## Mathematical Modeling

To be a more predictive science, pharmaceutics needs to embrace mathematical modeling of systems, however complex they may be. Mathematical modeling is one area that holds much promise for estimating the behavior of dose forms, such as in the release and diffusion of drugs in brain tissue (55). A thorough theoretical analysis can provide major benefits, including the following:

1. It can help to understand how the dose forms work, for example, why the drug is released at a particular rate from a controlled delivery system (56-58). The underlying mass transport mechanisms can be elucidated, and most importantly, the dominant physicochemical processes can be identified (59). Often a significant number of phenomena are involved, such as the wetting of a device's surface, water penetration into the dose form, drug dissolution, swelling of polymeric excipients, glassy to rubbery phase transitions, drug and excipient diffusion out of the dose form, polymer dissolution and/or degradation, and drug-polymer interactions, to mention just a few. Chapter 11 in volume 2 gives a detailed description of a number of these phenomena. If several of these processes occur in sequence and significantly differ in velocity, the much faster ones can be neglected. In the case where certain processes in the sequence are much slower than all the others, then these are release rate limiting.

It is valuable to know which phenomena are dominant in a particular system, because this knowledge simplifies device optimization and troubleshooting during product development, scale-up, and production. It has to be pointed out that the type of dose form, type and amount of drug and excipients, and even the type of preparation technique can significantly alter the relative importance of the involved physicochemical phenomena. Chapter 1 in volume 2 on sustained- and controlled-release drug delivery systems provides an overview on the most frequently used control mechanisms in advanced drug delivery systems, including matrix tablets for oral administration and biodegradable microparticles for parenteral use. Furthermore, a theoretical analysis based on comprehensive experimental in vivo results can give valuable insight into the phenomena that are governing the fate of the drug once it released into the human body. Obviously, this knowledge—the thorough understanding of the processes occurring within the dose form as well as within the living organism—provides the basis for an efficient improvement of the safety of the respective drug treatment, especially in the case of highly potent drugs with narrow therapeutic windows.

2. Mathematical modeling can significantly facilitate the development of new products and the optimization of existing ones. An appropriate mathematical theory allows for a quantitative prediction of the effects of different formulation and processing parameters on the resulting properties of the dose form, for example, the release rate of an incorporated drug. Figure 7 shows an example for such a simulation: the effects of the initial radius of propranolol HCl-loaded, hydroxypropyl methylcellulose (HPMC)-based

Time, h

Figure 7 Theoretical prediction (curves) of the effects of the initial radius of hydroxypropyl methylcellulose-based matrix tablets containing 5% propranolol HCl on the resulting drug release kinetics in phosphate buffer pH 7.4. The symbols represent independent experimental results, confirming the theoretical calculations. The initial tablet height is 2.6 mm, the initial tablet radius is varied from 1.0 to 2.5 to 6.5 mm (corresponding to the upper curve—open circles; middle curve— filled diamonds; and lower curve—open squares). Source: From Ref. 60.

Time, h

Figure 7 Theoretical prediction (curves) of the effects of the initial radius of hydroxypropyl methylcellulose-based matrix tablets containing 5% propranolol HCl on the resulting drug release kinetics in phosphate buffer pH 7.4. The symbols represent independent experimental results, confirming the theoretical calculations. The initial tablet height is 2.6 mm, the initial tablet radius is varied from 1.0 to 2.5 to 6.5 mm (corresponding to the upper curve—open circles; middle curve— filled diamonds; and lower curve—open squares). Source: From Ref. 60.

matrix tablets on the resulting drug release kinetics in phosphate buffer pH 7.4 are shown (curves) (60). The initial tablet height was constant (2.6 mm), whereas the initial tablet radius was varied from 1.0 to 6.5 mm (corresponding to the curves at the top to the bottom). Clearly, the resulting relative release rate significantly decreases when increasing the initial tablet radius, which can at least partially be attributed to the decreasing "surface area:volume ratio" of the system (and thus decreased relative surface area available for diffusion). The symbols in Figure 7 represent the independently determined experimental drug release kinetics from these matrix tablets. The good agreement between theory and experiment serves as an indication for the validity of this model for this type of drug delivery system. But also, the effects of other formulation and processing parameters on the resulting system properties can be theoretically predicted, including the initial drug and polymer content, type and amount of plasticizer, as well as size and shape of the dose form (61,62).

In silico simulations can, thus, be used to effectively replace or minimize the series of cost- and time-intensive "trial-and-error" experiments during product development. This is particularly useful in the case of controlled drug delivery systems with long-term release kinetics, for example, implants that are intended to provide appropriate drug levels during several months or years.

