Unconditional Positivity-Preserving and Energy Stable Schemes for a Reduced Poisson-Nernst-Planck System

The Poisson-Nernst-Planck (PNP) system is a widely accepted model for simulation of ionic channels. In this paper, we design, analyze, and numerically validate a second order unconditional positivity-preserving scheme for solving a reduced PNP system, which can well approximate the three dimensional ion channel problem. Positivity of numerical solutions is proven to hold true independent of the size of time steps and the choice of the Poisson solver. The scheme is easy to implement without resorting to any iteration method. Several numerical examples further confirm the positivity-preserving property, and demonstrate the accuracy, efficiency, and robustness of the proposed scheme, as well as the fast approach to steady states.     

Implicit-Explicit Runge-Kutta Schemes for the Boltzmann-Poisson System for Semiconductors

In this paper we develop a class of Implicit-Explicit Runge-Kutta schemes for solving the multi-scale semiconductor Boltzmann equation. The relevant scale which characterizes this kind of problems is the diffusive scaling. This means that, in the limit of zero mean free path, the system is governed by a drift-diffusion equation. Our aim is to develop a method which accurately works for the different regimes encountered in general semiconductor simulations: the kinetic, the intermediate and the diffusive one. Moreover, we want to overcome the restrictive time step conditions of standard time integration techniques when applied to the solution of this kind of phenomena without any deterioration in the accuracy. As a result, we obtain high order time and space discretization schemes which do not suffer from the usual parabolic stiffness in the diffusive limit. We show different numerical results which permit to appreciate the performances of the proposed schemes.     

Relaxation Schemes for the $M_1$ Model with Space-Dependent Flux: Application to Radiotherapy Dose Calculation

Because of stability constraints, most numerical schemes applied to hyperbolic systems of equations turn out to be costly when the flux term is multiplied by some very large scalar. This problem emerges with the $M_1$ system of equations in the field of radiotherapy when considering heterogeneous media with very disparate densities. Additionally, the flux term of the $M_1$ system is non-linear, and in order for the model to be well-posed the numerical solution needs to fulfill conditions called realizability. In this paper, we propose a numerical method that overcomes the stability constraint and preserves the realizability property. For this purpose, we relax the $M_1$ system to obtain a linear flux term. Then we extend the stencil of the difference quotient to obtain stability. The scheme is applied to a radiotherapy dose calculation example.     

An All-Speed Asymptotic-Preserving Method for the Isentropic Euler and Navier-Stokes Equations

The computation of compressible flows becomes more challenging when the Mach number has different orders of magnitude. When the Mach number is of order one, modern shock capturing methods are able to capture shocks and other complex structures with high numerical resolutions. However, if the Mach number is small, the acoustic waves lead to stiffness in time and excessively large numerical viscosity, thus demanding much smaller time step and mesh size than normally needed for incompressible flow simulation. In this paper, we develop an all-speed asymptotic preserving (AP) numerical scheme for the compressible isentropic Euler and Navier-Stokes equations that is uniformly stable and accurate for all Mach numbers. Our idea is to split the system into two parts: one involves a slow, nonlinear and conservative hyperbolic system adequate for the use of modern shock capturing methods and the other a linear hyperbolic system which contains the stiff acoustic dynamics, to be solved implicitly. This implicit part is reformulated into a standard pressure Poisson projection system and thus possesses sufficient structure for efficient fast Fourier transform solution techniques. In the zero Mach number limit, the scheme automatically becomes a projection method-like incompressible solver. We present numerical results in one and two dimensions in both compressible and incompressible regimes.     

Pseudo-Arclength Continuation Algorithms for Binary Rydberg-Dressed Bose-Einstein Condensates

We study pseudo-arclength continuation methods for both Rydberg-dressed Bose-Einstein condensates (BEC), and binary Rydberg-dressed BEC which are governed by the Gross-Pitaevskii equations (GPEs). A divide-and-conquer technique is proposed for rescaling the range/ranges of nonlocal nonlinear term/terms, which gives enough information for choosing a proper stepsize. This guarantees that the solution curve we wish to trace can be precisely approximated. In addition, the ground state solution would successfully evolve from one peak to vortices when the affect of the rotating term is imposed. Moreover, parameter variables with different number of components are exploited in curve-tracing. The proposed methods have the advantage of tracing the ground state solution curve once to compute the contours for various values of the coefficients of the nonlocal nonlinear term/terms. Our numerical results are consistent with those published in the literatures.     

