Martin Meere

Some design considerations for polymer-free drug-eluting stents: a mathematical approach

In this paper we provide the first model of drug elution from polymer-free arterial drug-eluting stents. The generalised model is capable of predicting drug release from a number of polymer-free systems including those that exhibit nanoporous, nanotubular and smooth surfaces. We derive analytical solutions which allow us to easily determine the important parameters that control drug release. Drug release profiles are provided, and we offer design recommendations so that the release profile may be tailored to achieve the desired outcome. The models presented here are not specific to drug-eluting stents and may also be applied to other biomedical implants that use nanoporous surfaces to release a drug.

Asymptotic analysis of a non-linear model for substitutional diffusion semiconductors

The simple non-Fickian model for the substitutional diffusion of an impurity in a III-V semiconductor proposed by Zahari and Tuck assumes that both the impurity and host atoms diffuse by a vacancy mechanism. Here we give an asymptotic analysis of the governing pair of coupled partial differential equations, obtaining analytical solutions for the impurity and vacancy distributions in systems for which the impurity diffusivity is much greater than that of the vacancies. Two problems are considered. The first is that in which the impurity concentration is initially zero and a prescribed concentration is given at the surface. The second problem models the diffusion of ion implanted impurity, for which initial impurity and vacancy distributions are specified and a condition of zero impurity flux is assumed at the surface. Both leading order and correction terms are obtained and in each case the solution predicts the anomalous ‘double profiles’ often observed in III-V systems. The ‘ion-implantation analysis’ also displays the observed phenomenon of ‘uphill’ diffusion against the impurity gradient close to the surface.

Minimizing the passive release of heparin-binding growth factors from an affinity-based delivery system.

We consider a mathematical model that describes the leakage of heparin-binding growth factors from an affinity-based delivery system. In the delivery system, heparin binds to a peptide which has been covalently cross-linked to a fibrin matrix. Growth factor in turn binds to the heparin, and growth factor release is governed by both binding and diffusion mechanisms, the purpose of the binding being to slow growth factor release. The governing mathematical model, which in its original formulation consists of six partial differential equations, is reduced to a system of just two equations. It is usually desirable that there be no passive release of growth factor from a device, with all of the growth factor being held in place via binding until such time as it is actively released by invading cells. However, there will inevitably be some passive release, and so it is of interest to identify conditions that will make this release as slow as possible. In this paper, we identify a parameter regime that ensures that at least a fraction of the growth factor will release slowly. It is found that slow release is assured if the matrix is prepared with the concentration of cross-linked peptide greatly exceeding the dissociation constant of heparin from the peptide, and with the concentration of heparin greatly exceeding the dissociation constant of the growth factor from heparin. Also, for the first time, in vitro experimental release data are directly compared with the theoretical release profiles generated by the model. We propose that the two stage release behaviour frequently seen in experiments is due to an initial rapid out-diffusion of free growth factor over a diffusion time scale (typically days), followed by a much slower release of the bound fraction over a time scale depending on both diffusion and binding parameters (frequently months).

A mathematical model for pulsatile release: controlled release of rhodamine B from UV-crosslinked thermoresponsive thin films.

A controlled drug delivery system fabricated from a thermoresponsive polymer was designed to obtain a pulsatile release profile which was triggered by altering the temperature of the dissolution medium. Two stages of release behaviour were found: fast release for a swollen state and slow (yet significant and non-negligible) release for a collapsed state. Six cycles of pulsatile release between 4 °C and 40 °C were obtained. The dosage of drug (rhodamine B) released in these cycles could be controlled to deliver approximately equal doses by altering the release time in the swollen state. However, for the first cycle, the swollen release rate was found to be large, and the release time could not be made short enough to prevent a larger dose than desired being delivered. A model was developed based on Ficks law which describes pulsatile release mathematically for the first time, and diffusion coefficients at different temperatures (including temperatures corresponding to both the fully swollen and collapsed states) were estimated by fitting the experimental data with the theoretical release profile given by this model. The effect of temperature on the diffusion coefficient was studied and it was found that in the range of the lower critical solution temperature (LCST), the diffusion coefficient increased with decreasing temperature. The model predicts that the effective lifetime of the system lies in the approximate range of 1-42 h (95% of drug released), depending on how long the system was kept at low temperature (below the LCST). Therefore this system can be used to obtain a controllable pulsatile release profile for small molecule drugs thereby enabling optimum therapeutic effects.

