Mathematical modeling of drug delivery

Abstract

Due to the significant advances in information technology mathematical modeling of drug delivery is a field of steadily increasing academic and industrial importance with an enormous future potential. The in silico optimization of novel drug delivery systems can be expected to significantly increase in accuracy and easiness of application. Analogous to other scientific disciplines, computer simulations are likely to become an integral part of future research and development in pharmaceutical technology. Mathematical programs can be expected to be routinely used to help optimizing the design of novel dosage forms. Good estimates for the required composition, geometry, dimensions and preparation procedure of various types of delivery systems will be available, taking into account the desired administration route, drug dose and release profile. Thus, the number of required experimental studies during product development can be significantly reduced, saving time and reducing costs. In addition, the quantitative analysis of the physical, chemical and potentially biological phenomena, which are involved in the control of drug release, offers another fundamental advantage: The underlying drug release mechanisms can be elucidated, which is not only of academic interest, but a pre-requisite for an efficient improvement of the safety of the pharmaco-treatments and for effective trouble-shooting during production. This article gives an overview on the current state of the art of mathematical modeling of drug delivery, including empirical/semi-empirical and mechanistic realistic models. Analytical as well as numerical solutions are described and various practical examples are given. One of the major challenges to be addressed in the future is the combination of mechanistic theories describing drug release out of the delivery systems with mathematical models quantifying the subsequent drug transport within the human body in a realistic way. Ideally, the effects of the design parameters of the dosage form on the resulting drug concentration time profiles at the site of action and the pharmacodynamic effects will become predictable.

Introduction

Mathematical modeling of drug delivery and predictability of drug release is a field of steadily increasing academic and industrial importance with an enormous future potential. Due to the significant advances in information technology, the in silico optimization of novel drug delivery systems can be expected to significantly improve in accuracy and easiness of application. Analogous to other scientific disciplines (e.g., aviation and aerospace), computer simulations are likely to become an integral part of future research and development in pharmaceutical technology. It is only a question of time when mathematical programs will be routinely used to help optimizing the design of novel dosage forms. Considering the desired type of administration, drug dose to be incorporated and targeted drug release profile, mathematical predictions will allow for good estimates of the required composition, geometry, dimensions and preparation procedure of the respective dosage forms. Thus, one of the major driving forces for the use of mathematical modeling in drug delivery is to save time and to reduce costs: The number of required experimental studies to develop a new and/or optimize an existing drug product can significantly be reduced.

In addition, the quantitative analysis of the physical, chemical and potentially biological phenomena, which are involved in the control of drug release, offers another fundamental advantage: The underlying drug release mechanisms can be elucidated. This knowledge is not only of academic interest, but a pre-requisite for an efficient improvement of the safety of new pharmaco-treatments. This is particularly true for highly potent drugs with narrow therapeutic windows. Furthermore, potential challenges encountered during production (trouble-shooting) can be much more efficiently addressed if the system is not treated as a “black box”, but if there is a thorough understanding of how drug released is controlled. It is decisive to know which device properties are crucial to provide the desired system performance.

Up to date, numerous mathematical theories have been described in the literature (Siepmann and Peppas, 2001, Siepmann and Goepferich, 2001, Arifin et al., 2006, Lin and Metters, 2006), but most of them still lack in accuracy and/or easiness of application. The “father” of mathematical modeling of drug delivery is Professor Takeru Higuchi. In 1961, he published his famous equation allowing for a surprisingly simple description of drug release from an ointment base exhibiting a considerable initial excess of non-dissolved drug within an inert matrix with film geometry (Higuchi, 1961a, Higuchi, 1961b). This was the beginning of the quantitative treatment of drug release from pharmaceutical dosage forms. Numerous models have been proposed since then, including empirical/semi-empirical as well as mechanistic realistic ones. In the first case, the mathematical treatment is (at least partially) purely descriptive and not based on real physical, chemical and/or biological phenomena. Consequently, no or very limited insight into the underlying drug release mechanisms can be gained. Furthermore, the predictive power of empirical/semi-empirical models is often low. This type of theories might for instance be useful if different types of drug release profiles are to be compared using a specific parameter (e.g., an apparent release rate constant for experimental design analysis). But great caution must be paid if mechanistic conclusions are drawn or quantitative predictions made. An exception are approaches based on artificial neural networks (ANNs), which can show good predictive power.

