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.

Application of high voltage, high frequency pulsed electromagnetic field on cortical bone tissue

Over the last few decades, electric and electromagnetic fields have achieved important role as stimulator and therapeutic facility in biology and medicine. In particular, low magnitude, low frequency, pulsed electromagnetic field has shown significant positive effect on bone fracture healing and some bone diseases treatment. Nevertheless, to date, little attention has been paid to investigate the possible effect of high frequency, high magnitude pulsed electromagnetic field (pulse power) on functional behaviour and biomechanical properties of bone tissue. Bone is a dynamic, complex organ, which is made of bone materials (consisting of organic components, inorganic mineral and water) known as extracellular matrix, and bone cells (live part). The cells give the bone the capability of self-repairing by adapting itself to its mechanical environment. The specific bone material composite comprising of collagen matrix reinforced with mineral apatite provides the bone with particular biomechanical properties in an anisotropic, inhomogeneous structure. This project hypothesized to investigate the possible effect of pulse power signals on cortical bone characteristics through evaluating the fundamental mechanical properties of bone material. A positive buck-boost converter was applied to generate adjustable high voltage, high frequency pulses up to 500 V and 10 kHz. Bone shows distinctive characteristics in different loading mode. Thus, functional behaviour of bone in response to pulse power excitation were elucidated by using three different conventional mechanical tests applying three-point bending load in elastic region, tensile and compressive loading until failure. Flexural stiffness, tensile and compressive strength, hysteresis and total fracture energy were determined as measure of main bone characteristics. To assess bone structure variation due to pulse power excitation in deeper aspect, a supplementary fractographic study was also conducted using scanning electron micrograph from tensile fracture surfaces. Furthermore, a non-destructive ultrasonic technique was applied for determination and comparison of bone elasticity before and after pulse power stimulation. This method provided the ability to evaluate the stiffness of millimetre-sized bone samples in three orthogonal directions. According to the results of non-destructive bending test, the flexural elasticity of cortical bone samples appeared to remain unchanged due to pulse power excitation. Similar results were observed in the bone stiffness for all three orthogonal directions obtained from ultrasonic technique and in the bone stiffness from the compression test. From tensile tests, no significant changes were found in tensile strength and total strain energy absorption of the bone samples exposed to pulse power compared with those of the control samples. Also, the apparent microstructure of the fracture surfaces of PP-exposed samples (including porosity and microcracks diffusion) showed no significant variation due to pulse power stimulation. Nevertheless, the compressive strength and toughness of millimetre-sized samples appeared to increase when the samples were exposed to 66 hours high power pulsed electromagnetic field through screws with small contact cross-section (increasing the pulsed electric field intensity) compare to the control samples. This can show the different load-bearing characteristics of cortical bone tissue in response to pulse power excitation and effectiveness of this type of stimulation on smaller-sized samples. These overall results may address that although, the pulse power stimulation can influence the arrangement or the quality of the collagen network causing the bone strength and toughness augmentation, it apparently did not affect the mineral phase of the cortical bone material. The results also confirmed that the indirect application of high power pulsed electromagnetic field at 500 V and 10 kHz through capacitive coupling method, was athermal and did not damage the bone tissue construction.

Solution methods for advection-diffusion-reaction equations on growing domains and subdomains, with application to modelling skin substitutes

Problems involving the solution of advection-diffusion-reaction equations on domains and subdomains whose growth affects and is affected by these equations, commonly arise in developmental biology. Here, a mathematical framework for these situations, together with methods for obtaining spatio-temporal solutions and steady states of models built from this framework, is presented. The framework and methods are applied to a recently published model of epidermal skin substitutes. Despite the use of Eulerian schemes, excellent agreement is obtained between the numerical spatio-temporal, numerical steady state, and analytical solutions of the model.

