Diffusion layer model dissolution

Fig. 15 Two of the simplest theories for the dissolution of solids (A) the interfacial barrier model, and (B) the diffusion layer model, in the simple form of Nemst [105] and Brunner [106] (dashed trace) and in the more exact form of Levich [104] (solid trace). c is the concentration of the dissolving solid, cs is the solubility, cb is the concentration in the bulk solution, and x is the distance from the solid-liquid interface of thickness h or 8, depending on how it is defined. (Reproduced with permission of the copyright owner, John Wiley and Sons, Inc., from Ref. 1, p. 478.)...

The diffusion layer model satisfactorily accounts for the dissolution rates of most pharmaceutical solids. Equation (43) has even been used to predict the dissolution rates of drugs in powder form by assuming approximate values of D (e.g., 10 5 cm2/sec), and h (e.g., 50 pm) and by deriving a mean value of A from the mean particle size of the powder [107,108]. However, as the particles dissolve, the wetted surface area, A, decreases in proportion to the 2/3 power of the volume of the powder. With this assumption, integration of Eq. (38) leads to the following relation, known as the Hixon-Crowell [109] cube root law ... 

Fortunately, in most cases, the salt form under serious consideration exhibits a faster dissolution rate than the corresponding parent drug at an equivalent pH. This dissolution phenomenon can be explained in light of the parameters that govern the dissolution rate, as found in the diffusion layer model of Brunner (1904) ... 

FIGURE 17.1 (a) Diffusion-layer model of dissolution, (b) Interfacial barrier model of dissolution. 

Two of the simplest theories to explain the dissolution rate of solutes are the interfacial barrier model and the diffusion-layer model (Figures 17.1 and 17.2). Both of these theories make the following two assumptions ... 

Early studies in this field of research formulated two main models for the interpretation of the dissolution mechanism the diffusion layer model and the... 

Figure 5.2 Schematic representation of the dissolution mechanisms according to (A) the diffusion layer model, and (B) the interfacial barrier model.

In the interfacial barrier model of dissolution it is assumed that the reaction at the solid-liquid interface is not rapid due to the high free energy of activation requirement and therefore the reaction becomes the rate-limiting step for the dissolution process (Figure 5.1), thus, drug dissolution is considered as a reaction-limited process for the interfacial barrier model. Although the diffusion layer model enjoys widespread acceptance since it provides a rather simplistic interpretation of dissolution with a well-defined mathematical description, the interfacial barrier model is not widely used because of the lack of a physically-based mathematical description. 

The diffusion layer theory is the most useful and best known model for transport-controlled dissolution and satisfactorily accounts for the dissolution rates of most pharmaceutical solids. In this model, the dissolution rate is controlled by the rate of diffusion of solute molecules across a thin diffusion layer. With increasing distance from the surface of the solid, the solute concentration decreases in a nonlinear manner across the diffusion layer. The dissolution process at steady state is described by the Noyes-Whitney equation ... 

Of the many theories proposed for studying the dissolution of solids, the simple diffusion model mostly suffices for pharmaceutical applications. The diffusion layer model assumes a thin stagnant diffusion layer (thickness = K) at the interface of the dissolving solid (x = 0) and the dissolution medium x = h), referred to as bulk (Figure 7.5). ... 

Distance from the solid sur ce Figure 7.5 Drug dissolution by the diffusion layer model. ... 

For dissolution of solid particles, the Hixson-Crowell cube-root law (Eq. 5.3) assumes that the thickness of the diffusion layer h is constant during dissolution. However, this is not necessarily true. In addition, most drug particles are nonspherical and nonuniform in size. Therefore, very often the dissolution mechanism of solid drug particles is actually much more complicated. Nevertheless, the Hixson-Crowell cube-root law provides the first approximation to model powder dissolution. 

In reality, the parameters 6 and 2 cannot be considered constant during the entire course of the dissolution process when poly-disperse powders are used and/or an initial phase of poor deaggregation of granules or poor wetting of formulation is encountered. In addition, the diffusion layer thickness appears to depend on particle size. For all aforementioned reasons, (5.5), (5.6), (5.7), and (5.8) have been proven adequate in modeling dissolution data only when the presuppositions of constancy of terms in (5.3) are fulfilled. 

FIGURE 2 Noyes-Whitney dissolution model where Ci is the drug solubility at the same conditions as the particle surface, Cb is the concentration in the bulk dissolution medium and A is the thickness of the boundary or diffusion layer. 

The effect of particle size and dissolution rate has been known since the pio-neeringworkofNoyes and Whitney (1897), and Hixson and Crowell (1931) subsequently derived a highly useful equation that expresses the rate of dissolution based on the cube root of the weight of the particles. When the Hixson-Crowell model is applied to micronized particles, for which the thickness of the aqueous diffusion layer around the dissolving particles is comparable to or larger than the radius of the particle, the change in particle radius with time is given by ... 

---