Editor Convidado: José Augusto Ferreira M. Filomena Teodoro, Pedro M. Lima, Neville J. Ford, Patricia M. Lumb, "Aproximação Numérica de Equações Funcionais Diferenciais do Tipo Misto" Pedro Lima, Filomena Teodoro, Neville Ford, Patricia Lumb, "Modelação Matemática da Propagação de Sinais no Sistema Nervoso" Olivier Pellegrino, Andreia Furtado, Eduarda Filipe, "Incertezas associadas a retas de calibração" José Carlos Santos, "A linearidade das transformadas de Lorenz" J. R. Branco, "Modelos matemáticos para o desenvolvimento de gliomas" Daniela Coelho, João Brandão, "Symmetric coordinates for an A2B2 molecule"  

Diffusion processes have traditionally been modeled using the classical parabolic advection-diffusion equation. However, as in the case of tracer transport in porous media, significant discrepancies between experimental results and numerical simulations have been reported in the literature. Therefore, in order to describe such anomalous behavior, known as non-Fickian diffusion, some authors have replaced the parabolic model with the continuous time random walk model, which has been very effective. Integro-differential models (IDMs) have been also proposed to describe non-Fickian diffusion in porous media. In this paper, we introduce and test a particular type of IDM by fitting breakthrough curves resulting from laboratory tracer transport. Comparisons with the traditional advection-diffusion equation and the continuous time random walk are also presented. Moreover, we propose and numerically analyze a stable and accurate numerical procedure for the two-dimensional IDM composed by a integro-differential equation for the concentration and Darcy's law for flow. In space, it is based on the combination of mixed finite element and finite volume methods over an unstructured triangular mesh. Copyright © 2015 John Wiley & Sons, Ltd.

A coupled non-Fickian model of a cardiovascular drug delivery system using a biodegradable drug-eluting stent is proposed. The numerical results are obtained using an implicit-explicit finite-element method. The influence of vessel stiffness on the transport of drug eluted from the stent is analysed. The results presented in this paper suggest new perspectives to adapt the drug delivery profile to the needs of the patient.

The changes caused by aging affect all body tissues. The vitreous humor, which fills the space between the lens and the retina, progressively liquefies and shrinks, eventually causing a posterior vitreous detachment. Retinal disorders caused by the breakdown of the blood retinal barrier are also associated with physiological aging. In this work a mathematical model of the pharmacokinetics of a drug eluted from an intravitreal biodegradable implant in an aging eye is presented. The influences of vitreous liquefaction and retina diseases are analyzed. The model is represented by a set of P.D.E’s describing the evolution of the concentration in a drug eluting implant, coupled with two systems of partial differential equations, describing the transport of drug in the vitreous chamber and the retina. Different scenarios of the physiology of vitreous and of inflammation’s degrees of the retina are numerically compared. The numerical results suggest a dominating influence of retina permeability.

In this paper the effect of plaque composition, on the accumulation of drug released by a drug eluting stent, is analyzed. The mathematical model is represented by two coupled systems of partial differential equations that describe the pharmacokinetics of drug in the stent coating and in the arterial wall. The influence of the stiffness and porosity of soft and hard plaques is studied. A case study based on optical coherence tomography images is also included.

In this paper we propose a numerical scheme for wave type equations with damping and space variable coefficients. Relevant equations of this kind arise for instance in the context of Maxwell's equations, namely, the electric potential equation and the electric field equation. The main motivation to study such class of equations is the crucial role played by the electric potential or the electric field in enhanced drug delivery applications. Our numerical method is based on piecewise linear finite element approximation and it can be regarded as a finite difference method based on non-uniform partitions of the spatial domain. We show that the proposed method leads to second order convergence, in time and space, for the kinetic and potential energies with respect to a discrete L^2-norm.

The aim of this paper is to study a finite difference method for quasilinear coupled problems of partial differential equations that presents numerically an unexpected second order convergence rate. The error analysis presented allows us to conclude that the finite difference method is supraconvergent. As the method studied in this paper can be seen as a fully discrete piecewise linear finite element method, we conclude the supercloseness of our approximations.

In this paper initial boundary value problems, defined using quasilinear diffusion equations of Volterra type, are considered. These equations arise for instance to describe diffusion processes in viscoelastic media whose behavior is represented by a Voigt–Kelvin model or a Maxwell model. A finite difference discretization defined on a general non-uniform grid with second order convergence order in space is proposed. The analysis does not follow the usual splitting of the global error using the solution of an elliptic equation induced by the integro-differential equation. The new approach enables us to reduce the smoothness required to the theoretical solution when the usual split technique is used. Non-singular and singular kernels are considered. Numerical simulations which show the effectiveness of the method are included.

A three-dimensional mathematical model for sorption/desorption by a cylindrical polymeric matrix with dispersed drug is proposed. The model is based on a system of partial differential equations coupled with boundary conditions over a moving boundary. We assume that the penetrant diffuses into a swelling matrix and causes a deformation, which induces a stress-driven diffusion and consequently a non-Fickian mass flux. A physically sound nonlinear dependence between strain and penetrant concentration is considered and introduced in a Boltzmann integral with a kernel computed from a Maxwell--Wiechert model. Numerical simulations show how the mechanistic behavior can have a role in drug delivery design.

Simulations show that the kinetics of permeant fluids in viscoelastic matrices depends on the rheological and chemical properties of the material. Fick's law fails to describe transport through viscoelastic materials because of the stress exerted on the incoming fluid which causes a delay. Reversible binding to immobilizing sites also retards permeation of molecules. The effects of mechanical properties and chemical affinities of materials on the transport of solutes are studied. An integro-partial-differential equation is used to model the transport. While the differential part of the equation is represented by an elliptic operator, the integral part describes the contributions of stress and reversible binding. The stability of the model is investigated. The steady-state flux and effective time constant are calculated. The lag time is also studied using multiple integration. Subsequent analyses reveal the dependence of the steady-state flux, the effective time constant, and the lag time on the Young modulus, the viscosity, and the binding/unbinding rates. The results presented in this paper make it possible to tune the mechanical and chemical properties to achieve a desired transport profile.