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Non Fickian diffusion in polymers and medical applications

Problem Description

The study of polymers is an emerging area in the interdisciplinary dialogue between mathematicians and material science. Polymers are long-chain molecules that consist of a number of small repeating units. They can be natural materials, as for example cellulose, natural rubber, DNA, or synthetic such as polystyrene and teflon. Polymers have a wide range of applications due to their durability, processability, transparency, electrical and thermal resistance. For this reason they are replacing traditional materials in a large variety of engineering applications. Also new applications in the field of biomaterials have grown enormously these last years. Among them we mention, for example, absorption and controlled release technology, where new synthetic polymers are used as separating membranes, encapsulating devices or impregnated matrices.

One unusual feature of release systems is the change of state in the polymer from a rubbery state, when the polymer is nearly saturated, to a glassy state, when the polymer is nearly dry. Polymer release systems involve an anomalous diffusive desorption which cannot be completely explained by Fickian models, as shown by experiments. There is no unifying explanation but most scientists agree that the change in diffusion coefficients is not enough to explain such anomalous phenomena. In recent years some attempts have been made to model mathematically such behaviors. The non-Fickian character of the diffusion is simulated by considering that there is a non local dependence of the dynamics upon the configuration. This has been taken into account by considering two systems of two nonlinear PDEs where the variables are the drug concentration and the stress. These systems represent the behavior of the concentration in the glassy and rubbery regions of the polymer which are defined by an evolving internal boundary. Interesting analytical results are obtained via an asymptotic analysis. However the simplifications imposed by this asymptotic analysis reduce the applicability of the model . Also the numerical simulations obtained from these models do not agree with experimental results because the profiles of mass uptake present very large gradients.

Research at LCM

In recent years several attempts have been made to model mathematically non-Fickian diffusion in polymers. For sorption phenomena we find two main approaches. In the first approach the diffusion in the glassy and rubbery regions is based on Fick´s law and the kinetics of the glass-rubber transition is taken into account; in the second approach the non local dependence of the dynamics upon the configuration of the polymer is considered by introducing a viscoelastic stress and a relaxation time which measures the time it takes one portion of the polymer to react to another portion. Much less attention has been placed on the mathematical modelling of the polymer desorption process involved in drug release. In the literature we find mainly mathematical models where a viscoelastic stress has been introduced in the flux. Implicit in these models is the idea that there is a global violation of Fick’s law in all the polymeric domain: the rubbery region and the glassy region. However the numerical simulations of mass uptake exhibit very sharp fronts which do not agree with experimental data. Experimentalists consider that in desorption phenomena there is not a global violation of Fick’s law and that the only obvious violation takes place at the moving boundary separating the two states.When medical applications are concerned the important measurable quantities are the flux through the exposed boundary and the total time of release. Accurate models are then crucial to provide information to design polymeric matrices with properties that closely match those required for a particular application.

In this project our aim is twofold:        

  1. To study from an analytical and numerical viewpoint a new mathematical  model for desorption based on the observation, by experimentalists, that the only obvious violation of Fick’s law takes place at the moving front separating the two states.
  2. To extend our model to design polymer lens (tridimensional  shells) used in drug delivery in the treatment of ophtalmic pathologies as for example glaucoma. 

Interdisciplinary dialogue will take place between the elements of the team: Chemical engineers which build polymers in laboratory, Ophthalmologists which specify the characteristics of treatment that is final time of drug release and total drug uptake.

Papers & Reports

  • [1] J.A. Ferreira, P. de Oliveira, D. Comissiong, A phenomenological model for desorption in polymers,  Preprint 06-35 of the Department of Mathematics, University of Coimbra

Project Team

  • João Ricardo Branco, LCM/CMUC and ISEC 
  • José Augusto Ferreira, LCM/CMUC
  • Paula de Oliveira, LCM/CMUC
  • Pascoal Martins da Silva, ISEC 
  • Helena Gil, Departament of Chemistry, FCTUC 
  • Joaquim Carlos Neto Murta, FMUC

Project Reference

FCT Research Project - PTDC/MAT/74548/2007