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Current tools for prediction of controlled release
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. Because of the noteworthy advances in information technology, the ’in-silico’ or ’ab-initio’ optimization of novel delivery systems can be required to essentially enhance in precision and effortlessness of utilization. Comparable to other fields (e.g., aeronautics and aviation), computer simulations are probably going to wind up plainly a necessary piece of future innovative work in pharmaceutical innovation. « It is just an issue of time when numerical projects will be routinely used to help optimizing the plan of novel dosage products » [9]. 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.
In this way, one of the real main thrusts for the utilization of mathematical modeling in drug delivery is to spare time and to diminish costs: the quantity of required experimental tests to build up as well as the optimization of a current drug delivery product can altogether be diminished. Moreover, the quantitative examination of the physical, chemical, and conceivably natural phenomena, which are associated with the control of drug release, offers another central preferred standpoint: the basic understanding on the release mechanisms can be illustrated. « This knowledge is not only of academic interest, but a pre-requisite for an efficient improvement of the safety of the new pharmaco-treatments (particularly for highly potent drugs with narrow therapeutic windows) » [9]. Moreover, potential difficulties experienced during production can be significantly more productively addressed if the system is not dealt as a ’black box’, but there is an intensive comprehension of how drug release is controlled. It is definitive to know which product properties are critical to give a desired performance.
As Siepmann (2008) reviewed, up to date, numerous mathematical theories have been described in the literature [77–79, 64], but most of them still lack in accuracy and/or easiness of application. The pioneer of mathematical modeling of drug delivery is consireded to be Professor Takeru Higuchi. In 1961, he published his famous equation allowing for a simple and surprising description of drug release from an ointment base exhibiting a considerable initial excess of dissolved drug within an inert matrix with film geometry [80, 81]. 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 theories’. In the first case, the mathematical treatment is (at least partially) purely descriptive and not based on real physical, chemical and/or biological phemonena. 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 [9]. This kind of models may for example be valuable if diverse types of drug release profiles are to be compared utilizing a particular parameter (e.g., a release rate constant for experimental design). However, extraordinary alert 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 [82–88]. 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 where the mechanisms of which can be understood [9]. For product development such mechanistic realistic mathematical models consider the quantitative prediction of the impacts of formulation and manufacture parameters (e.g., the underlying tablet heigth and radius) on the subsequent release. Accordingly, the required composition, size, shape and manufac-ture procedure of a novel dose frame with wanted properties turn out to be hypothetically predictable. In addition, challenges experienced during production are substantially less demanding to address while having a reasonable thought of how the system functions.
Theories considering polymer swelling
If polymer swelling is of importance of the control of drug release, e.g. as in the case of hydroxypropyl methylcellulose (HPMC)-based matrix tablets, the transition of the macro-molecules from the glassy (less mobile) to the rubbery (more mobile) state has to be con-sidered in the model [103]. The two most important consequences of significant polymer sweeling in a controlled release matrix system are: (i) the length of the diffusion pathways increases, resulting in decreasing drug concentration gradients (being the driving forces for diffusion) and, thus, potentially decreasing drug release rates; (ii) the mobility of the macro-molecules significantly increases, resulting in increased drug mobility and, thus potentially increasing drug release rates [9]. Fig. 1.17 schematically shows the physical phenomena which can be engaged with the control of drug release from a swellable delivery system.
Theories considering polymer swelling and drug dissolution
In practice, regularly considerably more more processes are simultaneously involved in the control of drug release from oral controlled release matrix tablets. Generally, the matrix former is water-soluble. Thus, also polymer dissolution must be taken into account [9]. Different comprehensive mathematical theories have been proposed aiming to describe this type of drug delivery systems [105, 77]. In the following, only one example will briefly be described. The reader is referred to the literature for more details [105, 77]. The so-called ’sequential layer model’ [106] takes into account the diffusion of water and drug with time- and position- dependent diffusivities, moving boundary conditions, the swelling of the system, polymer and drug dissolution, and radial and axial mass transfer within cylindrical tablets. The model was successfully fitted to drug release kinetics from matrices based on hydroxypropyl methylcellulose (HPMC) and HPMC derivatives.
Because the polymer dissolution must be taken into account, the complexity of partial differential equations increases, and numerical solution is required. Water and drug diffusion are considered based on Fick’s second law of diffusion for cylindrical geometry, taking into account axial and radial mass transport and concentration-dependent diffusivities [97]: ∂Ck = 1 ∂ · rDk · ∂Ck + ∂ · Dk · ∂Ck + ∂ · rDk · ∂Ck (1.19).
Choice of a system model for ab-initio modeling
Emusion technology is particularly suited for design and fabrication of delivery systems [113]. Traditionally, conventional emulsions consisiting of small spherical droplets of one liquid dispersed in another immiscible liquid, are used in industrial and medical applications for a variety of reasons: encapsulation and delivery of active components; modification of rheological properties; alteration of optical properties; lubrication; or modification of organoleptic attributes (among others). The two immiscible liquids are typically an oil phase and an aqueous phase, although other immiscible liquids can sometimes be used. The droplets in conventional emulsions usually have diameters in the range of 100 nm to 100 µm, and are coated by a single layer of active surface components (’emulsifiers’) that stabilize them against aggregation. For designing conventional emulsions, there are a limited number of ways (Fig. 2.1) that droplets can be designed to provide specifical functional properties, such as stability, rheology, lubricity, optical properties, encapsulation, and delivery [15].