An Iterative Discontinuous Galerkin Method for Solving the Nonlinear Poisson Boltzmann Equation

An iterative discontinuous Galerkin (DG) method is proposed to solve the nonlinear Poisson Boltzmann (PB) equation. We first identify a function space in which the solution of the nonlinear PB equation is iteratively approximated through a series of linear PB equations, while an appropriate initial guess and a suitable iterative parameter are selected so that the solutions of linear PB equations are monotone within the identified solution space. For the spatial discretization we apply the direct discontinuous Galerkin method to those linear PB equations. More precisely, we use one initial guess when the Debye parameter λ=O(1), and a special initial guess for λ≪1 to ensure convergence. The iterative parameter is carefully chosen to guarantee the existence, uniqueness, and convergence of the iteration. In particular, iteration steps can be reduced for a variable iterative parameter. Both one and two-dimensional numerical results are carried out to demonstrate both accuracy and capacity of the iterative DG method for both cases of λ=O(1) and λ≪1. The (m+1)th order of accuracy for L² and mth order of accuracy for H¹for P^m elements are numerically obtained.     

On Integral Equation and Least Squares Methods for Scattering by Diffraction Gratings

In this paper we consider the scattering of a plane acoustic or electromagnetic wave by a one-dimensional, periodic rough surface. We restrict the discussion to the case when the boundary is sound soft in the acoustic case, perfectly reflecting with TE polarization in the EM case, so that the total field vanishes on the boundary. We propose a uniquely solvable first kind integral equation formulation of the problem, which amounts to a requirement that the normal derivative of the Green's representation formula for the total field vanish on a horizontal line below the scattering surface. We then discuss the numerical solution by Galerkin's method of this (ill-posed) integral equation. We point out that, with two particular choices of the trial and test spaces, we recover the so-called SC (spectral-coordinate) and SS (spectral-spectral) numerical schemes of DeSanto et al., Waves Random Media, 8, 315-414, 1998. We next propose a new Galerkin scheme, a modification of the SS method that we term the SS^∗ method, which is an instance of the well-known dual least squares Galerkin method. We show that the SS^∗ method is always well-defined and is optimally convergent as the size of the approximation space increases. Moreover, we make a connection with the classical least squares method, in which the coefficients in the Rayleigh expansion of the solution are determined by enforcing the boundary condition in a least squares sense, pointing out that the linear system to be solved in the SS^∗ method is identical to that in the least squares method. Using this connection we show that (reflecting the ill-posed nature of the integral equation solved) the condition number of the linear system in the SS^∗ and least squares methods approaches infinity as the approximation space increases in size. We also provide theoretical error bounds on the condition number and on the errors induced in the numerical solution computed as a result of ill-conditioning. Numerical results confirm the convergence of the SS^∗ method and illustrate the ill-conditioning that arises.     

A Modified Crank-Nicolson Numerical Scheme for the Flory-Huggins Cahn-Hilliard Model

In this paper we propose and analyze a second order accurate numerical scheme for the Cahn-Hilliard equation with logarithmic Flory Huggins energy potential. A modified Crank-Nicolson approximation is applied to the logarithmic nonlinear term, while the expansive term is updated by an explicit second order Adams-Bashforth extrapolation, and an alternate temporal stencil is used for the surface diffusion term. A nonlinear artificial regularization term is added in the numerical scheme, which ensures the positivity-preserving property, i.e., the numerical value of the phase variable is always between -1 and 1 at a point-wise level. Furthermore, an unconditional energy stability of the numerical scheme is derived, leveraging the special form of the logarithmic approximation term. In addition, an optimal rate convergence estimate is provided for the proposed numerical scheme, with the help of linearized stability analysis. A few numerical results, including both the constant-mobility and solution-dependent mobility flows, are presented to validate the robustness of the proposed numerical scheme.     

