A numerical method for the fractional Fitzhugh–Nagumo monodomain model

A fractional FitzHugh–Nagumo monodomain model with zero Dirichlet boundary conditions is presented, generalising the standard monodomain model that describes the propagation of the electrical potential in heterogeneous cardiac tissue. The model consists of a coupled fractional Riesz space nonlinear reaction-diffusion model and a system of ordinary differential equations, describing the ionic fluxes as a function of the membrane potential. We solve this model by decoupling the space-fractional partial differential equation and the system of ordinary differential equations at each time step. Thus, this means treating the fractional Riesz space nonlinear reaction-diffusion model as if the nonlinear source term is only locally Lipschitz. The fractional Riesz space nonlinear reaction-diffusion model is solved using an implicit numerical method with the shifted Grunwald–Letnikov approximation, and the stability and convergence are discussed in detail in the context of the local Lipschitz property. Some numerical examples are given to show the consistency of our computational approach.

A semi-alternating direction method for a 2-D fractional FitzHugh–Nagumo monodomain model on an approximate irregular domain

A FitzHugh-Nagumo monodomain model has been used to describe the propagation of the electrical potential in heterogeneous cardiac tissue. In this paper, we consider a two-dimensional fractional FitzHugh-Nagumo monodomain model on an irregular domain. The model consists of a coupled Riesz space fractional nonlinear reaction-diffusion model and an ordinary differential equation, describing the ionic fluxes as a function of the membrane potential. Secondly, we use a decoupling technique and focus on solving the Riesz space fractional nonlinear reaction-diffusion model. A novel spatially second-order accurate semi-implicit alternating direction method (SIADM) for this model on an approximate irregular domain is proposed. Thirdly, stability and convergence of the SIADM are proved. Finally, some numerical examples are given to support our theoretical analysis and these numerical techniques are employed to simulate a two-dimensional fractional Fitzhugh-Nagumo model on both an approximate circular and an approximate irregular domain.

Exact solutions of coupled multispecies linear reaction–diffusion equations on a uniformly growing domain

Embryonic development involves diffusion and proliferation of cells, as well as diffusion and reaction of molecules, within growing tissues. Mathematical models of these processes often involve reaction–diffusion equations on growing domains that have been primarily studied using approximate numerical solutions. Recently, we have shown how to obtain an exact solution to a single, uncoupled, linear reaction–diffusion equation on a growing domain, 0 < x < L(t), where L(t) is the domain length. The present work is an extension of our previous study, and we illustrate how to solve a system of coupled reaction–diffusion equations on a growing domain. This system of equations can be used to study the spatial and temporal distributions of different generations of cells within a population that diffuses and proliferates within a growing tissue. The exact solution is obtained by applying an uncoupling transformation, and the uncoupled equations are solved separately before applying the inverse uncoupling transformation to give the coupled solution. We present several example calculations to illustrate different types of behaviour. The first example calculation corresponds to a situation where the initially–confined population diffuses sufficiently slowly that it is unable to reach the moving boundary at x = L(t). In contrast, the second example calculation corresponds to a situation where the initially–confined population is able to overcome the domain growth and reach the moving boundary at x = L(t). In its basic format, the uncoupling transformation at first appears to be restricted to deal only with the case where each generation of cells has a distinct proliferation rate. However, we also demonstrate how the uncoupling transformation can be used when each generation has the same proliferation rate by evaluating the exact solutions as an appropriate limit.

Estimating cell diffusivity and cell proliferation rate by interpreting IncuCyte ZOOM™ assay data using the Fisher-Kolmogorov model

