Drug diffusion from polymeric delivery devices : a problem with two moving boundaries

An existing model for solvent penetration and drug release from a spherically-shaped polymeric drug delivery device is revisited. The model has two moving boundaries, one that describes the interface between the glassy and rubbery states of polymer, and another that defines the interface between the polymer ball and the pool of solvent. The model is extended so that the nonlinear diffusion coefficient of drug explicitly depends on the concentration of solvent, and the resulting equations are solved numerically using a front-fixing transformation together with a finite difference spatial discretisation and the method of lines. We present evidence that our scheme is much more accurate than a previous scheme. Asymptotic results in the small-time limit are presented, which show how the use of a kinetic law as a boundary condition on the innermost moving boundary dictates qualitative behaviour, the scalings being very different to the similar moving boundary problem that arises from modelling the melting of an ice ball. The implication is that the model considered here exhibits what is referred to as ``non-Fickian'' or Case II diffusion which, together with the initially constant rate of drug release, has certain appeal from a pharmaceutical perspective.

Asymptotic and numerical results for a model of solvent-dependent drug diffusion through polymeric spheres

A model for drug diffusion from a spherical polymeric drug delivery device is considered. The model contains two key features. The first is that solvent diffuses into the polymer, which then transitions from a glassy to a rubbery state. The interface between the two states of polymer is modelled as a moving boundary, whose speed is governed by a kinetic law; the same moving boundary problem arises in the one-phase limit of a Stefan problem with kinetic undercooling. The second feature is that drug diffuses only through the rubbery region, with a nonlinear diffusion coefficient that depends on the concentration of solvent. We analyse the model using both formal asymptotics and numerical computation, the latter by applying a front-fixing scheme with a finite volume method. Previous results are extended and comparisons are made with linear models that work well under certain parameter regimes. Finally, a model for a multi-layered drug delivery device is suggested, which allows for more flexible control of drug release.

Delivering crushed tablets using thickened fluids : is drug diffusion into gastric fluids restricted?

Solid medications are often crushed and mixed with food or thickened water to aid drug delivery for those who cannot or prefer not to swallow whole tablets or capsules. Dysphagic patients have the added problem of being unable to safely swallow thin fluids so water thickened with polysaccharides is used to deliver crushed medications and ensure safe swallowing. It is postulated that these polysaccharide systems may restrict drug release by reducing the diffusion of the drug into gastric fluids. METHODS By using a vertical diffusion cell separated with a synthetic membrane, the diffusion of a model drug (atenolol) was studied from a donor system containing the drug dispersed into thickened water with xanthan gum (concentration range from 0.005%-2.2%) into a receptor system containing simulated gastric fluid (SGF) at 37°C. The amount of drug transferred was measured over 8 hours and diffusion coefficients estimated using the Higuchi model approach. RESULTS Atenolol diffusion decreased with increasing xanthan gum concentration up to 1.0%, above which diffusion remained around 300 μ2s-1. The rheological measurements captured the influence of the structure and conformation of the polysaccharide in water on the movement and availability of the drug in SGF. DISCUSSION Dose form administration for dysphagic patients’ needs special attention from general practitioners, pharmacist and patients. Improving drug release of crushed tablets from thickening agents requires a reduction in the diffusion pathway (e.g. by decreasing drop size radius). This approach could make the drug available in SGF in a short time without compromising the mechanical aspects of thickening agents that guarantee safe swallowing.

Mathematical modelling of controlled drug release from polymer micro-spheres : incorporating the effects of swelling, diffusion and dissolution via moving boundary problems