Some Random Batch Particle Methods for the Poisson-Nernst-Planck and Poisson-Boltzmann Equations

We consider in this paper random batch interacting particle methods for solving the Poisson-Nernst-Planck (PNP) equations, and thus the Poisson-Boltzmann (PB) equation as the equilibrium, in the external unbounded domain. To justify the simulation in a truncated domain, an error estimate of the truncation is proved in the symmetric cases for the PB equation. Then, the random batch interacting particle methods are introduced which are $\mathcal{O}(N)$ per time step. The particle methods can not only be considered as a numerical method for solving the PNP and PB equations, but also can be used as a direct simulation approach for the dynamics of the charged particles in solution. The particle methods are preferable due to their simplicity and adaptivity to complicated geometry, and may be interesting in describing the dynamics of the physical process. Moreover, it is feasible to incorporate more physical effects and interactions in the particle methods and to describe phenomena beyond the scope of the mean-field equations.     

Influence of endothelial glycocalyx degradation and surfactants on air embolism adhesion

BACKGROUND: Microbubble adherence to endothelial cells is enhanced after damage to the glycocalyx. The authors tested the hypothesis that exogenous surfactants delivered intravascularly have differential effects on the rate of restoration of blood flow after heparinase-induced degradation of the endothelial glycocalyx. METHODS: Air microbubbles were injected into the rat cremaster microcirculation after perfusion with heparinase or saline and intravascular administration of either saline or one of two surfactants. The surfactants were Pluronic F-127 (Molecular Probes, Eugene, OR) and Perftoran (OJSC SPC Perftoran, Moscow, Russia). Embolism dimensions and dynamics were observed using intravital microscopy. RESULTS: Significant results were that bubbles embolized the largest diameter vessels after glycocalyx degradation. Bubbles embolized smaller vessels in the surfactant treatment groups. The incidence of bubble dislodgement and the magnitude of distal displacement were smallest after glycocalyx degradation alone and largest after surfactant alone. The time to bubble clearance and restoration of blood flow was longest with heparinase alone and shortest with Pluronic F-127 alone. CONCLUSIONS: Degradation of the glycocalyx causes air bubbles to adhere to the endothelium more proximally in the arteriolar microcirculation. Surfactants added after glycocalyx degradation and before gas embolization promotes bubble lodging in the distal microcirculation. Surfactants may have a clinical role in reducing embolism bubble adhesion to endothelial cells undergoing glycocalyx disruption.     

Numerical Method of Fabric Dynamics Using Front Tracking and Spring Model

We use front tracking data structures and functions to model the dynamic evolution of fabric surface. We represent the fabric surface by a triangulated mesh with preset equilibrium side length. The stretching and wrinkling of the surface are modeled by the mass-spring system. The external driving force is added to the fabric motion through the "Impulse method" which computes the velocity of the point mass by superposition of momentum. The mass-spring system is a nonlinear ODE system. Added by the numerical and computational analysis, we show that the spring system has an upper bound of the eigen frequency. We analyzed the system by considering two spring models and we proved in one case that all eigenvalues are imaginary and there exists an upper bound for the eigen-frequency. This upper bound plays an important role in determining the numerical stability and accuracy of the ODE system. Based on this analysis, we analyzed the numerical accuracy and stability of the nonlinear spring mass system for fabric surface and its tangential and normal motion. We used the fourth order Runge-Kutta method to solve the ODE system and showed that the time step is linearly dependent on the mesh size for the system.     

The effect of ultrasonically induced cavitation on articular cartilage

Cavitation, the term used to describe bubble activity in fluids, is a destructive phenomenon encountered in fluid systems. The effect of cavitation on articular cartilage was investigated by ultrasonically inducing bubble activity on the surface of bovine specimens. Distinctive pits and craters, not present on control specimens, were observed using scanning electron microscopy on the damaged surface. Human osteoarthrotic articular cartilage specimens were removed during arthroplasty and examined using scanning electron microscopy. Craters and pits observed on the osteoarthrotic specimens were similar in appearance to those on the cavitated specimens. The mechanism of cavitation bubble collapse could be responsible for damage in vivo, thus providing articular cartilage with a degenerative pathway toward osteoarthrosis.     

