long time. In this case, the asymptotic long-time decay of the survival probability is a power law of the Mittag−Leffler functional form. When the barrier height is increased, the decay of the survival probability still remains nonexponential, in contrast to the ordinary Brownian motion case where the rate is given by the Smoluchowski limit of the well-known Kramers' expression. Interestingly, the reaction under dispersive diffusion is shown to exhibit strong dependence on the initial state of the system, thus predicting a strong dependence on the excitation wavelength for photoisomerization reactions in a dispersive medium. The theory also predicts a fractional viscosity dependence of the rate, which is often observed in the reactions occurring in complex environments.

Fractional damping: Stochastic origin and finite approximations

Fractional-order derivatives appear in various engineering applications including models for viscoelastic damping. Damping behavior of materials, if modeled using linear, constant coefficient differential equations, cannot include the long memory that fractional-order derivatives require. However, sufficiently great rnicrostructural disorder can lead, statistically, to macroscopic behavior well approximated by fractional order derivatives. The idea has appeared in the physics literature, but may interest an engineering audience. This idea in turn leads to an infinite-dimensional system without memory; a routine Galerkin projection on that infinite-dimensional system leads to a finite dimensional system of ordinary differential equations (ODEs) (integer order) that matches the fractional-order behavior over user-specifiable, but finite, frequency ranges. For extreme frequencies (small or large), the approximation is poor. This is unavoidable, and users interested in such extremes or in the fundamental aspects of true fractional derivatives must take note of it. However, mismatch in extreme frequencies outside the range of interest for a particular model of a real material may have little engineering impact.

Polymer melt dynamics: Microscopic roots of fractional viscoelasticity

The rheological properties of polymer melts and other complex macromolecular fluids are often successfully modeled by phenomenological constitutive equations containing fractional differential operators. We suggest a molecular basis for such fractional equations in terms of the generalized Langevin equation (GLE) that underlies the renormalized Rouse model developed by Schweizer [J. Chem. Phys. 91, 5802 (1989)]. The GLE describes the dynamics of the segments of a tagged chain under the action of random forces originating in the fast fluctuations of the surrounding polymer matrix. By representing these random forces as fractional Gaussian noise, and transforming the GLE into an equivalent diffusion equation for the density of the tagged chain segments, we obtain an analytical expression for the dynamic shear relaxation modulus G(t), which we then show decays as a power law in time. This power-law relaxation is the root of fractional viscoelastic behavior.