Langevin dynamics modeling of the water diffusion tensor in partially aligned collagen networks

In this work, a Langevin dynamics model of the diffusion of water in articular cartilage was developed. Numerical simulations of the translational dynamics of water molecules and their interaction with collagen fibers were used to study the quantitative relationship between the organization of the collagen fiber network and the diffusion tensor of water in model cartilage. Langevin dynamics was used to simulate water diffusion in both ordered and partially disordered cartilage models. In addition, an analytical approach was developed to estimate the diffusion tensor for a network comprising a given distribution of fiber orientations. The key findings are that (1) an approximately linear relationship was observed between collagen volume fraction and the fractional anisotropy of the diffusion tensor in fiber networks of a given degree of alignment, (2) for any given fiber volume fraction, fractional anisotropy follows a fiber alignment dependency similar to the square of the second Legendre polynomial of cos(θ), with the minimum anisotropy occurring at approximately the magic angle (θMA), and (3) a decrease in the principal eigenvalue and an increase in the transverse eigenvalues is observed as the fiber orientation angle θ progresses from 0◦ to 90◦. The corresponding diffusion ellipsoids are prolate for θ < θMA, spherical for θ ≈ θMA, and oblate for θ > θMA. Expansion of the model to include discrimination between the combined effects of alignment disorder and collagen fiber volume fraction on the diffusion tensor is discussed.

Effect of body position on measurements of diffusion capacity after exercise

Background--Pulmonary diffusing capacity for carbon monoxide (Dlco), alveolar capillary membrane diffusing capacity (Dm), and pulmonary capillary blood volume (Vc) are all significantly reduced after exercise. Objective--To investigate whether measurement position affects this impaired gas transfer. Methods--Before and one, two, and four hours after incremental cycle ergometer exercise to fatigue, single breath Dlco, Dm, and Vc measurements were obtained in 10 healthy men in a randomly assigned supine and upright seated position. Results--After exercise, Dlco, Dm, and Vc were significantly depressed compared with baseline in both positions. The supine position produced significantly higher values over time for Dlco (5.22 (0.13) v 4.66 (0.15) ml/min/mm Hg/l, p = 0.022) and Dm (6.78 (0.19) v 6.03 (0.19) ml/min/mm Hg/l, p = 0.016), but there was no significant position effect for Vc. There was a similar pattern of change over time for Dlco, Dm, and Vc in the two positions. Conclusions--The change in Dlco after exercise appears to be primarily due to a decrease in Vc. Although the mechanism for the reduction in Vc cannot be determined from these data, passive relocation of blood to the periphery as the result of gravity can be discounted, suggesting that active vasoconstriction of the pulmonary vasculature and/or peripheral vasodilatation is occurring after exercise.

Pore formation during dehydration of polycrystalline gypsum observed and quantified in a time-series synchrotron radiation based X-ray micro-tomography experiment

We conducted an in-situ X-ray micro-computed tomography heating experiment at the Advanced Photon Source (USA) to dehydrate an unconfined 2.3 mm diameter cylinder of Volterra Gypsum. We used a purpose-built X-ray transparent furnace to heat the sample to 388 K for a total of 310 min to acquire a three-dimensional time-series tomography dataset comprising nine time steps. The voxel size of 2.2 μm3 proved sufficient to pinpoint reaction initiation and the organization of drainage architecture in space and time. We observed that dehydration commences across a narrow front, which propagates from the margins to the centre of the sample in more than four hours. The advance of this front can be fitted with a square-root function, implying that the initiation of the reaction in the sample can be described as a diffusion process. Novel parallelized computer codes allow quantifying the geometry of the porosity and the drainage architecture from the very large tomographic datasets (20483 voxels) in unprecedented detail. We determined position, volume, shape and orientation of each resolvable pore and tracked these properties over the duration of the experiment. We found that the pore-size distribution follows a power law. Pores tend to be anisotropic but rarely crack-shaped and have a preferred orientation, likely controlled by a pre-existing fabric in the sample. With on-going dehydration, pores coalesce into a single interconnected pore cluster that is connected to the surface of the sample cylinder and provides an effective drainage pathway. Our observations can be summarized in a model in which gypsum is stabilized by thermal expansion stresses and locally increased pore fluid pressures until the dehydration front approaches to within about 100 μm. Then, the internal stresses are released and dehydration happens efficiently, resulting in new pore space. Pressure release, the production of pores and the advance of the front are coupled in a feedback loop.

