Paula de Oliveira

University of Coimbra | UC · Centre for Mathematics

Respiratory particles containing infectious pathogens are responsible for a large number of diseases. To define health politics and save lives, it is important to study their transmission mechanisms, namely the path of particles once expelled. This path depends on several driving factors as intrinsic properties of particles, environmental aspects a...

Drug and gene delivery to the eye, namely to the posterior segment of the eye, is one of the most challenging problems for ophthalmologists and pharmacologists. The reason lies in the fact that the eye is protected by multiple barriers that prevent the permeation of xenobiotics and consequently prevent drugs from permeating ocular tissues. Intravit...

Following the implantation of indwelling medical devices, bacteria inoculated during the surgery or coming from a preexistent focus of infection race for the medical surface where they attach. Adaptation to survive is a common feature of life, and microorganisms are not an exception. Bacteria form, in short periods of time, a habitat-the biofilm-wh...

To reduce the side effects of chemotherapy drugs, namely in cancer therapy, researchers of different fields are making tremendous efforts to design new drug systems that can be used to deliver the drug locally in a sustainable way. Polymeric nanocarriers are investigated to transport the drug to the target tissue where the cargo should be delivered...

These last years the insertion of implants and medical devices has emerged as a common surgical procedure. Following their implantation, the bacteria, inoculated during the surgery, or coming from a preexisting focus of infection, can colonize a significant proportion of them. The resistance of bacteria against antibiotics increases dramatically on...

The use of enhancers to increase drug release from medical devices and drug transport through tissues has been largely investigated. Researchers from different fields like polymer chemistry, materials science, pharmaceutics, bioengineering, and chemical engineering have addressed efforts to combine materials, stimuli and drugs to design effective d...

The main motivation of the present work is the numerical study of a system of Partial Differential Equations that governs drug transport, through a target tissue or organ, when enhanced by the simultaneous action of an electric field and a temperature rise. The electric field, while forcing charged drug molecules through the tissue or the organ, th...

In this paper we study a system of partial differential equations that couple nonfickian diffusion of one of two species with the fickian diffusion of a chemical or biological agent. This system can be used to describe the evolution a population (biological species, cells) that switch between two phenotypes: migratory and proliferative. This switch...

In this paper we consider the coupling between two quasilinear diffusion equations: the diffusion coefficient of the first equation depends on its solution and the diffusion and convective coefficients of the second equation depend on the solution of the first one. This system can be used to describe the drug evolution in a target tissue when the d...

Atherosclerosis in the coronary arteries is one of the leading causes of death in the world. Percutaneous coronary interventions (PCI) associated with the implantation of drug eluting stents (DES) is one of the most common forms of revascularization in patients with atherosclerotic coronary artery disease. The use of DES is considered as an effecti...

This paper focusses on the mathematical modeling of the ascorbic acid (antioxidant) release from a pectin edible film (packaging) to an agar hydrogel (food). The proposed model considers the viscoelastic properties of the polymeric film, the solid ascorbic acid dissolution inside the film, its degradation and diffusion in both the film and the hydr...

We consider a polymeric spherical platform containing a solid dispersed drug that is in contact with a solvent fluid. While swelling, a non-Fickian sorption of the solvent molecules occurs induced by the effect of the viscoelastic properties of the polymer. The solid drug in contact with the solvent fluid dissolves and a Fickian release of dissolve...

Efficacious drug delivery to the posterior chamber of the eye is a very challenging problem due to the many physiological barriers that protect the eye against the entry of exogenous substances. To avoid, or to limit, the action of these barriers several drug delivery routes are being investigated and used in clinical ophthalmology. To assist medic...

In this paper we consider general linear damped wave equations with memory. We establish energy estimates that under the assumption of exponentially bounded kernels, induce exponential decaying solutions. Numerical waves that mimic their continuous counterpart are also introduced using a finite element approach.

The use of enhancers to increase the drug molecules penetration into target tissues is a usual technique in drug delivery. In transdermal drug delivery, electric fields are often used to increase the drug transport through the skin. In this paper we study a drug delivery mechanism from a reservoir which is in contact with the skin. We assume that t...

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 m...

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 porosit...