In contrast, mechanistic mathematical theories are based on real phenomena, such as diffusion, dissolution, swelling, erosion, precipitation and/or degradation (Siepmann et al., 1998, Narasimhan, 2001, Frenning and Stromme, 2003, Lemaire et al., 2003, Zhou and Wu, 2003, Frenning et al., 2005, Raman et al., 2005). This type of models allows for the determination of system-specific parameters that can offer deeper insight into the underlying drug release mechanisms. For instance, the relative importance of several processes that are involved (e.g., drug diffusion and polymer swelling) can be estimated. The dosage form is not treated as a “black box”, but as a real drug delivery system the mechanisms of which can be understood. During product development such mechanistic realistic mathematical models allow for the quantitative prediction of the effects of formulation and processing parameters (e.g., the initial tablet height and radius) on the resulting drug release kinetics. Thus, the required composition, size, shape and preparation procedure of a novel dosage form with desired properties become theoretically predictable. In addition, challenges encountered during production are much easier to address when having a clear idea of how the system works.

When using and/or developing mathematical theories to quantify drug release from pharmaceutical dosage forms, the following aspects should carefully be taken into account:

• The accuracy of a mathematical theory generally increases with increasing model complexity: The more phenomena are taken into account, the more realistic the theory becomes. However, caution must be paid because too complex models are cumbersome to use. Too many system-specific parameters are required to allow for quantitative predictions. Thus, when developing a new mathematical theory for a particular drug delivery system great care must be taken to consider only the dominant physical, chemical and/or biological processes. If for instance several mass transport steps take place sequentially and if one of these processes is much slower than all others, only this step needs to be considered in the model.

• Theoretical calculations should always be compared to experimental results. Importantly, there are two different types of comparisons: The theory can either be fitted to experimental data, or theoretical predictions can be compared with independent experimental results. In the first case, one or more model parameters are optimized in such a way that the differences between the experimental results and the theoretical calculations are minimized. Especially if several model parameters are simultaneously fitted to the same set of experimental data great caution needs to be paid: The simultaneous adjustment of many model parameters generally leads to good agreement between theory and experiment, even if the theory is not appropriate. Ideally, only one model parameter should be fitted at a time, using a set of at least 12 experimental data points. In the case of fittings to experimentally measured drug release kinetics, it is furthermore important that the entire drug release profile is described, and not only one part of it (e.g., the early, intermediate or final phase). A much more reliable comparison (and indication for the validity of a mathematical theory for a specific type of drug delivery system) is that of theoretical predictions and independent experimental results. In this case, first all system-specific parameters are determined via fittings to different sets of experimental results. Once all required model parameters are known, the effects of different formulation and/or processing parameters on the systems’ properties (e.g., drug release kinetics) are predicted in silico. Then, the respective devices are prepared in reality and the predicted systems’ properties experimentally measured. If possible, not only one specific type of experimental results should be determined, but different device properties should be measured, such as the drug release kinetics, dry mass loss behavior, changes in wet weight as well as drug and excipient concentration profiles.

• There is no general mathematical theory that can be applied to all types of drug delivery systems. Certain models are applicable to only a very limited number of drug delivery systems, others have a much broader application spectrum.

• Even if a model shows good agreement between theoretical predictions and various types of independent experimental results, one should always be cautious and ready to abandon the theory if appropriate experimental evidence is given. A model describing drug delivery is always a simplification of the real system and its suitability is always restricted to certain cases.

The aim of this article is to give an overview on the current state of the art of empirical/semi-empirical and mechanistic realistic mathematical theories quantifying drug delivery and to provide an outlook into the future of this field of research. Due to the substantially high number of variables, no effort was made to present a uniform picture of the different systems of notation defined by the respective authors. The original nomenclatures are used and only some cases are modified by using more common abbreviations to avoid misunderstandings.