Background: Standard methods for quantifying IncuCyte ZOOM™ assays involve measurements that quantify how rapidly the initially-vacant area becomes re-colonised with cells as a function of time. Unfortunately, these measurements give no insight into the details of the cellular-level mechanisms acting to close the initially-vacant area. We provide an alternative method enabling us to quantify the role of cell motility and cell proliferation separately. To achieve this we calibrate standard data available from IncuCyte ZOOM™ images to the solution of the Fisher-Kolmogorov model. Results: The Fisher-Kolmogorov model is a reaction-diffusion equation that has been used to describe collective cell spreading driven by cell migration, characterised by a cell diffusivity, D, and carrying capacity limited proliferation with proliferation rate, λ, and carrying capacity density, K. By analysing temporal changes in cell density in several subregions located well-behind the initial position of the leading edge we estimate λ and K. Given these estimates, we then apply automatic leading edge detection algorithms to the images produced by the IncuCyte ZOOM™ assay and match this data with a numerical solution of the Fisher-Kolmogorov equation to provide an estimate of D. We demonstrate this method by applying it to interpret a suite of IncuCyte ZOOM™ assays using PC-3 prostate cancer cells and obtain estimates of D, λ and K. Comparing estimates of D, λ and K for a control assay with estimates of D, λ and K for assays where epidermal growth factor (EGF) is applied in varying concentrations confirms that EGF enhances the rate of scratch closure and that this stimulation is driven by an increase in D and λ, whereas K is relatively unaffected by EGF. Conclusions: Our approach for estimating D, λ and K from an IncuCyte ZOOM™ assay provides more detail about cellular-level behaviour than standard methods for analysing these assays. In particular, our approach can be used to quantify the balance of cell migration and cell proliferation and, as we demonstrate, allow us to quantify how the addition of growth factors affects these processes individually.

Survival probability for a diffusive process on a growing domain

We consider the motion of a diffusive population on a growing domain, 0 < x < L(t ), which is motivated by various applications in developmental biology. Individuals in the diffusing population, which could represent molecules or cells in a developmental scenario, undergo two different kinds of motion: (i) undirected movement, characterized by a diffusion coefficient, D, and (ii) directed movement, associated with the underlying domain growth. For a general class of problems with a reflecting boundary at x = 0, and an absorbing boundary at x = L(t ), we provide an exact solution to the partial differential equation describing the evolution of the population density function, C(x,t ). Using this solution, we derive an exact expression for the survival probability, S(t ), and an accurate approximation for the long-time limit, S = limt→∞ S(t ). Unlike traditional analyses on a nongrowing domain, where S ≡ 0, we show that domain growth leads to a very different situation where S can be positive. The theoretical tools developed and validated in this study allow us to distinguish between situations where the diffusive population reaches the moving boundary at x = L(t ) from other situations where the diffusive population never reaches the moving boundary at x = L(t ). Making this distinction is relevant to certain applications in developmental biology, such as the development of the enteric nervous system (ENS). All theoretical predictions are verified by implementing a discrete stochastic model.

Interfacial tension model for catalytically driven nanorods

We present an analysis of the interfacial tension model for the movement of the catalytically driven nanorod. The model considers the convective reaction-diffusion equation for the production and diffusion of oxygen around the bimetallic nanorod. We solve the equation and find the concentration difference, which drives the nanorod. We use our expression to calculate the force on the nanorod and find that the result is within 20% of the results found earlier [ W. Paxton et al., J. Am. Chem. Soc. 128, 14881 (2006) ] by an approximate method. Unlike the earlier results, our results are valid from short to long lengths of the nanorod.

A Model of Anomalous Enzyme-Catalyzed Gel Degradation Kinetics

We show that a model of target location involving n noninteracting particles moving subdiffusively along a line segment (a generalization of a model introduced by Sokolov et al. [Biophys. J. 2005, 89, 895.]) provides a basis for understanding recent experiments by Pelta et al. [Phys. Rev. Lett. 2007, 98, 228302.] on the kinetics of diffusion-limited gel degradation. These experiments find that the time t(c) taken by the enzyme thermolysin to completely hydrolyze a gel varies inversely as roughly the 3/2 power of the initial enzyme concentration [E]. In general, however, this time would be expected to vary either as [E](-1) or as [E](-2), depending on whether the Brownian diffusion of the enzyme to the site of cleavage took place along the network chains (1-d diffusion) or through the pore spaces (3-d diffusion). In our model, the unusual dependence of t(c) on [E] is explained in terms of a reaction-diffusion equation that is formulated in terms of fractional rather than ordinary time derivatives.

