Mathematical modeling of bioerodible, polymeric drug delivery systems

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Abstract

The aim of this article is to give an introduction into mathematical modeling approaches of bioerodible controlled drug delivery systems and to present the most important erosion theories reported in the literature. First, important parameters such as degradation and erosion are defined and physicochemical methods for their investigation are briefly presented. Then, phenomenological empirical models as well as models based on diffusion and chemical reaction theory are discussed. Due to the significant chemical and physicochemical differences among individual bioerodible polymers used for controlled drug delivery systems, various mathematical models have been developed to describe the chemical reactions and physical mass transport processes involved in erosion-controlled drug release. Various examples of practical applications of these models to experimental drug release data are given. For those involved in the design and development of biodegradable drug delivery systems this will help to choose the appropriate mathematical model for a specific drug release problem. Important selection criteria such as the desired predictive power and precision, but also the effort required to apply a model to a particular system will be discussed. Furthermore, before models can be used for drug release predictions certain parameters such as drug dissolution or polymer degradation rate constants, have to be known. The number of parameters to be determined significantly differs between the models. The practical benefit of carefully choosing the right model is that effects of composition and device geometry on the drug release kinetics can be predicted which can reduce laborious formulation studies to a minimum.

Introduction

Mathematical drug release modeling of bioerodible delivery systems is not as advanced as the modeling of diffusion or swelling-controlled devices, because it is generally more complex [1], [2], [3], [4]. Besides physical mass transport phenomena, chemical reactions such as polymer chain cleavage have to be considered in bioerodible systems. These reactions change the conditions for mass transfer processes continuously, thus rendering the mathematical treatment of erosion-controlled drug release rather difficult [5], [6], [7], [8]. This review presents the concepts of the most important theories and discusses briefly their major advantages and limitations. Only systems in which the drug is physically immobilized within a water-insoluble polymeric matrix are considered. Devices containing drugs which are covalently attached to the polymer chains are not treated. Both, empirical as well as diffusion and chemical reactions based mathematical models are discussed.

To avoid misunderstandings, first the terms ‘erosion’, ‘bioerosion’, ‘degradation’ and ‘biodegradation’ have to be defined. Unfortunately, many different definitions of these termini are found in the literature [9]. In this review, polymer degradation is the chain scission process by which polymer chains are cleaved into oligomers and monomers. Biodegradable designates materials, for which the degradation process is at least partially mediated by a biological system [10]. Erosion in contrast, is defined as the process of material loss from the polymer bulk. Such materials can be monomers, oligomers, parts of the polymer backbone or even parts of the polymer bulk. The term bioerodible indicates again that a biological system is involved in the kinetics of the process. By virtue of these definitions, the degradation of water-insoluble polymers is part of their erosion process. Among the many phenomena involved in erosion, such as water uptake and mass transfer, degradation is, therefore, the most important one.

Mathematical models reported in the literature describing erosion-controlled drug release can roughly be classified into two categories: (i) empirical models that usually assume a single zero-order process to control drug release rates; and (ii) models considering physicochemical phenomena such as diffusional mass transfer or chemical reaction processes. A subclass of the latter models are those that simulate polymer degradation as a random event using direct Monte Carlo techniques.

Depending on the composition of an erodible device (type of polymer, drug loading, additives) and geometry (size and shape), numerous mass transport phenomena and chemical reaction phenomena affect the resulting drug release kinetics such as:

(i) water intrusion into the device,

(ii) drug dissolution,

(iii) polymer degradation,

(iv) creation of aqueous pores,

(v) diffusion of drug and/or polymer degradation products inside the polymer matrix,

(vi) crystallization of polymer degradation products and/or drug within the system,

(vii) micro-environmental pH changes inside polymer matrix pores by degradation products,

(viii) diffusion of drug and/or polymer degradation products inside pores,

(ix) diffusion of hydrogen and/or hydroxide ions from the release medium into the device, altering the internal micro-environmental pH of the system,

(x) autocatalytic effects during polymer degradation,

(xi) osmotic effects,

(xii) polymer swelling,

(xiii) convection processes, and

(xiv) adsorption/desorption processes

to mention just a few. It is not reasonable to take all these phenomena into account, because this would lead to mathematical models too cumbersome for routine use. It is, therefore, crucial for any modeling approach to identify the dominating physical and chemical processes and to take only these into account. For sequential processes the slowest, and for parallel processes the fastest step are of prime interest. Due to the fundamental differences in the physicochemical properties of drugs and polymers used for erosion-controlled drug delivery, the dominating chemical reaction and/or physical mass transfer processes can significantly differ from system to system. In addition, the applied production technology and geometry (size and shape) of the device can alter the significance of the individual processes. Therefore, the preparation technique and the device design have to be considered when a mathematical model is chosen/developed to simulate or predict the erosion of a bioerodible matrix. Once an adequate mathematical model has been found or newly developed, it can be used to facilitate the optimization of the device and/or the development of related degradable matrices. For example, the effect of matrix composition (polymer to drug ratio etc.) and geometry (size and shape) on the drug release can be simulated. Thus, the required device design to achieve a certain, desired drug release profile can be predicted, minimizing the number of required experimental studies.

Due to the substantially high number of mathematical variables, no effort was made in this review to present a uniform picture of the different systems of notation defined by the respective authors. We rather used the original nomenclature which was modified only slightly to use either common abbreviations or to avoid misunderstandings.

Section snippets

Physicochemical characterization techniques

Due to the complexity of the physical and chemical phenomena involved, each drug delivery system should first be characterized precisely. The information obtained is essential for the development of a new, or the choice of an appropriate, already existing mathematical model. If no, or only few information are available on the physical processes occurring within the system during drug release, it is not possible to establish a powerful mathematical model being able to predict the effect of the

Empirical mathematical erosion models

In contrast to mechanistic mathematical models, empirical models quantifying drug release from erodible delivery systems are not based on the exact description of all the involved, real physical processes. Empirical models only describe the resulting, apparent drug release rates. For example, the superposition of various different mass transport phenomena such as water and drug diffusion, polymer swelling and polymer degradation can lead to overall zero-order drug release kinetics. In this

Diffusion and chemical reaction-based models

As discussed above, mechanistic mathematical models are based on the description of the real physical processes involved in controlling drug release. For bioerodible systems, diffusional mass transfer and chemical reactions are the most important phenomena to be taken into account. In contrast to the empirical models developed by Hopfenberg and Cooney these theories provide more information on the mechanisms controlling drug release. In this section, non-Monte Carlo-based models and Monte

Conclusions

The chemical reactions and mass transfer processes controlling drug release from bioerodible delivery systems strongly depend on the specific device characteristics, such as type of polymer, type of drug, size, shape, and composition. Consequently, it is a crucial point to choose the appropriate mathematical model for a specific delivery system. Important selection criteria include the desired predictive power and precision of the model, but also the effort required to apply the theory.

Acknowledgements

This work was supported by the European Commission (Marie Curie Individual Fellowship, Contract No. HPMF-CT-1999-00033) and the Deutsche Forschungsgemeinschaft (DFG) with grant GO 565/3-2.

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