Diffusion is one of the basic mass transport mechanisms, which is involved in the control of drug release from numerous drug delivery systems (14-16). Fick was the first to treat this phenomenon in a quantitative way (21), and the textbook of Crank (22) provides various solutions of Fick's second law for different device geometries and initial and boundary conditions. A very interesting introduction into this type of mass transport is given by Cussler (23).
Several types of species can diffuse in and out of controlled drug delivery systems, including water, drug, water-soluble excipients (e.g., polymers, plasticizers, and/or fillers), acids, and bases from the environmental bulk fluid as well as polymer degradation products. Often, the polymeric material is chosen in such a way that drug diffusion through the macromolecular network is the rate-limiting step. In these cases, the systems
Constant activity source
Constant activity source
Drug and malrin former
Drug and malrin former
Figure 2 Classification scheme for predominantly diffusion-controlled drug delivery systems. Only spherical dosage forms are illustrated, but the scheme is applicable to any type of geometry. Stars represent dissolved (individual) drug molecules, whereas black circles represent undissolved drug excess (e.g., crystals and/or amorphous aggregates).
are called primarily "diffusion controlled." According to their inner structure, three types of diffusion-controlled drug delivery systems can be distinguished (Fig. 2): reservoir systems, monolithic systems, and miscellaneous systems.
In these cases, the drug and the release rate-controlling material (generally a polymer or lipid) are physically separated: The drug is located at the center of the dosage form, constituting the "reservoir" or "drug depot," and the polymer or lipid surrounds this reservoir as a "release rate-controlling membrane." A clear "core-shell structure" is given. This can, for instance, be a coated pellet, capsule, or tablet. Please note that only spherical dosage forms are illustrated in Figure 2. However, this classification scheme is valid for any type of geometry. Depending on the ratio "initial drug concentration/drug solubility within the device," two types of systems can be distinguished (Fig. 2):
1. Reservoir devices with non-constant activity source: In this case, the initial drug loading is below drug solubility and drug molecules (or ions) that leave the dosage form are not replaced. Consequently, the resulting drug concentrations at the inner side of the release rate-controlling membrane decreases with time, resulting in decreasing drug-concentration gradients (which are the driving forces for diffusion). If the properties of the release rate-controlling membrane (in particular its thickness and permeability) do not change during drug release and if perfect sink conditions are provided throughout the experiment (negligible drug concentrations outside of the dosage form), first-order release kinetics result, irrespective of the geometry of the device.
2. Reservoir devices with constant activity source: In this case, the initial drug concentration is greater than drug solubility. Hence, dissolved and undissolved drugs coexist within the reservoir. The drug excess can, for instance, be drug crystals and/or amorphous aggregates. Please note that in Figure 2, the dissolved drug molecules are represented by stars, whereas the drug excess by black circles. Importantly, only dissolved (individualized) drug molecules are able to diffuse, not large drug crystals and/or amorphous aggregates. If the diffusion of the drug through the surrounding polymeric or lipidic membrane is release-rate limiting (and drug dissolution is comparatively fast), released drug molecules (or ions) are rapidly replaced by the (partial) dissolution of drug excess. Hence, constant drug concentrations at the inner membrane's surface result (saturated drug solutions). If perfect sink conditions are provided outside the system, constant concentration differences result as long as the drug concentration in the reservoir exceeds drug solubility. If, in addition, the properties of the release rate-controlling membrane (in particular its thickness and permeability) do not change during drug release, the drug-concentration gradients remain constant, leading to constant drug-release rates (zero-order kinetics), irrespective of the geometry of the device.
In these cases, the drug and matrix former are not clearly physically separated with a core-shell structure, but more or less homogeneously mixed (Fig. 2). Analogous to the reservoir systems, two types of devices can be distinguished according to the "initial drug concentration/drug solubility" ratio:
1. Monolithic solutions, in which the drug is molecularly dispersed within the matrix former. This type of controlled drug delivery system might, for instance, be highly suitable for poorly water-soluble drugs. Individualizing the drug molecules within a (often polymeric) matrix former avoids the dissolution step of the drug during drug release and might enhance the bioavailability. However, care must be taken that the systems are not metastable and poorly water-soluble drug crystals formed upon long-term storage.
2. Monolithic dispersions, which contain both dissolved and dispersed drug (stars and black circles in Fig. 2).
In contrast to the reservoir devices described above, the resulting drug-release kinetics depend on the device geometry in the case of monolithic systems. Under specific conditions (e.g., absence of polymer dissolution and polymer swelling, time- and position-independent diffusion coefficients), analytical solutions of Fick's second law can be derived, allowing for the calculation of the resulting drug-release kinetics as a function of the device geometry and dimensions (13,22). Also, easy-to-use (often short-time) approximations have been proposed (2). However, care must be taken not to violate any of the assumptions on which such simplified theories are based (14). One very interesting example for an easy-to-use approximation is the classical Higuchi equation (24,25). It has been derived for the geometry of thin ointment films with negligible edge effects and a significant initial excess of suspended drug particles (initial drug loading » drug solubility in the ointment base). The equation states that the cumulative absolute amount of drug released at time t, Mt, is proportional to the square root of time:
where A is the surface area of the film exposed to the skin or release medium, D is the drug diffusivity in the ointment base, and c0 and cs represent the initial drug concentration and the solubility of the drug in the ointment base, respectively. A major advantage of this equation is its simplicity. Importantly, it can also be applied to other controlled drug delivery systems than thin ointment films (e.g., polymeric patches). However, great care must be taken that none of the assumptions on which the derivation of this equation is based is violated, including the following ones:
• The initial drug concentration in the system is significantly greater than drug solubility. This assumption is crucial, because it provides the basis for the justification of the applied pseudo-steady-state approach allowing for major simplifications. Figure 3 illustrates this concept. The concentration-distance profiles of a drug that is homogeneously suspended within an ointment are shown. The release medium (or skin) providing perfect sink is on the right-hand side. The solid line represents the concentration profile after exposure of the ointment to this sink for a certain time t. Importantly, a sharp discontinuity is observed at distance h from the release medium. For this distance h, the concentration gradient is essentially constant, provided the initial drug concentration within the system, c0, is much greater than the solubility of the drug (c0 » cs) (pseudo steady state). The front, separating the ointment still containing both dissolved and dispersed drug from the ointment containing only dissolved drug, moves only very slowly inward (is virtually stationary), because it takes a long time for all the drug excess to be dissolved at a given position. After an additional time interval, At, the new concentration profile of the drug is indicated by the dashed line. Again, a sharp discontinuity and otherwise linear concentration profiles result.
• The device geometry is that of a thin film with negligible edge effects.
• The size of the drug particles is much smaller than the thickness of the film.
• The carrier material (e.g., ointment base) does not swell or dissolve.
• The diffusivity of the drug is constant (not dependent on time or position).
• Perfect sink conditions are maintained throughout the experiment.
Unfortunately, equation (1) is often misused and applied to controlled drug delivery systems, which do not fulfill all these assumptions. In these cases, any conclusion should be viewed with great caution. Even if the cumulative amount of drug that is released from a particular drug delivery system is proportional to the square root of time, this does not necessarily mean that the underlying drug-release mechanism is the same as in the ointment Higuchi studied. For example, the superposition of various other physicochem-ical phenomena (such as polymer swelling and dissolution, time- and position-dependent changes in the diffusion coefficients of water and/or drug) might result in an apparent square root of time kinetics. Later Higuchi also extended his classical equation to other geometries than thin films (26). However, these equations are implicit solutions and not as easy to apply as equation (1).
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