Effect of magnesium depletion and potassium and chlorothiazide on intracellular pH in the rat, studied by 31P NMR

1. Both dietary magnesium depletion and potassium depletion (confirmed by tissue analysis) were induced in rats which were then compared with rats treated with chlorothiazide (250 mg/kg diet) and rats on a control synthetic diet. 2. Brain and muscle intracellular pH was measured by using a surface coil and [31P]-NMR to measure the chemical shift of inorganic phosphate. pH was also measured in isolated perfused hearts from control and magnesium-deficient rats. Intracellular magnesium status was assessed by measuring the chemical shift of β-ATP in brain. 3. There was no evidence for magnesium deficiency in the chlorothiazide-treated rats on tissue analysis or on chemical shift of β-ATP in brain. Both magnesium and potassium deficiency, but not chlorothiazide treatment, were associated with an extracellular alkalosis. 4. Magnesium deficiency led to an intracellular alkalosis in brain, muscle and heart. Chlorothiazide treatment led to an alkalosis in brain. Potassium deficiency was associated with a normal intracellular pH in brain and muscle. 5. Magnesium depletion and chlorothiazide treatment produce intracellular alkalosis by unknown mechanism(s).

Problems in the assessment of magnesium depletion in the rat by in vivo 31P NMR

Prior in vitro studies, utilizing 31Pn uclear magnetic resonance (31PN MR) to measure the chemical shift (CT) of 0-ATP and lengthening of the phosphocreatine spin-spin (7"') relaxation time, suggested an assessment of their efficacy in measuring magnesium depletion in vivo. Dietary magnesium depletion (Me$) produced markedly lower magnesium in plasma (0.44 vs 1. I3 mmol/liter) and bone (1 30 vs 190 pmol/g) but much smaller changes in muscle (41 vs 45 pmol/g, P < 0.01), heart (42.5 vs 44.6 prnol/g), and brain (30 vs 32 pmollg). NMR experiments in anesthetized rats in a Bruker 7-T vertical bore magnet showed that in M e $ rats there was a significant change in brain j3-ATP shift (16.15 vs 16.03 ppm, P < 0.05). These chemical shifts gave a calculated free [Mg"] of 0.71 mM (control) and 0.48 mM (MgZ+$). In muscle the change in j3-ATP shift was not significant (Me$ 15.99 ppm, controls 15.96 ppm), corresponding to a calculated free M P of 0.83 and 0.95 mM, respectively. Phosphccreatine Tz (Carr-Purcell, spin-echo pulse sequence) was no different with M e $ in muscle in vivo (surface coil) (M$+$ 136, control 142 ms) or in isolated perfused hearts (Helmholtz coil) (control 83, M e $ 92 ms). 3'P NMR is severely limited in its ability to detect dietary magnesium depletion in vivo. Measurement of j3-ATP shift in brain may allow studies of the effects of interaction in group studies but does not allow prediction of an individual magnesium status.

Alternating direction implicit numerical method for the space and time fractional Bloch-Torrey equation in 3-D

Recently, some authors have considered a new diffusion model–space and time fractional Bloch-Torrey equation (ST-FBTE). Magin et al. (2008) have derived analytical solutions with fractional order dynamics in space (i.e., _ = 1, β an arbitrary real number, 1 < β ≤ 2) and time (i.e., 0 < α < 1, and β = 2), respectively. Yu et al. (2011) have derived an analytical solution and an effective implicit numerical method for solving ST-FBTEs, and also discussed the stability and convergence of the implicit numerical method. However, due to the computational overheads necessary to perform the simulations for nuclear magnetic resonance (NMR) in three dimensions, they present a study based on a two-dimensional example to confirm their theoretical analysis. Alternating direction implicit (ADI) schemes have been proposed for the numerical simulations of classic differential equations. The ADI schemes will reduce a multidimensional problem to a series of independent one-dimensional problems and are thus computationally efficient. In this paper, we consider the numerical solution of a ST-FBTE on a finite domain. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. A fractional alternating direction implicit scheme (FADIS) for the ST-FBTE in 3-D is proposed. Stability and convergence properties of the FADIS are discussed. Finally, some numerical results for ST-FBTE are given.

