Simulation of the USP drug delivery problem using CFD: experimental, numerical and mathematical aspects

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Abstract

The numerical simulation of the dissolution of drug-containing compacts in a stirred reactive medium is presented. This is of interest to the design of drug delivery systems in which the goal is to design compacts which release the drug according to certain desired release profiles. A moving boundary finite element approach is adopted to simulate dissolution of layered compacts made up of a number of layers of different acids which dissolve at different rates. The simulation results are compared to experimental measurements. Although a number of idealisations have been adopted in the numerical model, good agreement with experiment is achieved. A semi-analytic solution is also developed which leads to an expression for the mass flux from a dissolving cylinder. Results for this model are compared with the numerical and experimental data.

Introduction

The development of systems for the controlled delivery of drugs to the body is an area of some interest to researchers in pharmaceutical technology. The aim is to develop drug-containing compacts, which as they dissolve in the body, release the drug according to some desired drug release profile. Different therapies may require different drug release profiles. For example, the goal may be to design compacts that release drugs at a constant release rate or compacts for which the release rate increases over time. Designing drug compacts that can deliver such profiles is a challenging area in pharmaceutics and requires many hours of experimental research. The use of numerical simulation to assist in the design of drug delivery systems (DDSs) is still in its infancy, though in principle a software tool for simulating the drug release profiles of new designs could be very useful in reducing the amount of in vitro testing required. This paper describes a project1 whose aim has been to demonstrate the usefulness of numerical simulation in this scientific domain. We describe a simulation code which models the dissolution of a cylindrical compact in a US Pharmacopeia (USP) drug dissolution apparatus, in which the drug dissolves in a fluid stirred at a constant rotation rate. The simulation calculates the mass lost per unit time from the compact and the numerical results are compared against experimental results. The equation modelling the dissolution process is the unsteady diffusion–advection equation and the simulation domain focuses on the boundary layer close to the solid–liquid interface. As time advances and the drug dissolves, the simulation domain changes shape. The problem is solved using a moving-boundary finite element formulation. The model is tested against experimental results obtained for a single component compact of benzoic acid, and against a number of multi-layered compacts, made up of benzoic acid, salicylic acid and adipic acid layers.

Two distinct processes can be identified in this set-up. Firstly, the dissolution process at the solid–liquid interface and secondly the mixing of the drug concentration in the bulk solution. While our model captures the first process well, the mixing process is only crudely approximated. Nevertheless, with this approximation reasonable agreement with experimental results is obtained.

In this paper, the mathematics behind the numerical methods used are fully described, and comparison with experimental results is shown. Limitations of the model for dealing with more complex compacts are illustrated and a way forward for developing the simulation is discussed, so that more realistic drug compacts can be simulated in the future. The software is a finite element code, exploiting a pre-conditioned conjugate gradient solver and including modules for mesh moving and mesh adaptation around the moving boundary.

In addition, a semi-analytic model is developed, based on an analogy of dissolution of a cylinder with heat loss from an infinite plate. The model is developed to correct for curvature, an expression for the mass flux from such a cylinder is calculated for the USP apparatus and the results are compared with experimental data.

Section snippets

Background

In recent years, much research has concentrated on modelling drug release from a variety of DDSs, including simple compressed tablets, matrix tablets and more complex systems containing swellable/erodible polymers. Research has focused not only on modelling release from these formulations using established mathematical models but also on predicting drug release rates and drug dissolution profiles by developing models based on the physicochemical properties of the drug and excipients used and

Experimental apparatus

The experimental apparatus is shown in Fig. 1. The compact is fixed to a wax mould at a height of 3 mm from the base of the vessel.2 The compact is allowed to dissolve in a solution of 0.1 N HCl which is stirred at 50 rpm. At various points in time, a sample is taken from the bulk fluid, the mass that has dissolved into the bulk is measured and a profile of mass dissolved against time is calculated. Experiments

Numerical model

The model presented in Eq. (1) is simulated using a standard Galerkin finite element method, with linear triangular shape functions. A moving grid method similar to that described in [10] is employed. The mesh co-ordinates are functions of time, x(t), so that the concentration is discretised asc(x,t)=∑j=1Ncj(t)φj(x,t)where φj are the shape functions. The dependence of the co-ordinates on time means that a time correction term needs to be included in the time discretisationct=∑j=1Ndcj(t)dtφj

Results

The simulation focused on the diffusion boundary layer close to the solid–liquid interface. A simulation domain of length 0.2 mm in the r-direction and 11.6 mm in the z-direction was chosen and meshed with 1805 elements. Implicit time stepping was used with a time-step of 1 s. It was assumed that the concentration becomes fully mixed after 5 s. This is a reasonable estimate, given the dimensions of the vessel and the stirring rotation speed.

Fig. 3 shows the simulated mass of BA and SA released

Conclusions

The paper has investigated the extent to which numerical simulation and mathematical/semi-analytic models can be used to predict drug release profiles. It has demonstrated that good agreement with experimental data can be achieved on simple layered drug compacts, by simulating a simplified model of the dissolution process, with a crude approximation of the mixing process. Results from the semi-analytic model underestimated the mass flux from the pure Benzoic Acid compact by about 17%,

Acknowledgements

The authors would like to acknowledge the support of the European Union Fourth Framework Programme for Research (FP4), Elan Pharmaceutical Technologies and Hitachi Dublin Laboratory for this project. Most of the work described in this paper was carried out while Dr. Hurley and Dr. M. Crane were employees of Hitachi Dublin Laboratory.

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