Isotropic finite volume discretization

Finite volume methods traditionally employ dimension by dimension extension of the one-dimensional reconstruction and averaging procedures to achieve spatial discretization of the governing partial differential equations on a structured Cartesian mesh in multiple dimensions. This simple approach based on tensor product stencils introduces an undesirable grid orientation dependence in the computed solution. The resulting anisotropic errors lead to a disparity in the calculations that is most prominent between directions parallel and diagonal to the grid lines. In this work we develop isotropic finite volume discretization schemes which minimize such grid orientation effects in multidimensional calculations by eliminating the directional bias in the lowest order term in the truncation error. Explicit isotropic expressions that relate the cell face averaged line and surface integrals of a function and its derivatives to the given cell area and volume averages are derived in two and three dimensions, respectively. It is found that a family of isotropic approximations with a free parameter can be derived by combining isotropic schemes based on next-nearest and next-next-nearest neighbors in three dimensions. Use of these isotropic expressions alone in a standard finite volume framework, however, is found to be insufficient in enforcing rotational invariance when the flux vector is nonlinear and/or spatially non-uniform. The rotationally invariant terms which lead to a loss of isotropy in such cases are explicitly identified and recast in a differential form. Various forms of flux correction terms which allow for a full recovery of rotational invariance in the lowest order truncation error terms, while preserving the formal order of accuracy and discrete conservation of the original finite volume method, are developed. Numerical tests in two and three dimensions attest the superior directional attributes of the proposed isotropic finite volume method. Prominent anisotropic errors, such as spurious asymmetric distortions on a circular reaction-diffusion wave that feature in the conventional finite volume implementation are effectively suppressed through isotropic finite volume discretization. Furthermore, for a given spatial resolution, a striking improvement in the prediction of kinetic energy decay rate corresponding to a general two-dimensional incompressible flow field is observed with the use of an isotropic finite volume method instead of the conventional discretization. (C) 2014 Elsevier Inc. All rights reserved.

Progress in the understanding of Ni silicide formation for advanced MOS structures

Metallic silicides have been used as contact materials on source/drain and gate in metal-oxide semiconductor (MOS) structure for 40 years. Since the 65 nm technology node, NiSi is the preferred material for contact in microelectronic due to low resistivity, low thermal budget, and low Si consumption. Ni(Pt)Si with 10 at.% Pt is currently employed in recent technologies since Pt allows to stabilize NiSi at high temperature. The presence of Pt and the very low thickness (<10 nm) needed for the device contacts bring new concerns for actual devices. In this work, in situ techniques [X-ray diffraction (XRD), X-ray reflectivity (XRR), sheet resistance, differential scanning calorimetry (DSC)] were combined with atom probe tomography (APT) to study the formation mechanisms as well as the redistribution of dopants and alloy elements (Pt, Pd.) during the silicide formation. Phenomena like nucleation, lateral growth, interfacial reaction, diffusion, precipitation, and transient phase formation are investigated. The effect of alloy elements (Pt, Pd.) and dopants (As, B.) as well as stress and defects induced by the confinement in devices on the silicide formation mechanism and alloying element redistribution is examined. In particular APT has been performed for the three-dimensional (3D) analysis of MOSFET at the atomic scale. The advances in the understanding of the mechanisms of formation and redistribution are discussed. © 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Kinetics of ytterbium(III) extraction with Cyanex 272 using a constant interfacial cell with laminar flow

Studies have been made on the kinetics of ytterbium(III) with bis-(2,4,4-trimethylpentyl) phosphinic acid (Cyanex 272, HA) in n-heptane using a constant interfacial cell with laminar flow. The stiochiometry and the equilibrium constant of the extracted complex formation reaction between Yb3+ and Cyanex 272 are determined. The extraction rate is dependent of the stirring rate. This fact together with the Ea value suggests that the mass transfer process is a mixed chemical reaction-diffusion controlled at lower temperature, whereas it is entirely diffusion controlled at higher temperature. The rate equations for the ytterbium extraction with Cyanex 272 have been obtained. The rate-determining step is also made by predictions derived from interfacial reaction models, and through the approximate solutions of the flux equation, diffusion parameters and thickness of the diffusion film have been calculated.

Extraction kinetics of rare earth elements with sec-octylphenoxy acetic acid

The kinetics of RE (La, Gd, Er, Yb and Y) extraction with sec-octylphenoxy acetic acid was investigated using a constant interfacial area cell with laminar flow at 303 K. The natures of the extracted complexes have some effect on the extraction rate which is controlled by the reaction rate of M(III) and extractant molecules at two-phase interface for Er(III), Yb(III) and Y(III), by a mixed chemical reaction-diffusion for Gd(III) and a diffusion for La( III). The extractant molecules tend to adsorb at the interface. So an interfacial extraction reaction model was derived.