A two dimensional coupled nonlinear non-Fickian model for drug release from a biodegradable drug eluting stent into the arterial wall is studied. The influence of porosity and degradation of the polymer as well as the dissolution rate of the drug are analyzed. Numerical simulations that illustrate the kind of dependence of drug profiles on these pr...

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...

A mathematical model to simulate drug delivery from a viscoelastic erodible matrix is presented in this paper. The drug is initially distributed in the matrix that is in contact with water. The entrance of water in the material changes the molecular weight, and bulk erosion can be developed depending on how fast this entrance is and how fast degrad...

In this paper diffusion through a viscoelastic biodegradable material is studied. The phenomenon is described by a set of three coupled partial differential equations that take into account passive diffusion, stress driven diffusion and the degradation of the material. The stability properties of the model are studied. Erodible viscoelastic materia...

In this paper we propose a mathematical model to describe the evolution of glioma cells taking into account the viscoelastic properties of brain tissue. The mathematical model is established considering that the glioma cells are of two phenotypes: migratory and proliferative. The evolution of the migratory cells is described by a diffusion-reaction...

A coupled model of a cardiovascular drug delivery system using a biodegradable drug eluting stent is proposed. A reversible reaction between the drug and the binding sites in the arterial tissue is taken into account. The proposed model is sensitive to the different nature of the therapeutic compounds used. The numerical results are obtained using...

A mathematical model to simulate drug delivery from a vicoelastic erodible matrix is presented in this paper. The drug is initially distributed in the matrix which is in contact with water. The entrance of water in the material changes the molecular weight and bulk erosion can be developed depending on how fast is this entrance and how fast degrada...

In this paper a non linear mathematical model to describe absorption phenomena in polymers is proposed. The model is established assuming that the diffusing penetrant causes a deformation which induces a viscoelastic stress responsible for a convective field. This convective field is defined to represent an opposition of the polymer to the Fickian...

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 induc...

A mathematical model which simulates drug delivery through the cornea, from a therapeutic lens to the anterior chamber of the eye, is proposed. The model consists of three coupled systems of partial differential equations linked by interface conditions: drug diffusion in the therapeutic lens; diffusion and metabolic consumption in the cornea; diffu...

A two dimensional coupled model of drug delivery in the cardiovascular tissue using biodegradable drug eluting stents is developed. Qualitative behavior, stability analysis as well as simulations of the model have been presented. Numerical results computed with an implicit–explicit finite element method show a complete agreement with the expected p...

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 perme...

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-...

A two dimensional coupled model of drug delivery in the cardiovascular tissue using biodegradable drug eluting stents is developed. The qualitative behaviour of the model is analyzed. Numerical results computed with an Implicit Explicit Finite Element Method show a complete agreement with the expected physical behaviour

In this chapter we study the diffusion of a liquid agent into a polymeric matrix. We propose an initial-boundary value problem to model the process. Numerical methods are obtained for solving it. The stability and the convergence of the methods are studied.

The study of the dependence of fluxes, concentrations and response times, on the characteristic properties of drug delivery polymeric devices, plays an important role in the design of drug release platforms. The aim of this paper is to develop mathematical tools for an in-depth understanding of drug release tracking. The mathematical model presente...

In this paper we study initial boundary value problems that describe reaction–diffusion phenomena in viscoelastic materials. The mathematical model, represented by an integro-differential equation coupled with an ordinary differential equation, is analyzed from theoretical and numerical viewpoints.

In this paper we study initial boundary value problems that describe reaction–diffusion phenomena in viscoelastic materials. The mathematical model, represented by an integrodifferential equation coupled with an ordinary differential equation, is analyzed from theoretical and numerical viewpoints.

The goal of this paper is to present an overview of drug delivery from polymeric therapeutic lens to the anterior segment of the eye. Mathematical models that describe in vitro and in vivo drug delivery, from different types of lens, are presented. Healthy and pathological situations are addressed. Numerical simulations are included and compared wi...

The dynamics of diffusive and stress-induced transport in polymeric delivery systems was investigated. Partial and ordinary differential equations were first written to describe drug release behaviors in Maxwell and Maxwell–Voigt materials. The time constants governing the flux and concentration responses of a permeating species were determined fro...

