A mixed finite element method for nonlinear diffusion equations

We propose a mixed finite element method for a class of nonlinear diffusion equations, which is based on their interpretation as gradient flows in optimal transportation metrics. We introduce an appropriate linearization of the optimal transport problem, which leads to a mixed symmetric formulation. This formulation preserves the maximum principle in case of the semi-discrete scheme as well as the fully discrete scheme for a certain class of problems. In addition solutions of the mixed formulation maintain exponential convergence in the relative entropy towards the steady state in case of a nonlinear Fokker-Planck equation with uniformly convex potential. We demonstrate the behavior of the proposed scheme with 2D simulations of the porous medium equations and blow-up questions in the Patlak-Keller-Segel model.

Systemic effects of epidural Methylprednisolone injection on glucose tolerance in diabetic patients.

ABSTRACT: BACKGROUND: Several studies have shown that in diabetic patients, the glycemic profile was disturbed after intra-articular injection of corticosteroids. Little is known about the impact of epidural injection in such patients. The goal of this study was double, at first comparing the glycaemic profile in diabetic patients after a unique injection of 80 mg of acetate methylprednisolone either intra-articular or epidural and secondly to compare the amount of systemic diffusion of the drug after both procedures. METHODS: Seventeen patients were included. Glycemic changes were compared in 9 diabetic patients following intra-articular (4 patients) and epidural injections (5 patients). Epidural injections were performed using the sacral route under fluoroscopic control in patients with lumbar spinal stenosis. Diabetes control had to stable for more than 10 days and the renal function to be preserved. Blood glucose was monitored using a validated continuous measuring device (GMS, Medtronic) the day before and for two days following the injection. Results were expressed in the form of daily glycemic profiles and as by mean, peak and minimal values +/ SD. The urinary excretion of methylprednisolone after the 2 routes of injection was analyzed in 8 patients (4 in each group). Urine samples were cropped one hour before the injections, then 4 times during the first day and 3 times a week for 2 weeks. The measurements included the free and conjugated fraction RESULTS: The glycaemic profile remains unchanged with no significant changes in the group of the 5 diabetic patients receiving epidural injections. On the other end, the average peak and mean values were enhanced up to 3 mmol/l above baseline two days after the infiltration in the groups of the 4 diabetic patients infiltrated intra-articular. The mean urinary excretion of the steroid was about ten times higher in the intra-articular versus epidural group: 7000 ng/ml versus 700 ng/ml. Looking at each individual there were marked differences especially after intra-articular injections. CONCLUSION: This is the first study to show that a single epidural steroid injection of 80 mg depot methylprednisolone had no effect on the glycemic control in diabetic patients. The absence of glycemic control changes correlated well with the very low urinary excretion of the drug after epidural injection. Trial registration NCT01420497.

Stabilized Petrov-Galerkin methods for the convection-diffusion-reaction and the Helmholtz equations

We present two new stabilized high-resolution numerical methods for the convection–diffusion–reaction (CDR) and the Helmholtz equations respectively. The work embarks upon a priori analysis of some consistency recovery procedures for some stabilization methods belonging to the Petrov–Galerkin framework. It was found that the use of some standard practices (e.g. M-Matrices theory) for the design of essentially non-oscillatory numerical methods is not feasible when consistency recovery methods are employed. Hence, with respect to convective stabilization, such recovery methods are not preferred. Next, we present the design of a high-resolution Petrov–Galerkin (HRPG) method for the 1D CDR problem. The problem is studied from a fresh point of view, including practical implications on the formulation of the maximum principle, M-Matrices theory, monotonicity and total variation diminishing (TVD) finite volume schemes. The current method is next in line to earlier methods that may be viewed as an upwinding plus a discontinuity-capturing operator. Finally, some remarks are made on the extension of the HRPG method to multidimensions. Next, we present a new numerical scheme for the Helmholtz equation resulting in quasi-exact solutions. The focus is on the approximation of the solution to the Helmholtz equation in the interior of the domain using compact stencils. Piecewise linear/bilinear polynomial interpolation are considered on a structured mesh/grid. The only a priori requirement is to provide a mesh/grid resolution of at least eight elements per wavelength. No stabilization parameters are involved in the definition of the scheme. The scheme consists of taking the average of the equation stencils obtained by the standard Galerkin finite element method and the classical finite difference method. Dispersion analysis in 1D and 2D illustrate the quasi-exact properties of this scheme. Finally, some remarks are made on the extension of the scheme to unstructured meshes by designing a method within the Petrov–Galerkin framework.

A geometric mechanism of diffusion: Rigorous verification in a priori unstable Hamiltonian systems

In this paper we consider a representative a priori unstable Hamiltonian system with 2+1/2 degrees of freedom, to which we apply the geometric mechanism for diffusion introduced in the paper Delshams et al., Mem.Amer.Math. Soc. 2006, and generalized in Delshams and Huguet, Nonlinearity 2009, and provide explicit, concrete and easily verifiable conditions for the existence of diffusing orbits. The simplification of the hypotheses allows us to perform explicitly the computations along the proof, which contribute to present in an easily understandable way the geometric mechanism of diffusion. In particular, we fully describe the construction of the scattering map and the combination of two types of dynamics on a normally hyperbolic invariant manifold.