Antiretroviral postnatal prophylaxis to prevent HIV vertical transmission: present and future strategies

Introduction

Maternal antiretroviral therapy (ART) with viral suppression prior to conception, during pregnancy and throughout the breastfeeding period accompanied by infant postnatal prophylaxis (PNP) forms the foundation of current approaches to preventing vertical HIV transmission. Unfortunately, infants continue to acquire HIV infections, with half of these infections occurring during breastfeeding. A consultative meeting of stakeholders was held to review the current state of PNP globally, including the implementation of WHO PNP guidelines in different settings and identifying the key factors affecting PNP uptake and impact, with an aim to optimize future innovative strategies.

Discussion

WHO PNP guidelines have been widely implemented with adaptations to the programme context. Some programmes with low rates of antenatal care attendance, maternal HIV testing, maternal ART coverage and viral load testing capacity have opted against risk-stratification and provide an enhanced PNP regimen for all infants exposed to HIV, while other programmes provide infant daily nevirapine antiretroviral (ARV) prophylaxis for an extended duration to cover transmission risk throughout the breastfeeding period. A simplified risk stratification approach may be more relevant for high-performing vertical transmission prevention programmes, while a simplified non-risk stratified approach may be more appropriate for sub-optimally performing programmes given implementation challenges. In settings with concentrated epidemics, where the epidemic is often driven by key populations, infants who are found to be exposed to HIV should be considered at high risk for HIV acquisition. All settings could benefit from newer technologies that promote retention during pregnancy and throughout the breastfeeding period. There are several challenges in enhanced and extended PNP implementation, including ARV stockouts, lack of appropriate formulations, lack of guidance on alternative ARV options for prophylaxis, poor adherence, poor documentation, inconsistent infant feeding practices and in inadequate retention throughout the duration of breastfeeding.

Conclusions

Tailoring PNP strategies to a programmatic context may improve access, adherence, retention and HIV-free outcomes of infants exposed to HIV. Newer ARV options and technologies that enable simplification of regimens, non-toxic potent agents and convenient administration, including longer-acting formulations, should be prioritized to optimize the effect of PNP in the prevention of vertical HIV transmission.     

Influence of Endothelial Glycocalyx Degradation and Surfactants on Air Embolism Adhesion

Background: Microbubble adherence to endothelial cells is enhanced after damage to the glycocalyx. The authors tested the hypothesis that exogenous surfactants delivered intravascularly have differential effects on the rate of restoration of blood flow after heparinase-induced degradation of the endothelial glycocalyx.

Methods: Air microbubbles were injected into the rat cremaster microcirculation after perfusion with heparinase or saline and intravascular administration of either saline or one of two surfactants. The surfactants were Pluronic F-127 (Molecular Probes, Eugene, OR) and Perftoran (OJSC SPC Perftoran, Moscow, Russia). Embolism dimensions and dynamics were observed using intravital microscopy.

Results: Significant results were that bubbles embolized the largest diameter vessels after glycocalyx degradation. Bubbles embolized smaller vessels in the surfactant treatment groups. The incidence of bubble dislodgement and the magnitude of distal displacement were smallest after glycocalyx degradation alone and largest after surfactant alone. The time to bubble clearance and restoration of blood flow was longest with heparinase alone and shortest with Pluronic F-127 alone.     

Pathologic changes following transurethral canine prostatectomy with a cylindrically diffusing fiber

Transurethral laser prostatectomy was performed on eight mongrel dogs employing a cylindrically diffusing fiber delivery system and a 1.06 μ Nd:YAG laser. Each dog received 15,000 joules of laser energy delivered to the prostate in one continuous dose of 25 watts for 10 minutes. Gross and histopathologic examinations of serial sections of the prostate were performed postoperatively after intervals of 2 hours to 7 weeks. Grossly, a spherical zone of destruction averaging 2.8 cm in diameter was present in dogs except one. Histopathologic changes in the prostate consisted of acute coagulative necrosis with interstitial edema at 2 hours, becoming hemorrhagic by 24 hours. A prominent circular area of acute coagulative necrosis with progressively larger areas of liquefaction and hemorrhage was present in prostates harvested from 4 days to 1 week after lasing. Initial re-epithelization of the resulting cavity was observed at 3 weeks with nearly complete epithelialization 7 weeks after laser treatment. The simplified fiber placement and lack of postoperative complications in this small group of dogs suggest that the cylindrically diffusing fiber could offer significant advantages over laterally deflecting fibers for transurethral prostatectomies in the dog model. © 1994 Wiley-Liss, inc.     