In two previous papers the authors presented mathematical models that simulate the mass of drug delivered, in vitro Ferreira, Oliveira, Silva, Carreira, Gil and Murta (2010) and in vivo Ferreira, Oliveira, Silva and Murta (2011), from a therapeutic contact lens. In the present paper the time it takes to reach an equilibrium state is studied. A clos...

Mathematical models to describe drug concentration profiles of topi-cally administered drug in the anterior chamber aqueous humor have been proposed by several authors. The aim of this paper is to present a mathematical model to predict the drug concentration in the anterior chamber when a therapeutical contact lens with the drug is entrapped in na...

The controlled release of drug through polymeric membranes provides a mechanism with a wide range of applications. A review of the pharmaceutical literature has been carried on and two main classes of controlled drug release devices using polymeric matrices have been identified: ophthalmic therapeutic lenses that deliver drug to the cornea and tran...

In certain cases, quasilinear convection-diffusion-reaction equations range from parabolic to almost hyperbolic, depending on the ratio between convection and diffusion coefficients. From a numerical point of view, two main difficulties can arise related to the existence of layers and/or the non-smoothness of the coefficients of such equations. In...

This paper focuses on the release of an ophthalmic drug (flurbiprofen) from a loaded copolymer where the drug is simultaneously dispersed in the polymeric matrix and entrapped in particles. The copolymer is based in 2-hydroxyethyl methacrylate co-methacrylic acid and silicone is used to prepare the loaded particles. A mathematical model to simulate...

A phenomenological formulation is adopted to investigate desorption in polymers. The speed of the front is studied and the well-posedeness of the general model is analyzed. Numerical simulations illustrating the dynamics of the desorption process described by the proposed model are included.

The pulp and paper industry plays an important role in European economies. The chemical reactions that transform wood chips in pulp occur mainly in a complex moving bed reactor, the digester. Nowadays the use of mathematical models to simulate the transient behaviour of the digester in terms of temperature and compound concentrations represents a r...

This article focusses on the mathematical analysis of a delayed integro-differential model in which flux does not obey the classical Fick's law. The well-posedness of the integro-differential model in the Hadamard's sense is established. The dependence on the delay parameter of the total amount of desorpted/sorpted mass is studied. Numerical result...

A class of conservative numerical methods for solving hyperbolic nonhomogeneous scalar conservation laws is presented. Convergence and stability properties are studied. Particular attention is devoted to time depending point sources. Several numerical examples are presented.

In this paper we study numerical methods for solving integro-differential equations which generalize the well-known Fisher equation. The numerical methods are obtained considering the MOL (Method of Lines) approach. The stability and convergence of the methods are studied. Numerical results illustrating the theoretical results proved are also inclu...

In this article, we study continuous and discrete models to describe reaction transport systems with memory and long range interaction. In these models the transport process is described by a non-Brownian random walk model and the memory is induced by a waiting time distribution of the gamma type. Numerical results illustrating the behavior of the...

The effect of integral memory terms in the behavior of diffusion phenomena is studied. The energy functional associated with different models is analyzed and stability inequalities are established. Approximation methods for the computation of the solution of the integro-differential equations are constructed. Numerical results are included.

In this article the qualitative properties of numerical traveling wave solutions for integro- differential equations, which generalize the well known Fisher equation are studied. The integro-differential equation is replaced by an equivalent hyperbolic equation which allows us to characterize the numerical velocity of traveling wave solutions. Nume...

The aim of this paper is to study the role of explicitness, implicitness and order in the stability and qualitative properties of splitting methods for solving advection-reaction equations. Numerical pathologies produced by simulations are identified which allow the correction of wrong numerical reactive flows. Several numerical examples which show...

The development of mathematical models that describe industrial processes is playing an increasing role in industrial context, because such models can replace, in some cases, experimental simulation, in a more inexpensive and flexible way. In this paper is presented a preliminary study of a transient model of a continuous moving bed reactor - the d...

The aim of this paper is to study discretizations of convection-diffusion-reaction equations using splitting methods. Estimates for the physical splitting errors and the numerical splitting errors are established. These estimates lead to the selection of optimal sequences and coupling of physical phenomena and adequate use of implicitness and expli...

Notes for minicourse at the Workshop on Modeling and Simulation in Chemical Engineering -- Coimbra, 2003. In these notes some classical and modern splitting techniques are reviewed for transport-chemistry problems, modeled as time-dependent advection-di#usion-reaction equations. The material is largely based on the forthcoming book [6], where a mor...