Controlled drug delivery is a key topic in modern pharmacotherapy, where controlled drug delivery devices are required to prolong the period of release, maintain a constant release rate, or release the drug with a predetermined release profile. In the pharmaceutical industry, the development process of a controlled drug delivery device may be facilitated enormously by the mathematical modelling of drug release mechanisms, directly decreasing the number of necessary experiments. Such mathematical modelling is difficult because several mechanisms are involved during the drug release process. The main drug release mechanisms of a controlled release device are based on the device’s physiochemical properties, and include diffusion, swelling and erosion. In this thesis, four controlled drug delivery models are investigated. These four models selectively involve the solvent penetration into the polymeric device, the swelling of the polymer, the polymer erosion and the drug diffusion out of the device but all share two common key features. The first is that the solvent penetration into the polymer causes the transition of the polymer from a glassy state into a rubbery state. The interface between the two states of the polymer is modelled as a moving boundary and the speed of this interface is governed by a kinetic law. The second feature is that drug diffusion only happens in the rubbery region of the polymer, with a nonlinear diffusion coefficient which is dependent on the concentration of solvent. These models are analysed by using both formal asymptotics and numerical computation, where front-fixing methods and the method of lines with finite difference approximations are used to solve these models numerically. This numerical scheme is conservative, accurate and easily implemented to the moving boundary problems and is thoroughly explained in Section 3.2. From the small time asymptotic analysis in Sections 5.3.1, 6.3.1 and 7.2.1, these models exhibit the non-Fickian behaviour referred to as Case II diffusion, and an initial constant rate of drug release which is appealing to the pharmaceutical industry because this indicates zeroorder release. The numerical results of the models qualitatively confirms the experimental behaviour identified in the literature. The knowledge obtained from investigating these models can help to develop more complex multi-layered drug delivery devices in order to achieve sophisticated drug release profiles. A multi-layer matrix tablet, which consists of a number of polymer layers designed to provide sustainable and constant drug release or bimodal drug release, is also discussed in this research. The moving boundary problem describing the solvent penetration into the polymer also arises in melting and freezing problems which have been modelled as the classical onephase Stefan problem. The classical one-phase Stefan problem has unrealistic singularities existed in the problem at the complete melting time. Hence we investigate the effect of including the kinetic undercooling to the melting problem and this problem is called the one-phase Stefan problem with kinetic undercooling. Interestingly we discover the unrealistic singularities existed in the classical one-phase Stefan problem at the complete melting time are regularised and also find out the small time behaviour of the one-phase Stefan problem with kinetic undercooling is different to the classical one-phase Stefan problem from the small time asymptotic analysis in Section 3.3. In the case of melting very small particles, it is known that surface tension effects are important. The effect of including the surface tension to the melting problem for nanoparticles (no kinetic undercooling) has been investigated in the past, however the one-phase Stefan problem with surface tension exhibits finite-time blow-up. Therefore we investigate the effect of including both the surface tension and kinetic undercooling to the melting problem for nanoparticles and find out the the solution continues to exist until complete melting. The investigation of including kinetic undercooling and surface tension to the melting problems reveals more insight into the regularisations of unphysical singularities in the classical one-phase Stefan problem. This investigation gives a better understanding of melting a particle, and contributes to the current body of knowledge related to melting and freezing due to heat conduction.

Diffusion-sensitive magnetic resonance spectroscopy and imaging in biomedical sciences

We present a mini-review of the development and contemporary applications of diffusion-sensitive nuclear magnetic resonance (NMR) techniques in biomedical sciences. Molecular diffusion is a fundamental physical phenomenon present in all biological systems. Due to the connection between experimentally measured diffusion metrics and the microscopic environment sensed by the diffusing molecules, diffusion measurements can be used for characterisation of molecular size, molecular binding and association, and the morphology of biological tissues. The emergence of magnetic resonance was instrumental to the development of biomedical applications of diffusion. We discuss the fundamental physical principles of diffusion NMR spectroscopy and diffusion MR imaging. The emphasis is placed on conceptual understanding, historical evolution and practical applications rather than complex technical details. Mathematical description of diffusion is presented to the extent that it is required for the basic understanding of the concepts. We present a wide range of spectroscopic and imaging applications of diffusion magnetic resonance, including colloidal drug delivery vehicles; protein association; characterisation of cell morphology; neural fibre tractography; cardiac imaging; and the imaging of load-bearing connective tissues. This paper is intended as an accessible introduction into the exciting and growing field of diffusion magnetic resonance.

Detailed analysis of a conservative difference approximation for the time fractional diffusion equation

Diffusion equations that use time fractional derivatives are attractive because they describe a wealth of problems involving non-Markovian Random walks. The time fractional diffusion equation (TFDE) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α ∈ (0, 1). Developing numerical methods for solving fractional partial differential equations is a new research field and the theoretical analysis of the numerical methods associated with them is not fully developed. In this paper an explicit conservative difference approximation (ECDA) for TFDE is proposed. We give a detailed analysis for this ECDA and generate discrete models of random walk suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation. The stability and convergence of the ECDA for TFDE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.

A second order control-volume finite-element least-squares strategy for simulating diffusion in strongly anisotropic media

An unstructured mesh �nite volume discretisation method for simulating di�usion in anisotropic media in two-dimensional space is discussed. This technique is considered as an extension of the fully implicit hybrid control-volume �nite-element method and it retains the local continuity of the ux at the control volume faces. A least squares function recon- struction technique together with a new ux decomposition strategy is used to obtain an accurate ux approximation at the control volume face, ensuring that the overall accuracy of the spatial discretisation maintains second order. This paper highlights that the new technique coincides with the traditional shape function technique when the correction term is neglected and that it signi�cantly increases the accuracy of the previous linear scheme on coarse meshes when applied to media that exhibit very strong to extreme anisotropy ratios. It is concluded that the method can be used on both regular and irregular meshes, and appears independent of the mesh quality.