Two types of mathematical theories can be distinguished: empirical models and mechanistic realistic models. Empirical models are purely descriptive and do not allow for a better understanding of the underlying physical, chemical, and/or biological phenomena. Furthermore, they cannot be used to quantitatively predict the effects of formulation and processing parameters on the resulting system properties. In contrast, mechanistic theories are based on real phenomena, such as diffusion, dissolution, swelling, and/or dissolution/erosion. They allow for the determination of realistic parameters characterizing the respective dose form, for example, the diffusion coefficient of the drug within a matrix former or the degradation rate constant of a polymeric excipient. On the basis of these parameters, further insight into the underlying mass transport phenomena can be gained. For instance, the relative importance of the involved processes can be determined.

Because of the large variety of drugs, excipients, and dose forms that are used, there is no universal mathematical theory valid for all types of systems. In each particular case, it must be determined which processes are involved and—if possible—which of them are dominant. This type of analysis must be based on comprehensive experimental results, otherwise, no reliable conclusions can be drawn. It has to be pointed out that obtaining good agreement between a fitted theory and a set of experimental results is not a proof for the validity of a mathematical theory. Fitting a model to experimental results implies that one or more model parameters are adjusted to obtain the least-deviations "theory-experiment." Especially if a significant number of parameters are simultaneously fitted to the same set of experimental data, caution has to be paid when drawing conclusions. To evaluate the validity of a mathematical model for a particular type of drug-loaded dose form, the theory should be used to predict the effects of different formulation and/or processing parameters on the resulting system properties and these theoretical predictions should be compared with independent experimental results. Also, care should be taken when applying a mathematical theory to a specific dose form so that no major assumptions on which the model is based are violated. As an example, the famous Higuchi equation (10) is unfortunately often misused and applied to drug delivery systems for which it is not valid.

Yet, there is a significant lack of mechanistic realistic mathematical theories, which appropriately describe both the physicochemical phenomena occurring within the dose form and the subsequent fate of the drug within the human body (55). This can at least partially be explained by the complexity of the resulting set of mathematical equations considering all the involved physical, chemical, and biological processes (63). Also, a large variety of phenomena can be of importance in vivo for the fate of the drug, including diffusion and convection within the extra- and intracellular space, reversible and irreversible binding to extracellular matrix, drug metabolism, passive and active uptake into living cells (e.g., by "simple" diffusion and/or by receptor-mediated internalization), release from endolysosomes into the cytosol of the cells, uptake into the cell nuclei, and uptake/elimination/distribution from/into the blood stream and/or lymphatic system. Figure 8 shows as an example some of the processes that can be decisive for the drug upon direct administration into brain tissue. Importantly, drug transport in vivo can be highly anisotropic (direction dependent), because the human organism is not one homogeneous mass. Major advances in this research area allowing for a better understanding of the underlying drug transport mechanisms have been achieved by the working groups of Nicholson (64) and of Haller and Saltzman (65,66). However, there is a significant need for comprehensive and mechanistic mathematical theories relating formulation and processing parameters of dose forms to the resulting drug concentrations at the site of action and the pharmacodynamic effects of the treatment.

An often underestimated aspect when characterizing dose forms in vitro is the importance of the type of environment the systems are exposed to. This includes, for instance, the physical state of the medium (liquid or gel), degree of agitation, pH and ionic strength of the medium, the maintenance or absence of sink conditions, the presence/absence of enzymes and/or macrophages, etc. For example, the drug release patterns from PLGA parenteral controlled drug delivery systems can strongly depend on the pH of the surrounding environment, because ester hydrolysis is catalyzed by

Figure 8 Schematic presentation of some of the processes that can be involved in drug transport within human brain tissue, including diffusion and convection through the extracellular spaces, permeation through capillaries, systemic elimination, internalization, and metabolism. The black circles represent drug molecules in the interstitial space. Source: From Ref. 63.

Figure 8 Schematic presentation of some of the processes that can be involved in drug transport within human brain tissue, including diffusion and convection through the extracellular spaces, permeation through capillaries, systemic elimination, internalization, and metabolism. The black circles represent drug molecules in the interstitial space. Source: From Ref. 63.

protons (67). Also, the osmotic pressure of the release medium can be of crucial importance for the drug release patterns from oral coated dose forms, in which crack formation is mandatory to allow for drug release. Appropriate mathematical modeling can be of great help in this perspective if the theory adequately takes into account the effects of the environmental conditions on the systems' properties. This allows for an appropriate correlation of the in vitro and in vivo results and thus for a significantly facilitated product development and improved safety of the drug treatment.

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