In vitro Comparison of Disk Diffusion and Agar Dilution Antibiotic Susceptibility Test Methods for Neisseria gonorrhoeae

At present, most Neisseria gonorrhoeae testing is done with ß-lactamase and agar dilution tests with common therapeutic agents. Generally, in bacteriological diagnosis laboratories in Argentina, study of antibiotic susceptibility of N.gonorrhoeae is based on ß-lactamase determination and agar dilution method with common therapeutic agents. The National Committee for Clinical Laboratory Standards (NCCLS) has recently described a disk diffusion test that produces results comparable to the reference agar dilution method for antibiotic susceptibility of N.gonorrhoeae, using a dispersion diagram for analyzing the correlation between both techniques. We obtained 57 gonococcal isolates from patients attending a clinic for sexually transmitted diseases in Tucumán, Argentina. Antibiotic susceptibility tests using agar dilution and disk diffusion techniques were compared. The established NCCLS interpretive criteria for both susceptibility methods appeared to be applicable to domestic gonococcal strains. The correlation between the MIC's and the zones of inhibition was studied for penicillin, ampicillin, cefoxitin, spectinomycin, cefotaxime, cephaloridine, cephalexin, tetracycline, norfloxacin and kanamycin. Dispersion diagrams showed a high correlation between both methods.

On Lundh's percolation diffusion

A collection of spherical obstacles in the unit ball in Euclidean space is said to be avoidable for Brownian motion if there is a positive probability that Brownian motion diffusing from some point in the ball will avoid all the obstacles and reach the boundary of the ball. The centres of the spherical obstacles are generated according to a Poisson point process while the radius of an obstacle is a deterministic function. If avoidable configurations are generated with positive probability Lundh calls this percolation diffusion. An integral condition for percolation diffusion is derived in terms of the intensity of the point process and the function that determines the radii of the obstacles.

An introduction to model-independent diffusion magnetic resonance imaging.

ABSTRACT: q-Space-based techniques such as diffusion spectrum imaging, q-ball imaging, and their variations have been used extensively in research for their desired capability to delineate complex neuronal architectures such as multiple fiber crossings in each of the image voxels. The purpose of this article was to provide an introduction to the q-space formalism and the principles of basic q-space techniques together with the discussion on the advantages as well as challenges in translating these techniques into the clinical environment. A review of the currently used q-space-based protocols in clinical research is also provided.

MR spectroscopy of the human brain with enhanced signal intensity at ultrashort echo times on a clinical platform at 3T and 7T.

Recently, the spin-echo full-intensity acquired localized (SPECIAL) spectroscopy technique was proposed to unite the advantages of short TEs on the order of milliseconds (ms) with full sensitivity and applied to in vivo rat brain. In the present study, SPECIAL was adapted and optimized for use on a clinical platform at 3T and 7T by combining interleaved water suppression (WS) and outer volume saturation (OVS), optimized sequence timing, and improved shimming using FASTMAP. High-quality single voxel spectra of human brain were acquired at TEs below or equal to 6 ms on a clinical 3T and 7T system for six volunteers. Narrow linewidths (6.6 +/- 0.6 Hz at 3T and 12.1 +/- 1.0 Hz at 7T for water) and the high signal-to-noise ratio (SNR) of the artifact-free spectra enabled the quantification of a neurochemical profile consisting of 18 metabolites with Cramér-Rao lower bounds (CRLBs) below 20% at both field strengths. The enhanced sensitivity and increased spectral resolution at 7T compared to 3T allowed a two-fold reduction in scan time, an increased precision of quantification for 12 metabolites, and the additional quantification of lactate with CRLB below 20%. Improved sensitivity at 7T was also demonstrated by a 1.7-fold increase in average SNR (= peak height/root mean square [RMS]-of-noise) per unit-time.

On Lundh's percolation diffusion

A collection of spherical obstacles in the unit ball in Euclidean space is said to be avoidable for Brownian motion if there is a positive probability that Brownian motion diffusing from some point in the ball will avoid all the obstacles and reach the boundary of the ball. The centres of the spherical obstacles are generated according to a Poisson point process while the radius of an obstacle is a deterministic function. If avoidable con gurations are generated with positive probability Lundh calls this percolation di usion. An integral condition for percolation di ffusion is derived in terms of the intensity of the point process and the function that determines the radii of the obstacles.

Anti-basal ganglia antibodies and Tourette's syndrome: a voxel-based morphometry and diffusion tensor imaging study in an adult population.

Anti-basal ganglia antibodies (ABGAs) have been suggested to be a hallmark of autoimmunity in Gilles de la Tourette's syndrome (GTS), possibly related to prior exposure to streptococcal infection. In order to detect whether the presence of ABGAs was associated with subtle structural changes in GTS, whole-brain analysis using independent sets of T(1) and diffusion tensor imaging MRI-based methods were performed on 22 adults with GTS with (n = 9) and without (n = 13) detectable ABGAs in the serum. Voxel-based morphometry analysis failed to detect any significant difference in grey matter density between ABGA-positive and ABGA-negative groups in caudate nuclei, putamina, thalami and frontal lobes. These results suggest that ABGA synthesis is not related to structural changes in grey and white matter (detectable with these methods) within frontostriatal circuits.