3 research outputs found
Recent developments in non-Fickian diffusion : a new look at viscoelastic materials
Tese do Programa Inter-Universitário de Doutoramento em Matemática, apresentada à Faculdade de Ciências e Tecnologia da Universidade de CoimbraThe aim of this dissertation is to fully understand from a mathematical point of view the two coupled
processes of sorption of a fluid by a viscoelastic material and the successive or simultaneous
desorption of the fluid with solved molecules of a chemical compound which is dispersed in the
material. These two coupled processes have a central role in several areas of Life Sciences and
Material Sciences namely in Controlled Drug Delivery.
When a penetrant fluid diffuses into a viscoelastic material, such as a polymer, it is well known
that the process cannot be completely described by Fick’s classical law of diffusion. The reason
lies in the fact that as the fluid diffuses into the material, it causes a deformation which induces a
stress driven diffusion that act as a barrier to the fluid penetration. Thus a modified flux must be
considered, resulting from the sum of the Fickian flux and a non-Fickian flux. We propose a new
interpretation of this non-Fickian mass flux as being related to a convective field which represents
an opposition of the polymer to the incoming penetrant fluid.
To study the complete problem of sorption coupled with desorption, we progressively address
more complex models. We begin by studying the process of sorption in Chapter 1, we generalize
the model to a more abstract formulation in Chapter 2, and we study a numerical method for the
abstract formulation in Chapter 3. The complete problem of sorption coupled with desorption is
addressed in Chapter 4.
The first sorption model studied is based on an integro-differential equation, coupled with
initial and boundary conditions. The non-linear dependence between strain and the incoming fluid
concentration is considered and introduced in a Boltzmann integral with a kernel computed from
a Maxwell-Wiechert model. To illustrate the behavior of the model we solve it numerically on a
general nonuniform grid in space and a uniform grid in time. We exhibit numerical simulations
that give some insight of the dependence of the solution on the different parameters that describe
the viscoelastic properties of the polymer.
This lead us to a generalization of this model by considering a class of integro-differential
equations of Volterra type. We establish the well posedness, in the Hadamard sense, of the initial
boundary value problem. The stability analysis is separated in two cases, non-singular kernels and
weakly singular kernels.
An implicit explicit difference scheme, which can be seen as a fully discrete piecewise linear
finite element method, is proposed to discretize the general model. Stability and convergence
results for the method are established showing that it is second order convergent in space and first
order convergent in time. The numerical analysis of the method does not follow the usual splitting
of the global error using the solution of an elliptic equation induced by the integro-differential
equation. A new approach, that enable us to reduce the smoothness required for the theoretical
solution, is used. The results are established for both non-singular and weakly singular kernels.
A tridimensional model of the whole process of sorption and desorption is presented in Chapter
4. A viscoelastic matrix with a dispersed drug, or a chemical compound, is considered. The model
is based on a system of partial differential equations coupled with boundary conditions over a
moving boundary. We combine non-Fickian sorption of a penetrant fluid , non-Fickian desorption
of the fluid with dispersed drug, with non-linear dissolution of a drug agent and polymer swelling.
An Implicit-Explicit numerical scheme is used to numerically solve the model and some plots are
presented to illustrate the behavior of the approximations.
Experimental rheological information of the polymer-solvent matrix system can be easily introduced
in the models studied in this dissertation because all the parameters can be measured or
estimated according to well-known theories of viscoelastic materials. This makes the models suitable
for both data fitting and quantitative prediction of drug release kinetics, opening new routes
of research in Material Science
A 3D Model for Mechanistic Control of Drug Release
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