The major advantage of using the conventional emulsion design approach to create functional properties in emulsions is that they are relatively straightforward and inexpensive to implement, which is important for many industrial applications. On the other hand, during the past decade, there have been major advances in the utilization of structural design principles to fabricate emulsions with novel functional properties and enhanced functionality [15], such as layering (tailor made coatings around the particles), clustering (control of the aggregation state of the droplets) and embedding (comprising droplets within larger particles of a different material). These novel approaches allow to create functional attributes that cannot be achieved using conventional emulsions (See Fig. 2.2).
Since that these emulsion-based delivery systems can potentially depict an increasing number of scenarios depending on different levels of complexity, it is forecasted that they will offer an interesting framework of possibilities in order to implement a reverse engineering.
Because of the enormous ’know-how’ developed in our laboratory during the last years, regarding the preparation of highly concentrated emulsions (thesis of Emilio Paruta, 2010) and also mass transfer within these systems (thesis of Hala Fersadou, 2011), we consider that they are presented as a relevant model system for the start-up of the reverse engineering that this thesis proposes. Highly concentrated emulsions allow a high amount of dispersed phase, and a high ability to encapsulate active substances. As conventional emulsions, they can be of the oil-in-water (O/W) or water-in-oil (W/O) type. The different polarities of domains enable to solubilize hydrophilic and/or lipophilic molecules. Even though identified along as complex systems, these emulsions are very flexible regarding composition, size distribution, phase equilibria, interface type, morphology and viscosity. Therefore, they gather very good features to be applied a reverse engineering and to perform experimental validations.
A characteristic feature of these emulsions is their gel-like texture, leading to a significant improvement of emulsion stability, and offering a wide range of possibilities cocerning rheological properties and droplet size diameter. By way of illustration, in the following pages, a publication concerning the effects of formulation and processing on the rheological properties of oil-in-water highly concentrated emulsions is shown.
Interfacial tension measurements
Studies of adsorption of ABA-block polymeric surfactants at oil-water interfaces are scarce [38,39,34]. Comparison with extensive studies at air-water shows in general an analogous behavior at both interfaces [49]. We measured the effect on the interfacial tension (γ) of the stabilizers in aqueous solutions as function of the concentration. The results are plotted in Fig. 3. Generally, as the concentration of stabilizer increased, a lowering in γ occurred until a critical concentration (CC) above which no further decrease is observed. For all stabilizers, the CC values were found to be of the order of 1–10 g/L. These CC values are likely related to the formation layer at the inter-face, from which, further increases in the bulk concentration do not cause any more adsorption and as consequence, γ remains constant. Assuming that higher concentrations would not change this value, an average for γ using the last points (γ CC) has been taken to characterize every system.
The more hydrophobic ABA block stabilizers; P105 and F127, with 56 and 65 PO units respectively, were the most effective on decreasing the interfacial tension (until 3–4 mN/m). As a general trend, longer POP blocks decrease interfacial tension to a larger extent because POP is more surface active than POE [21]. This is clearly observed when comparing these stabilizers with those of shorter POP blocks: L64 and F68, of 13 EO and 76 units respectively. Nonetheless, longer POE blocks can also result in a decrease of the interfacial tension because of an increased repulsion between the POE chains [7]. This would explain the higher reduction when using P105 instead of F127 or L64 instead of F68 in spite of having a similar number of central POP blocks. Con-versely, the LMW stabilizer Tween 80, having a structure combining a single hydrocarbon group (oleate) and a bulky non-ionic polar group (sorbitan ring), led to an intermediate value of 6.63 mN/m. Surpris-ingly, the amphiphilic polysaccharides DexP10 and DexP15 with much less hydrophobic nature (25–37 phenoxy subunits) decreased γ to a same extent (up to 11–12 mN/m) than Pluronics L64 and F68. Possible interactions between the hydrophile glucose chains could also explain the higher reduction when using the less substituted DexP10.
Concentration of stabilizer
The range of used concentrations at this work is shown in Fig. 4. Every point represents the initial mass concentration of stabilizer in the continuous phase at different ϕ (from 0.850 to 0.938). These values range from 60 to 180 g/L. In general, it is assumed that block copoly-mers and low-molecular-weight surfactants in a selective solvent form micelles via a so-called closed association process, characterized by a certain CMC, below which only molecularly dissolved stabilizer is present in solution, usually as unimers [42]. On the other hand, am-phiphilic polysaccharides have shown that they can also behave like classical associative polymers [44]. Generally, it is assumed that at CMC and slightly above it, micelles are still loose but with further increase in amphiphile concentration in the medium, the unimer-micelle equili-brium shifts towards micelle formation [18]. Although the CMC is de-fined as a single concentration point, the micellization and micelle structure transitions can occur in a relatively broad range of con-centrations in the proximity of the CMC [51]. This implies micelles become more tight and stable and decrease their size during this tran-sition. It could be reasonable to think that previous CC determined by tensiometry should be related to those CMC reported on the literature (see Table 1). However, caution must be always applied because find-ings from different authors might not be transferable. In practice, a certain CMC range with some notable uncertainty could be detected, and large differences are often noted between CMC values determined by different methods [1] because their sensitivity to quantity of mole-cularly dispersed unimers present may vary or because of the presence of impurities. For any case, in this work, we assume and consider that all concentrations employed at this work are generally higher than these both CC and CMC. Therefore, we hypothesize that the excess of concentration above the CC, will be turned into a higher amount of micelles with probably different shapes existing in the aqueous solu-tion.