The Geometry Behind Numerical Solvers of the Poisson-Boltzmann Equation

Electrostatics interactions play a major role in the stabilization of biomolecules: as such, they remain a major focus of theoretical and computational studies in biophysics. Electrostatics in solution is strongly dependent on the nature of the solvent and on the ions it contains. While methods that treat the solvent and ions explicitly provide an accurate estimate of these interactions, they are usually computationally too demanding to study large macromolecular systems. Implicit solvent methods provide a viable alternative, especially those based on Poisson theory. The Poisson-Boltzmann equation (PBE) treats the system in a mean field approximation, providing reasonable estimates of electrostatics interactions in a solvent treated as continuum. In the first part of this paper, we review the theory behind the PBE, including recent improvement in which ions size and dipolar features of solvent molecules are taken into account explicitly. The PBE is a non linear second order differential equation with discontinuous coefficients, for which no analytical solution is available for large molecular systems. Many numerical solvers have been developed that solve a discretized version of the PBE on a mesh, either using finite difference, finite element, or boundary element methods. The accuracy of the solutions provided by these solvers highly depend on the geometry of their underlying meshes, as well as on the method used to embed the physical system on the mesh. In the second part of the paper, we describe a new geometric approach for generating unstructured tetrahedral meshes as well as simplifications of these meshes that are well fitted for solving the PBE equation using multigrid approaches.     

Numerical Simulations of Hydrodynamics of Nematic Liquid Crystals: Effects of Kinematic Transports

In this paper, we investigate the effects of kinematic transports on the nematic liquid crystal system numerically and theoretically. The model we used is a "1+2" elastic continuum model simplified from the Ericksen-Leslie system. The numerical experiments are carried out by using a Legendre-Galerkin spectral method which can preserve the energy law in the discrete form. Based on this highly accurate numerical approach we find some interesting and important relationships between the kinematic transports and the characteristics of the flow. We make some analysis to explain these results. Several significant scaling properties are also verified by our simulations.     

Numerical Solution of 3D Poisson-Nernst-Planck Equations Coupled with Classical Density Functional Theory for Modeling Ion and Electron Transport in a Confined Environment

We have developed efficient numerical algorithms for solving 3D steady-state Poisson-Nernst-Planck (PNP) equations with excess chemical potentials described by the classical density functional theory (cDFT). The coupled PNP equations are discretized by a finite difference scheme and solved iteratively using the Gummel method with relaxation. The Nernst-Planck equations are transformed into Laplace equations through the Slotboom transformation. Then, the algebraic multigrid method is applied to efficiently solve the Poisson equation and the transformed Nernst-Planck equations. A novel strategy for calculating excess chemical potentials through fast Fourier transforms is proposed, which reduces computational complexity from $\mathcal{O}$($N^2$) to $\mathcal{O}$($NlogN$), where $N$ is the number of grid points. Integrals involving the Dirac delta function are evaluated directly by coordinate transformation, which yields more accurate results compared to applying numerical quadrature to an approximated delta function. Numerical results for ion and electron transport in solid electrolyte for lithium-ion (Li-ion) batteries are shown to be in good agreement with the experimental data and the results from previous studies.     

Ultrafast imaging of tissue ablation by a XeCl excimer laser in saline.

To determine the temporal evolution of laser induced tissue ablation, arterial wall specimens with either hard calcified or fatty plaques and normal tissue were irradiated in a 0.9% saline solution using a XeCl excimer laser (wavelength 308 nm, energy fluence 7 J/cm2, pulse width 30 ns) through a 600 microns fused silica fiber pointing perpendicular either at a 0.5 mm distance or in direct contact to the vascular surface. Radiation of a pulsed dye laser (wavelength 580 nm) was used to illuminate the tissue surface. The ablation process and the arising bubble above the tissue surface were recorded with a CCD camera attached to a computer based image-processing system. Spherical cavitation bubbles and small tissue particles emerging from the irradiated area have been recorded. The volume of this bubble increased faster for calcified plaques than for normal tissue.