Mathematical modeling and simulation of drug release from microspheres: Implications to drug delivery systems

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Abstract

This article aims to provide a comprehensive review of existing mathematical models and simulations of drug release from polymeric microspheres and of drug transport in adjacent tissues. In drug delivery systems, mathematical modeling plays an important role in elucidating the important drug release mechanisms, thus facilitating the development of new pharmaceutical products by a systematic, rather than trial-and-error, approach. The mathematical models correspond to the known release mechanisms, which are classified as diffusion-, swelling-, and erosion-controlled systems. Various practical applications of these models which explain experimental data are illustrated. The effect of γ-irradiation sterilization on drug release mechanism from erosion-controlled systems will be discussed. The application of existing models to nanoscale drug delivery systems specifically for hydrophobic and hydrophilic molecules is evaluated. The current development of drug transport modeling in tissues utilizing computational fluid dynamics (CFD) will also be described.

Introduction

For decades, polymeric systems have been used for pharmaceutical applications, especially to provide controlled release of drugs. Drug–polymer systems may also be useful in protecting the drug from biological degradation prior to its release. The development of these device starts with the use of non-biodegradable polymers, which rely on the diffusion process, and subsequently progresses to the use of biodegradable polymers, in which swelling and erosion take place.

Based on the physical or chemical characteristics of polymer, drug release mechanism from a polymer matrix can be categorized in accordance to three main processes (systems) [1], which are:

  • 1.

    Drug diffusion from the non-degraded polymer (diffusion-controlled system).

  • 2.

    Enhanced drug diffusion due to polymer swelling (swelling-controlled system).

  • 3.

    Drug release due to polymer degradation and erosion (erosion-controlled system).

In all three systems, diffusion is always involved. For a non-biodegradable polymer matrix, drug release is due to the concentration gradient by either diffusion or matrix swelling (enhanced diffusion). For biodegradable polymer matrix, release is normally controlled by the hydrolytic cleavage of polymer chains that lead to matrix erosion, even though diffusion may be still dominant when the erosion is slow. This categorization allows mathematical models to be developed in different ways for each type of system.

Mathematical modeling of drug release provides insights concerning mass transport and chemical processes involved in drug delivery system as well as the effect of design parameters, such as the device geometry and drug loading, on drug release mechanism. Thus, the optimized device design for a required drug release profile can be predicted using a systematic approach with a minimum number of experimental studies.

This review presents the concepts contained in important and readily available mathematical models for controlled release primarily from microspheres. Mathematical models for cylindrical geometry, especially for swelling- and erosion-controlled systems, will also be discussed as real systems are readily available in this geometry. It also discusses recent improvements and the major advantages and limitations of each model. The original notations are not retained in this review; instead a common notation is used to facilitate understanding and comparison between the models. The penultimate section of this review discusses the implication of phenomena at the nanoscale and the last section focuses on coupling to transport in tissue.

Section snippets

Mathematical models for diffusion-controlled systems

For diffusion-controlled microspheres, drug release profile is obtained by solving Fick's second law of diffusion subject to appropriate boundary conditions. For one-dimensional drug release from a microsphere, the second Fick's law of diffusion is given by:Ct=1r2r[Dr2Cr]where D and C are the diffusion coefficient and drug concentration in the polymer matrix. The boundary conditions are influenced by the mass transfer process at the surface and the volume of the surrounding system. Based

Mathematical models for swelling-controlled systems

The idea of using a swelling polymer is to provide more control over the release of drug, especially when its diffusivity in polymer is very low. For this purpose, a swellable device is commonly made using a hydrophilic polymer so that water is able to imbibe into the polymer matrix and cause polymer disentanglement. The level of polymer disentanglement as a function of polymer concentration is illustrated in Fig. 4. The imbibing water into the polymer matrix decreases the polymer concentration

Mathematical models for erosion-controlled systems

Bioerodible polymers are versatile materials for a variety of biomedical applications, especially for drug delivery systems, since their chemistry and surfaces can be tailored to stabilize macromolecular agents and enhance the tissue site-targeting. More importantly, the erosion kinetics can be tailored by careful selection of polymer and a variety of techniques of encapsulation to control the drug release profile. In the simplest manner, the erosion kinetics can be altered by modifying

Irradiation effect on drug release profile from polymeric microspheres

Irradiation sterilization, i.e., γ-irradiation, of pharmaceutical products is popular in recent years due to its ease of operation. Conventional sterilization processes, such as heat and chemical sterilizations often lead to degradation of polymer backbones and toxicological problems respectively. Dose of 25 kGy (2.5 Mrad) is essentially accepted to be satisfactory for pharmaceutical products sterilization. However, γ-irradiation sterilization is also known to alter the properties of drug

Going to nanoscale: the implication on drug release mechanism

Biodegradable microparticulate systems have been well-studied for controlled release of various types of drugs, both hydrophobic and hydrophilic. In recent years, there is an obvious trend of increasing interest in development of nanoparticles for drug delivery applications especially in intravenous delivery. Nanoparticles may range from sizes 10 nm to 1000 nm [90] and include liposomes, micelles, polymer–drug conjugates, and polymer particles [91]. Novel fabrication processes have also been

Simulation of drug delivery in tissue: linking drug release profile and tissue elimination kinetics to predict temporal and spatial drug transport

In recent years, in vitro release profile of drug from controlled release devices has been combined with state-of-the-art computational fluid dynamics simulation to predict the spatial and temporal variation of drug transport in the living tissues. Macroscopically, the tissue, which can be either normal or tumor tissue, is ideally assumed as an isotropic porous medium, which is described by Darcy's law [120] for the balance of linear momentum in the tissue interstitium. The full-form of the

Conclusions

The mathematical model and simulation of drug release from polymeric microspheres have developed and evolved to various approaches and concepts. Based on the nature of the polymeric matrices used and the behavior during drug release, these models can be distinctly categorized to diffusion-controlled, swelling-controlled, and erosion-controlled systems. For all types of systems, chemical reactions and mass transfer processes, which are affected by polymer and drug type, device size, shape,

Acknowledgements

This work was supported by Science and Engineering Research Council (SERC), Singapore and National University of Singapore under the grant number R279-000-208-305. The authors are grateful to Professor Kenneth A. Smith (Department of Chemical Engineering, MIT) for helpful discussion on this project. We would also thank J. Xie and L.K. Lim for their technical support in the course of this study.

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    This review is part of the Advanced Drug Delivery Reviews theme issue on “Computational Drug Delivery", Vol. 58/12-13, 2006.

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