Kinetics of cerium(IV) extraction from H2SO4-HF medium with Cyanex 923

Studies of the extraction kinetics of cerium(IV) from H2SO4-HF solutions with Cyanex 923 in n-heptane have been carried out using a constant interfacial area cell with laminar flow. The experimental hydrodynamic conditions were chosen so that the contribution of diffusion to the measured rate of reaction was minimized. The data were analyzed in terms of pseudo-first order constants. The results were compared with those of the system without HF. It was concluded that the addition of HF reduces the activation energy for the forward rate from 46.2 to 36.5 U mol(-1) while it has an opposite effect on the activation energy for the reverse process(the activation energy increased from 23.3 to 90.8 U mol(-1)). Thus, HF can accelerate the rate of cerium(IV) extraction. At the same time, the extraction rate is controlled by a mixed chemical reaction-diffusion rather than by a chemical reaction alone. A rate equation has also been obtained.

Uniformly convergent finite element and finite difference methods for singularly perturbed ordinary differential equations

This thesis is concerned with uniformly convergent finite element and finite difference methods for numerically solving singularly perturbed two-point boundary value problems. We examine the following four problems: (i) high order problem of reaction-diffusion type; (ii) high order problem of convection-diffusion type; (iii) second order interior turning point problem; (iv) semilinear reaction-diffusion problem. Firstly, we consider high order problems of reaction-diffusion type and convection-diffusion type. Under suitable hypotheses, the coercivity of the associated bilinear forms is proved and representation results for the solutions of such problems are given. It is shown that, on an equidistant mesh, polynomial schemes cannot achieve a high order of convergence which is uniform in the perturbation parameter. Piecewise polynomial Galerkin finite element methods are then constructed on a Shishkin mesh. High order convergence results, which are uniform in the perturbation parameter, are obtained in various norms. Secondly, we investigate linear second order problems with interior turning points. Piecewise linear Galerkin finite element methods are generated on various piecewise equidistant meshes designed for such problems. These methods are shown to be convergent, uniformly in the singular perturbation parameter, in a weighted energy norm and the usual L2 norm. Finally, we deal with a semilinear reaction-diffusion problem. Asymptotic properties of solutions to this problem are discussed and analysed. Two simple finite difference schemes on Shishkin meshes are applied to the problem. They are proved to be uniformly convergent of second order and fourth order respectively. Existence and uniqueness of a solution to both schemes are investigated. Numerical results for the above methods are presented.

A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs

We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with "polynomial" nonlinearities and additive noise, considered as abstract evolution equations in some Hilbert space. It is shown that if Hörmander's bracket condition holds at every point of this Hilbert space, then a lower bound on the Malliavin covariance operatorμt can be obtained. Informally, this bound can be read as "Fix any finite-dimensional projection on a subspace of sufficiently regular functions. Then the eigenfunctions of μt with small eigenvalues have only a very small component in the image of Π." We also show how to use a priori bounds on the solutions to the equation to obtain good control on the dependency of the bounds on the Malliavin matrix on the initial condition. These bounds are sufficient in many cases to obtain the asymptotic strong Feller property introduced in [HM06]. One of the main novel technical tools is an almost sure bound from below on the size of "Wiener polynomials," where the coefficients are possibly non-adapted stochastic processes satisfying a Lips chitz condition. By exploiting the polynomial structure of the equations, this result can be used to replace Norris' lemma, which is unavailable in the present context. We conclude by showing that the two-dimensional stochastic Navier-Stokes equations and a large class of reaction-diffusion equations fit the framework of our theory.

How simple grazing rules can lead to persistent boundaries in vegetation communities.

Natural landscape boundaries between vegetation communities are dynamically influenced by the selective grazing of herbivores. Here we show how this may be an emergent property of very simple animal decisions, without the need for any sophisticated choice rules etc., using a model based on biased diffusion. Animal grazing intensity is coupled with plant competition, resulting in reaction-diffusion dynamics, from which stable boundaries spontaneously emerge. In the model, animals affect their resources by both consumption and trampling. It is assumed that forage consists of two heterogeneously distributed competing resource species, one that is preferred (grass) over the other (heather) by the animals. The solutions to the resulting system of differential equations for three cases a) optimal foraging, b) random walk foraging and c) taxis-diffusion are presented. Optimal and random foraging gave unrealistic results, but taxis-diffusion accorded well with field observations. Persistent boundaries between patches of near-monoculture vegetation were predicted, with these boundaries drifting in response to overall grazing pressure (grass advancing with increased grazing and vice versa). The reaction-taxis-diffusion model provides the first mathematical explanation for such vegetation mosaic dynamics and the parameters of the model are open to experimental testing.