In this paper we study convergence of numerical discretizations of hyperbolic nonhomogeneous scalar conservation laws. Particular attention is devoted to point source problems. Standard numerical methods, obtained by a direct discretization of the differential form, fail to converge, even in the linear case. We consider the equation in integral for...

In this paper we study a class of numerical methods used to solve two-point boundary value problems on nonuniform grids. Particular attention is devoted to positive solutions, i.e. conditions under which the solutions of the problem are positive. Applications to steady states of air pollution problems are also referred to.

In this paper we study a class of numerical methods used to solve two-point boundary-value problems on nonuniform grids. Particular attention is devoted to numerical oscillations which are quantified for different methods. Numerical experiments are also included.

In this paper, two adaptive gridding algorithms, based on the Method of Lines, are proposed for the numerical solution of the PDE which describes the energy balance of a fixed bed system. A standard time-step control of the stiff solver—based on Richardson extrapolation—is coupled, in both algorithms, with a spatial adapting strategy. In one of the...

In this paper attention is focused on convergence properties on nonuniform grids of numerical discretizations of first- and second-order spatial derivatives which occur, for example, in the transport and heat equations. The direct study of the discretization error equations lead us to the establishment of expressions for the global discretization e...

This paper completes a series of three papers concerned with automatic selection of multistep methods with stability regions fitted with the eigenvalues of a Jacobian matrix of an ordinary differential system. Here the treatment is extended to cover the case of purely imaginary eigenvalues. The class of multistep methods has k steps, order k+l and...

A class of second-derivative two-step methods, S(2), with order 5 and depending on two parameters – which are used to adapt the stability region while minimizing the truncation error coefficient – is constructed. An analysis of two subclasses of S(2)–A(0) stable methods and methods with finite stability interval - with their stability and accuracy...

Regridding methods has become an important tool in the integration of PDE systems whose solutions exhibit sharp transitions in spatial derivatives. This paper improves the results presented in an earlier contribution of the author and F. Oliveira (1988). Theoretical justifications of finite differences regridding criteria for the transport and heat...

In this paper we study some contractivity properties of second-derivative linear multistep methods—(SD)LMM—when a test equation of type y′ = λ(t)y is used. Necessary and sufficient conditions for a (SD)LMM to be contractive in some interval of the negative real axis are constructed. A class of A0-contractive (SD)LMMs with any number of k steps, ord...

In this paper we study some properties of contractivity regions of second derivative multistep methods. These properties lead to the design of classes of Ao and A contractive methods.

We present necessary and sufficient conditions for a multiderivative multistep method to be contractive in a sector, and in disk of the complex plane.

Classes of multistep methods with k steps, order k+1 and depending on a certain number of free parameters, one of them representing the size of the real interval of stability are constructed. A criterion to select automatically multistep methods of such classes, which are fitted with the eigenvalues of the jacobian matrix of a differential system i...

In the context of finite difference approximations and semi-discretization methods, empirical criteria of adaptive gridding, based on the concentration of the nodes in the regions of high spatial derivatives and motion of them at a prescribed velocity, are generally used. The purpose of the present paper is to give a contribution to overcome the ga...

In this paper we present a study of consistency, stability and convergence properties of linear multiderivative multistep variable stepsize variable formula methods.

In Structural Optimization one often finds the problem of minimizing the weight of a plate under some geometric constraints and considering that it must not rupt. We study in this paper the dependence on the thickness of a solution h∗ of the three-dimensional optimization problem, namely the convergence of h∗ to a solution of the two-dimensional op...

The paper studies the analytical and numerical behaviours of some non Brownian models for diffusion phenomena. These models have been introduced in the literature to overcome the gap between experimental data and numerical simulations. From analytical point of view stability results leading to the well-posedness in the Hadamard sense of the initial...

The evolution in time of European options is usually studied using the Black-Scholes formula. This formula is obtained from the equivalence between the Black-Scholes equation and a heat equation. The solution of the last equation presents infinite speed of propagation which induces the same property for European options. In this paper we study inte...

We use the concept of monotone matrix to construct positive numerical solutions of convection-diffusion-reaction equations. Steady and evolution problems are studied. Numerical examples are also included.

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