Transport Mechanisms within Controlled Drug Delivery Systems

Often, the active agent is embedded within a polymeric matrix that hinders instantaneous drug dissolution/release (16,17). Different types of macromolecules and device geometries (shapes and sizes) can be used. Practical examples include poly(anhydride)-based, flat cylinders and poly (lactic-co-glycolic acid) (PLGA)-based, spherical microparticles. Depending on the composition, dimension, and sometimes even preparation method of the system, different physical and chemical phenomena can be involved in the control of the resulting drug release kinetics, including: (i) water penetration into the system; (ii) drug dissolution; (iii) dissolution/degradation of the polymer chains; (iv) precipitation and re-dissolution of polymer degradation products; (v) structural changes within the system occurring during drug release, such as creation/closure of water-filled pores; (vi) changes in micro-pH (e.g. generation of acidic microclimates in PLGA-based delivery systems); (vii) diffusion of drug and/or degradation products of the polymer out of the device with constant or time/position-dependent diffusion coefficients; (viii) osmotic effects; (ix) convectional processes; as well as (x) adsorption/desorption phenomena. In contrast to oral controlled drug delivery systems, significant swelling of the polymer (a phenomenon that can also be effectively used to control drug release) must be avoided due to the limited space in the crane: Intracranially administered devices that remarkably swell would generate mechanical pressure on the surrounding (highly sensitive) brain tissue and, thus, lead to serious side effects.

The physical and chemical processes involved in the control of drug release from a particular delivery system can affect each other in a rather complex way. For example, PLGA-based devices can show autocatalytic effects: Water penetration into PLGA-based microparticles and implants is much faster than the subsequent ester bond cleavage (18). Thus, the entire drug delivery system is rapidly wetted and polymer degradation occurs throughout the device, generating shorter chain acids (and alcohols). Due to concentration gradients these species diffuse out of the dosage form. In addition, bases from the surrounding environment diffuse into the drug delivery system, neutralizing the generated acids. However, diffusional processes are generally slow and the rate at which the acids are produced within the dosage forms can be higher than the rate at which they are neutralized. Consequently, the micro-pH within the system can significantly drop (19,20). As ester bond cleavage is catalyzed by protons, this leads to accelerated polymer degradation and drug release (21). It has to be pointed out that device characteristics, for example, porosity and size, can significantly affect the relative diffusion rates of the involved acids and bases and, thus, alter the underlying drug release mechanisms (22). Consequently, systems with identical chemical composition that are prepared using different methods (e.g. water-in-oil-in-water vs. oil-in-water solvent extraction/evaporation techniques for microparticles) resulting in porous versus non-porous devices can exhibit very different drug release patterns due to altered drug release mechanisms.

To get deeper insight into the physical and chemical processes that govern drug release within a particular controlled delivery system, adequate, mechanistic mathematical theories can be used. Obviously, empirical models are not suitable for this purpose. The application of semi-empirical theories should be viewed with great caution as the conclusions can be misleading. Examples for appropriate mechanistic theories describing drug transport in biodegradable controlled delivery systems include those developed by Goepferich et al. (21-30). These models combine Monte Carlo simulations (quantifying polymer degradation) with Fick's second law (describing drug diffusion). Importantly, the theories are applicable to both, surface as well as bulk eroding polymeric systems [31]. Furthermore, they have been extended to describe the erosion of composite matrices made of bulk and surface eroding polymers [e.g. poly(D,L-lactic acid) and poly[bis (p-carboxyphenoxy) propane - sebacic acid]] (28).

Siepmann et al. (32) proposed a mathematical theory considering: (i) polymer degradation (based on Monte Carlo simulations); (ii) drug diffusion (based on Fick's second law); and - optionally - (iii) limited drug solubilities within spherical microparticles. Figure 1 shows a schematic presentation of such a system for mathematical analysis. To minimize computation time, it is assumed that there are no concentration gradients in the direction of the angle 9. Thus, a two-dimensional grid (Fig. 1B) can be defined, which upon rotation around the z-axis describes the three-dimensional structure of the microparticle. Considering symmetry planes at z = 0 and r = 0, the mathematical analysis can be further reduced to only one quarter of the two-dimensional circle (Fig. 2A). Before exposure to the release medium, each pixel represents either non-degraded polymer or drug. Knowing the initial drug loading of the microparticles as well as the initial drug distribution within the system, direct Monte Carlo techniques can be used to define which pixel represents non-degraded polymer and which pixel represents drug. Figure 2A shows an example for a homogeneous initial drug distribution. Importantly, all pixels are defined in such a way that they have the same height, but different widths. The coordinates are chosen to assure that the volumes of the rings, which are described by the rectangular pixels upon rotation around the z-axis, are all identical. This results in about equal numbers of cleavable ester bonds within each ring. Thus, the probability with which the polymer pixels erode within a certain time period after contact with water can be assumed to be similar (being essentially a function of the number of cleavable polymer bonds). As polymer degradation is a random process, not all pixels degrade exactly at the same time point. Each pixel is characterized by an individual, randomly distributed "lifetime" (tlifetime), which can be calculated as in Equation 1, as a function of the random variable e (integer between 0 and 99):