Measurement constructs to explore innovation diffusion in construction

Although the drivers of innovation have been studied extensively in construction, greater attention is required on how innovation diffusion can be effectively assessed within this complex and interdependent project-based industry. The authors draw on a highly cited innovation diffusion model by Rogers (2006) and develop a tailored conceptual framework to guide future empirical work aimed at assessing innovation diffusion in construction. The conceptual framework developed and discussed in this paper supports a five-stage process model of innovation diffusion namely: 1) knowledge and idea generation, 2) persuasion and evaluation; 3) decision to adopt, 4) integration and implementation, and 5) confirmation. As its theoretical contribution, this paper proposes three critical measurements constructs which can be used to assess the effectiveness of the diffusion process. These measurement constructs comprise: 1) nature and introduction of an innovative idea, 2) organizational capacity to acquire, assimilate, transform and exploit an innovation, and 3) rates of innovation facilitation and adoption. The constructs are interpreted in the project-based context of the construction industry, extending the contribution of general management theorists. Research planned by the authors will test the validity and reliability of the constructs developed in this paper.

A note on a reaction-diffusion model describing the bone morphogen protein gradient in Drosophila embryonic patterning

In this article, we consider the Eldar model [3] from embryology in which a bone morphogenic protein, a short gastrulation protein, and their compound react and diffuse. We carry out a perturbation analysis in the limit of small diffusivity of the bone morphogenic protein. This analysis establishes conditions under which some elementary results of [3] are valid.

Near-Wall Turbulent Pressure Diffusion Modeling and Influence in Three-Dimensional Secondary Flows

The purpose of this paper is to develop a second-moment closure with a near-wall turbulent pressure diffusion model for three-dimensional complex flows, and to evaluate the influence of the turbulent diffusion term on the prediction of detached and secondary flows. A complete turbulent diffusion model including a near-wall turbulent pressure diffusion closure for the slow part was developed based on the tensorial form of Lumley and included in a re-calibrated wall-normal-free Reynolds-stress model developed by Gerolymos and Vallet. The proposed model was validated against several one-, two, and three-dimensional complex flows.

Efficient solution of two-sided nonlinear space-fractional diffusion equations using fast Poisson preconditioners

We develop a fast Poisson preconditioner for the efficient numerical solution of a class of two-sided nonlinear space fractional diffusion equations in one and two dimensions using the method of lines. Using the shifted Gr¨unwald finite difference formulas to approximate the two-sided(i.e. the left and right Riemann-Liouville) fractional derivatives, the resulting semi-discrete nonlinear systems have dense Jacobian matrices owing to the non-local property of fractional derivatives. We employ a modern initial value problem solver utilising backward differentiation formulas and Jacobian-free Newton-Krylov methods to solve these systems. For efficient performance of the Jacobianfree Newton-Krylov method it is essential to apply an effective preconditioner to accelerate the convergence of the linear iterative solver. The key contribution of our work is to generalise the fast Poisson preconditioner, widely used for integer-order diffusion equations, so that it applies to the two-sided space fractional diffusion equation. A number of numerical experiments are presented to demonstrate the effectiveness of the preconditioner and the overall solution strategy.

A banded preconditioner for the two-sided, nonlinear space-fractional diffusion equation

The method of lines is a standard method for advancing the solution of partial differential equations (PDEs) in time. In one sense, the method applies equally well to space-fractional PDEs as it does to integer-order PDEs. However, there is a significant challenge when solving space-fractional PDEs in this way, owing to the non-local nature of the fractional derivatives. Each equation in the resulting semi-discrete system involves contributions from every spatial node in the domain. This has important consequences for the efficiency of the numerical solver, especially when the system is large. First, the Jacobian matrix of the system is dense, and hence methods that avoid the need to form and factorise this matrix are preferred. Second, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. In this paper, we show how an effective preconditioner is essential for improving the efficiency of the method of lines for solving a quite general two-sided, nonlinear space-fractional diffusion equation. A key contribution is to show, how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach.

A preconditioned Lanczos method for space-fractional reaction-diffusion equations on two-dimensional unstructured meshes

We consider a two-dimensional space-fractional reaction diffusion equation with a fractional Laplacian operator and homogeneous Neumann boundary conditions. The finite volume method is used with the matrix transfer technique of Ilić et al. (2006) to discretise in space, yielding a system of equations that requires the action of a matrix function to solve at each timestep. Rather than form this matrix function explicitly, we use Krylov subspace techniques to approximate the action of this matrix function. Specifically, we apply the Lanczos method, after a suitable transformation of the problem to recover symmetry. To improve the convergence of this method, we utilise a preconditioner that deflates the smallest eigenvalues from the spectrum. We demonstrate the efficiency of our approach for a fractional Fisher’s equation on the unit disk.