Numerical methods for solving the multi-term time-fractional wave-diffusion equations

In this paper, the multi-term time-fractional wave diffusion equations are considered. The multiterm time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.

Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain

Generalized fractional partial differential equations have now found wide application for describing important physical phenomena, such as subdiffusive and superdiffusive processes. However, studies of generalized multi-term time and space fractional partial differential equations are still under development. In this paper, the multi-term time-space Caputo-Riesz fractional advection diffusion equations (MT-TSCR-FADE) with Dirichlet nonhomogeneous boundary conditions are considered. The multi-term time-fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0, 1], [1, 2] and [0, 2], respectively. These are called respectively the multi-term time-fractional diffusion terms, the multi-term time-fractional wave terms and the multi-term time-fractional mixed diffusion-wave terms. The space fractional derivatives are defined as Riesz fractional derivatives. Analytical solutions of three types of the MT-TSCR-FADE are derived with Dirichlet boundary conditions. By using Luchko's Theorem (Acta Math. Vietnam., 1999), we proposed some new techniques, such as a spectral representation of the fractional Laplacian operator and the equivalent relationship between fractional Laplacian operator and Riesz fractional derivative, that enabled the derivation of the analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations. © 2012.

Numerical techniques for the variable order time fractional diffusion equation

In this paper we consider the variable order time fractional diffusion equation. We adopt the Coimbra variable order (VO) time fractional operator, which defines a consistent method for VO differentiation of physical variables. The Coimbra variable order fractional operator also can be viewed as a Caputo-type definition. Although this definition is the most appropriate definition having fundamental characteristics that are desirable for physical modeling, numerical methods for fractional partial differential equations using this definition have not yet appeared in the literature. Here an approximate scheme is first proposed. The stability, convergence and solvability of this numerical scheme are discussed via the technique of Fourier analysis. Numerical examples are provided to show that the numerical method is computationally efficient. Crown Copyright © 2012 Published by Elsevier Inc. All rights reserved.

Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain

Multi-term time-fractional differential equations have been used for describing important physical phenomena. However, studies of the multi-term time-fractional partial differential equations with three kinds of nonhomogeneous boundary conditions are still limited. In this paper, a method of separating variables is used to solve the multi-term time-fractional diffusion-wave equation and the multi-term time-fractional diffusion equation in a finite domain. In the two equations, the time-fractional derivative is defined in the Caputo sense. We discuss and derive the analytical solutions of the two equations with three kinds of nonhomogeneous boundary conditions, namely, Dirichlet, Neumann and Robin conditions, respectively.

Sensitivity of the NMR density matrix to pulse sequence parameters : a simplified analytic approach

We present a formalism for the analysis of sensitivity of nuclear magnetic resonance pulse sequences to variations of pulse sequence parameters, such as radiofrequency pulses, gradient pulses or evolution delays. The formalism enables the calculation of compact, analytic expressions for the derivatives of the density matrix and the observed signal with respect to the parameters varied. The analysis is based on two constructs computed in the course of modified density-matrix simulations: the error interrogation operators and error commutators. The approach presented is consequently named the Error Commutator Formalism (ECF). It is used to evaluate the sensitivity of the density matrix to parameter variation based on the simulations carried out for the ideal parameters, obviating the need for finite-difference calculations of signal errors. The ECF analysis therefore carries a computational cost comparable to a single density-matrix or product-operator simulation. Its application is illustrated using a number of examples from basic NMR spectroscopy. We show that the strength of the ECF is its ability to provide analytic insights into the propagation of errors through pulse sequences and the behaviour of signal errors under phase cycling. Furthermore, the approach is algorithmic and easily amenable to implementation in the form of a programming code. It is envisaged that it could be incorporated into standard NMR product-operator simulation packages.

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.

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.