(being characteristic for the type and physical state of the polymer). As soon as a pixel comes into contact with water, its "lifetime" starts to decrease. After the latter has expired, the pixel is assumed to erode instantaneously and to be converted into a water-filled pore. Once the initial condition (Fig. 2A) and the specific "life times" of all polymer pixels are defined, it is possible to determine the status of each pixel (representing drug, non-degraded polymer or a water-filled pore) at any time point. Figure 2B shows an example for the composition and structure of a microparticle at a specific time point during drug release. This structural information is of major where t.

average

where t.

average is the average "lifetime" of the pixels, and l is a constant

y

N.

/

\

/

----

j

____

-

/

V

Figure 1 Schematic presentation of a single bioerodible microparticle for mathematical analysis: (A) Three-dimensional geometry; (B) two-dimensional cross-section with two-dimensional pixel grid used for numerical analysis. Source: From Ref. 32.

Figure 2 Principle of the Monte Carlo-based approach to simulate polymer degradation and diffusional drug release; schematic structure of the system: (A) At time t = 0 (before exposure to the release medium); and (B) during drug release. Gray, dotted, and white pixels represent non-degraded polymer, drug, and pores, respectively. Source: From Ref. 32.

Figure 2 Principle of the Monte Carlo-based approach to simulate polymer degradation and diffusional drug release; schematic structure of the system: (A) At time t = 0 (before exposure to the release medium); and (B) during drug release. Gray, dotted, and white pixels represent non-degraded polymer, drug, and pores, respectively. Source: From Ref. 32.

importance, because it allows to calculate the porosity of the microparticles at any time point in radial and axial direction [s(z,t) and s(r,t)j (Eq. 2, 3):

with s being the "status function" of the pixel xij at time t, defined as Equations 4 and 5:

where, nz and nr represent the number of pixels in the axial and radial direction at r and z, respectively. Using Equations 2-5, the time- and direction-dependent porosities within the microparticles can be calculated at any gridpoint. This is essential information for the accurate calculation of the time-, position-, and direction-dependent diffusivities (Eq. 6, 7):

where Dcrit represents a critical diffusion coefficient, being characteristic for a specific drug-polymer combination. These equations are combined with Fick's second law of diffusion and the resulting set of Partial Differential Equations is solved numerically. Fitting the model to sets of experimentally measured drug release kinetics from 5-fluorouracil-loaded, PLGA-based microparticles (which are used for the treatment of brain tumors), good agreement between experiment and theory was obtained (Fig. 3). Based on these calculations important information on the underlying drug release mechanisms could be gained: It was shown that the initial high drug release rate from these systems (also called "burst effect") can primarily be attributed to pure drug diffusion (at early time points the diffusion pathways are short, thus, drug release is rapid). The second release phase (with an about constant release rate) can be attributed to a combination of two processes: drug diffusion and polymer erosion. The increase in the length of the diffusion pathways with time (which should result in a decrease in the drug release rate) is compensated by the increase in drug mobility within the polymeric matrix (due to the decreasing chain length of the macromolecules upon ester hydrolysis and, thus, increased macromolecular mobility). The final, again more rapid drug release phase can be attributed to the disintegration of the microparticles: As soon as a critical minimal polymer

Time, d

Figure 3 Experiment (symbols) and theory (curve): Fitting of a Monte Carlo-based mathematical model to experimentally determined 5-fluorouracil release from PLGA-based microparticles. Source: From Ref. 32.

Time, d

Figure 3 Experiment (symbols) and theory (curve): Fitting of a Monte Carlo-based mathematical model to experimentally determined 5-fluorouracil release from PLGA-based microparticles. Source: From Ref. 32.

molecular weight is reached, the mechanical stability of the system is no more guaranteed. Consequently, the relative surface area of the system increases and the diffusion pathway lengths decrease.

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