Effect of volumetric fraction (ϕ) on plateau modulus (G°N)
The results from scanning measurements into the linear domain of the samples can be visualized in Fig. 9. Similarly to other authors [30,8,19], it was observed the typical behavior of ideal elastic materials [14]; the elastic modulus G’ was independent of frequency, covering several orders of magnitude, with G’ > G in the whole amplitude do-main. In Fig. 6, we show the evolution of Plateau modulus as function of ϕ. In general, all ABA-block or ABn stabilizers showed a dramatic increase of G’, specially at very high ϕ. However, for emulsions pre-pared by Tween 80, this fact was much more moderated. On the other hand, the amphiphilic dextrans DexP10 and DexP15 exhibited compar-ables values of G°N to those obtained by using the commercial ABA-
Table of contents :
1 TOWARDS A REVERSE ENGINEERING APPROACH
1.1 Introduction
1.2 Research challenges in chemical product design
1.3 State-of-the-art of controlled release technologies
1.3.1 Carriers developed for controlled release
1.3.2 General analysis of release mechanisms
1.4 Current tools for prediction of controlled release
1.5 Empirical and semi/empirical mathematical models
1.5.1 Peppas equation
1.5.2 Hopfenberg model
1.5.3 Cooney model
1.5.4 Artificial neural networks
1.6 Mechanistic realistic theories
1.6.1 Theories based on Fick’s law of diffusion
1.6.2 Theories considering polymer swelling
1.6.3 Theories considering polymer swelling and drug dissolution
1.6.4 Theories considering polymer erosion/degradation
1.7 A reverse engineering methodology
1.7.1 Motivation
1.7.2 A proposed reverse engineering framework
2 MODELING
2.1 Introduction
2.1.1 Choice of a system model for ab-initio modeling
2.1.2 Objectives
2.2 Diffusion in highly concentrated emulsion systems
2.3 Theoretical estimation of mass transfer parameters
2.3.1 Permeability of surfactant layer
2.3.2 Interfacial mass transfer coefficient
2.3.3 Diffusion coefficient in liquids
2.3.4 Partition coefficient for a solute between two liquid phases
2.4 Computer-aided molecular design (CAMD) techniques
2.4.1 UNIQUAC Functional-group Activity Coefficients (UNIFAC) model
2.4.2 Modified UNIFAC (Dortmund) model
2.4.3 Revision and Extension 6 of Modified UNIFAC (Dortmund) model
2.4.4 Flash algorithm to predict L-L equilibrium
2.4.5 Atomic and Bond Contributions of van der Waals volume (VABC) .
2.4.6 Estimation of liquid mixture viscosity by UNIFAC-VISCO
2.4.7 Estimation of liquid molar volume at the normal boiling point by Tyn and Callus Method
2.4.8 Estimation of critical volume by Joback Method
2.5 Mass transfer model conception
3 SIMULATION
3.1 Introduction
3.2 Logical Architecture Diagram
3.3 Cartography of materials
3.3.1 Active ingredient
3.3.2 Dispersed phase
3.3.3 Continuous phase
3.3.4 Surfactants
3.4 Objetives of a sensitivity analysis
3.4.1 Preparation of Virtual Design of Experiments (VDOE)
3.4.2 Preliminar analysis of VDOE : 64 case studies
3.4.3 Main effects and interaction plots of VDOE : 300 case studies
3.5 Conclusions of sensitivity analysis
4 EXPERIMENTAL VALIDATIONS
4.1 Introduction
4.2 Predicted and experimental distribution coefficients
4.2.1 Comparison of UNIFAC methods
4.2.2 Effect of salt content and blank/dilution media
4.2.3 Effect of material of the cartography and dispersed phase volume .
4.2.4 Conclusions
4.3 Predicted and experimental mixture viscosities
4.3.1 Conclusions
4.4 Predicted and experimental release experiments
4.4.1 Preparation and characterization of emulsions
4.4.2 Experimental tests of controlled release
4.4.3 Comparison of predicted and experimental diffusion coefficients .
4.4.4 Analysis of storage moduli
4.4.5 Model limitations and possible useful extensions
4.4.6 Screening of most accurate predicted scenarios
4.4.7 Principal component analysis
5 FINAL CONCLUSIONS
5.1 Summary of results and conclusions
5.2 Implications to future research
References