Asymptotic analysis of a model for substitutional-interstitial diffusion

A substitutional-interstitial model for impurity diffusion in semi-conductors is discussed. In particular we consider a surface-source problem and obtain asymptotic solutions in the limit of the surface concentration of impurity being much greater than the equilibrium vacancy concentration. In the absence of vacancy generation, a double error function impurity curve is obtained. These double profiles reproduce some of the qualitative features of diffusion in many III–V semiconductor systems. We also discuss how vacancy generation modifies the analysis and show that in the limit of high vacancy generation, the problem becomes one of linear diffusion with the diffusion curves then being single error function complements.

An experimental and theoretical assessment of quantum dot cytotoxicity

Quantum dots (QDs) are a class of semiconductor nanoparticles that possess a unique set of size-tunable optical properties. The potential applications of QDs in biological and medical applications are enormous – some notable examples being in high-resolution cellular imaging, cancer tumour targeting and drug delivery. However, the mechanisms for QD-cell interactions are at best partially understood, and QD cytotoxicity is an ongoing concern. In particular, it remains unclear how QD uptake by cells and subsequent cell fate are influenced by QD parameters such as size, composition, concentration, and exposure time. To help resolve this complex issue in a systematic manner, we have developed here one of the first mathematical models that describes the toxic effects of QDs on cells. The model consists of a system of ordinary differential equations describing (among other things) the transition of healthy cells to an apoptotic or necrotic state induced by QD toxicity. We also experimentally investigated the behaviour of a cell population subsequent to exposure to various types of CdTe QDs. In a population of identical cells exposed to QDs of similar size (2–5 nm), it was found that some of the cells entered apoptosis, others entered necrosis, and others demonstrated no response at all. The toxicity of the various QDs was conveniently quantitatively assessed using the parameters appearing in the mathematical model, and satisfactory agreement between theory and experiment was found.

Modelling gas motion in a rapid-compression machine

In this paper, a model which describes the behaviour of the pressure, density and temperature of a gas mixture in a rapid compression machine is developed and analyzed. The model consists of a coupled system of nonlinear partial differential equations, and both formal asymptotic and numerical solutions are presented. Using asymptotic techniques, a simple discrete algorithm which tracks the time evolution of the pressure, temperature and density of the gas in the chamber core is derived. The results which this algorithm predict are in good agreement with experimental data.

Asymptotic Analysis of Some Two-Dimensional Diffusion Problems

In this paper we discuss two-dimensional surface source and implant problems for a substitutional-interstitial diffusion model. We present asymptotic solutions in the limit of the surface concentration of impurity (or peak concentration of the implant) being far greater than the equilibrium vacancy concentration. Using leading order composite solutions we plot contours of constant impurity concentration. Some of these contours differ markedly from those of the corresponding linear problem, having the ‘bird’s beak’ shape which is frequently observed in experiments. We also discuss a two-dimensional surface source problem for a vacancy model.

Some Mathematical Models of Intermolecular Autophosphorylation

Intermolecular autophosphorylation refers to the process whereby a molecule of an enzyme phosphorylates another molecule of the same enzyme. The enzyme thereby catalyses its own phosphorylation. In the present paper, we develop two generic models of intermolecular autophosphorylation that also include dephosphorylation by a phosphatase of constant concentration. The first of these, a solely time-dependent model, is written as one ordinary differential equation that relies upon mass-action and Michaelis-Menten kinetics. Beginning with the enzyme in its dephosphorylated state, it predicts a lag before the enzyme becomes significantly phosphorylated, for suitable parameter values. It also predicts that there exists a threshold concentration for the phosphorylation of enzyme and that for suitable parameter values, a continuous or discontinuous switch in the phosphorylation of enzyme are possible. The model developed here has the advantage that it is relatively easy to analyse compared with most existing models for autophosphorylation and can qualitatively describe many different systems. We also extend our time-dependent model of autophosphorylation to include a spatial dependence, as well as localised binding reactions. This spatio-temporal model consists of a system of partial differential equations that describe a soluble autophosphorylating enzyme in a spherical geometry. We use the spatio-temporal model to describe the phosphorylation of an enzyme throughout the cell due to an increase in local concentration by binding. Using physically realistic values for model parameters, our results provide a proof-of-concept of the process of activation by local concentration and suggest that, in the presence of a phosphatase, this activation can be irreversible.

A Mathematical Model for Drug Delivery

We consider a model for local drug re-distribution in a tissue that incorporates the effects of diffusion and reversible binding with immobile sites within the tissue. The model tracks the evolution of the concentration in the tissue of free drug, specifically and non-specifically bound drug, and specific binding sites. We reduce the model to a scalar nonlinear diffusion equation for the total drug. The model is used to investigate tissue residence time for strongly bound drugs by considering a problem with uniform initial drug concentration and perfect sink boundary conditions. The behaviour predicted by the model has potential implications for the design of local drug delivery systems if the drug is strongly bound.